diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzikxc" "b/data_all_eng_slimpj/shuffled/split2/finalzzikxc" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzikxc" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nIn this paper, we study the optimal control for the following stochastic\nevolution equation with jumps\n\\begin{eqnarray} \\label{eq:1.1}\n \\left\\{\n \\begin{aligned}\n d X (t) = & \\ [ A (t) X (t) + b ( t, X (t), u(t)) ] d t\n+ [B(t)X(t)+g( t, X (t), u(t)) ]d W(t)\n \\\\&\\quad +\\int_E \\sigma (t, e,X(t-),u(t))\\tilde \\mu(de,dt), \\\\\nX (0) = & \\ x , \\quad t \\in [ 0, T ]{\\color{blue},}\n \\end{aligned}\n \\right.\n\\end{eqnarray}\nwith the cost functional\n\\begin{equation}\\label{eq:1.2}\nJ(u(\\cdot))= {\\mathbb E} \\bigg [ \\int_0^T l ( t, X (t), u (t) ) d t\n+ \\Phi ( X (T) ) \\bigg ],\n\\end{equation}\nand state constraint\n\\begin{equation} \\label{eq:3.3}\n{\\mathbb E} [ \\phi(X(T)) ] =0 ,\n\\end{equation}\nin the framework of a Gelfand triple $V \\subset H= H^*\\subset V^{*},$ where $ H$ and $V$ are\ntwo given Hilbert spaces. Here on a given filtrated probability space $(\\Omega, \\mathscr{F},\\{\n{\\mathscr F}_t\\}_{0\\leq t\\leq T}, P),$ $W$ is a one-dimensional Brownian motion and $\\tilde \\mu$\nis a Poisson random martingale measure on a fixed nonempty Borel measurable\nsubset ${E}$ of $\\mathbb R^1,$\n$A:[0,T]\\times \\Omega \\longrightarrow {\\mathscr L} (V, V^*)$, $B\n : [0,T]\\times \\Omega \\longrightarrow {\\mathscr L} (V, H ),$ $b:[0,T]\\times \\Omega\\times H\n \\times U_{ad}\\longrightarrow H$, $g:[0,T]\\times\\Omega\\times H\n \\times U_{ad} \\longrightarrow H$ and $\\sigma:[0,T]\\times \\Omega \\times\n E\\times H \\times U_{ad}\\longrightarrow H$ are given random mappings, where the control variable $u$ takes value in a nonempty convex subset $U_{ad}$ of a real Hilbert space $U$. Here we denote by $\\mathscr{L}(V,V^*)$ the space of bounded\nlinear transformations of V into $V^*$, by ${\\mathscr L} (V, H)$ the space of bounded\nlinear transformations of $H$ into $V.$\n An adapted solution of\n \\eqref{eq:1.1} is a $V$-valued, $\\{{\\mathscr F}_t\\}_{0\\leq t\\leq T}$-adapted process $X(\\cdot)$ which satisfies \\eqref{eq:1.1} under some appropriate sense.\n The optimal control problem is to find an admissible control to minimize the cost functional \\eqref{eq:1.2}\n over the set of admissible controls.\n\n\nOne of the basic method to solve stochastic optimal control\nproblems is the stochastic maximum principle whose objective is to\nestablish necessary (as well as sufficient) optimality conditions of\ncontrols. For optimal control problems of infinite dimensional\nstochastic systems, many works are concerned with the\nstochastic systems and the corresponding stochastic maximum\nprinciples, see e.g.( \\cite{HuPe901, Ben830, Zhou93, Alhu10,\nAlhu101, Alhu111, Alhu112, LuZh12,Guat11, DuMeng}.\n\n In contrast, there have not been\n a\n number of results on the optimal control for stochastic partial\ndifferential equations driven by jump processes. In 2005, {\\O}ksendal, Proske, Zhang \\cite{Ok} studied the optimal control problem of quasilinear semielliptic SPDEs driven by Poisson random measure and gave sufficient maximum principle results, not necessary ones. In 2017, Tang and Meng \\cite{Tangmeng2016} studied the optimal control problem\nfor a controlled stochastic evolution equation \\eqref{eq:1.1} with the\ncost functional \\eqref{eq:1.2}, where the control domain is assumed to be convex. \\cite{Tangmeng2016} adopt the convex variation method and the first adjoint duality analysis to show a necessary maximum principle. And Under the convexity assumption of the Hamiltonian and the terminal cost, a sufficient maximum principle for this optimal problem which is the so-called verification theorem is obtained\n\n The purpose of this paper is to establish the\n maximum principle\n for the optimal control problem where the state process is driven by a controlled stochastic evolution equation \\eqref{eq:1.1} with the cost\n functional \\eqref{eq:1.2} and the\n state constraint \\eqref{eq:3.3} by Ekland variational\nprinciple, combining the convex variation method\nand the duality technique.\n\n\nThe paper is organized as follows. In section 2 we formulate the problem and give\nvarious assumptions used throughout the paper. In section 3, we present a penalized optimal control problem. Section 4 is devoted to derive necessary optimality\nconditions in the form of stochastic maximum principles in a unified\nway. Some basic results on the SEE and the BSEE with jump are given in the Appendix which will been\nused in this paper.\n\\section{Problem formulation}\n\nIn this section, we introduce basic notation and standing assumptions, and state an optimal control\nproblem with state constraint under a stochastic evolution equation with jumps\nin Hilbert space, which was considered by Tang and Meng\\cite{Tangmeng2016}.\n\n\nLet $(\\Omega, \\mathscr{F}, \\mathbb P)$ be a complete probability space\nequipped with a one-dimensional standard Brownian motion $\\{W(t),\n0\\leq t\\leq T\\}$ and a stationary Poisson point process\n$\\{\\eta_t\\}_{t\\geq 0}$ defined on a fixed nonempty Borel measurable\nsubset ${E}$ of $\\mathbb R^1$.\n Denote by $\\mathbb E[\\cdot]$ the expectation\nunder the probability $\\mathbb P.$\n We denote by $\\mu(de,dt)$\n the counting measure induced by $\\{\\eta_t\\}_{t\\geq 0}$ and\n by $\\nu(de)$ the corresponding\n characteristic measure. Then the compensatory\n random martingale measure is denoted by\n $\\tilde{\\mu}(de, dt):={\\mu}(de,\ndt)-\\nu(de)dt$ which is assumed to be independent of the Brownian\nmotion $\\{W(t),\n0\\leq t\\leq T\\}$.\n Furthermore, we assume that\n$\\nu({E})<\\infty$. Let $\\{{\\mathscr F}_t\\}_{0\\leq t\\leq T}$ be the\nP-augmentation of the natural filtration generated by\n$\\{{W_t}\\}_{t\\geq 0}$ and $\\{\\eta_t\\}_{t\\geq 0}$.\n By $\\mathscr{P}$ we denote the\npredictable $\\sigma$ field on $\\Omega\\times [0, T]$ and\nby $\\mathscr B(\\Lambda)$\n the Borel $\\sigma$-algebra of any topological space $\\Lambda.$\n Let $X$ be a separable Hilbert space with norm $\\|\\cdot\\|_X$.\n Denote by $M^{\\nu,2}( E; X)$ the set of all $X$-valued measurable\n functions $r=\\{r(e), e\\in E\\}$ defined on the measure\n space $(E, \\mathscr B(E); v)$\n such that\n$\\|r\\|_{M^{\\nu,2}( E; X)}\\triangleq\n\\sqrt{{\\int_E\\|r(e)\\|_X^2v(de)}}<~\\infty,$ by\n${M}_\\mathscr{F}^{\\nu,2}{([0,T]\\times E; X)}$ the set of all\n$\\mathscr{P}\\times {\\mathscr B}(E)$-measurable $X$-valued processes\n$r=\\{r(t,\\omega,e),\\\n(t,\\omega,e)\\in[0,T]\\times\\Omega\\times E\\}$ such that\n$\\|r\\|_{{M}_\\mathscr{F}^{\\nu,2}{([0,T]\\times E; X)}}\\triangleq\n\\sqrt{{\\mathbb E\\bigg[\\displaystyle\\int_0^T\\displaystyle\\int_E\\displaystyle\\|r(t,e)\\|_X^2\n\\nu(de)dt\\bigg]}}<~\\infty,$\n by\n$M_{\\mathscr{F}}^2(0,T;X)$ the set of all ${\\mathscr{F}}_t$-adapted\n$X$-valued processes $f=\\{f(t,\\omega),\\\n(t,\\omega)\\in[0,T]\\times\\Omega\\}$ such that\n$\\|f\\|_{M_{\\mathscr{F}}^2(0,T;X)}\n\\triangleq\\sqrt{\\mathbb E\\bigg[\\displaystyle\\int_0^T\\|f(t)\\|_X^2dt\\bigg]}<\\infty,$\nby $S_{\\mathscr{F}}^2(0,T;X)$ the set of all\n${\\mathscr{F}}_t$-adapted $X$-valued c\\`{a}dl\\`{a}g processes\n$f=\\{f(t,\\omega),\\ (t,\\omega)\\in[0,T]\\times\\Omega\\}$ such that\n$\\|f\\|_{S_{\\mathscr{F}}^2(0,T;X)}\\triangleq\\sqrt{\n\\mathbb E\\bigg[\\displaystyle\\sup_{0\n\\leq t \\leq T}\\|f(t)\\|_X^2}\\bigg]<+\\infty,$ by\n$L^2(\\Omega,{\\mathscr{F}},\\mathbb P;X)$ the set of all $X$-valued random\nvariables $\\xi$ on $(\\Omega,{\\mathscr{F}},\n\\mathbb P)$ such that\n$\\|\\xi\\|_{L^2(\\Omega,{\\mathscr{F}},\n\\mathbb P};X)\\triangleq\n\\sqrt{\\mathbb E[\\|\\xi\\|_X^2]}<\\infty.$\nThroughout this paper, we let $C$ and $K$ be two generic positive constants, which may be different from line to line.\n\n\n\n\n\nIn what follows, we set up a Gelfand triple $(V, H, V^*)$, based on which the state process and the adjoint process is defined.\nIndeed, the state process is governed by a SEE with jumps, while the adjoint process is governed by a BSEE with jumps. We provide the existence, uniqueness\nand continuous dependence theorems for SEEs with jumps and BSEEs with jumps in the appendix.\n\nLet $V$ and $H$ be two separable (real) Hilbert spaces such that $V$\nis densely embedded in $H$. We identify $H$ with its dual space by the Riesz mapping. Then we can take $H$ as a pivot space and get a Gelfand triple $(V, H,\nV^*)$ such that $V \\subset H = H^*\\subset V^{*}$. Let $(\\cdot,\\cdot)_H$ denote the inner product in $H$, and $\\la\\cdot,\\cdot\\ra$ denote the duality product between\n$V$ and $V^{*}$. Moreover, we write $\\mathscr{L}(V,V^*)$ for the space of bounded linear transformations of V into $V^*$.\n\n\n\nThe state process is governed by the following controlled SEE with jumps in the Gelfand triple $(V, H, V^*)$:\n\\begin{eqnarray} \\label{eq:4.1}\n \\left\\{\n \\begin{aligned}\n d X (t) = & \\ [ A (t) X (t) + b ( t, X (t), u(t)) ] d t\n+ [B(t)X(t)+g( t, X (t), u(t)) ]d W(t)\n \\\\&\\quad +\\int_E \\sigma (t, e,X(t-),u(t))\\tilde \\mu(de,dt), \\\\\nX (0) = & \\ x , \\quad t \\in [ 0, T ]{\\color{blue},}\n \\end{aligned}\n \\right.\n\\end{eqnarray}\nwhere the space of controls $U_{ad}$ is given by a nonempty closed convex subset of a separable real Hilbert space $U$.\n\n\\begin{definition}\nA stochastic process $u(\\cdot)$ is an admissible control, if $u(t)\\in U_{ad}$ for almost $t\\in [0, T]$ and $ u(\\cdot)\\in\nM_{\\mathbb F}^2(0, T;U)$. The set of all admissible controls is denoted by ${\\cal A}$.\n\\end{definition}\n\nThe cost functional is given by\n\\begin{equation}\\label{eq:2.2}\nJ(u(\\cdot))= {\\mathbb E} \\bigg [ \\int_0^T l ( t, x (t), u (t) ) d t\n+ \\Phi ( x (T) ) \\bigg ].\n\\end{equation}\nWe assume that the control system (\\ref{eq:4.1})-(\\ref{eq:2.2}) is subject to the following state constraint\n\\begin{equation} \\label{eq:2.555}\n{\\mathbb E} [ \\phi(X(T)) ] =0 .\n\\end{equation}\nHere the coefficients $(A,B, b,g,\\sigma, l,\\Phi, \\phi)$ of the control system (\\ref{eq:4.1})-(\\ref{eq:3.3})\n\\begin{assumption}\\label{ass:2.5}\n\\begin{enumerate}\n\\item[]\n\\item[(i)]\n\nThe operator processes $A:[0,T]\\times \\Omega \\longrightarrow {\\mathscr L} (V, V^*)$ and $B\n : [0,T]\\times \\Omega \\longrightarrow {\\mathscr L} (V, H)$\n are weakly predictable; i.e.,\n $ \\langle A(\\cdot)x, y \\rangle$ and $(B(\\cdot)x, y)_H$\n are both predictable process for every $x, y\\in V, $\n and satisfy the coercive condition, i.e., there exist\n some constants $ C, \\alpha>0$ and $\\lambda$ such that for any $x\\in V$ and each $(t,\\omega)\\in [0,T]\\times \\Omega,$\n \\begin{eqnarray}\n \\begin{split}\n - \\langle A(t)x, x \\rangle +\\lambda ||x||_H^2\\geq \\alpha\n ||x||_V^2+||Bx||_H^2,\n \\end{split}\n \\end{eqnarray}\n and \\begin{eqnarray} \\label{eq:3.3}\n\\sup_{( t, \\omega ) \\in [0, T] \\times \\Omega} \\| A ( t,\\omega ) \\|_{{\\mathscr L} ( V, V^* )}\n +\\sup_{( t, \\omega ) \\in [0, T] \\times \\Omega} \\| B ( t,\\omega ) \\|_{{\\mathscr L} ( V, H )} \\leq C \\ .\n\\end{eqnarray}\n\n\\item[(ii)]$b, g: [ 0, T ] \\times \\Omega \\times H \\times {\\mathscr U} \\rightarrow H$ are $\\mathscr P\\times\n \\mathscr B(H)\\times \\mathscr B(\\mathscr U)\/\\mathscr B(H) $ measurable\n mappings and $\\sigma:[0,T]\\times \\Omega \\times\n E\\times H \\times \\mathscr U \\longrightarrow H$ is a $\\mathscr P\\times \\mathscr B(E)\\times\n \\mathscr B(H)\\times \\mathscr B(U)\/\\mathscr B(H)$-measurable mapping such that $b ( \\cdot, 0, 0 ), g ( \\cdot, 0, 0 ) \\in {\nM}^2_{\\mathscr F} ( 0, T; H ), \\sigma(\\cdot, \\cdot, 0,0)\\in {M}_\\mathscr{F}^{\\nu,2}{([0,T]\\times E; H)}.$ Moreover, for almost all $( t, \\omega, e ) \\in [ 0, T ] \\times \\Omega \\times E$, $b$, $g$ and $\\sigma$ are G\\^ateaux differentiable in $(x,u)$ with continuous bounded G\\^ateaux derivatives\n$b_x, g_x,\\sigma_x, b_u, g_u$ and $\\sigma_u$;\n\\item[(iii)]\n$l:[ 0, T ] \\times \\Omega \\times H \\times {\\mathscr U} \\rightarrow \\mathbb R $ is a\n ${\\mathscr P} \\otimes {\\mathscr B} (H) \\otimes {\\mathscr B} ({\\mathscr U})\/ {\\mathscr B} ({\\mathbb R})$-measurable mapping\n and $\\Phi,\\phi: \\Omega \\times H \\rightarrow {\\mathbb R}$\nis a ${\\mathscr F}_T\\otimes {\\mathscr B} (H) \/ {\\mathscr B} ({\\mathbb R})$-measurable\nmapping.\nFor almost all $( t, \\omega ) \\in [ 0, T ] \\times \\Omega$, $l$ is continuous G\\^ateaux\ndifferentiable in $(x,u)$\nwith continuous G\\^ateaux derivatives $l_x$ and $l_u$, and $\\Phi$ and $\\phi$\n are G\\^ateaux differentiable in $x$\nwith continuous\n G\\^ateaux derivative $\\Phi_x$ and\n $\\phi_x$.\nMoreover, for almost all $( t, \\omega ) \\in [ 0, T ] \\times \\Omega$, there exists a constant $C > 0$ such that for all $( x, u ) \\in H \\times {\\mathscr U}$\n\\begin{eqnarray*}\n| l ( t, x, u ) |\n\\leq C ( 1 + \\| x \\|^2_H + + \\| u \\|_U^2 ) ,\n\\end{eqnarray*}\n\\begin{eqnarray*}\n&& \\| l_x ( t, x,u) \\|_H +\n+ \\| l_u ( t, x, u ) \\|_U \\leq C ( 1 + \\| x \\|_H + \\| u \\|_U ) ,\n\\end{eqnarray*}\nand\n\\begin{eqnarray*}\n& | \\Phi (x) | \\leq C ( 1 + \\| x \\|^2_H) , | \\phi (x) | \\leq C ( 1 + \\| x \\|^2_H) \\\\\n& \\| \\Phi_x (x) \\|_H \\leq C ( 1 + \\| x \\|_H),\\| \\phi_x (x) \\|_H \\leq C ( 1 + \\| x \\|_H).\n\\end{eqnarray*}\n\\end{enumerate}\n\\end{assumption}\n\n\n\n\n\nUnder Assumption \\ref{ass:2.5}, it can be shown from Lemma \\ref{thm:3.1} that for any $u(\\cdot)\\in {\\cal A},$\nthe state equation \\eqref{eq:4.1} admits a unique solution $X(\\cdot)\\in M_{\\mathscr{F}}^2(0,T;V)\\bigcap S_{\\mathscr{F}}^2(0,T;H)$. We also denote this solution as\n$X^u(\\cdot)$ whenever we want to emphasis its dependence on the control $u(\\cdot)$. Then we call\n$X(\\cdot)$ the state process corresponding to the control process $u(\\cdot)$ and $(u(\\cdot); X(\\cdot))$\nthe admissible pair. Furthermore, from Assumption \\ref{ass:2.5} and the a priori estimate \\eqref{eq:3.4}, we can easily validate that\n\\begin{eqnarray*}\n|J(u(\\cdot))|<\\infty.\n\\end{eqnarray*}\n\nNow we state formally the optimal control problem\n\\begin{problem}\\label{pro:2.1}\nFind an admissible control $\\bar{u}(\\cdot)$ such that\n\\begin{eqnarray*}\\label{eq:b7}\nJ(\\bar{u}(\\cdot))=\\inf_{u(\\cdot)\\in {\\cal A}}J(u(\\cdot)),\n\\end{eqnarray*}\nsubject to \\eqref{eq:4.1} and \\eqref{eq:2.555}, where the cost functional is given by \\eqref{eq:2.2}.\n\\end{problem}\n\nAny $\\bar{u}(\\cdot)\\in {\\cal A}$ satisfying the above is called an optimal control process of Problem \\ref{pro:2.1};\nthe corresponding state process $\\bar{X}(\\cdot)$ is called an optimal state process; correspondingly,\n$(\\bar{u}(\\cdot); \\bar{X}(\\cdot))$ is called an optimal pair of Problem \\ref{pro:2.1}.\n\n\\section{Penalized optimal control problem}\n\nIn this section, we relate the original constrained control problem with one without state constraint.\n\nThe results relies on the following Ekeland's principle.\n\n\\begin{lemma}[Ekeland's principle, \\cite{ekeland1974variational}]\nLet $(S,d)$ be a complete metric space and $\\rho (\\cdot ):S\n\\rightarrow {\\mathbb R}$ be lower-semicontinuous and bounded from\nbelow. For $\\varepsilon \\geq 0$, suppose $u^{\\varepsilon }\\in S$\nsatisfies\n\\begin{equation*}\n\\rho ( u^\\varepsilon ) \\leq \\inf_{u\\in S} \\rho (u) + \\varepsilon .\n\\end{equation*}\nThen for any $\\lambda >0$, there exists $u^\\lambda \\in S$ such that\n\\begin{eqnarray*}\n\\rho ( u^\\lambda ) \\leq \\rho ( u^\\varepsilon ) , \\quad d (\nu^\\lambda, u^\\varepsilon ) \\leq \\lambda ,\n\\end{eqnarray*}\nand\n\\begin{eqnarray*}\n\\rho ( u^\\lambda ) \\leq \\rho (u) + \\frac{\\varepsilon}{\\lambda} d\n(u^\\lambda, u) , {\\mbox { for all }} u \\in S.\n\\end{eqnarray*}\n\\end{lemma}\n\n Define\na metric $d$ on the admissible\n controls set $\\cal A$ as\n\\begin{eqnarray}\\label{eq:3.13}\nd (u_1 (\\cdot), u_2 (\\cdot)) \\triangleq \\bigg \\{ {\\mathbb E} \\bigg [ \\int_0^T||u_1 (t) - u_2 (t)||^2_U dt\\bigg] \\bigg \\}^{\\frac{1}{2}} ,\n\\quad \\forall u_1(\\cdot), u_2 (\\cdot) \\in \\cal A.\n\\end{eqnarray}\n\n\nWe can assume that\n$\\cal A$ is a bounded closed convex set in the sense of \\eqref{eq:3.13},\nthe unbounded case can be reduced to the\nbounded case.\n\n\n\n\n\n\nUnder this assumption of boundedness and closedness\nof $\\cal A$,\nwe have the following basic\nlemma which will be used in the sequence.\n\n\n\n\n\\begin{lemma}\\label{lem:4.2}\n$(\\Lambda, d)$ is a complete metric space.\n\\end{lemma}\n\n\\begin{proof}\nSince the control space $U$ is a Hilbert space $M_{\\mathbb{F}}^2(0,T;U)$ is also a Hilbert space under \\eqref{eq:3.13}. Therefore, $\\cal A$ is complete under the distance defined by \\eqref{eq:3.13}.\nsince $\\cal A$ is a closed subset of $M_{\\mathbb{F}}^2(0,T;U).$\nThe proof is complete.\n\\end{proof}\n\n\nThe next lemma shows that a mapping from the control process in $\\cal A$ to the state process in ${\\cal M}^2_{\\mathbb F} (0, T)$, to be defined below, is bounded and continuous.\nTo simplify our notation, we write\n\\begin{eqnarray}\n{\\cal M}^2_{\\mathscr F} (0, T) \\triangleq S_ {\\mathscr{F}}^2(0,T;H) \\cap M_{\\mathscr{F}}^2(0,T;V)\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n|| X(\\cdot)||_{{\\cal M}^2_{\\mathscr F} (0, T)} \\triangleq \\sqrt{||X(\\cdot)||^2_{S^2_{\\mathscr F}(0,T;H)}\n+ ||X(\\cdot)||^2_{M^2_{\\mathscr F}(0,T;V)} }.\n\\end{eqnarray}\nThe next lemma shows that a mapping from the control process in ${\\cal A}$ to the state process in $ M_{\\mathscr{F}}^2(0,T;V)$ is bounded and continuous.\n\n\n\n\\begin{lemma}\\label{lem:4.7}\nLet Assumption \\ref{ass:2.5} be satisfied. Then the mapping ${\\cal I}: ({\\cal A}, d) \\rightarrow ({\\cal M}_{\\mathscr{F}}^2(0,T), || \\cdot ||_{{\\cal M}_{\\mathscr{F}}^2(0,T)})$\ndefined by\n\\begin{eqnarray*}\n{\\cal I} (u (\\cdot)) = X^u(\\cdot)\n\\end{eqnarray*}\nis bounded and continuous.\n\\end{lemma}\n\n\\begin{proof}\nBy the a priori estimate of SEE (Lemma \\ref{thm:3.2}), it can be shown that for any $u(\\cdot)\\in \\Lambda$,\n\\begin{eqnarray}\\label{eq:5.7}\n|| X^u(\\cdot) ||^2_{{\\cal M}_{\\mathscr{F}}^2(0,T)}\n&\\leq& K \\bigg \\{ {\\mathbb E} [ ||x||^2_H ] + {\\mathbb E} \\bigg [ \\int_0^T|| u(t)|| _{U}^{2}dt \\bigg ] + 1 \\bigg \\} \\nonumber \\\\\n&\\leq& K.\n\\end{eqnarray}\nHere $K$ is a positive constant independent of $u(\\cdot)$ and may change from line to line.\n\nOn the other hand, let $\\{v_n(\\cdot)\\}_{n \\geq 1}$ be a sequence in $\\cal A$ such that it converges an admissible $v(\\cdot)\\in \\cal A $ under the metric $d$. Suppose that $X_n(\\cdot)$, for each $n = 1, 2, \\cdots$,\nand $X(\\cdot)$ are the state processes corresponding to $v_n(\\cdot)$ and $v(\\cdot)$, respectively. By making use of the a priori estimate of SEE (Lemma \\ref{thm:3.2}), we can deduce that\n\\begin{eqnarray}\\label{eq5.9}\n&&|| X^{v_n}(\\cdot) - X^v(\\cdot)||^2_{{\\cal M}_{\\mathscr{F}}^2(0,T)}\n\\\\&\\leq& K \\bigg\\{{\\mathbb E} \\bigg [\\int_{0}^{T}||b(t,X^{v}(t), v_n(t))-b(t,X^{ v}(t), v(t) )||^2_H dt\n \\bigg]+\n {\\mathbb E} \\bigg [\\int_{0}^{T}||g(t,X^{v}(t), v_n(t))-g(t,X^{ v}(t), v(t) )||^2_H dt\n \\bigg]\\nonumber\n \\\\&&\\quad\\quad+\n {\\mathbb E} \\bigg [\\int_{0}^{T}||\\sigma(t,X^{v}(t), v_n(t))-\\sigma (t,X^{ v}(t), v(t) )||^2_{M^{\\nu,2}( E; H)} dt\n \\bigg]\\bigg\\}\n \\nonumber \\\\\n&\\leq& K {\\mathbb E} \\bigg [\\int_{0}^{T}||v_n(t)- v(t)||^2_U dt\\bigg] \\nonumber \\\\\n&=& K d^2(v_n(\\cdot), v(\\cdot)).\n\\end{eqnarray}\nSending $n\\rightarrow\\infty$ in (\\ref{eq5.9}) yields\n\\begin{eqnarray}\\label{eq:5.11}\n|| X^{v_n}(\\cdot) - X^v(\\cdot)||^2_{{\\cal M}_{\\mathscr{F}}^2(0,T)} \\rightarrow 0 .\n\\end{eqnarray}\nThis validates the continuity of ${\\cal I}$.\n\\end{proof}\n\n\\begin{lemma} \\label{lem:4.4}\nLet Assumption \\ref{ass:2.5} be satisfied. Then the cost functional $J(u(\\cdot))$ is bounded and continuous on $\\cal A$ under the metric \\eqref{eq:3.13}.\n\\end{lemma}\n\n\\begin{proof}\nFor any $u(\\cdot)\\in {\\cal A}$, under Assumption \\ref{ass:2.5} and from Lemma \\ref{lem:4.7} we have\n\\begin{eqnarray}\n|J(u(\\cdot))| &\\leq& {\\mathbb E} \\bigg [ \\int_0^T |l(t,X^u(t),u(t))|dt + |\\Phi(X^u(T))| \\bigg ] \\nonumber \\\\\n&\\leq& K \\bigg [ 1 + ||X^u(\\cdot)||^2_{M_{\\mathscr{F}}^2(0,T;V)}\n+ ||u(\\cdot)||_{M_{\\mathscr{F}}^2(0,T;U)}^2 + ||X(T)||^2_{L^2(\\Omega,{\\mathscr{F}_T},\n\\mathbb P;H)} \\bigg] \\nonumber \\\\\n&\\leq& K \\bigg [ 1 + ||X^u(\\cdot)||^2_{{\\cal M}^2_{\\mathscr F}(0,T)} + ||{ u}(\\cdot)||_{M^2_{\\mathscr F}(0,T;U)}^2 \\bigg] \\nonumber \\\\\n&\\leq& K .\n\\end{eqnarray}\nHere $K$ is a positive constant independent of $u(\\cdot)$ and may change from line to line. This implies the cost\nfunctional $J(u(\\cdot))$ is bounded on ${\\cal A}$.\n\nTo show the continuity of the cost functional, as in the proof of Lemma \\ref{lem:4.7} we pick up the sequence $\\{v_n(\\cdot)\\}_{n \\geq 1}$\nand its converging point $v(\\cdot)$ in $\\cal A$ as well as the corresponding state processes $X_n(\\cdot)$ and $X(\\cdot)$.\nThus using Lemma \\ref{lem:4.7} and the Lebesgue dominated convergence theorem, we obtain\n\\begin{eqnarray}\\label{eq:3.4}\nJ(v^n(\\cdot))\\rightarrow J(v(\\cdot)) , \\quad \\mbox {as} \\ n \\rightarrow \\infty .\n\\end{eqnarray}\nThe completes the proof.\n\\end{proof}\n\nDefine a penalized cost functional associated with Problem \\eqref{pro:2.1} as\n\\begin{equation}\\label{eq:3.2000}\nJ^\\varepsilon(v(\\cdot)) \\triangleq \\bigg\\{ \\big [ J(v(\\cdot)) -J(\\bar u(\\cdot)) + \\varepsilon \\big ]^2+ \\big | {\\mathbb E} [ \\phi( X^v(T))] \\big |^2 \\bigg\\}^{\\frac{1}{2}} ,\n\\quad \\forall \\varepsilon>0 .\n\\end{equation}\nIt is worthwhile to point out that we will study this functional over $\\cal A$.\n\n\\begin{lemma}\\label{lem:4.5}\n$J^\\varepsilon(v(\\cdot))$ is bounded and continuous on ${\\cal A}$ under the metric \\eqref{eq:3.13}.\n\\end{lemma}\n\n\\begin{proof}\nThe proof can be obtained by Lemma \\ref{lem:4.4} and Lemma \\ref{lem:4.7} immediately.\n\\end{proof}\n\nNow we introduce an auxiliary optimal control problem without state constraint:\n\n\\begin{problem}[$(SC)^\\varepsilon$]\nFind an admissible control such that\n\\begin{eqnarray}\n\\inf_{v(\\cdot)\\in {\\cal A}} J^\\varepsilon(v(\\cdot)),\n\\end{eqnarray}\nwhere the state process is given by (\\ref{eq:4.1}) and the cost functional $J^\\varepsilon(v(\\cdot))$ is given by (\\ref{eq:3.4}).\n\\end{problem}\n\nFrom the definition of the penalized cost functional (\\ref{eq:3.2000}), we see that\n\\begin{eqnarray}\nJ^\\varepsilon(\\bar u(\\cdot)) = \\varepsilon\n \\leq \\inf_{v(\\cdot)\\in {\\cal A}} J^\\varepsilon(v(\\cdot))+\\varepsilon.\n\\end{eqnarray}\nAn application of Ekeland's variational principle shows that there is a $u^\\varepsilon(\\cdot)\\in {\\cal A}$ such that\n\\begin{eqnarray}\\label{eq:5.1}\n\\left\\{\n\\begin{aligned}\n& J^\\varepsilon(u^\\varepsilon(\\cdot))\\leq J^\\varepsilon(\\bar u(\\cdot))=\\varepsilon , \\\\\n& d(u^\\varepsilon(\\cdot),\\bar u(\\cdot))\\leq \\varepsilon^{\\frac{1}{2}}, \\\\\n& J^\\varepsilon(v(\\cdot))- J^\\varepsilon(u^\\varepsilon(\\cdot))\\geq -\\varepsilon^{\\frac{1}{2}}d(u^\\varepsilon(\\cdot),v(\\cdot)), \\quad \\forall v(\\cdot)\\in {\\cal A}.\n\\end{aligned}\n\\right.\n\\end{eqnarray}\nDefine a convex perturbed control of ${u}^{\\varepsilon }\\left( \\cdot \\right)$ as\n\\begin{eqnarray}\\label{eq:5.2}\nu^{\\varepsilon ,\\rho}\\left( \\cdot \\right) \\triangleq {u}^{\\varepsilon }(\\cdot )+\\rho\nu\\left( \\cdot \\right)-u^{\\varepsilon }(\\cdot ) ),\n\\end{eqnarray}\nwhere $u\\left( \\cdot \\right)$ is an arbitrary admissible control in ${\\cal A}$ and $0\\leq \\rho\\leq 1$. It is easy to verify that\n$u^{\\varepsilon,\\rho }\\left( \\cdot \\right)$ is also in ${\\cal A}$. Suppose that $X^{\\epss,\\rho}(\\cdot)$ and\n$X^{\\epss}(\\cdot)$ are the state processes corresponding to $u^{\\epss, \\rho}(\\cdot)$ an $ u^\\epss(\\cdot)$, respectively.\nBy \\eqref{eq:5.1} and the fact\n\\begin{eqnarray}\\label{eq:5.3}\nd\\left( u^{\\varepsilon ,\\rho}\\left( \\cdot \\right) ,{u}^{\\varepsilon\n}\\left( \\cdot \\right) \\right) \\leq C\\rho,\n\\end{eqnarray}\nwe have\n\\begin{eqnarray}\\label{eq:3.9}\nJ^\\varepsilon(u^{\\varepsilon,\\rho}(\\cdot))-J^\\varepsilon(u^\\varepsilon(\\cdot))\n\\geq {-\\varepsilon ^{\\frac{1}{2}}d\\left( u^{\\varepsilon ,\\rho}\\left(t\\right),{u}^{\\varepsilon}\\left( t\\right) \\right) }\n\\geq -\\varepsilon^{\\frac{1}{2}}C\\rho.\n\\end{eqnarray}\n\nOn the other hand, from the definition of $J^\\varepsilon(\\bar u(\\cdot))$, we have\n\\begin{eqnarray}\\label{eq:3.17}\nJ^\\varepsilon(u^{\\varepsilon,\\rho}(\\cdot))-J^\\varepsilon(u^\\varepsilon(\\cdot))\n&=& \\frac{[J^\\varepsilon(u^{\\varepsilon,\\rho}(\\cdot))]^2-[J^\\varepsilon(u^\\varepsilon(\\cdot))]^2}\n{J^\\varepsilon(u^{\\varepsilon,\\rho}(\\cdot))+J^\\varepsilon(u^\\varepsilon(\\cdot))} \\nonumber \\\\\n&=& \\frac {J(u^{\\varepsilon,\\rho}(\\cdot))+J(u^\\varepsilon(\\cdot))-2J(\\bar u(\\cdot))+2\\varepsilon}\n{J^\\varepsilon(u^{\\varepsilon,\\rho}(\\cdot))+J^\\varepsilon(u^\\varepsilon(\\cdot))}\\times[J(u^{\\varepsilon,\\rho}(\\cdot))-J(u^\\varepsilon(\\cdot))] \\nonumber \\\\\n&& +\\frac {{\\mathbb E}[\\phi(X^{\\varepsilon,\\rho}(T))]+{\\mathbb E}[\\phi(X^\\varepsilon(T))]}{J^\\varepsilon(u^{\\varepsilon,\\rho}(\\cdot))+J^\\varepsilon(u^\\varepsilon(\\cdot))}\n\\times \\big \\{ {\\mathbb E}[\\phi(X^{\\varepsilon,\\rho}(T))]-{\\mathbb E}[\\phi(X^\\varepsilon(T))] \\big \\} \\nonumber \\\\\n&=& {\\lambda}^{\\varepsilon,\\rho}[{J(u^{\\varepsilon,\\rho}(\\cdot))-J(u^\\varepsilon(\\cdot))}]\n+\\mu^{\\varepsilon,\\rho} \\big \\{ {\\mathbb E}[\\phi(X^{\\varepsilon,\\rho}(T))]-{\\mathbb E}[\\phi(X^\\varepsilon(T))] \\big \\} ,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n{\\lambda}^{\\varepsilon,\\rho}\\triangleq\\frac {J(u^{\\varepsilon,\\rho}(\\cdot))+J(u^\\varepsilon(\\cdot))-2J(\\bar\nu(\\cdot))+2\\varepsilon}{J^\\varepsilon(u^{\\varepsilon,\\rho}(\\cdot))+J^\\varepsilon(u^\\varepsilon(\\cdot))}\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n\\mu^{\\varepsilon,\\rho}\\triangleq\\frac{{\\mathbb E}[\\phi(X^{\\varepsilon,\\rho}(T))]+{\\mathbb E}[\\phi(X^\\varepsilon(T))]}\n{J^\\varepsilon(u^{\\varepsilon,\\rho}(\\cdot))+J^\\varepsilon(u^\\varepsilon(\\cdot))} .\n\\end{eqnarray}\nFrom \\eqref{eq:5.3}, we have\n\\begin{eqnarray} \\label{eq:5.8}\n\\lim_{\\rho\\rightarrow 0} d\\left( u^{\\varepsilon ,\\rho}\\left( \\cdot \\right) ,{u}^{\\varepsilon}\\left( \\cdot \\right) \\right) =0\n\\end{eqnarray}\nThen it follows from Lemma \\ref{lem:4.4} and Lemma \\ref{lem:4.5} that\n\\begin{eqnarray} \\label{eq:5.9}\n\\lim_{\\rho\\rightarrow 0} ||X^{\\epss,\\rho}(\\cdot)-X^{ \\epss}(\\cdot)||_{{\\cal M}^2_{\\mathscr F}(0,T)}^2=0\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n\\lim_{\\rho\\rightarrow 0} J^{\\epss}(u^{\\epss,\\rho}(\\cdot))= J^{\\epss}(u^{\\epss}(\\cdot)) .\n\\end{eqnarray}\nConsequently,\n\\begin{eqnarray}\\label{eq:5.10}\n\\lim_{\\rho\\rightarrow 0} { \\lambda}^{\\varepsilon,\\rho}= {\\lambda}^{\\varepsilon}, \\quad \\lim_{\\rho \\rightarrow 0} \\mu^{\\varepsilon,\\rho}= \\mu^{\\varepsilon} ,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n{\\lambda}^{\\varepsilon}\\triangleq\\frac {J(u^{\\varepsilon}(\\cdot))-J(\\bar u(\\cdot))+\\varepsilon}{J^\\varepsilon(u^\\varepsilon(\\cdot))}\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n\\mu^{\\varepsilon}\\triangleq\\frac{{\\mathbb E}[\\phi(X^{\\varepsilon}(0))]}{J^\\varepsilon(u^{\\varepsilon}(\\cdot))} .\n\\end{eqnarray}\nNote that\n\\begin{eqnarray}\n|{\\lambda}^\\epss|^2+|\\mu^\\epss|^2=1.\n\\end{eqnarray}\n\nTherefore, there exists a subsequence $\\{({\\lambda}^\\epss, \\mu^\\epss)\\}_{\\epss > 0}$ ( still denoted also by $\\{({\\lambda}^\\epss, \\mu^\\epss)\\}_{\\epss > 0}$, such that\n\\begin{eqnarray} \\label{eq:5.13}\n\\lim_{\\epss\\rightarrow 0} {\\lambda}^{\\varepsilon}= {\\lambda}, \\quad \\lim_{\\epss \\rightarrow0} \\mu^{\\varepsilon}= \\mu ,\n\\end{eqnarray}\nand\n\\begin{eqnarray} \\label{eq:5.14}\n|{\\lambda}|^2+|\\mu|^2=1.\n\\end{eqnarray}\n\n\n\\section{Stochastic Maximum Principle}\n\nIn this section, we first drive a variational formula for the penalized cost functional $J^\\epss(u(\\cdot))$.\n\nTo simplify our notation, we write partial derivatives of $b,g \\sigma$ and $l$ as\n\\begin{eqnarray*}\n& \\varphi_x^{\\epss,\\rho}(t) \\triangleq \\varphi_x (t,{X}^{\\varepsilon,\\rho}(t),{u}^{\\varepsilon,\\rho}(t)), \\\\\n& \\varphi_x^{\\epss}(t) \\triangleq \\varphi_a (t,{X}^{\\varepsilon}(t),{u}^{\\varepsilon}(t)), \\\\\n& \\bar\\varphi_x(t) \\triangleq \\varphi_a (t,\\bar{X}(t),\\bar{u}(t)),\n\\end{eqnarray*}\nwhere $\\varphi = b, g,\\sigma$ and $l$.\n\nDefine the Hamiltonian ${\\cal H}: [ 0, T ] \\times \\Omega \\times H \\times {\\mathscr U} \\times H\\times H \\times M^{\\nu,2}( E; H)\\times \\mathbb R\n\\rightarrow {\\mathbb R}$ by\n\\begin{eqnarray}\\label{eq:5.3}\n{\\cal H} ( t, x, u, p, q, r(\\cdot),\\lambda ) := \\left ( b ( t, x, u ), p \\right )_H\n+\\left( g ( t, x, u), q \\right)_H\n+\\int_{E}\\left( \\sigma ( t,e, x, u), r(t,e) \\right)_H\\nu(de)\n+ \\lambda l ( t, x, u ) .\n\\end{eqnarray}\nUsing Hamiltonian ${\\cal H}$,\nthe adjoint equation \\eqref{eq:4.4}\ncan be written in the following form:\n\n\\begin{eqnarray}\\label{eq:5.4}\n\\begin{split}\n \\left\\{\\begin{array}{ll}\nd\\bar p(t)=&-\\bigg[A^*(t)\\bar p(t)+B(t)^*\\bar q(t)+\\bar {\\cal H}_{x} (t)\\bigg]dt+\\bar q(t)dW(t)+\\displaystyle\n\\int_{{E}}\\bar r(t, e)\\tilde{\\mu}(de, dt),\n~~~~0\\leqslant t\\leqslant T,\n\\\\ \\bar p(T)=&\\Phi_x( \\bar X(T)),\n \\end{array}\n \\right.\n \\end{split}\n \\end{eqnarray}\nwhere we denote\n\\begin{eqnarray}\\label{eq:5.6}\n\\bar {\\cal H} (t) \\triangleq {\\cal H} ( t,\n\\bar x (t), \\bar u (t), \\bar p (t),\n\\bar q (t),\n\\bar r(t,\\cdot) ).\n\\end{eqnarray}\n\nSimilarly, for notational simplify, we write partial derivatives of $H$ as\n\\begin{eqnarray*}\n&{\\cal H}_a^{\\epss, \\rho}(t) \\triangleq\n{\\cal H}_a (t,{X}^{\\epss, \\rho}(t),{u}^{\\epss, \\rho}(t),{p}^{\\epss, \\rho}(t),q^{\\epss, \\rho}(t),r^{\\epss, \\rho}(t,\\cdot),{\\lambda}^{\\epss, \\rho}), \\\\\n&{\\cal H}_a^{\\epss}(t) \\triangleq H_a (t,{X}^{\\epss}(t),{u}^{\\epss}(t),\n{p}^{\\epss}(t),q^{\\epss}(t),r^{\\epss}(t,\\cdot),\n{\\lambda}^{\\epss}), \\\\\n&{\\bar{\\cal H}}_a (t) \\triangleq H_a (t,\\bar{X}(t),\\bar{u}(t),\\bar{p}(t),{\\bar q}(t), \\bar r(t,\\cdot),{\\lambda}) .\n\\end{eqnarray*}\nwhere $a = x$ or $u$.\n\nFor the admissible pair $({u}^{\\varepsilon, \\rho}(\\cdot);{X}^{\\varepsilon,\\rho}(\\cdot))$ and $({u}^{\\varepsilon}(\\cdot);\n{X}^{\\varepsilon}(\\cdot))$ and the optimal pair $(\\bar{u}(\\cdot);\\bar{X}(\\cdot))$, the corresponding adjoint processes are denoted by\n$\\{ (p^{\\epss,\\rho}(t), q^{\\epss,\\rho}(t),\nr^{\\epss,\\rho}(t,\\cdot)), 0 \\leq t \\leq T \\}$, $\\{ (p^{\\epss}(t), q^{\\epss}(t),\nr^{\\epss,\\rho}(t)), 0 \\leq t \\leq T\\}$ and $\\{ {\\bar p}(t), \\bar q(t),\n \\bar r(t,\\cdot)),0 \\leq t \\leq T \\}$.\nWe now define the adjoint equations for $\\{ (p^{\\epss,\\rho}(t), q^{\\epss,\\rho}(t),\nr^{\\epss,\\rho}(t,\\cdot)), 0 \\leq t \\leq T \\}$, $\\{ (p^{\\epss}(t), q^{\\epss}(t),\nr^{\\epss,\\rho}(t)), 0 \\leq t \\leq T\\}$ and $\\{ {\\bar p}(t), \\bar q(t),\n \\bar r(t,\\cdot)),0 \\leq t \\leq T \\}$ as\n\\begin{eqnarray}\\label{eq:4.4}\n\\left\\{\n\\begin{aligned}\nd p^{\\epss,\\rho}(t)=&-\\bigg[A^*(t) p^{\\epss,\\rho}(t)+B(t)^* q^{\\epss,\\rho}(t)+ {\\cal H}_{x}^{\\epss,\\rho} (t)\\bigg]dt+ q^{\\epss,\\rho}(t)dW(t)+\\displaystyle\n\\int_{{E}}r^{\\epss,\\rho}(t, e)\\tilde{\\mu}(de, dt),\n~~~~0\\leqslant t\\leqslant T,\n\\\\ p^{\\epss,\\rho}(T)=&\\lambda^{\\varepsilon, \\rho}\\Phi_x( X^{\\varepsilon,\\rho}(T))+\\mu\n^{\\varepsilon, \\rho}\\phi_y(X^{\\varepsilon,\\rho}(T)),\n\\end{aligned}\n\\right.\n\\end{eqnarray}\n\\begin{eqnarray}\\label{eq:4.5}\n\\left\\{\n\\begin{aligned}\nd p^{\\epss}(t)=&-\\bigg[A^*(t) p^{\\epss}(t)+B(t)^* q^{\\epss}(t)+ {\\cal H}_{x}^{\\epss} (t)\\bigg]dt+ q^{\\epss}(t)dW(t)+\\displaystyle\n\\int_{{E}}r^{\\epss}(t, e)\\tilde{\\mu}(de, dt),\n~~~~0\\leqslant t\\leqslant T,\n\\\\ p^{\\epss}(T)=\n&\\lambda^{\\varepsilon}\\Phi_x( X^{\\varepsilon}(T))+\\mu\n^{\\varepsilon}\\phi_y(X^{\\varepsilon}(T)),\n\\end{aligned}\n\\right.\n\\end{eqnarray}\nand\n\\begin{eqnarray}\\label{eq:4.6}\n\\begin{split}\n \\left\\{\\begin{array}{ll}\nd\\bar p(t)=&-\\bigg[A^*(t)\\bar p(t)+B(t)^*\\bar q(t)+\\bar {\\cal H}_{x} (t)\\bigg]dt+\\bar q(t)dW(t)+\\displaystyle\n\\int_{{E}}\\bar r(t, e)\\tilde{\\mu}(de, dt),\n~~~~0\\leqslant t\\leqslant T,\n\\\\ \\bar p(T)=&\\lambda\\Phi_x( \\bar X(T))\n+\\mu\\phi_x(\\bar X(T)),\n \\end{array}\n \\right.\n \\end{split}\n \\end{eqnarray}\nrespectively. In fact, the adjoint equations \\eqref{eq:4.4}, \\eqref{eq:4.5} and \\eqref{eq:4.6} are three linear BSEEs satisfying Assumptions \\ref{ass:3.1} and\n\\ref{ass:3.2}.\nHence by Lemma \\ref{lem:1.3}, it is easy to check that these three adjoint equations have unique solutions, respectively.\n\n\\begin{lemma}\\label{lem:4.1}\nUnder Assumptions \\ref{ass:2.5}, the following convergence results hold\n\\begin{eqnarray}\\label{eq:6.16}\n\\begin{split}\n&\\lim _{\\rho\\rightarrow 0} {\\mathbb E} \\bigg [ \\sup_{0\\leq t\\leq T}\\|p^{\\varepsilon,\\rho}(t)-{p}^{\\varepsilon}(t)\\|^2_H \\bigg ]\n+ {\\mathbb E} \\bigg [ \\int_{0}^ T \\|p^{\\varepsilon, \\rho}(t)-{p}^{\\varepsilon}(t)\\|^2_V d t\n+ {\\mathbb E} \\bigg [ \\int_{0}^ T \\|q^{\\varepsilon, \\rho}(t)-{q}^{\\varepsilon}(t)\\|^2_H d t\\bigg ]\n\\\\ &\\quad \\quad+ {\\mathbb E} \\bigg [ \\int_{0}^ T \\|r^{\\varepsilon, \\rho}(t,\\cdot)\n-{r}^{\\varepsilon}(t,\\cdot)\\|^2_{{M}_\\mathscr{F}^{\\nu,2}{([0,T]\\times E; H)}} d t\\bigg]= 0 ,\n\\end{split}\n\\end{eqnarray}\nand\n\\begin{eqnarray} \\label{eq:6.13}\n\\begin{split}\n&\\lim _{\\epss\\rightarrow 0} {\\mathbb E} \\bigg [ \\sup_{0\\leq t\\leq T}\\|p^{\\varepsilon}(t)\n-{\\bar p}(t)\\|^2_H \\bigg ]\n+ {\\mathbb E} \\bigg [ \\int_{0}^ T \\|p^{\\varepsilon}(t)-{\\bar p}(t)\\|^2_V d t\n+ {\\mathbb E} \\bigg [ \\int_{0}^ T \\|q^{\\varepsilon}(t)-{\\bar q}(t)\\|^2_H d t\\bigg ]\n\\\\ &\\quad \\quad+ {\\mathbb E} \\bigg [ \\int_{0}^ T \\|r^{\\varepsilon}(t,\\cdot)\n-{\\bar r}^{}(t,\\cdot)\\|^2_{{M}_\\mathscr{F}^{\\nu,2}{([0,T]\\times E; H)}} d t\\bigg]=0\n\\end{split}\n\\end{eqnarray}\n\\end{lemma}\n\n\\begin{proof}\nBy the continuous dependence theorem of BSEE (i.e., Lemma \\ref{lem:1.4}), we derive\n\\begin{eqnarray}\n\\begin{split}\n&{\\mathbb E} \\bigg [ \\sup_{0\\leq t\\leq T}\\|p^{\\varepsilon,\\rho}(t)-{p}^{\\varepsilon}(t)\\|^2_H \\bigg ]\n+ {\\mathbb E} \\bigg [ \\int_{0}^ T \\|p^{\\varepsilon, \\rho}(t)-{p}^{\\varepsilon}(t)\\|^2_V d t\n+ {\\mathbb E} \\bigg [ \\int_{0}^ T \\|q^{\\varepsilon, \\rho}(t)-{q}^{\\varepsilon}(t)\\|^2_H d t\\bigg ]\n\\\\ &\\quad \\quad+ {\\mathbb E} \\bigg [ \\int_{0}^ T \\|r^{\\varepsilon, \\rho}(t,\\cdot)\n-{r}^{\\varepsilon}(t,\\cdot)\\|^2_{{M}_\\mathscr{F}^{\\nu,2}{([0,T]\\times E; H)}} d t\\bigg] \\nonumber \\\\\n& \\leq K \\bigg \\{ {\\mathbb E} \\bigg[ \\int_0^T ||(b_x^{\\varepsilon,\\rho}(t)\n-b_x^{\\epss}(t))\\cdot p^{\\epss}(t)\n+(g_x^{\\varepsilon,\\rho}(t)\n-g_x^{\\epss}(t))\\cdot q^{\\epss}(t)+\n\\int_E(\\sigma_x^{\\varepsilon,\\rho}(t,e)\n-\\sigma_x^{\\epss}(t,e))\\cdot r^{\\epss}(t,e)\n\\nu (de)\n\\\\&\\quad\\quad+\n {\\lambda}^{\\varepsilon,\\rho}l_x^{\\varepsilon,\\rho}(t)-{\\lambda} ^{\\varepsilon}l_x^{\\varepsilon}(t)||^2_H d t \\bigg ]+\\mathbb E\\bigg[||\\lambda^{\\varepsilon, \\rho}\\Phi_x( X^{\\varepsilon,\\rho}(T))+\\mu\n^{\\varepsilon, \\rho}\\phi_y(X^{\\varepsilon}(T))-\n\\lambda^{\\varepsilon}\\Phi_x( X^{\\varepsilon}(T))-\\mu\n^{\\varepsilon}\\phi_y(X^{\\varepsilon}(T))||_H.\\bigg]\n\\bigg\\}\n \\end{split}\n\\end{eqnarray}\nThen using \\eqref{eq:5.9} and \\eqref{eq:5.10} gives the desired result \\eqref{eq:6.16}.\nThe proof of \\eqref{eq:6.13} is similar and omitted here.\n\\end{proof}\n\n\nIn the next lemma, we give a representation of the difference $J^{\\varepsilon}(u^{\\varepsilon,\\rho}(\\cdot))-J^\\epss( u^\\epss(\\cdot))$\nin terms of the Hamiltonian $H$, the adjoint process $(p^{\\epss,\\rho}(\\cdot),\nq^{\\epss,\\rho}(\\cdot), r^{\\epss,\\rho}(\\cdot,\\cdot))$ and other relevant expressions associated with the admissible\npair $({u}^{\\epss, \\rho}(\\cdot);{X}^{\\epss,\\rho}(\\cdot))$.\n\n\\begin{lemma}\\label{lem:4.3}\nUnder Assumptions \\ref{ass:2.5}, it holds\n\\begin{eqnarray}\\label{eq:6.15}\nJ^{\\varepsilon}(u^{\\varepsilon,\\rho}(\\cdot))-J^\\epss(u^\\epss(\\cdot))\n&=& {\\mathbb E} \\bigg [ \\int_0^T \\big\\{ {\\cal H}^{\\epss, \\rho} (t)\n- {\\cal H} (t,{X}^{\\epss}(t),\n{u}^{\\epss}(t),{p}^{\\epss,\\rho}(t),q^{\\epss, \\rho}(t),r^{\\epss, \\rho}(t,\\cdot),{\\lambda}^{\\epss, \\rho}) \\nonumber \\\\\n&& -{\\cal H}_x^{\\epss, \\rho}(t) \\cdot(X^{\\epss,\\rho}(t)-X^{\\epss}(t)) \\big\\} dt \\bigg ] \\nonumber \\\\\n&& + \\mu^{\\varepsilon,\\rho} {\\mathbb E} \\big [ \\phi^{\\epss,\\rho}(X^{\\epss,\\rho}(T))\n-\\phi^{\\epss}(X^{\\epss}(T))\n- \\phi_x(X^{\\epss,\\rho}(T))\n\\cdot(X^{\\epss,\\rho}(T)-X^{\\epss}(T)) \\big ] \\nonumber \\\\\n&& + {\\lambda}^{\\varepsilon,\\rho} {\\mathbb E} \\big [ \\Phi^{\\epss,\\rho}(X^{\\epss,\\rho}(T))\n-\\Phi^{\\epss}(X^{\\epss}(T))\n- \\Phi_x(X^{\\epss,\\rho}(T))\\cdot(X^{\\epss, \\rho}(T)-y^{\\epss}(T)) \\big ].\n\\end{eqnarray}\n\\end{lemma}\n\n\\begin{proof}\nFrom the definition of the Hamiltonian $\\cal H$ and $J^\\epss( u(\\cdot))$ (see \\eqref{eq:3.17}), we deduce\n\\begin{eqnarray}\\label{eq:4.10}\nJ^{\\varepsilon}(u^{\\varepsilon,\\rho}(\\cdot))-J^\\epss( u^\\epss(\\cdot))\n&=& {\\lambda}^{\\varepsilon,\\rho}[{J(u^{\\varepsilon,\\rho}(\\cdot))-J(u^\\varepsilon(\\cdot))}]\n+\\mu^{\\varepsilon,\\rho} {\\mathbb E} \\big [ \\phi(X^{\\varepsilon,\\rho}(T)) - \\phi(X^\\varepsilon(T)) \\big ] \\nonumber \\\\\n&=& {\\mathbb E} \\bigg [ \\int_0^T \\bigg\\{ {\\cal H}^{\\epss, \\rho} (t) - {\\cal H} (t,{X}^{\\epss}(t),\n{u}^{\\epss}(t),{p}^{\\epss,\\rho}(t),q^{\\epss, \\rho}(t),r^{\\epss, \\rho}(t,\\cdot),{\\lambda}^{\\epss, \\rho}) \\nonumber \\\\\n&& - ( p^{\\epss,\\rho}(t), b^{\\epss,\\rho}(t)-b^{\\epss}(t)))_H - (q^{\\epss,\\rho}(t), g^{\\epss, \\rho} (t)-g^{\\epss} (t))_H \\nonumber\n\\\\&&-\\int_{E}\\bigg[(r^{\\epss,\\rho}(t,e)), \\sigma^{\\epss,\\rho}(t,e)-\\sigma^{\\epss} (t,e))_H\\nu(de)\\bigg]\\bigg\\} dt \\bigg ] \\nonumber \\\\\n&& + \\mu^{\\varepsilon,\\rho} {\\mathbb E} \\big [ \\phi(X^{\\epss, \\rho}(T))-\\phi(X^{\\epss}(T)) \\big ]\n+ {\\lambda}^{\\varepsilon,\\rho} {\\mathbb E} \\big [ \\Phi(X^{\\epss, \\rho}(T))-\\Phi(X^{\\epss}(T)) \\big ] .\n\\end{eqnarray}\nOn the other hand,\n\\begin{eqnarray}\n \\left\\{\n \\begin{aligned}\n d (X^{\\epss,\\rho} (t)-X^{\\epss} (t))\n = & \\ [ A (t)(X^{\\epss,\\rho} (t)-X^{\\epss} (t))\n + (b ( t, X^{\\epss,\\rho} (t), u^{\\epss,\\rho}(t))-b ( t, X^{\\epss} (t), u^{\\epss}(t))) ] d t\n\\\\&+ [ B(t)(X^{\\epss,\\rho} (t)-X^{\\epss} (t))\n + (g( t, X^{\\epss,\\rho} (t), u^{\\epss,\\rho}(t))-g ( t, X^{\\epss} (t), u^{\\epss}(t))) ]d W(t)\n \\\\&+\\int_E [\\sigma( t,e, X^{\\epss,\\rho} (t), u^{\\epss,\\rho}(t))-\\sigma ( t,e, X^{\\epss} (t), u^{\\epss}(t))) ]\\tilde \\mu(de,dt), \\\\\nX ^{\\epss,\\rho}(0)-X^\\epss(0) = & \\ 0, \\quad t \\in [ 0, T ]\n \\end{aligned}\n \\right.\n\\end{eqnarray}\nThen applying It\\^{o} formula to $( p^{\\epss,\\rho}(t), X^{\\epss, \\rho}(t)-X^{\\epss}(t) )_H$ gives\n\\begin{eqnarray}\\label{eq:4.12}\n&& {\\mathbb E} \\bigg [ \\int_0^T \\bigg\\{ (p^{\\epss,\\rho}(t), b^{\\epss, \\rho} (t)-b^{\\epss} (t))_H+(q^{\\epss,\\rho}(t), g^{\\epss, \\rho} (t)-g^{\\epss} (t))_H\n+\\int_{E}(r^{\\epss,\\rho}(t,e), \\sigma^{\\epss, \\rho} (t,e)-\\sigma^{\\epss} (t,e))_H\\nu(de) \\bigg\\} dt \\bigg ] \\nonumber \\\\\n&& = {\\mathbb E} \\bigg [ \\int_0^T {\\cal H}_x^{\\epss, \\rho}(t) \\cdot(X^{\\epss,\n\\rho}(t)-X^{\\epss}(t)) dt \\bigg ] \\nonumber +\\mu^{\\epss,\\rho} {\\mathbb E} \\big [ \\phi_x( X^{\\epss}(T))\\cdot(X^{\\epss, \\rho}(T)-X^{\\epss}(T)) \\big ]\n\\\\&&~~~~+{\\lambda}^{\\epss,\\rho} {\\mathbb E} \\big [ \\Phi_x( X^\\epss(T))\\cdot(X^{\\epss, \\rho}(T)-X^{\\epss}(T)) \\big ].\n\\end{eqnarray}\nPutting \\eqref{eq:4.12} into \\eqref{eq:4.10} leads to the desired representation \\eqref{eq:6.15}.\n\\end{proof}\n\nWe have the following basic Lemma.\n\n\n\\begin{lemma}\\label{lem:3.2}\nUnder Assumptions \\ref{ass:2.5}, it follows that\n\\begin{eqnarray}\n\\| X^{\\varepsilon,\\rho}(\\cdot)\n-{X}^\\varepsilon(\\cdot)\\|_{{\\cal M}^2_{\\mathbb F}(0,T)}^2 = O (\\rho^2),\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n\\| X^\\varepsilon(\\cdot)- {\\bar X}(\\cdot)\\|_{{\\cal M}^2_{\\mathbb F}(0,T)}^2 = O (\\varepsilon^2).\n\\end{eqnarray}\n\\end{lemma}\n\n\\begin{proof}\nBy the continuous dependence theorem of BSEE (Lemma \\ref{lem:1.4}) and the uniform boundedness of the G\\^{a}teaux derivative $b_u$, we have\n\\begin{eqnarray*}\n&&\\| X^{\\varepsilon,\\rho}(\\cdot)\n-{X}^\\varepsilon(\\cdot)\\|_{{\\cal M}^2_{\\mathbb F}(0,T)}^2\n\\\\&\\leq& K {\\mathbb E} \\bigg [ \\int_0^T\n\\bigg\\{\\|b (t, {X}^\\varepsilon(t), u^{\\varepsilon,\\rho}(t)) - b^\\varepsilon (t)\\big\\|^2_Hdt\n+\\| g(t, {y}^\\varepsilon(t), z^\\varepsilon(t), u^{\\varepsilon,\\rho}(t)) - g^\\varepsilon (t)\\big\\|^2_H\n\\\\&&\\quad\\quad+\\int_E\\bigg[\\|\\sigma (t, e,{X}^\\varepsilon(t), u^{\\varepsilon,\\rho}(t)) - \\sigma^\\varepsilon (t)\\big\\|^2_H\\bigg]\\nu(de)\\bigg\\}dt\\bigg] \\\\\n&\\leq& K {\\mathbb E} \\bigg [ \\int_0^T \\|u^{\\varepsilon,\\rho}(t)-{u^\\epss}(t)\\|^2_Udt\\bigg] \\\\\n&=& K \\rho^2 {\\mathbb E} \\bigg[ \\int_0^T \\|v(t)-{u^\\varepsilon}(t)\\|^2_Udt\\bigg] \\\\\n&\\leq& K \\rho^2 \\\\\n&=& O(\\rho^2).\n\\end{eqnarray*}\nHere $K$ is a generic positive constant and might change from line to line.\n\nIn the same vein, we deduce\n\\begin{eqnarray*}\n\\| X^\\varepsilon(\\cdot)- {\\bar X}(\\cdot)\\|_{{\\cal M}^2_{\\mathbb F}(0,T)}^2\n&\\leq& K {\\mathbb E} \\bigg[ \\int_0^T \\|u^{\\varepsilon}(t)-{\\bar u}(t)\\|^2_Udt\\bigg] \\\\\n&=& K d^2(u^{\\varepsilon}(t),{\\bar u}(t)).\\\\\n&\\leq& K \\varepsilon^2\\\\\n&=& O(\\epss).\n\\end{eqnarray*}\nThe proof is complete.\n\\end{proof}\n\nNow we state the variational formula for the cost functional $J^\\epss(\\cdot)$.\n\n\\begin{theorem}\\label{them:3.1}\nUnder Assumptions \\ref{ass:2.5}, it follows that\nfor any admissible control $v(\\cdot),$ the cost functional $J(u(\\cdot))$ is\nG\\^{a}teaux differentiable at $ u^\\epss(\\cdot)$ in the direction $v(\\cdot)- u^\\epss(\\cdot)$\nand the corresponding G\\^{a}teaux derivative $J'$ is given by\n\\begin{eqnarray}\\label{eq:4.16}\n\\frac{d}{d\\rho}J^\\epss( u^\\epss(\\cdot)+\\rho(v(\\cdot)- u^\\epss(\\cdot)))|_{\\rho=0}\n&=& \\lim_{\\rho\\rightarrow 0} \\frac{J^\\epss(u^\\epss(\\cdot)+\\rho(v(\\cdot)- u^\\epss(\\cdot)))-J^\\epss(u^\\epss(\\cdot))}{\\rho} \\nonumber \\\\\n&=& {\\mathbb E} \\bigg [ \\int_0^T( {\\cal H}_u^\\epss(t), v(t)-{u}^\\epss(t))_Udt \\bigg ]\\nonumber\n\\\\\n&\\geq& -C\\varepsilon ^{\\frac{1}{2}}.\n\\end{eqnarray}\nHere $\\rho >0$ is a sufficiently small positive constant.\n\\end{theorem}\n\n\\begin{proof}\nBy \\eqref{eq:6.15}, we have\n\\begin{eqnarray}\\label{eq:6.23}\n&& J^\\epss( u^\\epss(\\cdot)+\\rho(v(\\cdot)- u^\\epss(\\cdot)))-J^\\epss(u^\\epss(\\cdot)) = I + II ,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray*}\nI &\\triangleq& {\\mathbb E} \\bigg [ \\int_0^T \\big\\{ {\\cal H}^{\\epss, \\rho} (t)\n- {\\cal H} (t,{X}^{\\epss}(t),\n{u}^{\\epss}(t),{p}^{\\epss,\\rho}(t),q^{\\epss, \\rho}(t),r^{\\epss, \\rho}(t,\\cdot),{\\lambda}^{\\epss, \\rho}) \\nonumber \\\\\n&& -{\\cal H}_x^{\\epss, \\rho}(t) \\cdot(X^{\\epss,\\rho}(t)-X^{\\epss}(t))\n -{\\cal H}_u^{\\epss, \\rho}(t) \\cdot(u^{\\epss,\\rho}(t)-u^{\\epss}(t))\\big\\} dt \\bigg ] \\nonumber \\\\\n&& + \\mu^{\\varepsilon,\\rho} {\\mathbb E} \\big [ \\phi^{\\epss,\\rho}(X^{\\epss,\\rho}(T))\n-\\phi^{\\epss}(X^{\\epss}(T))\n- \\phi_x(X^{\\epss,\\rho}(T))\n\\cdot(X^{\\epss,\\rho}(T)-X^{\\epss}(T)) \\big ] \\nonumber \\\\\n&& + {\\lambda}^{\\varepsilon,\\rho} {\\mathbb E} \\big [ \\Phi^{\\epss,\\rho}(X^{\\epss,\\rho}(T))\n-\\Phi^{\\epss}(X^{\\epss}(T))\n- \\Phi_x(X^{\\epss,\\rho}(T))\\cdot(X^{\\epss, \\rho}(T)-y^{\\epss}(T)) \\big ].\n\\end{eqnarray*}\nand\n\\begin{eqnarray*}\nII &\\triangleq& {\\mathbb E} \\bigg [ \\int_0^T {\\cal H}_u^{\\epss, \\rho}(t) \\cdot(u^{\\epss, \\rho}(t)-u^{\\epss}(t)) d t \\bigg ]\n\\end{eqnarray*}\nRecalling Lemma \\ref{lem:3.2} and Assumption \\ref{ass:2.5} and using the Taylor Expansion for $H$ and the\ndominated convergence theorem, we obtain\n\\begin{eqnarray}\\label{eq:4.14}\nI = o(\\rho).\n\\end{eqnarray}\nOn the other hand, similarly, using Lemma \\ref{lem:4.1}, Lemma \\ref{lem:3.2} and Assumption \\ref{ass:2.5} and using the Taylor Expansion for $H$ and the\ndominated convergence theorem, we deduce\n\\begin{eqnarray}\\label{eq:4.19}\nII =\\rho{\\mathbb E} \\bigg [ \\int_0^T( {\\cal H}_u^\\epss(t), v(t)-{u}^\\epss(t))_Udt \\bigg ]+o(\\rho)\n\\end{eqnarray}\nHence, putting \\eqref{eq:4.14} and\n \\eqref{eq:4.19}into \\eqref{eq:6.23} and combing\n\\eqref{eq:3.9}, by the\ndominated convergence theorem we conclude that\n\\begin{eqnarray}\n\\frac{d}{d\\rho}J^\\epss( u^\\epss(\\cdot)+\\rho(v(\\cdot)- u^\\epss(\\cdot)))|_{\\rho=0}\n&=& \\lim_{\\rho\\rightarrow 0} \\frac{J^\\epss(u^\\epss(\\cdot)+\\rho(v(\\cdot)- u^\\epss(\\cdot)))-J^\\epss(u^\\epss(\\cdot))}{\\rho} \\nonumber \\\\\n&=& {\\mathbb E} \\bigg [ \\int_0^T( {\\cal H}^\\epss_u(t), v(t)-{u}^\\epss(t))_U d t \\bigg ]\n\\geq - C \\varepsilon^{\\frac{1}{2}} .\n\\end{eqnarray}\n\\end{proof}\n\nNow we are ready to give the necessary condition of optimality for the existence of the optimal control of Problem \\ref{pro:2.1}.\n\n\\begin{theorem}\nLet Assumptions \\ref{ass:2.5} be satisfied. Let $(\\bar{u}(\\cdot); \\bar{X}(\\cdot))$ be an optimal pair of Problem \\ref{pro:2.1}.\nThen there exist a $({\\lambda},\\mu)$ satisfying $|{\\lambda}|^2+|\\mu|^2=1$ such that\n\\begin{eqnarray}\\label{eq:4.20}\n( {\\cal H}_u(t,{\\bar X}(t),{\\bar u}(t),{\\bar p}(t),\\bar q(t),\\bar r(t,\\cdot), {\\lambda}), u-{\\bar u}(t))_U\\geq 0, \\quad \\forall u\\in U_{ad}, \\quad \\mbox{a.e.} \\ \\mbox{a.s.}.\n\\end{eqnarray}\nHere $\\{ {\\bar p}(t), \\bar q(t),\n \\bar r(t,\\cdot)),0 \\leq t \\leq T \\}$ be the solution of the corresponding adjoint equation \\eqref{eq:6.10} associated\nwith $(\\bar{u}(\\cdot); \\bar{X}(\\cdot))$.\n\\end{theorem}\n\n\\begin{proof}\nFrom \\eqref{eq:5.14}, there exists a pair $({\\lambda},\\mu)$ satisfying $|{\\lambda}|^2+|\\mu|^2=1$. Note that\n\\begin{eqnarray}\\label{eq:4.22}\n\\lim_{\\epss \\rightarrow 0} d\\left( u^{\\varepsilon }\\left( \\cdot \\right) ,{\\bar u}\\left( \\cdot \\right) \\right) =0\n\\end{eqnarray}\nFrom \\ref{lem:4.1}, Lemma \\ref{lem:3.2} and Assumption \\ref{ass:2.5} and \\eqref{eq:5.13}, sending $\\varepsilon$ to $0$ on the both sides of \\eqref{eq:4.16} and using\nthe dominated convergence theorem, we conclude that\n\\begin{eqnarray}\n{\\mathbb E} \\bigg [ \\int_0^T( {\\cal H}_u(t,{\\bar X}(t),{\\bar u}(t),{\\bar p}(t),\\bar q(t),\\bar r(t,\\cdot), {\\lambda}), v(t)-{\\bar u}(t))_Udt \\bigg ] \\geq 0, \\quad \\forall v (\\cdot) \\in \\cal A,\n\\end{eqnarray}\nwhich implies that\n\\eqref{eq:4.20} holds.\nThis completes the proof.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\bibliographystyle{amsplain}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\vspace{1mm}\n\n\n\n\n\\section*{Appendix}\n\n\\setcounter{equation}{0}\n\\renewcommand{\\theequation}{A.\\arabic{equation}}\n\\renewcommand{\\thedefinition}{A.\\arabic{definition}}\n\\renewcommand{\\thetheorem}{A.\\arabic{theorem}}\n\\renewcommand{\\thelemma}{A.\\arabic{lemma}}\n\\renewcommand{\\theassumption}{A.\\arabic{assumption}}\n\nIn this appendix, we introduce some preliminary results of SEEs and BSEEs, including\nexistence, uniqueness and continuous dependence theorems.\n\nConsider a SEE in the Gelfand triple $(V, H, V^*)$:\n\\begin{eqnarray} \\label{eq:3.1}\n \\left\\{\n \\begin{aligned}\n d X (t) = & \\ [ A (t) X (t) + b ( t, X (t)) ] d t\n+ [B(t)X(t)+g( t, { X (t)}) ]d W(t)\n \\\\&\\quad +\\int_E \\sigma (t,e, X(t-))\\tilde \\mu(de,dt), \\\\\nX (0) = & \\ x \\in H , \\quad t \\in [ 0, T ],\n \\end{aligned}\n \\right.\n\\end{eqnarray}\nwhere $A,B,b,g$ and $\\sigma $ are given random mappings\nwhich satisfy the following\nstandard assumptions.\n\n\\begin{assumption} \\label{ass:3.1}\n The operator processes $A:[0,T]\\times \\Omega \\longrightarrow {\\mathscr L} (V, V^*)$ and $B\n : [0,T]\\times \\Omega \\longrightarrow {\\mathscr L} (V, H)$\n are weakly predictable; i.e.,\n $ \\langle A(\\cdot)x, y \\rangle$ and $(B(\\cdot)x, y)_H$\n are both predictable process for every $x, y\\in V, $\n and satisfy the coercive condition, i.e., there exist\n some constants $ C, \\alpha>0$ and $\\lambda$ such that for any $x\\in V$ and each $(t,\\omega)\\in [0,T]\\times \\Omega,$\n \\begin{eqnarray}\n \\begin{split}\n - \\langle A(t)x, x \\rangle +\\lambda ||x||_H^2 \\geq \\alpha\n ||x||_V^2+||Bx||_H^2{\\color{blue},}\n \\end{split}\n \\end{eqnarray}\n and \\begin{eqnarray}\n\\sup_{( t, \\omega ) \\in [0, T] \\times \\Omega} \\| A ( t,\\omega ) \\|_{{\\mathscr L} ( V, V^* )}\n +\\sup_{( t, \\omega ) \\in [0, T] \\times \\Omega} \\| B ( t,\\omega ) \\|_{{\\mathscr L} ( V, H )} \\leq C \\ .\n\\end{eqnarray}\n\\end{assumption}\n\n\\begin{assumption} \\label{ass:3.2}\n The mappings $b:[0,T]\\times \\Omega\\times H \\longrightarrow H$ and $g:[0,T]\\times\\Omega\\times H \\longrightarrow H$ are both $\\mathscr P\\times\n \\mathscr B(H)\/\\mathscr B(H) $-measurable\n such that $b(\\cdot,0), g(\\cdot,0)\\in M_{\\mathscr{F}}^2(0,T;H)\n $; the mapping $\\sigma:[0,T]\\times \\Omega \\times\n E\\times H \\longrightarrow H$ is $\\mathscr P\\times\\mathscr B(E) \\times\n \\mathscr B(H)\/\\mathscr B(H) $-measurable\n such that $\\sigma(\\cdot,\\cdot, 0)\\in {M}_\\mathscr{F}^{\\nu,2}{([0,T]\\times E; H)}$.\n And there exists a constant $C$ such that\n for all $x, \\bar x\\in V$ and a.s.$(t,\\omega)\\in [0,T]\\times \\Omega,$\n \\begin{eqnarray}\n \\begin{split}\n ||b(t,x)-b(t,x)||_H+ ||g(t,x)-g(t,x)||_H\n +||\\sigma(t,\\cdot, x)-\\sigma(t,\\cdot,x)||\n _{M^{\\nu,2}( E; H)} \\leq C||x-\\bar x||_H.\n \\end{split}\n \\end{eqnarray}\n\n\\end{assumption}\n\n\\begin{definition}\n\\label{defn:c1}\nA $V$-valued, $\\{{\\mathscr F}_t\\}_{0\\leq t\\leq T}$-adapted process $X(\\cdot)$ is said to be a solution to the\nSEE \\eqref{eq:3.1}, if $X (\\cdot) \\in { M}_{\n\\mathscr F}^2 ( 0, T; V )$ such that for every $\\phi \\in V$\nand a.e. $( t, \\omega ) \\in [0, T ] \\times \\Omega$, it holds that\n\\begin{eqnarray}\n\\left\\{\n\\begin{aligned}\n( X (t), \\phi )_H =& \\ ( x, \\phi )_H + \\int_0^t \\left < A (s) X (s), \\phi \\right > d s\n+\\int_0^t ( b ( s, X (s){\\color{blue})}, \\phi )_H d s \\\\\n& + \\int_0^t ( B(s)X(s)+ g ( s, X (s) ), \\phi )_H d W (s)\n\\\\& + \\int_0^t \\int_{E} (\\sigma ( s,e, X (s-) ), \\phi )_H d \\tilde \\mu (de,ds), \\quad t \\in [ 0, T ] , \\\\\nX(0) =& \\ x\\in H ,\n\\end{aligned}\n\\right.\n\\end{eqnarray}\nor alternatively, $X (\\cdot)$ satisfies the following It\\^o's equation in $V^*$:\n\\begin{eqnarray}\n\\left\\{\n\\begin{aligned}\nX (t)=& \\ x+ \\int_0^t A (s) X (s)d s\n+\\int_0^t b ( s, X (s))d s + \\int_0^t\n[B(s)X(s)+ g ( s, X (s) )] d W (s)\n\\\\& + \\int_0^t \\int_{E} \\sigma ( s,e, X (s-) ) d \\tilde \\mu (de,ds), \\quad t \\in [ 0, T ] , \\\\\nX(t) =& \\ x\\in H .\n\\end{aligned}\n\\right.\n\\end{eqnarray}\n\\end{definition}\n\n\n\nNow we state our main result.\n\n\\begin{lemma} \\label{thm:3.1}\n Let Assumptions \\ref{ass:3.1}-\\ref{ass:3.2} be\n satisfied by any given coefficients\n $(A,B,b,g,\\sigma)$ of the SEE \\eqref{eq:3.1}. Then for any initial\n value $X(0)=x,$ the\n SEE \\eqref{eq:3.1} has a unique\n solution $X(\\cdot)\\in M_{\\mathscr{F}}^2(0,T;V) \\bigcap S_{\\mathscr{F}}^2(0,T;H).$\n\\end{lemma}\nTo prove this theorem, we first show the following\nresult on the continuous dependence of the solution to the SEE \\eqref{eq:3.1}.\n\n \\begin{lemma} \\label{thm:3.2}\n Let $ X(\\cdot)$ be a solution to\n the SEE \\eqref{eq:3.1}\n with the initial value $X(0)=x$ and\n the coefficients $(A,B, b,g,\\sigma)$\n which satisfy Assumptions \\ref{ass:3.1}-\\ref{ass:3.2}. Then\n the following estimate holds:\n\\begin{eqnarray}\\label{eq:3.4}\n\\begin{split}\n&{\\mathbb E} \\bigg [ \\sup_{0 \\leq t \\leq T} \\| X (t) \\|_H^2 \\bigg\n]\n+ {\\mathbb E} \\bigg [ \\int_0^T \\| X (t) \\|_V^2 d t \\bigg ] \\\\\n& \\leq K \\bigg \\{ ||x||_H^2 + {\\mathbb E}\n\\bigg [ \\int_0^T \\| b ( t, 0) \\|_H^2 d t \\bigg ] + {\\mathbb E}\n\\bigg [ \\int_0^T \\| g ( t, 0) \\|_H^2 d t \\bigg ]\n+ {\\mathbb E} \\bigg [ \\int_{0}^T\\int_E \\| \\sigma (t,e,0) \\|^2_H \\nu(de)d t \\bigg ] \\bigg \\}.\n\\end{split}\n\\end{eqnarray}\nFurthermore, suppose that $ \\bar X(\\cdot)$ is a solution to\n the SEE \\eqref{eq:3.1}\n with the initial value $\\bar X(0)=\\bar x \\in H$ and the coefficients $(A,B, \\bar b, \\bar g,\\bar\\sigma)$\n satisfying Assumptions \\ref{ass:3.1}-\\ref{ass:3.2},\n then we have\n\n \\begin{eqnarray}\\label{eq:3.5}\n&& {\\mathbb E} \\bigg [ \\sup_{0 \\leq t \\leq T} \\| X (t) - {\\bar X} (t) \\|_H^2 \\bigg ]\n+ {\\mathbb E} \\bigg [ \\int_0^T \\| X (t) - {\\bar X} (t) \\|_V^2 d t \\bigg ] \\nonumber \\\\\n&& \\leq K \\bigg \\{ \\|x-\\bar x\\|^2_H\n+ {\\mathbb E} \\bigg [ \\int_0^T \\| b ( t, {\\bar X} (t) )\n- {\\bar b} ( t, {\\bar X} (t) ) \\|_H^2 d t \\bigg ] \\\\\n&&+ {\\mathbb E} \\bigg [ \\int_0^T \\| g ( t, {\\bar X} (t) ) - {\\bar g}( t, {\\bar X} (t) ) \\|_H^2 d t \\bigg ]\n+ {\\mathbb E} \\bigg [ \\int_0^T \\int_{E}\\| \\sigma ( t, e, {\\bar X} (t)) - {\\bar \\sigma}( t, e, {\\bar X} (t) ) \\|_H^2 \\nu(de)d t \\bigg ] \\bigg \\} .\\nonumber\n\\end{eqnarray}\n \\end{lemma}\n\n\n\nNext we consider a BSEE in the Gelfand triple $(V, H, V^*)$:\n\\begin{eqnarray}\\label{eq:2.7}\n\\left\\{\n\\begin{aligned}\ndY(t)=& \\ [ A^*(t)Y(t)+B^*(t)Z(t)\n+f(t, Y(t), Z(t), R(t,\\cdot)) ] dt +Z(t)dW(t)+\\int_{E}R(t,e)\\tilde\\mu(dt,de),\\\\\nY(T)=& \\ \\xi,\n\\end{aligned}\n\\right.\n\\end{eqnarray}\nwhere $(A^*,B^*,f, \\xi)$ are given random mappings. Here $A^*$ and $B^*$\nare the adjoint operators of $A$ and\n$B$, respectively. Furthermore, we assume that the coefficients $(A^*, B^*, f, \\xi)$\nsatisfy the following conditions:\n\n\n\\begin{assumption} \\label{ass:3.1}\n The operator processes $A^*:[0,T]\\times \\Omega \\longrightarrow {\\mathscr L} (V, V^*)$ and $B^*\n : [0,T]\\times \\Omega \\longrightarrow {\\mathscr L} (V, H)$\n are weakly predictable; i.e.,\n $ \\langle A^*(\\cdot)x, y \\rangle$ and $(B^*(\\cdot)x, y)_H$\n are both predictable process for every $x, y\\in V, $\n and satisfy the coercive condition, i.e., there exist\n some constants $ C, \\alpha>0$ and $\\lambda$ such that for any $x\\in V$ and each $(t,\\omega)\\in [0,T]\\times \\Omega,$\n \\begin{eqnarray}\n \\begin{split}\n - \\langle A^*(t)x, x \\rangle +\\lambda ||x||_H \\geq \\alpha\n ||x||_V+||B^*x||_H{\\color{blue},}\n \\end{split}\n \\end{eqnarray}\n and \\begin{eqnarray}\n\\sup_{( t, \\omega ) \\in [0, T] \\times \\Omega} \\| A^* ( t,\\omega ) \\|_{{\\mathscr L} ( V, V^* )}\n +\\sup_{( t, \\omega ) \\in [0, T] \\times \\Omega} \\| B^* ( t,\\omega ) \\|_{{\\mathscr L} ( V, H )} \\leq C \\ .\n\\end{eqnarray}\n\\end{assumption}\n\n\\begin{assumption} \\label{ass:3.2}\n The mapping $\\xi: \\Omega \\rightarrow H$ is ${\\cal F}_T$-measurable such that\n $\\xi\\in L^2(\\Omega,{\\mathscr{F}_T},\n \\mathbb P;H).$\n The mappings $f:[0,T]\\times \\Omega\\times H \\times H\\times\n M^{\\nu,2}( E; H) \\longrightarrow $ are both $\\mathscr P\\times\n \\mathscr B(H)\\times \\mathscr B(H)\n \\times\n \\mathscr B(M^{\\nu,2}( E; H))\/\\mathscr B(H) $-measurable\n such that $f(\\cdot,0,0,0)\\in M_{\\mathscr{F}}^2(0,T;H)\n $.\n And there exists a constant $C$ such that\n for all $$(t,y,z,r,\\bar{y},\\bar{z},\n \\bar r)\n\\in [0, T]\\times H\\times H\\times M^{\\nu,2}( E; H)\\times H\\times H \\times\nM^{\\nu,2}( E; H)$$ and a.s.$(t,\\omega)\\in [0,T]\\times \\Omega,$\n \\begin{eqnarray}\n \\begin{split}\n ||f(t,y,z,r)-b(t,y,z,r)||_H \\leq C\\bigg\\{||y-\\bar y||_H+||z-\\bar z||_H+\n ||r-\\bar r||_{M^{\\nu,2}( E; H)}\\bigg\\}.\n \\end{split}\n \\end{eqnarray}\n\\end{assumption}\n\n\n\n\nIf the coefficients $(A^*, B^*,f,\\xi)$ satisfy Assumptions \\ref{ass:3.1} and\n\\ref{ass:3.2}, they are said to be a generator of BSEE \\eqref{eq:2.7}.\n\n\\begin{definition}\\label{defn:c1}\nA $(V \\times H \\times M^{\\nu,2}( E; H) )$-valued, ${\\mathbb F}$-adapted process $( Y (\\cdot), Z (\\cdot), R (\\cdot, \\cdot) )$\nis called a solution to the BSEE \\eqref{eq:2.7}, if $Y (\\cdot) \\in { M}_{\\mathscr F}^2 ( 0, T; V )$,\n$Z (\\cdot) \\in { M}_{\\mathscr F}^2 ( 0, T; H )$ and $R (\\cdot, \\cdot) \\in { M}_{\\mathscr F}^{\\nu, 2}(0,T; H)$ such that for every $\\phi \\in V$ and\na.e. $( t, \\omega ) \\in [ 0, T ] \\times \\Omega$, it holds that\n\\begin{eqnarray}\\label{eq:c5}\n( Y (t), \\phi )_H &=& (\\xi, \\phi)_H\n- \\int_t^T \\Big\\langle A^* (s) Y (s) +B^*(s)Z(s)+f ( s, Y (s), Z (s), Y ( s ), R(s, \\cdot) ), \\phi \\Big\\rangle d t \\nonumber \\\\\n&& - \\int_t^T ( Z (s), \\phi )_H d W (s) -\\int_t^T\\int_E (R(s,e), \\phi)_H\\tilde \\mu(ds,de), \\quad t \\in [0, T] ,\n\\end{eqnarray}\nor alternatively, $( Y (\\cdot), Z (\\cdot), R (\\cdot, \\cdot) )$ satisfies the following It\\^{o}'s equation in $V^*$:\n\\begin{eqnarray}\nY(t)&=& \\xi -\\int_t^T \\big[A^* (s) Y (s) d s+ B^* (s) Z (s)+f ( t, {Y} (s), {Z} (s), R(s,\\cdot) ) \\big] d s \\nonumber \\\\\n&&- \\int_t^T Z (s) d W (s)-\\int_t^T \\int_E R (s,e) d \\tilde \\mu(ds,de) , \\quad t \\in [0, T] .\n\\end{eqnarray}\n\\end{definition}\n\n\\begin{lemma}[{\\bf Existence and Uniqueness of BSEE \\cite{Me}}]\\label{lem:1.3}\nFor any generator $(A^*,B^*, f,\\xi)$, BSEE \\eqref{eq:2.7} has a unique solution $(Y(\\cdot), Z(\\cdot), R(\\cdot,\\cdot)).$ Moreover, $Y(\\cdot)\\in S_{\\mathbb{ F}}^2(0,T;H)$.\n\\end{lemma}\n\n\\begin{lemma}[{\\bf Continuous Dependence Theorem of BSEE}]\\label{lem:1.4}\nLet $(A^*, B^*, f,\\xi)$ and $(A^*,\nB^*, \\bar{f},\\bar{\\xi})$ be two generators of BSEE \\eqref{eq:2.7}. Suppose that $(Y(\\cdot),Z(\\cdot), R(\\cdot,\\cdot))$ and $(\\bar{Y}(\\cdot),\\bar{Z}(\\cdot),\n\\bar R(\\cdot,\\cdot))$ are the\nsolutions of BSEE \\eqref{eq:2.7} corresponding to $(A^*,B^*, f,\\xi)$ and $(A^*,B^*, \\bar f,\\bar{\\xi})$, respectively. Then\n\\begin{eqnarray}\\label{eq:2.14}\n&& {\\mathbb E} \\bigg [ \\sup_{t \\in [0, T]}\\|Y(t) -\\bar{Y}(t)\\|_H^2 \\bigg ] + {\\mathbb E} \\bigg [ \\int_{0}^T\\|Y(t)-\\bar{Y}(t)\\|_V^2dt \\bigg ]\n+ {\\mathbb E} \\bigg [ \\int_0^T\\|Z(t)-\\bar{Z}(t)\\|^2_Hdt \\bigg ]\n\\nonumber\n\\\\&&~~~+{\\mathbb E} \\bigg [ \\int_0^T\\int_E\\|R(t,e)-\\bar{R}(t,e)\\|^2_H\n\\nu(de)dt \\bigg ] \\nonumber \\\\\n&& \\leq K \\bigg \\{ {\\mathbb E} [ \\|\\xi-\\bar{\\xi}\\|_H^2 ] + {\\mathbb E} \\bigg [ \\int_0^T\\|f(t,\\bar{Y}{(t),\\bar{Z}(t),\n\\bar R(t,\\cdot)})-\\bar{f}(t, \\bar{Y}{(t),\\bar{Z}(t)},\n\\bar R(t,\\cdot))\\|^2_Hdt \\bigg ] \\bigg \\} ,\n\\end{eqnarray}\nwhere $K$ is a positive constant depending only on $T$ and the constants $C,\\alpha, {\\lambda}$ in Assumption \\ref{ass:3.1}.\n\nIn particular, if $(A^*, B^*,\\bar{f},\\bar{\\xi})=(A^*, B^*, 0,0)$, the following a priori estimate holds\n\\begin{eqnarray}\\label{eq:2.15}\n&&{\\mathbb E} \\bigg [ \\sup_{t \\in [0, T]}\\|Y(t)\\|^2_H \\bigg ] + {\\mathbb E} \\bigg [ \\int_{0}^T\\|Y(t)\\|_V^2 d t \\bigg ] + {\\mathbb E} \\bigg [ \\int_0^T\\|Z(t)\\|^2_Hdt\n + {\\mathbb E} \\bigg [ \\int_0^T\\int_{E}\\|R(t,e)\\|^2_H\\nu(de)dt\\bigg ]\n\\\\&&\\leq K \\bigg\\{ {\\mathbb E} [ \\|\\xi\\|^2_H ] + {\\mathbb E} \\bigg [ \\int_0^T\\|f(t,0,0,0) \\|^2_Hdt \\bigg ] \\bigg \\}.\n\\end{eqnarray}\n\\end{lemma}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\nThe theory of Discrete Exterior Calculus (DEC) is a relatively recent discretization \\cite{HiraniThesis} of the classical theory of \nExterior Differential Calculus, a theory developed by \nE. Cartan \\cite{Cartan} which has been a fundamental tool in Differential Geometry and Topology for over a century. \nThe aim of DEC is to solve partial differential equations preserving their geometrical and physical\nfeatures as much as possible.\nThere are only a few papers about implementions of DEC to solve certain PDEs, such as \nthe Darcy flow and Poisson's equation \\cite{Hirani_K_N}, \nthe Navier-Stokes equations \\cite{Mohamedetal},\nthe simulation of elasticity, plasticity and failure of isotropic materials \\cite{Dassiosetal}, some comparisons with \nthe finite differences and finite volume methods on regular flat meshes \\cite{Griebel_R_S}, as well as applications in digital geometry processing \\cite{Craneetal}.\n\nIn this paper, we describe a local formulation of DEC which is reminiscent of that of the Finite Element Method (FEM) since,\nonce the local systems of equations have been established, they can be assembled into a global linear system.\nThis local formulation is also efficient and helpful in understanding various features of DEC that can otherwise remain unclear while dealing \nwith an entire mesh. We will, therefore, take a local approach when recalling all the objects required by DEC \\cite{Esqueda1}.\nOur main results are the following: \n\\begin{itemize}\n\\item We develop a local formulation of DEC analogous to that of FEM, which\nallows a natural treatment of heterogeneous material properties assigned to subdomains (element by element) and \neliminates the need of dealing with it through ad hoc modifications of the global discrete Hodge star operator. \n\n\\item Guided by the local formulation,\nwe also deduce a natural way to approximate the flux\/gradient-vector of a discretized function, as well as the anisotropic flux vector.\nWe carry out a comparison of the formulas defining the flux in both DEC and Finite Element Method with linear interpolation functions (FEML).\n\n\\item From the local formulation, we deduce the local DEC-discretization of the anisotropic Poisson equation. More precisely, in Exterior Differential Calculus the anisotropy tensor acts by {\\em pullback} on the differential of the unknown function. Here, we deduce how the anisotropy tensor acts on primal 1-forms.\nWe also carry out an algebraic comparison of the DEC and FEML local formulations of the anisotropic Poisson equation.\n\n\\item We present three numerical examples of the approximate solutions to the stationary anisotropic Poisson equation on different domains using DEC and FEML. The numerical examples show numerical convergence and a competitive performance \nof DEC, as well as a computational cost similar to that of FEML. In fact, the numerical solutions with both methods on fine meshes are identical, and\nDEC shows a slightly better performance than FEML on coarse meshes.\n\\end{itemize}\n\nThe paper is organized as follows.\nIn Section \\ref{sec: preliminaries}, we describe the local versions of the discrete derivative operator,\nthe dual mesh and the discrete Hodge star operator. \nIn Section \\ref{sec: anisotropy}, we deduce the natural way of computing flux vectors in DEC (which turns out to be equivalent to the FEML procedure), as well as the anisotropic flux vectors. \nIn Section \\ref{sec: comparison}, \nwe present the local DEC formulation of the 2D anisotropic Poisson equation\nand compare it with the local system of FEML, proving that the diffusion terms are identical while the source terms are discretized differently due to a different area-weight assignment for the nodes. \nIn Section \\ref{sec: extension}, we re-examine some of the local DEC quantities.\nIn Section \\ref{sec: examples}, we present and compare numerical examples of DEC and FEML approximate solutions to the 2D anisotropic \nPoisson equation on different domains with meshes of various resolutions. \nIn Section \\ref{sec: conclusions} we summarize the contributions of this paper.\n\n\n{\\em Acknowledgements}. The second named author was partially supported by a CONACyT grant,\nand would like to thank \nthe International Centre for Numerical Methods in Engineering (CIMNE) and the University of Swansea\nfor their hospitality. We gratefully acknowledge the support of\nNVIDIA Corporation with the donation of the Titan X Pascal GPU used for this research.\n\n\\section{Preliminaries on DEC from a local viewpoint}\\label{sec: preliminaries}\n\nLet us consider a primal mesh made up of a single (positively oriented) triangle.\n \\begin{center}\n \\includegraphics[width=0.4\\textwidth]{Triangle01.png}%\n \\captionof{figure}{Triangle $[v_1,v_2,v_3]$.}\n \\end{center}\n\\subsection{Boundary operator}\nThere is a well known boundary operator \n\\begin{equation}\n\\partial [v_1,v_2,v_3]=[v_2,v_3]-[v_1,v_3]+[v_1,v_2],\\label{eq: boundary 2-1}\n\\end{equation}\nwhich describes the boundary of the triangle as an alternated sum of its oriented edges $[v_1,v_2]$, $[v_2,v_3]$ and $[v_3,v_1]$. \nSimilarly, one can compute the \nboundary of each edge\n\\begin{eqnarray}\n\\partial [v_1,v_2] &=& [v_2]-[v_1], \\nonumber\\\\ \n\\partial [v_2,v_3] &=& [v_3]-[v_2], \\label{eq: boundary 1-0}\\\\ \n\\partial [v_3,v_1] &=& [v_1]-[v_3]. \\nonumber\n\\end{eqnarray}\nIf we consider \n\\begin{itemize} \n\\item the symbol $[v_1,v_2,v_3]$ as a basis vector of a 1-dimensional vector space,\n\\item the symbols $[v_1,v_2]$, $[v_2,v_3]$, $[v_3,v_1]$ as an ordered basis of a 3-dimensional vector space, \n\\item the symbols $[v_1]$, $[v_2]$, $[v_3]$ as an ordered basis of a 3-dimensional vector space,\n\\end{itemize}\nthen the map (\\ref{eq: boundary 2-1}), which sends the oriented triangle to a sum of its oriented edges, is represented by the matrix\n\\[\\left(\\begin{array}{r}\n1 \\\\ \n1 \\\\ \n1\n\\end{array}\\right) ,\\]\nwhile the map (\\ref{eq: boundary 1-0}), which sends the oriented edges to sums of their oriented vertices, is represented by the matrix\n\\[\\left(\\begin{array}{rrr}\n-1 & 0 & 1 \\\\ \n1 & -1 & 0 \\\\ \n0 & 1 & -1\n\\end{array} \\right).\\]\n\n\\subsection{Discrete derivative}\\label{subsec: discrete derivative}\nIt has been argued that the DEC discretization of the differential of a function is given by the transpose of the matrix of the \nboundary operator on edges (see \\cite{HiraniThesis, Esqueda1}). More precisely, suppose we have a function discretized by its values at the vertices\n\\[f\\sim \\left(\\begin{array}{l}\nf_1\\\\\nf_2\\\\\nf_3\n \\end{array}\n\\right).\\]\nIts discrete derivative, according to DEC, is \n\\begin{eqnarray*}\n\\left(\\begin{array}{rrr}\n-1 & 0 & 1 \\\\ \n1 & -1 & 0 \\\\ \n0 & 1 & -1\n\\end{array} \\right)^T\n\\left(\\begin{array}{l}\nf_1\\\\\nf_2\\\\\nf_3\n \\end{array}\n\\right)\n&=&\n\\left(\\begin{array}{rrr}\n-1 & 1 & 0 \\\\ \n0 & -1 & 1 \\\\ \n1 & 0 & -1\n\\end{array} \\right)\n\\left(\\begin{array}{l}\nf_1\\\\\nf_2\\\\\nf_3\n \\end{array}\n\\right)\\\\\n&=&\n\\left(\\begin{array}{l}\nf_2-f_1\\\\\nf_3-f_2\\\\\nf_1-f_3\n \\end{array}\n\\right) .\n\\end{eqnarray*}\nIndeed, such differences are rough approximations of the directional derivatives of $f$. For instance, $f_2-f_1$ is a rough approximation of the directional derivative of $f$ at $v_1$ in the direction of the vector $v_2-v_1$, i.e.\n\\[f_2-f_1 \\approx df_{v_1}(v_2-v_1) .\\]\nIt is precisely in this sense that, according to DEC, \n\\begin{itemize}\n\\item the value $f_2-f_1$ is assigned to the edge $[v_1,v_2]$, \n\\item the value $f_3-f_2$ is assigned to the edge $[v_2,v_3]$, \n\\item and the value $f_1-f_3$ is assigned to the edge $[v_3,v_1]$.\n\\end{itemize}\nLet\n\\[D_0:=\\left(\\begin{array}{rrr}\n-1 & 1 & 0 \\\\ \n0 & -1 & 1 \\\\ \n1 & 0 & -1\n\\end{array} \\right).\n\\]\n\n\\subsection{Dual mesh}\n\nThe dual mesh of the primal mesh consisting of a single triangle is constructured as follows:\n\\begin{itemize}\n \\item To the 2-dimensional triangular face $[v_1,v_2,v_3]$ will correspond the 0-dimensional point given by the circumcenter $c$ of the triangle.\n \\begin{center}\n \\includegraphics[width=0.4\\textwidth]{Triangle02.png}%\n \\captionof{figure}{Circumcenter $c$ of the triangle $[v_1,v_2,v_3]$.}\n \\end{center}\n\n \\item To the 1-dimensional edge $[v_1,v_2]$ will correspond the 1-dimensional straight line segment $[p_1,c]$ joining the midpoint $p_1$ of the edge $[v_1,v_2]$ to the circumcenter $c$. \n Similarly for the other edges.\n \\begin{center}\n \\includegraphics[width=0.4\\textwidth]{Triangle03.png}%\n \\captionof{figure}{Dual segment $[p_1,c]$ of the edge $[v_1,v_2]$.}\n \\end{center}\n\n \\item To the 0-dimensional vertex\/node $[v_1]$ will correspond the 2-dimensional quadrilateral $[v_1,p_1,c,p_3]$.\n \\begin{center}\n \\includegraphics[width=0.4\\textwidth]{Triangle04.png}%\n \\captionof{figure}{Dual quadrilateral $[v_1,p_1,c,p_3]$ of the vertex $[v_1]$.}\n \\end{center}\n\n\\end{itemize}\n\n\\subsection{Discrete Hodge star}\n\nFor the Poisson equation in 2D, we need two matrices: one relating original edges to dual edges, and another relating vertices to dual cells.\n\\begin{itemize}\n\\item The discrete Hodge star map $M_1$ applied to the discrete differential of a discretized function $f\\sim (f_1,f_2,f_3)$ \nis given as follows:\n\\begin{itemize}\n \\item the value $f_2-f_1$ assigned to the edge $[v_1,v_2]$ is changed to the new value \n \\[{{\\rm length}[p_1,c]\\over {\\rm length}[v_1,v_2]}(f_2-f_1)\\]\n assigned to the segment $[p_1,c]$;\n \\item the value $f_3-f_2$ assigned to the edge $[v_2,v_3]$ is changed to the new value \n \\[{{\\rm length}[p_2,c]\\over {\\rm length}[v_2,v_3]}(f_3-f_2)\\]\n assigned to the segment $[p_2,c]$;\n \\item the value $f_1-f_3$ assigned to the edge $[v_3,v_1]$ is changed to the new value \n \\[{{\\rm length}[p_3,c]\\over {\\rm length}[v_3,v_1]}(f_1-f_3)\\]\n assigned to the edge $[p_3,c]$.\n\\end{itemize}\nIn other words, \n\\[M_1=\\left(\\begin{array}{ccc}\n{{\\rm length}[p_1,c]\\over {\\rm length}[v_1,v_2]} & 0 & 0 \\\\ \n0 & {{\\rm length}[p_2,c]\\over {\\rm length}[v_2,v_3]} & 0 \\\\ \n0 & 0 & {{\\rm length}[p_3,c]\\over {\\rm length}[v_3,v_1]}\n\\end{array} \\right).\\]\n\n\n\\item Similarly, the discrete Hodge star map $M_0$ on values on vertices is given as follows\n\\begin{itemize}\n \\item the value $f_1$ assigned to the vertex $[v_1]$ is changed to the new value \n \\[{\\rm Area}[v_1,p_1,c,p_3]f_1\\]\n assigned to the quadrilateral $[v_1,p_1,c,p_3]$;\n \\item the value $f_2$ assigned to the vertex $[v_2]$ is changed to the new value\n \\[{\\rm Area}[v_2,p_2,c,p_1]f_2\\]\n assigned to the quadrilateral $[v_2,p_2,c,p_1]$;\n \\item the value $f_3$ assigned to the vertex $[v_3]$ is changed to the new value\n \\[{\\rm Area}[v_3,p_3,c,p_2]f_2\\]\n assigned to the quadrilateral $[v_3,p_3,c,p_2]$.\n\\end{itemize}\nIn other words,\n\\[M_0=\\left(\\begin{array}{ccc}\n{\\rm Area}[v_1,p_1,c,p_3] & 0 & 0 \\\\ \n0 & {\\rm Area}[v_2,p_2,c,p_1] & 0 \\\\ \n0 & 0 & {\\rm Area}[v_3,p_3,c,p_2]\n\\end{array} \\right).\\]\n\\end{itemize}\n\n\n\n\\section{Flux and anisotropy}\\label{sec: anisotropy}\nIn this section, we deduce the DEC formulae for the local flux, the local anisotropic flux and the local anisotropy operator \nfor primal 1-forms.\n\n\\subsection{The flux in local DEC}\\label{subsec: local DEC flux}\nWe wish to find a natural construction for the discrete flux (discrete gradient vector) of a discrete function.\nRecall from Vector Calculus that the directional derivative of a differentiable function $f:\\mathbb{R}^2\\longrightarrow \\mathbb{R}$ at a point $p\\in \\mathbb{R}^2$ in the direction of $w\\in\\mathbb{R}^2$ is defined by\n\\begin{eqnarray*}\ndf_p(w)&:=&\\lim_{t\\rightarrow 0}{f(p+tw)-f(p)\\over t}\\\\\n&=&\\nabla f(p)\\cdot w.\n\\end{eqnarray*}\nThus, we have three Vector Calculus identities\n\\begin{eqnarray*}\n df_{v_1}(v_2-v_1) &=& \\nabla f(v_1)\\cdot (v_2-v_1) ,\\\\\n df_{v_2}(v_3-v_2) &=& \\nabla f(v_2)\\cdot (v_3-v_2),\\\\\n df_{v_3}(v_1-v_3) &=& \\nabla f(v_3)\\cdot (v_1-v_3) .\n\\end{eqnarray*}\nAs in subsection \\ref{subsec: discrete derivative}, the rough approximations to directional derivatives of a function $f$ in the directions of the (oriented) edges are given as follows\n\\begin{eqnarray*}\n df_{v_1}(v_2-v_1) &\\approx& f_2-f_1 ,\\\\\n df_{v_2}(v_3-v_2) &\\approx& f_3-f_2 ,\\\\\n df_{v_3}(v_1-v_3) &\\approx& f_1-f_3 .\n\\end{eqnarray*}\nThus, if we want to find a discrete gradient vector $W_1$ of $f$\nat the point $v_1$, we need to solve the equations of approximations\n\\begin{eqnarray}\nW_1\\cdot (v_2-v_1) &=& f_2-f_1 \\label{eq: W1-1}\\\\\nW_1\\cdot (v_3-v_1) &=& f_3-f_1.\\label{eq: W1-2}\n\\end{eqnarray} \nIf\n\\begin{eqnarray*}\nv_1&=&(x_1,y_1),\\\\\nv_2&=&(x_2,y_2),\\\\\nv_3&=&(x_3,y_3),\n\\end{eqnarray*}\nthen\n\\begin{eqnarray*}\nW_1\n&=&\n\\left({f_1y_2-f_1y_3-f_2y_1+f_2y_3+f_3y_1-f_3y_2\\over x_1y_2-x_1y_3-x_2y_1+x_2y_3+x_3y_1-x_3y_2}\n, -{f_1x_2-f_1x_3-f_2x_1+f_2x_3+f_3x_1-f_3x_2\\over x_1y_2-x_1y_3-x_2y_1+x_2y_3+x_3y_1-x_3y_2}\\right)^T\n\\end{eqnarray*}\n\n\nNow, if we were to find \na discrete gradient vector $W_2$ of $f$\nat the point $v_2$, we need to solve the equations\n\\begin{eqnarray}\nW_2\\cdot (v_1-v_2) &=& f_1-f_2\\label{eq: W2-1}\\\\\nW_2\\cdot (v_3-v_2) &=& f_3-f_2.\\nonumber\n\\end{eqnarray} \nThe vectors $W_2$ solving these equations is actually equal to $W_1$. \nIndeed, consider\n\\begin{eqnarray*}\nf_3-f_1 &=& W_1\\cdot (v_3-v_1) \\\\\n&=& W_1\\cdot (v_3-v_2+v_2-v_1)\\\\\n&=& W_1\\cdot (v_3-v_2)+W_1\\cdot (v_2-v_1)\\\\\n&=& W_1\\cdot (v_3-v_2)+f_2-f_1,\n\\end{eqnarray*} \nso that\n\\begin{equation}\n W_1\\cdot (v_3-v_2) = f_3-f_2 .\\label{eq: W1-3}\n\\end{equation} \nThus, adding up (\\ref{eq: W1-1}) and (\\ref{eq: W2-1}) we get\n\\begin{equation}\n(W_1-W_2)\\cdot (v_2-v_1)=0.\\label{eq: W1-W2-1}\n\\end{equation}\nSubtracting (\\ref{eq: W1-2}) from (\\ref{eq: W1-3}) we get\n\\begin{equation}\n(W_1-W_2)\\cdot (v_3-v_2)=0.\\label{eq: W1-W2-2}\n\\end{equation}\nSince $v_2-v_1$ and $v_3-v_2$ are linearly independent \nand the two inner products in (\\ref{eq: W1-W2-1}) and (\\ref{eq: W1-W2-2}) vanish,\n\\[W_1-W_2=0.\\]\n\nAnalogously, the corresponding gradient vector $W_3$ of $f$ at the vertex $v_3$ is equal to $W_1$.\nThis means that the three approximate gradient vectors at the three vertices coincide. \nLet us call this unique vector $W$.\nNote that discrete flux $W$ satisfies \n\\begin{eqnarray*}\nW\\cdot (v_2-v_1) &=& f_2-f_1,\\\\\nW\\cdot (v_3-v_1) &=& f_3-f_1,\\\\\nW\\cdot (v_3-v_2) &=& f_3-f_2 .\n\\end{eqnarray*}\nThis means that the primal 1-form discretizing $df$ can be obtained by the dot products of the discrete flux $W$ with the \nvectors of the triangle's edges.\n\n\\vspace{.1in}\n\n{\\bf Remark}.\nMore generally, we can see that {\\em any vector which is constant on the triangle, naturally gives a primal 1-form on the edges of the triangle \nby means of its dot products with the triangle's edge-vectors}.\n\n\n\n\\subsubsection{Comparison of DEC and FEML local fluxes}\n\nThe local flux (gradient) of $f$ in FEML is given by\n\\[\n\\left(\n\\begin{array}{ccc}\n{\\partial N_1\\over\\partial x} & {\\partial N_2\\over\\partial x} & {\\partial N_3\\over\\partial x} \\\\ \n{\\partial N_1\\over\\partial y} & {\\partial N_2\\over\\partial y} & {\\partial N_3\\over\\partial y}\n\\end{array} \n\\right)\n\\left(\n\\begin{array}{c}\nf_1 \\\\ \nf_2 \\\\ \nf_3\n\\end{array} \n\\right),\n\\]\nwhere\n\\begin{eqnarray*}\nN_1 &=& {1\\over 2A}[(y_2-y_3)x+(x_3-x_2)y+x_2y_3-x_3y_2] ,\\\\\nN_2 &=& {1\\over 2A}[(y_3-y_1)x+(x_1-x_3)y+x_3y_1-x_1y_3] ,\\\\\nN_3 &=& {1\\over 2A}[(y_1-y_2)x+(x_2-x_1)y+x_1y_2-x_2y_1] ,\n\\end{eqnarray*}\nand\n\\begin{eqnarray*}\nA &=& {1\\over 2} [(x_2y_3-x_3y_2) -(x_1y_3-x_3y_1)+(x_1y_2-x_2y_1)]\n\\end{eqnarray*}\nis the area of the triangle.\nExplicitly\n\\begin{eqnarray*}\n\\left(\n\\begin{array}{ccc}\n{\\partial N_1\\over\\partial x} & {\\partial N_2\\over\\partial x} & {\\partial N_3\\over\\partial x} \\\\ \n{\\partial N_1\\over\\partial y} & {\\partial N_2\\over\\partial y} & {\\partial N_3\\over\\partial y}\n\\end{array} \n\\right)\n&=&\n{1\\over 2A}\n\\left(\n\\begin{array}{ccc}\ny_2-y_3 & y_3-y_1 & y_1-y_2 \\\\ \nx_3-x_2 & x_1-x_3 & x_2-x_1\n\\end{array} \n\\right) \n\\end{eqnarray*}\nso that the FEML flux is given by\n\\[\n\\left(\n{[(y_2-y_3)f_1+(y_3-y_1)f_2+(y_1-y_2)f_3]\\over 2A} ,\n{[(x_3-x_2)f_1+(x_1-x_3)f_2+(x_2-x_1)f_3]\\over 2A}\n \\right)^T,\n\\]\nand we can see that its formula coincides with that of the DEC flux.\n\n\n\\subsection{The anisotropic flux vector in local DEC} \\label{subsec: discrete anisotropic flux}\n\nWe will now discuss how to discretize anisotropy in 2D DEC.\nLet $K$ denote the symmetric anisotropy tensor \n\\[K=\\left(\\begin{array}{cc}\nk_{11} & k_{12} \\\\ \nk_{12} & k_{22}\n\\end{array} \\right)\\]\nand recall the anisotropic Poisson equation\n\\[-\\nabla\\cdot (K\\, \\nabla f) = q.\\]\nAs in Subsection \\ref{subsec: local DEC flux}, we wish to find a vector $W'$ which will play the role of a discrete version of the {\\em anisotropic flux vector} $K \\nabla f$.\n\nFirst observe that, since $K$ is symmetric, for any $w\\in\\mathbb{R}^2$\n\\begin{eqnarray*}\n(K\\nabla f(p))\\cdot w \n &=& \n \\nabla f(p)\\cdot (K^Tw)\\\\\n &=& \n \\nabla f(p)\\cdot (Kw)\\\\\n &=&\n df_p(Kw)\\\\\n &=&\n (df_p\\circ K)(w)\\\\\n &=:&\n (K^*df_p)(w),\n\\end{eqnarray*}\nwhere $K^*df_p$ is called the {\\em pullback} of $df_p$ by $K$. \nThese identities mean that in order to discretize the anisotropic flux we need to understand the discretization\nof the linear functional $df_p\\circ K$. \nLet us suppose that {\\em $K$ is constant on our triangle}.\nAs before, we have three natural vectors on the triangle, \n\\begin{eqnarray*}\nw_1 &=& v_2-v_1,\\\\\nw_2 &=& v_3-v_2,\\\\\nw_3 &=& v_1-v_3.\n\\end{eqnarray*}\nGiven the vector $Kw_1$, we have the option to write it down as a linear combination of two of the three aforementioned vectors.\nSince $w_1$ is being used already, we use the other two vectors, i.e.\n\\begin{eqnarray*}\nKw_1 &=& \\lambda_1 w_2 + \\mu_1 w_3,\n\\end{eqnarray*}\nfor some $\\lambda_1,\\mu_1\\in\\mathbb{R}$.\nSimilarly,\n\\begin{eqnarray*}\nKw_2 &=& \\lambda_2 w_3 + \\mu_2 w_1,\\\\\nKw_3 &=& \\lambda_3 w_1 + \\mu_3 w_2,\n\\end{eqnarray*}\nfor some $\\lambda_2,\\mu_2,\\lambda_3,\\mu_3\\in\\mathbb{R}$.\nThese equations can be solved for $\\lambda_1,\\lambda_2,\\lambda_3,\\mu_1,\\mu_2,\\mu_3$.\nNow\n\\begin{eqnarray*}\n\\left(K\\nabla f(v_3)\\right)\\cdot w_1 \n &=&\n df_{v_3}(Kw_1)\\\\\n &=&\n df_{v_3}(\\lambda_1 w_2 + \\mu_1 w_3) \\\\\n &=&\n \\lambda_1 df_{v_3}(w_2) + \\mu_1 df_{v_1}(w_3) \\\\\n &=&\n \\lambda_1 df_{v_3}(v_3-v_2) + \\mu_1 df_{v_3}(v_1-v_3) \\\\\n &=&\n \\lambda_1 df_{v_3}(-(v_2-v_3)) + \\mu_1 df_{v_3}(v_1-v_3) \\\\\n &=&\n -\\lambda_1 df_{v_3}(v_2-v_3) + \\mu_1 df_{v_3}(v_1-v_3) .\n\\end{eqnarray*}\nSimilarly, for the vectors $w_2$ and $w_3$ we have the identities \n\\begin{eqnarray*}\n\\left(K\\nabla f(v_1)\\right)\\cdot w_2 &=& \\lambda_2 df_{v_1}(w_3) + \\mu_2 df_{v_1}(w_1),\\\\\n\\left(K\\nabla f(v_2)\\right)\\cdot w_3 &=& \\lambda_3 df_{v_3}(w_1) +\\mu_3 df_{v_3}(w_2).\n\\end{eqnarray*}\nThese equations lead to the three equations of approximations\n\\begin{eqnarray}\nW'\\cdot w_1\n & = &\n \\lambda_1 (f_3-f_2) + \\mu_1 (f_1-f_3), \\nonumber \\\\\nW'\\cdot w_2\n & = &\n \\lambda_2 (f_1-f_3) + \\mu_2 (f_2-f_1), \\label{eq: system equations 1 for anisotropic flux vector}\\\\\nW'\\cdot w_3\n & = &\n \\lambda_3 (f_2-f_1) + \\mu_3 (f_3-f_2). \\nonumber\n\\end{eqnarray}\nwhere $W'$ is the vector that should approximate $K\\nabla f(v_3)$, $K\\nabla f(v_3)$ and $K\\nabla f(v_3)$.\nThus, in order to find the discrete version $W'$ of the anisotropic flux vector $K\\nabla f$ on the triangle, we need to solve \nthe system (\\ref{eq: system equations 1 for anisotropic flux vector})\n\nThe system (\\ref{eq: system equations 1 for anisotropic flux vector}) has a unique solution. Indeed, since\n\\[w_1+w_2+w_3=0,\\] \nthen\n\\[Kw_1+Kw_2+Kw_3=0,\\] \ni.e.\n\\begin{eqnarray*}\n(\\lambda_3+\\mu_2-\\lambda_2-\\mu_1) w_1 + (\\lambda_1+\\mu_3-\\lambda_2-\\mu_1) w_2\n =0.\n\\end{eqnarray*}\nSince $w_1$ and $w_2$ are linearly independent\n\\begin{eqnarray*}\n\\lambda_3+\\mu_2-\\lambda_2-\\mu_1&=&0\\\\\n\\lambda_1+\\mu_3-\\lambda_2-\\mu_1&=&0.\n\\end{eqnarray*}\ni.e.\n\\begin{eqnarray*}\n\\lambda_3&=&\\lambda_2+\\mu_1-\\mu_2\\\\\n\\mu_3&=&-\\lambda_1+\\lambda_2+\\mu_1.\n\\end{eqnarray*}\nThus, making the appropriate substitutions, we see that\nthe third equation in (\\ref{eq: system equations 1 for anisotropic flux vector}) is dependent on the first two independent equations,\nand there is a unique vector $W'$ that solves the system.\n\nFor the sake of completeness, the values of the parameters are:\n\\begin{eqnarray*}\n\\lambda_1\n &=&\n {[k_{11}(x_2-x_1)+k_{12}(y_2-y_1)](y_1-y_3) -[k_{12}(x_2-x_1)+k_{22}(y_2-y_1)](x_1-x_3)\n \\over (x_3-x_2)(y_1-y_3)-(x_1-x_3)(y_3-y_2)}\\\\ \n &=&\n -{J(w_3)\\cdot K(w_1) \\over 2A},\n \\\\\n\\mu_1\n &=&\n -{[[k_{11}(x_2-x_1)+k_{12}(y_2-y_1)](y_3-y_2)-[k_{12}(x_2-x_1)+k_{22}(y_2-y_1)](x_3-x_2) \n \\over (x_3-x_2)(y_1-y_3)-(x_1-x_3)(y_3-y_2)}\\\\ \n &=&\n {J(w_2)\\cdot K(w_1) \\over 2A},\n \\\\\n\\lambda_2\n &=&\n {[k_{11}(x_3-x_2)+k_{12}(y_3-y_2)](y_2-y_1) -[k_{12}(x_3-x_2)+k_{22}(y_3-y_2)](x_2-x_1)\n \\over (x_1-x_3)(y_2-y_1)-(x_2-x_1)(y_1-y_3)}\\\\ \n &=&\n -{J(w_1)\\cdot K(w_2) \\over 2A},\n \\\\\n\\mu_2\n &=&\n -{[[k_{11}(x_3-x_2)+k_{12}(y_3-y_2)](y_1-y_3)-[k_{12}(x_3-x_2)+k_{22}(y_3-y_2)](x_1-x_3) \n \\over (x_1-x_3)(y_2-y_1)-(x_2-x_1)(y_1-y_3)}\\\\ \n &=&\n {J(w_3)\\cdot K(w_2) \\over 2A},\n \\\\\n\\lambda_3\n &=&\n {[k_{11}(x_1-x_3)+k_{12}(y_1-y_3)](y_3-y_2) -[k_{12}(x_1-x_3)+k_{22}(y_1-y_3)](x_3-x_2)\n \\over (x_2-x_1)(y_3-y_2)-(x_3-x_2)(y_2-y_1)}\\\\ \n &=&\n -{J(w_2)\\cdot K(w_3) \\over 2A},\n \\\\\n\\mu_3\n &=&\n -{[[k_{11}(x_1-x_3)+k_{12}(y_1-y_3)](y_2-y_1)-[k_{12}(x_1-x_3)+k_{22}(y_1-y_3)](x_2-x_1) \n \\over (x_2-x_1)(y_3-y_2)-(x_3-x_2)(y_2-y_1)}\\\\ \n &=&\n {J(w_1)\\cdot K(w_3) \\over 2A}, \n\\end{eqnarray*}\nwhere \n\\begin{eqnarray*}\nJ\n &=&\n \\left(\\begin{array}{cc}\n 0 & -1 \\\\ \n 1 & 0\n \\end{array} \\right)\n\\end{eqnarray*}\nis the $90^\\circ$ counter-clockwise rotation,\nand\n\\begin{eqnarray*}\nW'\n &=&\n\\left(\n\\begin{array}{c}\n{k_{11}[f_1(y_2-y_3)+f_2(y_3-y_1)+f_3(y_1-y_2)]\n+k_{12}[f_1(x_3-x_2)+f_2(x_1-x_3)+f_3(x_2-x_1)]\n\\over x_1y_2-x_1y_3-x_2y_1+x_2y_3+x_3y_1-x_3y_2} \\\\\n{k_{12}[f_1(y_2-y_3)+f_2(y_3-y_1)+f_3(y_1-y_2)]\n+k_{22}[f_1(x_3-x_2)+f_2(x_1-x_3)+f_3(x_2-x_1)]\n\\over x_1y_2-x_1y_3-x_2y_1+x_2y_3+x_3y_1-x_3y_2}\n\\end{array}\n\\right)\\\\\n&=&\n\\left(\\begin{array}{cc}\nk_{11} & k_{12} \\\\ \nk_{12} & k_{22}\n\\end{array} \\right)\n\\left(\n\\begin{array}{c}\n{[(y_2-y_3)f_1+(y_3-y_1)f_2+(y_1-y_2)f_3]\\over 2A} \\\\ \n{[(x_3-x_2)f_1+(x_1-x_3)f_2+(x_2-x_1)f_3]\\over 2A}\n\\end{array} \n\\right)\n\\end{eqnarray*}\nwhich is, in fact, {\\em the image under $K$ of the discrete isotropic flux} and shows the consistency of our {\\em local} reasoning. \nAlso observe that this formula is the same as that of the FEML anisotropic flux.\n\n\\subsection{Anisotropy on primal 1-forms}\nThe system (\\ref{eq: system equations 1 for anisotropic flux vector}) can be rewritten in matrix form as follows\n\\begin{eqnarray}\n\\left(\\begin{array}{c}\nW'\\cdot w_1 \\\\ \nW'\\cdot w_2 \\\\ \nW'\\cdot w_3\n\\end{array} \\right)\n &=&\n\\left(\n\\begin{array}{ccc}\n0 & \\lambda_1 & \\mu_1 \\\\ \n\\mu_2 & 0 & \\lambda_2 \\\\ \n\\lambda_3 & \\mu_3 & 0\n\\end{array} \n\\right)\n \\left(\\begin{array}{l}\nf_2-f_1\\\\\nf_3-f_2\\\\\nf_1-f_3\n \\end{array}\n\\right) \\nonumber\\\\\n &=&\n\\left(\n\\begin{array}{ccc}\n0 & \\lambda_1 & \\mu_1 \\\\ \n\\mu_2 & 0 & \\lambda_2 \\\\ \n\\lambda_3 & \\mu_3 & 0\n\\end{array} \n\\right)\n\\left(\\begin{array}{rrr}\n-1 & 1 & 0 \\\\ \n0 & -1 & 1 \\\\ \n1 & 0 & -1\n\\end{array} \\right)\n \\left(\\begin{array}{l}\nf_1\\\\\nf_2\\\\\nf_3\n \\end{array}\n\\right) \\nonumber\\\\\n &=&\n\\left(\n\\begin{array}{ccc}\n0 & \\lambda_1 & \\mu_1 \\\\ \n\\mu_2 & 0 & \\lambda_2 \\\\ \n\\lambda_3 & \\mu_3 & 0\n\\end{array} \n\\right)\nD_0[f] .\\label{eq: dual of W}\n\\end{eqnarray}\nRecalling the Remark at the end of Subsection \\ref{subsec: local DEC flux}, the matrix identity (\\ref{eq: dual of W})\nstates that the primal 1-form dual to the anisotropic flux vector $W'$ is given by the local DEC discretization \nof the anisotropy tensor \n\\begin{eqnarray*}\nK^{DEC}\n &=&\n\\left(\n\\begin{array}{ccc}\n0 & \\lambda_1 & \\mu_1 \\\\ \n\\mu_2 & 0 & \\lambda_2 \\\\ \n\\lambda_3 & \\mu_3 & 0\n\\end{array} \n\\right)\\\\\n &=&\n{1 \\over 2A}\n\\left(\n\\begin{array}{ccc}\n0 & -J(w_3)\\cdot K(w_1) & J(w_2)\\cdot K(w_1) \\\\ \nJ(w_3)\\cdot K(w_2) & 0 & -J(w_1)\\cdot K(w_2) \\\\ \n-J(w_2)\\cdot K(w_3) & J(w_1)\\cdot K(w_3) & 0\n\\end{array} \n\\right).\n\\end{eqnarray*}\nacting on the primal 1-form $D_0[f]$.\n\n{\\bf Remark}. The matrix $K^{DEC}$ is the local DEC discretization on primal 1-forms of the pullback operator \n$K^*$ on 1-forms. In this case, the discretization of\n$K^*df:=df\\circ K$.\n\n\n\\subsubsection{Geometric interpretation of the entries of $K^{DEC}$}\nLet us examine $\\lambda_1$ in the anisotropic case.\nConsider the following figure\n \\begin{center}\n \\includegraphics[width=0.6\\textwidth]{DiscretizedAnisotropyOperator03.png}%\n \\captionof{figure}{Geometric interpretation of the entries of the anisotropy tensor discretization $K^{ DEC}$}\\label{fig: anisotropy triangle 3}\n \\end{center}\nWe have\n\\begin{eqnarray*}\n\\lambda_1\n &= & \n -{J(w_3)\\cdot K(w_1) \\over 2A}\\\\\n &= & \n {1\\over 2A} |-J(w_3)| |K(w_1)| \\cos(\\beta) \\\\\n &= & \n {1\\over 2A} |w_3| |K(w_1)| \\cos(\\alpha+\\pi\/2) \\\\\n &= & \n -{1\\over 2A} |w_3| |K(w_1)| \\sin(\\alpha) \\\\\n &=&\n -{A'\\over A},\n\\end{eqnarray*}\nwhere $A'$ is the area of the red triangle and\nwe have used a well known formula for the area of a triangle in terms of an inner angle. Thus\n$\\lambda_1$ is the negative of the quotient of the area $A'$ of the red triangle and the area $A$ of the original triangle.\nThe calculations for the other entries are similar.\n\n\n\\subsubsection{Isotropic case}\nNow, let us assume $K=k\\,{\\rm Id}_{2\\times 2}$ on the triangle.\nThe previous calculations show that\n\\[K^{DEC}=\nk\\left(\\begin{array}{ccc}\n0 & -1 & -1 \\\\ \n-1 & 0 & -1 \\\\ \n-1 & -1 & 0\n\\end{array} \\right).\\]\nNote that, in this case,\n\\[K^{DEC} D_0= k\\, D_0.\\]\n\n\n\n\n\n\\section{2D anisotropic Poisson equation}\\label{sec: comparison}\nIn this section, we describe the local DEC discretization of the 2D anisotropic Poisson equation and compare it to that of FEML.\n\n\\subsection{Local DEC discretization of the 2D anisotropic Poisson equation}\nThe anisotropic Poisson equation reads as follows\n\\[-\\nabla\\cdot (K\\, \\nabla f) = q,\\]\nwhere $f$ and $q$ are two functions on a certain domain in $\\mathbb{R}^2$.\nIn terms of the exterior derivative $d$ and the Hodge star operator $\\star$ it reads as follows\n\\[-\\star d \\star (K^*df) = q\\]\nwhere $K^*df:=df\\circ K$\nand $K=K^T$.\nFollowing the discretization of the discretized divergence operator \\cite{Esqueda1},\nthe corresponding local DEC discretization of the anisotropic Poisson equation is\n\\[-M_0^{-1} \\, \\left(-D_0^T\\right) \\, M_1\\, K^{DEC} \\, D_0 \\, [f] = [q], \\]\nor equivalently \n\\begin{equation}\nD_0^T \\, M_1\\, K^{DEC}\\, D_0 \\, [f] = M_0 \\, [q]. \\label{eq: local DEC discretized anisotropic Poisson equation}\n\\end{equation} \nIn order to simplify the notation,\nconsider the lengths and areas defined in the Figure \\ref{fig: triangulo}.\n \\begin{center}\n \\includegraphics[width=0.35\\textwidth]{WellCenteredTriangle02-eps-converted-to.pdf}%\n \\captionof{figure}{Triangle }\\label{fig: triangulo}\n \\end{center}\nNow, the discretized equation (\\ref{eq: local DEC discretized anisotropic Poisson equation}) looks as follows:\n\\[\n\\left(\\begin{array}{ccc}\n-1 & 0 & 1\\\\\n1 & -1 & 0\\\\\n0 & 1 & -1\n \\end{array}\n\\right)\n\\left(\\begin{array}{ccc}\n{l_1\\over L_1} & 0 & 0\\\\\n0 & {l_2\\over L_2} & 0\\\\\n0 & 0 & {l_3\\over L_3} \n \\end{array}\n\\right)\n\\left(\n\\begin{array}{ccc}\n0 & \\lambda_1 & \\mu_1 \\\\ \n\\mu_2 & 0 & \\lambda_2 \\\\ \n\\lambda_3 & \\mu_3 & 0\n\\end{array} \n\\right)\n\\left(\\begin{array}{rrr}\n-1 & 1 & 0 \\\\ \n0 & -1 & 1 \\\\ \n1 & 0 & -1\n\\end{array} \\right)\n\\left(\\begin{array}{c}\nf_1 \\\\ \nf_2 \\\\ \nf_3\n\\end{array} \\right)\n =\n\\left(\\begin{array}{c}\nA_1q_1 \\\\ \nA_2q_2 \\\\ \nA_3q_3\n\\end{array} \\right).\n\\]\nThe diffusive term matrix \n\\begin{eqnarray*}\n&&\n\\left(\\begin{array}{ccc}\n-1 & 0 & 1\\\\\n1 & -1 & 0\\\\\n0 & 1 & -1\n \\end{array}\n\\right)\n\\left(\\begin{array}{ccc}\n{l_1\\over L_1} & 0 & 0\\\\\n0 & {l_2\\over L_2} & 0\\\\\n0 & 0 & {l_3\\over L_3} \n \\end{array}\n\\right)\n\\left(\n\\begin{array}{ccc}\n0 & \\lambda_1 & \\mu_1 \\\\ \n\\mu_2 & 0 & \\lambda_2 \\\\ \n\\lambda_3 & \\mu_3 & 0\n\\end{array} \n\\right)\n\\left(\\begin{array}{rrr}\n-1 & 1 & 0 \\\\ \n0 & -1 & 1 \\\\ \n1 & 0 & -1\n\\end{array} \\right)\\\\\n&=&\n\\left(\\begin{array}{ccc}\n-{\\lambda_3l_3\\over L_3}-{\\mu_1l_1\\over L_1} & {\\lambda_1l_1\\over L_1}+{(\\lambda_3-\\mu_3)l_3\\over L_3} \n & -{(\\lambda_1-\\mu_1)l1\\over L_1}+{\\mu_3l_3\\over L3} \\\\ \n{\\mu_1l_1\\over L_1}-{(\\lambda_2-\\mu_2)l_2\\over L_2} & -{\\lambda_1l1\\over L_1} -{\\mu_2l_2\\over L_2}\n & {(\\lambda_1-\\mu_1)l_1\\over L_1}+{\\lambda_2l_2\\over L_2}\\\\ \n{(\\lambda_2-\\mu_2)l_2\\over L_2} + {\\lambda_3l_3\\over L_3} & {\\mu_2l_2\\over L_2}-{(\\lambda_3-\\mu_3)l_3\\over L_3} \n & -{\\lambda_2l_2\\over L_2}-{\\mu_3l_3\\over L_3}\n\\end{array} \n\\right)\n\\end{eqnarray*}\nis actually symmetric (see Subsection \\ref{subsubsec: diffusive term DEC FEML}).\n\n\n\n\\subsection{Local FEML-Discretized 2D anisotropic Poisson equation}\n\n\n\nThe diffusive elemental matrix in FEM (frequently called \"stiffness matrix\") on an element $e$ is given by\n\\[K_e=\\int B^tDBdA,\\]\nwhere $D$ is the matrix representing the anisotropic diffusion tensor $K$ in this paper, \nand the matrix $B$ is given explicitly by\n\\[\nB=\\left(\n\\begin{array}{ccc}\n{\\partial N_1\\over\\partial x} & {\\partial N_2\\over\\partial x} & {\\partial N_3\\over\\partial x} \\\\ \n{\\partial N_1\\over\\partial y} & {\\partial N_2\\over\\partial y} & {\\partial N_3\\over\\partial y}\n\\end{array} \n\\right)\n=\n{1\\over 2A}\n\\left(\n\\begin{array}{ccc}\ny_2-y_3 & y_3-y_1 & y_1-y_2 \\\\ \nx_3-x_2 & x_1-x_3 & x_2-x_1\n\\end{array} \n\\right) . \n\\]\nSince the matrix $B$ is constant on an element of the mesh, the integral is easy to compute. Thus, the difussive matrix $K_e$ for a linear triangular element (FEML) is given by\n\\begin{eqnarray*}\nK_e\n &=&\n\\int B^T DB dA \\\\\n &=&\n B^T D B A_e \\\\\n &=&\n {1\\over 4A_e}\n\\left(\n\\begin{array}{ccc}\ny_2-y_3 & x_3-x_2 \\\\ \ny_3-y_1 & x_1-x_3 \\\\\ny_1-y_2 & x_2-x_1\n\\end{array} \n\\right)\n \\left(\\begin{array}{cc}\n k_{11} & k_{12} \\\\ \n k_{12} & k_{22}\n \\end{array} \\right)\n\\left(\n\\begin{array}{ccc}\ny_2-y_3 & y_3-y_1 & y_1-y_2 \\\\ \nx_3-x_2 & x_1-x_3 & x_2-x_1\n\\end{array} \n\\right)\n\\end{eqnarray*}\nNow, let us consider the first diagonal entry of the local FEML anisotropic Poisson diffusive matrix $K_e$, \n\\begin{eqnarray*}\n (K_e)_{11} \n &=&\n{1\\over 4A}(k_{11}(y_2-y_3)^2 +(k_{12}+k_{12})(y_2-y_3)(x_3-x_2) + k_{22}(x_3-x_2)^2)\\\\\n &=& \n{1\\over 4A} (\\begin{array}{cc}\n -(y_3-y_2), & x_3-x_2\n \\end{array} )\n \\left(\\begin{array}{cc}\n k_{11} & k_{12} \\\\ \n k_{12} & k_{22}\n \\end{array} \\right)\n \\left(\\begin{array}{c}\n -(y_3-y_2) \\\\ \n x_3-x_2\n \\end{array} \\right)\\\\\n &=& \n{1\\over 4A} (\\begin{array}{cc}\n x_3-x_2, & y_3-y_2\n \\end{array} )\n \\left(\\begin{array}{cc}\n 0 & 1 \\\\ \n -1 & 0\n \\end{array} \\right)\n \\left(\\begin{array}{cc}\n k_{11} & k_{12} \\\\ \n k_{12} & k_{22}\n \\end{array} \\right)\n \\left(\\begin{array}{cc}\n 0 & -1 \\\\ \n 1 & 0\n \\end{array} \\right)\n \\left(\\begin{array}{c}\n x_3-x_2 \\\\ \n y_3-y_2\n \\end{array} \\right)\\\\\n &=&\n{1\\over 4A} (J(v_3-v_2))^T K J(v_3-v_2)\\\\\n &=&\n{1\\over 4A} J(v_3-v_2)\\cdot K (J(v_3-v_2)),\n\\end{eqnarray*}\nwhere \n\\begin{eqnarray*}\nJ\n &=&\n \\left(\\begin{array}{cc}\n 0 & -1 \\\\ \n 1 & 0\n \\end{array} \\right)\n\\end{eqnarray*}\nis the $90^\\circ$ counter-clockwise rotation.\nIn this notation, the diffusive term in local FEML is given as follows\n{\\small\n\\[\n{1\\over 4A}\n\\left(\\begin{array}{ccc}\nJ(v_3-v_2)\\cdot K(J(v_3-v_2)) & J(v_3-v_2)\\cdot K(J(v_1-v_3)) & J(v_3-v_2)\\cdot K(J(v_2-v_1)) \\\\ \nJ(v_1-v_3)\\cdot K(J(v_3-v_2)) & J(v_1-v_3)\\cdot K(J(v_1-v_3)) & J(v_1-v_3)\\cdot K(J(v_2-v_1)) \\\\ \nJ(v_2-v_1)\\cdot K(J(v_3-v_2)) & J(v_2-v_1)\\cdot K(J(v_1-v_3)) & J(v_2-v_1)\\cdot K(J(v_2-v_1))\n\\end{array}\\right) . \\]\n}\n\n\n\\subsection{Comparison between local DEC and FEML discretizations}\nFor the sake of brevity, \nwe are only going to compare the entries of the first row and first column of each formulation.\nConsider the various lengths, areas and angles given in the triangle of Figure \\ref{fig: circumscribed triangle 01}.\n \\begin{center}\n \\includegraphics[width=0.33\\textwidth]{WellCenteredTriangle03-eps-converted-to.pdf}%\n \\hspace{12mm}\n \\includegraphics[width=0.33\\textwidth]{WellCenteredTriangle02-eps-converted-to.pdf}%\n \\captionof{figure}{Circumscribed triangle.}\\label{fig: circumscribed triangle 01}\n \\end{center}\nWe have the following:\n\\begin{eqnarray*}\n \\pi &=& 2(\\alpha_1+\\alpha_2+\\alpha_3),\\\\\n {2l_i\\over L_i} &=& \\tan(\\alpha_i),\\\\\n {l_i\\over R} &=& \\sin(\\alpha_i),\\\\\n {L_i\\over 2R} &=& \\cos(\\alpha_i),\\\\\n A_1&=& {L_1l_1\\over 4} + {L_3l_3\\over 4},\\\\\n A_2&=& {L_1l_1\\over 4} + {L_2l_2\\over 4},\\\\\n A_3&=& {L_2l_2\\over 4} + {L_3l_3\\over 4}.\n\\end{eqnarray*}\n\n\\subsubsection{The diffusive term}\\label{subsubsec: diffusive term DEC FEML}\nWe claim that\n\\[{J(v_3-v_2)\\cdot K(J(v_3-v_2))\\over 4A}\n= \n-{\\lambda_3l_3\\over L_3}-{\\mu_1l_1\\over L_1}.\n\\]\nIndeed,\n\\begin{eqnarray*}\n\\lambda_3\n &=&\n -{J(v_3-v_2)\\cdot K( v_1-v_3)\\over 2A},\\\\\n\\mu_1\n &=&\n {J(v_3-v_2)\\cdot K( v_2-v_1)\\over 2A}.\n\\end{eqnarray*}\nThus\n\\begin{eqnarray*}\n-{\\lambda_3l_3\\over L_3}-{\\mu_1l_1\\over L_1}\n &=& \n {J(v_3-v_2)\\cdot K(v_1-v_3)\\over 2A}{\\tan(\\alpha_3)\\over 2}\n -{J(v_3-v_2)\\cdot K( v_2-v_1)\\over 2A}{\\tan(\\alpha_1)\\over 2}\\\\\n &=& \n {1\\over 4A}J(v_3-v_2)\\cdot K((v_1-v_3)\\tan(\\alpha_3)- (v_2-v_1)\\tan(\\alpha_1)).\n\\end{eqnarray*}\nAll we have to do is show that\n\\[(v_1-v_3)\\tan(\\alpha_3)- (v_2-v_1)\\tan(\\alpha_1) = J(v_3-v_2).\\]\nNote that, since $J(v_3-v_2)$ is orthogonal to $v_3-v_2$, $J(v_3-v_2)$ must be parallel to $c - {v_2+v_3\\over 2}$. Thus, \n\\begin{equation}\nJ(v_3-v_2) = {L_2\\over l_2}\\left(c - {v_2+v_3\\over 2}\\right). \\label{eq: J(w2)}\n\\end{equation}\nNow we are going to express $c$ in terms of $v_1,v_2,v_3$.\nLet us consider \n\\[c-v_1 = a (v_2-v_1) + b(v_3-v_1)\\]\nwhere $a,b$ are coefficients to be determined.\nTaking inner products with $(v_2-v_1)$ and $(v_3-v_1)$ we get the two equations\n\\begin{eqnarray*}\nR\\cos(\\alpha_1)\n &=&\n aL_1+bL_3\\cos(\\alpha_1+\\alpha_3),\\\\ \nR\\cos(\\alpha_3)\n &=&\n aL_1\\cos(\\alpha_1+\\alpha_3)+bL_3. \n\\end{eqnarray*}\nSolving for $a$ and $b$\n\\begin{eqnarray*}\na\n &=&\n {\\sin(\\alpha_3)\\over 2\\cos(\\alpha_1)\\sin(\\alpha_1+\\alpha_3)},\\\\\nb\n &=&\n {\\sin(\\alpha_1)\\over 2\\cos(\\alpha_3)\\sin(\\alpha_1+\\alpha_3)}.\n\\end{eqnarray*}\nSubstituting all the relevant quantities in (\\ref{eq: J(w2)}) we have, for instance, that the coefficient of $(v_2-v_1)$ is\n\\begin{eqnarray*}\n2{\\cos(\\alpha_2)\\over \\sin(\\alpha_2)}\\left({\\sin(\\alpha_3)\\over 2\\cos(\\alpha_1)\\sin(\\alpha_1+\\alpha_3)}-{1\\over 2}\\right)\n &=&\n{\\cos(\\alpha_2)\\over \\sin(\\alpha_2)}\\left({\\sin(\\alpha_3)-\\cos(\\alpha_1)\\sin(\\alpha_1+\\alpha_3)\\over \\cos(\\alpha_1)\\sin(\\alpha_1+\\alpha_3)}\\right)\\\\ \n &=&\n{\\cos(\\alpha_2)\\over \\sin(\\alpha_2)}\\left({\\sin(\\alpha_3)-\\cos(\\alpha_1)(\\sin(\\alpha_1)\\cos(\\alpha_3)+\\sin(\\alpha_3)\\cos(\\alpha_1))\\over \\cos(\\alpha_1)\\sin(\\pi\/2-\\alpha_2)}\\right)\\\\ \n &=&\n{\\sin(\\alpha_1)\\over \\sin(\\alpha_2)}\\left({\\sin(\\alpha_3)\\sin(\\alpha_1)-\\cos(\\alpha_1)\\cos(\\alpha_3)\\over \\cos(\\alpha_1)}\\right)\\\\ \n &=&\n \\tan(\\alpha_1){-\\cos(\\alpha_1+\\alpha_3)\\over\\sin(\\alpha_2)}\\\\ \n &=&\n \\tan(\\alpha_1){-\\cos(\\pi\/2-\\alpha_2)\\over\\sin(\\alpha_2)}\\\\ \n &=&\n \\tan(\\alpha_1){-\\sin(\\alpha_2)\\over\\sin(\\alpha_2)}\\\\ \n &=&\n -\\tan(\\alpha_1), \n\\end{eqnarray*}\nand similarly for the coefficient of $(v_1-v_3)$.\nThe calculations for the remaining entries are similar.\n\nThus, the local DEC and FEML diffusive terms of the 2D anisotropic Poisson equation coincide.\n\n\n\\subsubsection{The source term}\nAs already observed in \\cite{Esqueda1}, the right hand sides of the local DEC and FEML systems are different\n\\[\\left(\\begin{array}{c}\nA_1q_1\\\\\nA_2q_2\\\\\nA_3q_3\n \\end{array}\n\\right)\n\\not =\n{A\\over 3}\\left(\\begin{array}{l}\nq_1\\\\\nq_2\\\\\nq_3\n \\end{array}\n\\right).\\]\nWhile FEML uses a barycentric subdivision to calculate the areas associated to each node\/vertex,\nDEC uses a circumcentric subdivision. Eventually, this leads the DEC discretization to a better approximation of the solution\n(on coarse meshes). \n\n\n\n\\section{Some remarks about DEC quantities}\\label{sec: extension}\n\n\n\\subsection{The discrete Hodge star quantities revisited} \\label{subsec: determinants} \nThe numbers appearing in the local DEC matrices can be expressed both in terms of determinants and in terms of trigonometric functions.\nMore precisely, \n\\begin{eqnarray*}\n A_1\n&=& {1\\over 4}\\left[\\det\\left(\\begin{array}{ccc}\nx_1 & y_1 & 1\\\\\nx_c & y_c & 1\\\\\nx_2 & y_2 & 1\n \\end{array}\n\\right)\n+\\det\\left(\\begin{array}{ccc}\nx_3 & y_3 & 1\\\\\nx_c & y_c & 1\\\\\nx_1 & y_1 & 1\n \\end{array}\n\\right)\n\\right]\n \\quad =\\quad {R^2\\over 4}(\\sin(2\\alpha_1)+\\sin(2\\alpha_3)),\\\\\n A_2\n &=& {1\\over 4}\\left[\\det\\left(\\begin{array}{ccc}\nx_1 & y_1 & 1\\\\\nx_c & y_c & 1\\\\\nx_2 & y_2 & 1\n \\end{array}\n\\right)\n+\\det\\left(\\begin{array}{ccc}\nx_2 & y_2 & 1\\\\\nx_c & y_c & 1\\\\\nx_3 & y_3 & 1\n \\end{array}\n\\right)\n\\right]\n\\quad = \\quad {R^2\\over 4}(\\sin(2\\alpha_1)+\\sin(2\\alpha_2)),\\\\\n A_3\n &=& {1\\over 4}\\left[\\det\\left(\\begin{array}{ccc}\nx_2 & y_2 & 1\\\\\nx_c & y_c & 1\\\\\nx_3 & y_3 & 1\n \\end{array}\n\\right)\n+\\det\\left(\\begin{array}{ccc}\nx_3 & y_3 & 1\\\\\nx_c & y_c & 1\\\\\nx_1 & y_1 & 1\n \\end{array}\n\\right)\n\\right]\n\\quad = \\quad {R^2\\over 4}(\\sin(2\\alpha_2)+\\sin(2\\alpha_3)),\\\\\n{l_1\\over L_1}\n &=&\n{1\\over L_1^2}\n\\det\\left(\\begin{array}{ccc}\nx_1 & y_1 & 1\\\\\nx_c & y_c & 1\\\\\nx_2 & y_2 & 1\n \\end{array}\n\\right)\n\\quad = \\quad {\\tan(\\alpha_1)\\over 2},\\\\\n{l_2\\over L_2}\n &=&\n{1\\over L_2^2}\n\\det\\left(\\begin{array}{ccc}\nx_2 & y_2 & 1\\\\\nx_c & y_c & 1\\\\\nx_3 & y_3 & 1\n \\end{array}\n\\right)\n \\quad = \\quad {\\tan(\\alpha_2)\\over 2},\\\\\n{l_3\\over L_3}\n &=&\n{1\\over L_3^2}\n\\det\\left(\\begin{array}{ccc}\nx_3 & y_3 & 1\\\\\nx_c & y_c & 1\\\\\nx_1 & y_1 & 1\n \\end{array}\n\\right)\n \\quad = \\quad {\\tan(\\alpha_3)\\over 2}.\n\\end{eqnarray*}\nThese expressions are valid regardless of the location of the circumcenter and can, indeed, take negative values.\nhe angles that are measured in the scheme can be negative as in the obtuse triangle of Figure \\ref{fig: negative angles}\n \\begin{center}\n \\includegraphics[width=0.3\\textwidth]{NonWellCenteredTriangle01a-eps-converted-to.pdf}%\n \\hspace{8mm}\n \\includegraphics[width=0.3\\textwidth]{NegativeAngles01-eps-converted-to.pdf}%\n \\captionof{figure}{Negative (exterior) angles measured in an obtuse triangle.}\\label{fig: negative angles}\n \\end{center}\nand some quantities can be zero or negative.\nFor instance, if \n\\[\\alpha_2={\\pi\\over 2}-2\\alpha_1,\\]\nhen\n\\[A_1=0.\\]\n\n\n\\subsection{Area weights assigned to vertices}\nIn order to understand how local DEC assigns area weights to vertices differently from FEML, let us\nconsider the obtuse triangle shown in Figure \\ref{fig: negative angles}\nLet $p_1,p_2,p_3$ be the middle points of the segments\n$[v_2,v_3],[v_2,v_3],[v_3,v_1]$ respectively. \nAs shown in Figure \\ref{fig: obtuse triangle 01}, the triangle $[v_1,p_3,c]$ lies completely outside of the triangle $[v_1,v_2,v_3]$. Geometrically, \nthis implies that its area must be assigned a negative sign, which is confirmed \nby the determinant formulas of Subsection \\ref{subsec: determinants}. On the other hand, the triangle $[v_1,p_1,c]$ will have positive area. \nThus, their sum gives us the area $A_1$ in Figure \\ref{fig: obtuse triangle 01}.\n \\begin{center}\n \\includegraphics[width=0.28\\textwidth]{NonWellCenteredTriangle03a-eps-converted-to.pdf}%\n \\hspace{3mm}\n \\includegraphics[width=0.28\\textwidth]{NonWellCenteredTriangle04a-eps-converted-to.pdf}%\n \\hspace{3mm}\n \\includegraphics[width=0.28\\textwidth]{NonWellCenteredTriangle05a-eps-converted-to.pdf}%\n \\captionof{figure}{Area weight assigned to $v_1$.}\\label{fig: obtuse triangle 01}\n \\end{center}\nThe area $A_3$ is computed similarly, where the triangle $[p_3,v_3,c]$ is assigned negative area (see Figure \\ref{fig: obtuse triangle 02}).\n \\begin{center} \n \\includegraphics[width=0.28\\textwidth]{NonWellCenteredTriangle06a-eps-converted-to.pdf}%\n \\hspace{3mm}\n \\includegraphics[width=0.28\\textwidth]{NonWellCenteredTriangle07a-eps-converted-to.pdf}%\n \\hspace{3mm}\n \\includegraphics[width=0.28\\textwidth]{NonWellCenteredTriangle08a-eps-converted-to.pdf}%\n \\captionof{figure}{Area weight assigned to $v_3$.}\\label{fig: obtuse triangle 02}\n \\end{center}\nNote that for $A_2$, the two triangles $[p_1,v_3,c]$ and $[v_2,p_2,c]$ both have positive areas (see Figure \\ref{fig: obtuse triangle 03}). \n \\begin{center}\n \\includegraphics[width=0.28\\textwidth]{NonWellCenteredTriangle09a-eps-converted-to.pdf}%\n \\hspace{2mm}\n \\includegraphics[width=0.28\\textwidth]{NonWellCenteredTriangle10a-eps-converted-to.pdf}%\n \\hspace{2mm}\n \\includegraphics[width=0.28\\textwidth]{NonWellCenteredTriangle11a-eps-converted-to.pdf}%\n \\captionof{figure}{Area weight assigned to $v_2$.}\\label{fig: obtuse triangle 03}\n \\end{center}\n\n \n\\section{Numerical Examples}\\label{sec: examples}\n\n\nIn this section, we present three examples in order to illustrate the performance of DEC resulting from the local formulation and its implementation. \nIn all cases, we solve the anisotropic Poisson equation.\nThe FEML methodology that we have used in the comparison can be consulted \\cite{Onate,Zienkiewicz1,Botello}.\n\n\n\\subsection{First example: Heterogeneity}\nThis example is intended to highlight how Local DEC deals effectively with heterogeneous materials. \nConsider the region in the plane given in Figure \\ref{fig: square example geometry}.\n\\begin{figure}[h]\n\t\\centering\n\t\\includegraphics[width=0.38\\textwidth]{Fig_12_SquareWithConditions.png}%\n\t\\captionof{figure}{Square and inner circle with different conditions.}\\label{fig: square example geometry}\n\\end{figure}\n\\begin{itemize}\n\t\\item The difussion constant for the region labelled mat1 is $k=12$ and its source term is $q=20$.\n\t\\item The difussion constant for the region labelled mat2 is $k=6$ and its source term is $q=5$.\n\\end{itemize}\nThe meshes used in this example are shown in Figure \\ref{fig: first example meshes} and vary from coarse to very fine.\n\\begin{figure}[h]\n\t\\centering\n\t\\subfigure[]{\\includegraphics[width=0.25\\textwidth]{Fig_13_Square_m1.png}}\\hspace{5mm}\n\t\\subfigure[]{\\includegraphics[width=0.25\\textwidth]{Fig_13_Square_m2.png}}\\hspace{5mm}\n\t\\subfigure[]{\\includegraphics[width=0.25\\textwidth]{Fig_13_Square_m3.png}}\n\n\t\\subfigure[]{\\includegraphics[width=0.25\\textwidth]{Fig_13_Square_m4.png}}\\hspace{5mm}\n\t\\subfigure[]{\\includegraphics[width=0.25\\textwidth]{Fig_13_Square_m5.png}}\\hspace{5mm}\n\t\\subfigure[]{\\includegraphics[width=0.25\\textwidth]{Fig_13_Square_m6.png}}\n\n\t\\captionof{figure}{Six of the meshes used in the first example.}\\label{fig: first example meshes}\n\\end{figure}\n\nThe numerical results for the maximum temperature value are exemplified in Table \\ref{table: square example numerical results}.\n\\begin{center}\n\t\\begin{tabular}{|c|c|c|c|c|c|c|}\\hline\n\t\t\\multirow{2}{*}{\\rm Mesh}&\\multirow{2}{*}{\t\\# {\\rm nodes}}&\\multirow{2}{*}{\\# {\\rm elements}}\t&\t\\multicolumn{2}{c|}{\\mbox{Max. Temp. Value}}&\t\\multicolumn{2}{c|}{\\mbox{Max. Flux Magnitude}} \\\\\n\t\t&&&{\\rm DEC} \t&{\\rm FEML}&{\\rm DEC} \t&{\\rm FEML}\t \\\\\\hline\n\t\tFigure \\ref{fig: first example meshes}(a) &\n\t\t49 &\n\t\t80 &\n\t\t5.51836 &\n\t\t5.53345 &\n\t\t13.837 &\n\t\t13.453 \\\\\n\t\tFigure \\ref{fig: first example meshes}(b) &\n\t\t98 &\n\t\t162 &\n\t\t5.65826 &\n\t\t5.66648 &\n\t\t14.137 &\n\t\t14.024 \\\\\n\t\tFigure \\ref{fig: first example meshes}(c) &\n\t\t258 &\n\t\t466 &\n\t\t5.70585 &\n\t\t5.71709 &\n\t\t14.858 &\n\t\t14.770 \\\\\n\t\tFigure \\ref{fig: first example meshes}(d) &\n\t\t1,010 &\n\t\t1,914 &\n\t\t5.72103 &\n\t\t5.72280 &\n\t\t15.008 &\n\t\t15.006 \\\\\n\t\tFigure \\ref{fig: first example meshes}(e) &\n\t\t3,813 &\n\t\t7,424 &\n\t\t5.72725 &\n\t\t5.72725 &\n\t\t15.229 &\n\t\t15.228 \\\\\n\t\tFigure \\ref{fig: first example meshes}(f) &\n\t\t13,911 &\n\t\t27,420 &\n\t\t5.72821 &\n\t\t5.72826 &\n\t\t15.342 &\n\t\t15.337 \\\\\n\t\t&\n\t\t50,950 &\n\t\t101,098 &\n\t\t5.72841 &\n\t\t5.72842 &\n\t\t15.395 &\n\t\t15.396 \\\\\n\t\t&\n\t\t135,519 &\n\t\t269,700 &\n\t\t5.72845 &\n\t\t5.72845 &\n\t\t15.420 &\n\t\t15.417 \\\\\n\t\t&\n\t\t298,299 &\n\t\t594,596 &\n\t\t5.72848 &\n\t\t5.72848 &\n\t\t15.430 &\n\t\t15.429 \\\\\n\t\t&\n\t\t600,594 &\n\t\t1,198,330 &\n\t\t5.72848 &\n\t\t5.72848 &\n\t\t15.433 &\n\t\t15.433 \\\\\n\t\t&\n\t\t1,175,238 &\n\t\t2,346,474 &\n\t\t5.72849 &\n\t\t5.72849 &\n\t\t15.43724 &\n\t\t15.43724 \\\\\n\t\t\\hline\n\t\\end{tabular}\n\t\\captionof{table}{Numerical simulation results of the first example.}\\label{table: square example numerical results} \n\\end{center}\nThe temperature and flux-magnitude distribution fields are shown in Figure \\ref{fig: square example temperature field}. \n\\begin{figure}[h]\n\t\\centering\n\t\\subfigure[Contour Fill of Temperatures]{\\includegraphics[width=0.4\\textwidth]{Fig_14_Square_contour_temp.png}}\\hspace{3mm}\n\t\\subfigure[Contour Fill of Flux vectors on Elems]{\\includegraphics[width=0.4\\textwidth]{Fig_14_Square_contour_flux.png}}\n\t\\captionof{figure}{Temperature and flux-magnitude distribution fields of the first example.}\\label{fig: square example temperature field}\n\\end{figure}\n\nFigure \\ref{fig: first example diametral graphs} shows the graphs of the temperature and the flux-magnitude along a horizontal line crossing the inner circle for the first two meshes.\n\\begin{figure}\n\t\\centering\n\t\\subfigure[]{\\includegraphics[width=0.41\\textwidth]{Fig_15_Square_m1_diametral_temp.png}}\\hspace{2mm}\n\t\\subfigure[]{\\includegraphics[width=0.41\\textwidth]{Fig_15_Square_m1_diametral_flux.png}}\n\t\\subfigure[]{\\includegraphics[width=0.41\\textwidth]{Fig_15_Square_m2_diametral_temp.png}}\\hspace{2mm}\n\t\\subfigure[]{\\includegraphics[width=0.41\\textwidth]{Fig_15_Square_m2_diametral_flux.png}}\n\t\\subfigure[]{\\includegraphics[width=0.41\\textwidth]{Fig_15_Square_m3_diametral_temp.png}}\\hspace{2mm}\n\t\\subfigure[]{\\includegraphics[width=0.41\\textwidth]{Fig_15_Square_m3_diametral_flux.png}}\n\t\\captionof{figure}{Temperature and Flux magnitude graphs of the first example along a cross-section of the domain for different meshes.}\\label{fig: first example diametral graphs}\n\\end{figure}\n\n\n\n\\newpage\n\n\\subsection{Second example: Anisotropy}\nLet us solve the Poisson equation in a circle of radius one centered at the origin $(0,0)$ under the following conditions (see Figure \\ref{fig: second example geometry}): \n\\begin{itemize}\n\t\\item heat anisotropic diffusion constants $K_x = 1.5, K_y=1.0$;\n\t\\item material angle $30^\\circ$;\n\t\\item source term $q= 1$;\n\t\\item Dirichlet boundary condition $u=10$.\n\\end{itemize}\n\\begin{figure}[h]\n\t\\centering\n\t\\includegraphics[width=0.3\\textwidth]{Fig_16_CircleWithConditions.png}\n\t\\captionof{figure}{Disk of radius one.}\\label{fig: second example geometry}\n\\end{figure}\nThe meshes used in this example are shown in Figure \\ref{fig: second example meshes} and vary from very coarse to very fine.\n\\begin{figure}[h]\n\t\\centering\n\t\\subfigure[]{\\includegraphics[width=0.25\\textwidth]{Fig_17_CircleMesh1.png}}\\hspace{4mm}\n\t\\subfigure[]{\\includegraphics[width=0.25\\textwidth]{Fig_17_CircleMesh2.png}}\\hspace{4mm}\n\t\\subfigure[]{\\includegraphics[width=0.25\\textwidth]{Fig_17_CircleMesh3.png}}\n\t\\subfigure[]{\\includegraphics[width=0.25\\textwidth]{Fig_17_CircleMesh4.png}}\\hspace{4mm}\n\t\\subfigure[]{\\includegraphics[width=0.25\\textwidth]{Fig_17_CircleMesh5.png}}\\hspace{4mm}\n\t\\subfigure[]{\\includegraphics[width=0.25\\textwidth]{Fig_17_CircleMesh6.png}}\n\n\t\\captionof{figure}{Six firsts meshes used for unit disk.}\\label{fig: second example meshes}\n\\end{figure}\nThe numerical results for the maximum temperature value ($u(0,0)=10.2$) are exemplified in Table \\ref{table: numerical results} where \na comparison with the Finite Element Method with linear interpolation functions (FEML) is also shown. \n\\begin{center}\n\t\\begin{tabular}{|c|c|c|c|c|c|c|}\\hline\n\t\t\\multirow{2}{*}{\\rm Mesh}&\\multirow{2}{*}{\t\\# {\\rm nodes}}&\\multirow{2}{*}{\\# {\\rm elements}}\t&\t\\multicolumn{2}{c|}{\\mbox{Temp. Value at $(0,0)$}}&\t\\multicolumn{2}{c|}{\\mbox{Flux Magnitude at $(-1,0)$}} \\\\\n\t\t&&&{\\rm DEC} \t&{\\rm FEML}&{\\rm DEC} \t&{\\rm FEML}\t \\\\\\hline\n\t\t{\\rm Figure \\,\\,\\,\\ref{fig: second example meshes}(a)}&\t17\t&\t20\t\t&\t10.20014&\t\t\t10.19002\t\t& 0.42133 & 0.43865\\\\\n\t\t{\\rm Figure \\,\\,\\,\\ref{fig: second example meshes}(b)}&\t41\t&\t56\t\t&\t10.20007&\t\t\t10.19678\t\t& 0.48544 & 0.49387\\\\\n\t\t{\\rm Figure \\,\\,\\,\\ref{fig: second example meshes}(c)}&\t201\t&\t344\t\t&\t10.20012&\t\t\t10.20158\t\t& 0.52470 & 0.52428\\\\\n\t\t{\\rm Figure \\,\\,\\,\\ref{fig: second example meshes}(d)}&\t713\t&\t1304\t&\t10.20000&\t\t\t10.19969\t\t& 0.54143 & 0.54224\\\\\n\t\t{\\rm Figure \\,\\,\\,\\ref{fig: second example meshes}(e)}&\t2455&\t4660\t&\t10.20000&\t\t\t10.19990\t\t& 0.54971 & 0.55138\\\\\n\t\t{\\rm Figure \\,\\,\\,\\ref{fig: second example meshes}(f)}& \t8180&\t15862\t&\t10.20000&\t\t\t10.20002\t\t& 0.55326 & 0.55409\\\\\n\t\t& 20016\t&\t39198\t&\t10.20000&\t\t\t10.19999\t\t& 0.55470 & 0.55520\\\\\n\t\t& 42306\t&\t83362\t&\t10.20000&\t\t\t10.20000\t\t& 0.55540 & 0.55572\\\\\\hline\n\t\\end{tabular} \n\t\\captionof{table}{Temperature value at the point $(0,0)$ and Flux magnitude value at the point $(-1,0)$ of the numerical simulations for the second example.}\\label{table: numerical results} \n\\end{center}\nThe temperature distribution and Flux magnitude fields for the finest mesh are shown in Figure \\ref{fig: second example temperature field}.\n\\begin{figure}[h]\n\t\\centering\n\t\\subfigure[Contour Fill of Temperatures]{\\includegraphics[width=0.48\\textwidth]{Fig_18_CircleContourTemp.png}}\\hspace{2mm}\n\t\\subfigure[Contour Fill of Flux vectors on Elems]{\\includegraphics[width=0.48\\textwidth]{Fig_18_CircleContourFlux.png}}\n\t\\captionof{figure}{Temperature distribution and Flux magnitude fields for the finest mesh of the second example.}\\label{fig: second example temperature field}\n\\end{figure}\n\nFigures \\ref{fig: second example diametral graphs}(a), \\ref{fig: second example diametral graphs}(b) and \\ref{fig: second example diametral graphs}(c) \nshow the graphs of the temperature and flux magnitude values along a diameter of the circle for the different meshes of Figures \\ref{fig: second example meshes}(a), \\ref{fig: second example meshes}(b)\nand \\ref{fig: second example meshes}(c) respectively.\n\\begin{figure}\n\t\\centering\n\t\\subfigure[]{\\includegraphics[width=0.45\\textwidth]{Fig_19_CircleTempCrossSection01.png}}\\hspace{2mm}\n\t\\subfigure[]{\\includegraphics[width=0.45\\textwidth]{Fig_19_CircleFluxCrossSection01.png}}\\vspace{1mm}\n\t\\subfigure[]{\\includegraphics[width=0.45\\textwidth]{Fig_19_CircleTempCrossSection02.png}}\\hspace{2mm}\n\t\\subfigure[]{\\includegraphics[width=0.45\\textwidth]{Fig_19_CircleFluxCrossSection02.png}}\\vspace{1mm}\n\t\\subfigure[]{\\includegraphics[width=0.45\\textwidth]{Fig_19_CircleTempCrossSection03.png}}\\hspace{2mm}\n\t\\subfigure[]{\\includegraphics[width=0.45\\textwidth]{Fig_19_CircleFluxCrossSection03.png}}\\vspace{1mm}\n\t\\captionof{figure}{Temperature and Flux magnitude graphs of the second example along a diameter of the circle for different meshes: \n\t\tmesh in Figure \\ref{fig: second example meshes}(a), a-Temperature, b-Flux; \n\t\tmesh in Figure \\ref{fig: second example meshes}(b), c-Temperature, d-Flux; \n\t\tmesh in Figure \\ref{fig: second example meshes}(c), e-Temperature, f-Flux; \n\t}\\label{fig: second example diametral graphs}\n\\end{figure}\n\n\n\\newpage\n\n\\subsection{Third example: Heterogeneity and anisotropy}\nLet us solve the Poisson equation in a circle of radius on the following domain (see Figure \\ref{fig: third example geometry}) with various material properties.\nThe geometry of the domain is defined by segments of ellipses passing through the given points which have centers at the origin $(0,0)$. \n\\begin{figure}[h]\n\t\\centering\n\t\\includegraphics[width=0.4\\textwidth]{Fig_20_HuevoGeometry.png}\n\t\\captionof{figure}{Egg-like domain with different materials.}\\label{fig: third example geometry}\n\\end{figure}\n\\begin{center}\n\t\\begin{tabular}{|c|c|c|c|c|c|}\\hline\n\t\tPoint &$x$ &$y$ &Point &$x$ &$y$ \\\\\\hline\n\t\ta &-5 &0 &A &0 &-4 \\\\\n\t\tb &-4 &0 &B &0 &-3 \\\\\n\t\tc &-3 &0 &C &0 &-2 \\\\\n\t\td &-1 &0 &D &0 &-1 \\\\\n\t\te &1 &0 &E &0 &1 \\\\\n\t\tf &6 &0 &F &0 &2 \\\\\n\t\tg &7 &0 &G &0 &3 \\\\\n\t\th &8 &0 &H &0 &4 \\\\\\hline\n\t\\end{tabular}\n\\end{center}\n\\begin{itemize}\n\t\\item The Dirichlet boundary condition is $u=10$ and material properties (anisotropic heat diffusion constants, material angles and source terms) are given according to Figure \\ref{fig: third example dirichlet condition} and the table below.\n\t\\begin{figure}[h]\n\t\t\\centering\n\t\t\\includegraphics[width=0.38\\textwidth]{Fig_21_HuevoWithConditions.png}%\n\t\t\\captionof{figure}{Dirichlet condition.}\\label{fig: third example dirichlet condition}\n\t\\end{figure}\n\t\n\t\\begin{center} \n\t\t\\begin{tabular}{|l|c|c|c|c|}\\hline\n\t\t\t& $K_x$\t&$K_y$&\tangle&\t$q$\\\\\\hline\n\t\t\tDomain mat1&\t5\t&25\t&30\t&15\\\\\\hline\n\t\t\tDomain mat2&\t25\t&5\t&0\t&5\\\\\\hline\n\t\t\tDomain mat3&\t50\t&12&\t45&\t5\\\\\\hline\n\t\t\tDomain mat4&\t10\t&35\t&0\t&5\\\\\\hline\n\t\t\\end{tabular}\n\t\\end{center}\n\t\n\\end{itemize}\nThe meshes used in this example are shown in Figure \\ref{fig: third example meshes}.\n\\begin{figure}[h]\n\t\\centering\n\t\\subfigure[]{\\includegraphics[width=0.35\\textwidth]{Fig_22_HuevoMesh1.png}}\\hspace{5mm}\n\t\\subfigure[]{\\includegraphics[width=0.35\\textwidth]{Fig_22_HuevoMesh2.png}}\n\t\\subfigure[]{\\includegraphics[width=0.35\\textwidth]{Fig_22_HuevoMesh3.png}}\\hspace{5mm}\n\t\\subfigure[]{\\includegraphics[width=0.35\\textwidth]{Fig_22_HuevoMesh4.png}}\n\t\\captionof{figure}{Meshes for layered egg-like figure.}\\label{fig: third example meshes}\n\\end{figure}\nThe numerical results for the maximum temperature value ($u(0,0)=10.2$) are exemplified in Table \\ref{table: numerical results huevo} where \na comparison with the Finite Element Method with linear interpolation functions (FEML) is also shown. \n\\begin{center}\n\t\\begin{tabular}{|c|c|c|c|c|c|c|}\\hline\n\t\t\\multirow{2}{*}{\\rm Mesh}&\\multirow{2}{*}{\t\\# {\\rm nodes}}&\\multirow{2}{*}{\\# {\\rm elements}}\t&\t\\multicolumn{2}{c|}{\\mbox{Max. Temp. Value}}&\t\\multicolumn{2}{c|}{\\mbox{Max. Flux Magnitude}} \\\\\n\t\t&&&{\\rm DEC} \t&{\\rm FEML}&{\\rm DEC} \t&{\\rm FEML}\t \\\\\\hline\n\t\tFigure \\ref{fig: third example meshes}(a) &\n\t\t342 &\n\t\t616 &\n\t\t2.79221 &\n\t\t2.79854 &\n\t\t18.41066 &\n\t\t18.40573 \\\\\n\t\tFigure \\ref{fig: third example meshes}(b) &\n\t\t1,259 &\n\t\t2,384 &\n\t\t2.83929 &\n\t\t2.84727 &\n\t\t18.93838 &\n\t\t18.91532 \\\\\n\t\tFigure \\ref{fig: third example meshes}(c) &\n\t\t4,467 &\n\t\t8,668 &\n\t\t2.85608 &\n\t\t2.85717 &\n\t\t19.13297 &\n\t\t19.13193 \\\\\n\t\tFigure \\ref{fig: third example meshes}(d) &\n\t\t14,250 &\n\t\t28,506 &\n\t\t2.85994 &\n\t\t2.86056 &\n\t\t19.20982 &\n\t\t19.20909 \\\\\n\t\t&\n\t\t20,493 &\n\t\t40,316 &\n\t\t2.86120 &\n\t\t2.86177 &\n\t\t19.23120 &\n\t\t19.23457 \\\\\n\t\t&\n\t\t60,380 &\n\t\t119,418 &\n\t\t2.86219 &\n\t\t2.86231 &\n\t\t19.26655 &\n\t\t19.26628 \\\\\n\t\t&\n\t\t142,702 &\n\t\t283,162 &\n\t\t2.86249 &\n\t\t2.86256 &\n\t\t19.28045 &\n\t\t19.28028 \\\\\n\t\t&\n\t\t291,363 &\n\t\t579,360 &\n\t\t2.86263 &\n\t\t2.86267 &\n\t\t19.28727 &\n\t\t19.28755 \\\\\n\t\t&\n\t\t495,607 &\n\t\t986,724 &\n\t\t2.86275 &\n\t\t2.86269 &\n\t\t19.29057 &\n\t\t19.29081 \\\\\n\t\t&\n\t\t1,064,447 &\n\t\t2,122,160 &\n\t\t2.86272 &\n\t\t2.86273 &\n\t\t19.29385 &\n\t\t19.29389 \\\\\n\t\t&\n\t\t2,106,077 &\n\t\t4,202,536 &\n\t\t2.86274 &\n\t\t2.86274 &\n\t\t19.29618 &\n\t\t19.29615 \\\\\n\t\t&\n\t\t4,031,557 &\n\t\t8,049,644 &\n\t\t2.86275 &\n\t\t2.86275 &\n\t\t19.29763 &\n\t\t19.29765 \\\\\n\t\t\\hline\n\t\\end{tabular} \n\t\\captionof{table}{Maximum temperature and Flux magnitude values in the numerical simulations of the third example.}\\label{table: numerical results huevo} \n\\end{center}\nThe temperature distribution and Flux magnitude fields for the finest mesh are shown in Figure \\ref{fig: third example temperature field}.\n\\begin{figure}\n\t\\centering\n\t\\subfigure[Contour Fill of Temperatures]{\\includegraphics[width=0.44\\textwidth]{Fig_23_HuevoContourTemp.png}}\\hspace{2mm}\n\t\\subfigure[Contour Fill of Flux vectors on Elems]{\\includegraphics[width=0.44\\textwidth]{Fig_23_HuevoContourFlux.png}}\n\t\\captionof{figure}{Temperature distribution and Flux magnitude fields for the finest mesh of the third example.}\\label{fig: third example temperature field}\n\\end{figure}\nFigure \\ref{fig: third example diametral graphs}\nshows the graphs of the temperature and flux magnitude values along a diameter of the circle for different meshes of Figure \n\\ref{fig: third example meshes}\n\\begin{figure}\n\t\\centering\n\t\\subfigure[]{\\includegraphics[width=0.42\\textwidth]{Fig_24_HuevoTempCrossSection01.png}}\\hspace{2mm}\n\t\\subfigure[]{\\includegraphics[width=0.42\\textwidth]{Fig_24_HuevoFluxCrossSection01.png}}\n\t\\subfigure[]{\\includegraphics[width=0.42\\textwidth]{Fig_24_HuevoTempCrossSection02.png}}\\hspace{2mm}\n\t\\subfigure[]{\\includegraphics[width=0.42\\textwidth]{Fig_24_HuevoFluxCrossSection02.png}}\n\t\\subfigure[]{\\includegraphics[width=0.42\\textwidth]{Fig_24_HuevoTempCrossSection03.png}}\\hspace{2mm}\n\t\\subfigure[]{\\includegraphics[width=0.42\\textwidth]{Fig_24_HuevoFluxCrossSection03.png}}\n\t\\captionof{figure}{Temperature and Flux magnitude graphs of the third example along a cross-section of the domain for different meshes: \n\t\tMesh in Figure \\ref{fig: third example meshes}(a), a-Temperature, b-Flux; \n\t\tMesh in Figure \\ref{fig: third example meshes}(b), c-Temperature, d-Flux; \n\t\tMesh in Figure \\ref{fig: third example meshes}(c), e-Temperature, f-Flux; \n\t}\\label{fig: third example diametral graphs}\n\\end{figure}\n\n\\newpage\n\n{\\bf Remark}. As can be seen from the previous examples, DEC behaves well\non coarse meshes. \nAs expected, the results of DEC and FEML are similar for fine meshes. \nWe would also like to point out the the computational costs of DEC and FEML are very similar.\n\n\n\n\\section{Conclusions}\\label{sec: conclusions}\n\nDEC is a relatively recent discretization scheme for PDE's which takes into account the geometric and analytic \nfeatures of the operators and the domains involved. \nThe main contributions of this paper are the following:\n\\begin{enumerate}\n \\item We have made explicit the local formulation of DEC, i.e. on each triangle of the mesh. As is customary, the local pieces can be assembled, which facilitates the implementation of DEC by the interested reader. \n Furthermore, the profiles of the assembled DEC matrices are equal to those of assembled FEML matrices.\n\n\\item Guided by the local formulation,\nwe have deduced a natural way to approximate the flux\/gradient vector of a discretized function as well as the anisotropic flux vector.\nWe have shown that the formulas defining the flux in DEC and FEML coincide. \n\n\\item We have deduced how the anisotropy tensor acts on primal 1-forms.\n\n\\item We have deduced the local DEC formulation of the 2D anisotropic Poisson equation, and\nhave proved that the DEC and FEML diffusion terms are identical, while the source terms are not \n-- due to the different area-weight allocation for the nodes.\n\n\n\\item Local DEC allows a simple treatment of heterogeneous material properties assigned to subdomains (element by element), \nwhich eliminates the need of dealing with it through ad hoc modifications of the global discrete Hodge star operator matrix. \n\n\\end{enumerate}\n\n\\vspace{.1in}\n\nOn the other hand we would like to point the following features:\n\\begin{itemize}\n\\item \n The area weights assigned to the nodes of the mesh when solving the 2D anisotropic Poisson equation can even be negative (when a triangle has an inner angle greater that $120^\\circ$), in stark contrast to the FEML formulation.\n\n\\item The computational cost of DEC is similar to that of FEML. While the numerical results of DEC and FEML on fine meshes are virtually identical, the DEC solutions are better than those of FEML on coarse meshes. Furthermore, DEC solutions display numerical convergence.\n\n\\end{itemize}\n\n\\vspace{.1in}\n\nOur future work will include the DEC discretization of convective terms and DEC on 2-dimensional simplicial surfaces in 3D. \nPreliminary results on both problems are promising and competitive with FEML. \n\n\\bigskip\n\n{\\small\n\\renewcommand{\\baselinestretch}{0.5}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nSequential Monte Carlo (SMC) is a class of algorithms that enable simulation from a target distribution of interest. These algorithms are based on defining a series of distributions, and generating samples from each distribution in turn. SMC was initially used in the analysis of state-space models. In this setting there is a time--evolving hidden state of interest, inference about which is based on a set of noisy observations \\citep{gordon1993,liuchen1998,SMCMiP,fearnhead2002}. The sequence of distributions are defined to be the posterior distributions of the state at consecutive time-points given the observations up to those time points. More recent work has looked at developing SMC methods that can analyse state-space models which have unknown fixed parameters. Such methods introduce steps into the algorithm to allow the support of the sample of parameter values to change over time, for example by using ideas from kernel density estimation \\citep{SMCMPliuwest}, or MCMC moves \\citep{gilks2001,storvik2002,fearnhead2002}.\n\nMost recently, SMC methods have been applied as an alternative to MCMC for standard Bayesian inference problems. \\citep{neal2001,chopin2002,delmoral2006,fearnhead2008}. In this paper the focus will be on methods for sampling from the posterior distribution of a set of parameters of interest. SMC methods for this class of targets introduce an artificial sequence of distributions that run from the prior to the posterior and sample recursively from these using a combination of Importance Sampling and MCMC moves. This approach to sampling has been demonstrated empirically to often be more effective than using a single MCMC chain \\citep{jasra2007,jasra2008}. There are heuristic reasons for why this may true in general: the annealing of the target and spread of samples over the support means that SMC is less likely to be become trapped in posterior modes.\n\nSimply invoking an untuned MCMC move within an SMC algorithm would likely lead to poor results because the move step would not be effective in combating sample depletion. The structure of SMC means that at the time of a move there is a sample from the target readily available, this can be used to compute posterior moments and inform the shape of the proposal kernel as in \\cite{jasra2008a}; however, further refinements can lead to even better performance. Such refinements include the scaling of estimated target moments by an optimal factor, see \\cite{roberts2001} for example. For general targets and proposals no theoretical results for the choice of scaling exist, and this has led to the recent popularity of adaptive MCMC \\citep{haario1998,andrieu2001,roberts2009,craiu2009,andrieu2008}. In this paper the idea of adapting the MCMC kernel within an SMC algorithm will be explored.\n\nTo date there has been little work at adapting SMC methods. Exceptions include the method of \\cite{jasra2008a}, whose method assumes a likelihood tempered sequence of target densities (see \\cite{neal2001}) and the adaptation procedure both chooses this sequence online, as well as computing the variance of a random walk proposal kernel used for particle dynamics. \\cite{cornebise2008} also considers adapting the proposal distribution within SMC for state-space models. Assuming that the proposal density belongs to a parametric family with parameter $\\theta$, their method proceeds by simulating a number of realisations for each of a range of values of $\\theta$ and selecting the value that minimises the empirical Shannon entropy of the importance weights; new samples are then re--proposed using this approximately optimal value. Further related work includes that of \\cite{douc2005a} and \\cite{cappe2008} on respectively population Monte Carlo and adaptive importance sampling.\n\nThe aims of this paper are to introduce a new adaptive SMC algorithm (ASMC) that automatically tunes MCMC move kernels and chooses between different proposal densities and to provide theoretical justification of the method. The algorithm is based on having a distribution of kernels and their tuning parameters at each iteration. Each current sample value, called a particle, is moved using an MCMC kernel drawn from this distribution. By observing the expected square jumping distance \\citep{craiu2009,sherlock2009} for each particle it is possible to learn which MCMC kernels are mixing better. The information thus obtained can then used to update the distribution of kernels. The key assumption of the new approach is that the optimal MCMC kernel for moving particles does not change much over the iterations of the SMC algorithm. As will be discussed, and shown empirically, in section \\ref{sect:resultsASMC} this can often be achieved by appropriate parameterisation of a family of MCMC kernels.\n\nThe structure of the paper is as follows. In the next section, the model of interest will be introduced and followed by a review of MCMC and SMC approaches. Then in Section \\ref{sect:newmethod}, the new adaptive SMC will be presented. Guidelines on implementing the algorithm as well as some theory on the convergence will be presented in Section \\ref{sect:theoryASMC}. In Section \\ref{sect:resultsASMC} the method will be evaluated using simulated data. The results show that the proposed method can successfully choose both an appropriate MCMC kernel and an appropriate scaling for the kernel. The paper ends with a discussion. \n\n\\section{Model}\n\nThe focus of this paper will be on Bayesian inference for parameters, $\\theta$, from a model where independent identically distributed data is available. Note that the ideas behind the proposed adaptive SMC algorithm can be applied more generally (see section \\ref{sect:discuss_extASMC}). Let $\\pi(\\theta)$ denote the prior for $\\theta$ and $\\pi(y|\\theta)$ the probability density for the observations. The aim will be to calculate the posterior density,\n\\begin{equation} \\label{eq:1}\n \\pi(\\theta|y_{1:n})\\propto\\pi(\\theta)\\prod_{i=1}^n\\pi(y_i|\\theta),\n\\end{equation}\nwhere, here and throughout, $\\pi$ will be used to denote a probability density, and $y_{1:t}$ means $y_1,\\ldots,y_t$. \n\nIn general, $\\pi(\\theta|y_{1:n})$ is analytically intractable and so to compute posterior functionals of interest, for example expectations, Monte Carlo simulation methods are often employed. Sections \\ref{sect:MCMCintro} and \\ref{sect:SMCintro} provide a brief description of two such Monte Carlo approaches.\n\n\\subsection{MCMC\\label{sect:MCMCintro}}\n\nAn MCMC transition kernel, $K_h$, is an iterative rule for generating samples from a target probability density, for example a posterior. $K_h$ comprises a proposal kernel, here and throughout denoted $q_h$ (the subscript $h$ indicates dependence on a tuning parameter) and an acceptance ratio that depends on the target and, in general, the proposal densities (see \\cite{MCMCiP,gamermanlopes} for reviews of MCMC methodology). The most generally applicable MCMC method is Metropolis--Hastings, see Algorithm \\ref{alg:met_hast} \\citep{metropolis1953,hastings1970}.\n\n\\begin{algorithm}\n \\caption{Metropolis--Hastings Algorithm \\citep{metropolis1953,hastings1970}}\n \\label{alg:met_hast}\n \\begin{algorithmic}[1]\n \\STATE Start with an initial sample, $\\theta^{(0)}$, drawn from any density, $\\pi_0$.\n \\FOR{$j=1,2,\\ldots$}\n \\STATE Propose a move to a new location, $\\tilde\\theta$, by drawing a sample from $q_h(\\theta^{(i-1)},\\tilde\\theta)$.\n \\STATE Accept the move (ie set $\\theta^{(i)} = \\tilde\\theta$) with probability,\n \\begin{equation}\\label{eqn:MHacc}\n \\min\\left\\{1,\\frac{\\pi(\\tilde\\theta|y_{1:n})}{\\pi(\\theta^{(i-1)}|y_{1:n})}\\frac{q_h(\\tilde\\theta,\\theta^{(i-1)})}{q_h(\\theta^{(i-1)},\\tilde\\theta)}\\right\\},\n \\end{equation}\n else set $\\theta^{(i)} = \\theta^{(i-1)}$.\n \\ENDFOR\n \\end{algorithmic}\n\\end{algorithm}\n\nProbably the simplest MH algorithm is the random walk Metropolis (RWM). The proposal kernel for RWM is a symmetric density centred on the current state, the most common example being a multivariate normal, $q_h(\\theta^{(i-1)},\\tilde\\theta)=\\mathcal{N}(\\tilde\\theta;\\theta^{(i-1)},h^2\\hat\\Sigma_\\pi)$, where $\\hat\\Sigma_\\pi$ is an estimate of the target covariance. Both the values of $\\hat\\Sigma_\\pi$ and $h$ are critical to the performance of the algorithm. If $\\hat\\Sigma_\\pi$ does not accurately estimate the posterior covariance matrix, then the likely directions of the random walk moves will likely be inappropriate. On the other hand, a value of $h$ that is too small will lead to high acceptance rates, but the samples will be highly correlated. If $h$ is too large then the algorithm will rarely move, which in the worst case scenario could lead to a degenerate sample. \n\nThese observations on the r\\^ole of $h$ point to the idea of an \\emph{optimal scaling}, a $h$ somewhere between the extremes that promotes the best mixing of the algorithm. In the case of elliptically symmetric unimodal targets, an optimal random walk scaling can sometimes be computed numerically; this class of targets includes the Multivariate Gaussian \\citep{sherlock2009}. Other theoretical results include optimal acceptance rates which are derived in the limit as the dimension of $\\theta$, $d\\rightarrow\\infty$ (see \\cite{roberts2001} for examples of targets and proposals). In general however, there are no such theoretical results. \n\nOne way of circumventing the need for analytical optimal scalings is to try to learn them online \\citep{andrieu2001,atchade2005}, this can include learning both a good scaling, $h$, and estimating the target covariance, $\\hat\\Sigma_\\pi$ \\citep{haario1998}. Recent research in adaptive MCMC has generated a number of new algorithms (see for example \\cite{andrieu2008,roberts2009,craiu2009}), though some care must be taken to ensure that the resulting chain has the correct ergodic distribution.\n\n\n\\subsection{Sequential Monte Carlo\\label{sect:SMCintro}}\n\nAn alternative approach to generating samples from a posterior is to use sequential Monte Carlo (SMC, see \\cite{delmoral2006} for a review). The main idea behind SMC is to introduce a sequence of densities leading from the prior to the target density of interest and to iteratively update an approximation to these densities. For the application considered here, it is natural to define these densities as $\\pi_t(\\theta)=\\pi(\\theta|y_{1:t})$ for $t=1,\\ldots,n$; this `data tempered' schedule will be used in the sequel. The approximations to each density are defined in terms of a set of weighted particles, $\\{\\theta_t^{(j)},w_t^{(j)}\\}_{j=1}^M$, produced so that as $M\\rightarrow\\infty$, Monte Carlo sums converge to their `correct' expectations:\n\\begin{equation*}\\label{eqn:properweight}\n \\lim_{M\\rightarrow\\infty} \\left\\{\\frac{\\sum_{j=1}^Mw_t^{(j)}f(\\theta_t^{(j)})}{\\sum_{i=1}^Mw_t^{(i)}}\\right\\} = \\mathbb{E}_{\\pi_t(\\theta_t)}[f(\\theta_t)],\n\\end{equation*}\nfor all $\\pi_t$--integrable functions, $f$. One step of an SMC algorithm can involve importance reweighting, resampling and moving the particles via an MCMC kernel \\citep{gilks2001,chopin2002}. For concreteness, this paper will focus on the iterated batch importance sampling (IBIS) algorithm of \\cite{chopin2002}. \n\nThe simplest way to update the particle approximation in model (\\ref{eq:1}) is to let $\\theta_{t}^{(j)}=\\theta_{t-1}^{(j)}$ and\n$w_t^{(j)}=w_{t-1}^{(j)}\\pi(y_t|\\theta_{t}^{(j)})$. However such an algorithm will degenerate for large $t$, as eventually only one particle will have non-negligible weight. Within IBIS, resample--move steps (sometimes referred to here as simply `move steps') are introduced to alleviate this. In a move step, the particles are first resampled so that the expected number of copies of particle $\\theta_t^{(j)}$ is proportional to $w_t^{(j)}$. This process produces multiple copies of some particles. In order to create particle diversity, each resampled particle is moved by an MCMC kernel. The MCMC kernel is chosen to have stationary distribution $\\pi_t$. The resulting particles are then assigned a weight of $1\/M$.\n\nThe decision of whether to apply a resample-move step within IBIS is based on the effective sample size (ESS, see \\cite{kong1994,liuchen1998}). The ESS is a measure of variability of the particle weights; using this to decide whether to resample is justified by arguments within \\cite{liu1995} and \\cite{liu1998}. Full details of IBIS are given in Algorithm \\ref{alg:IBIS}.\n\n\\begin{algorithm}\n \\caption{Chopin's IBIS algorithm}\n \\label{alg:IBIS}\n \\begin{algorithmic}[1]\n \\STATE Initialise from the prior $\\{\\theta_0^{(j)},w_0^{(j)}\\}_{j=1}^M\\sim\\pi_0$.\n \\FOR{$t=1,\\ldots,n$}\n \\STATE Assume current $\\{\\theta_{t-1}^{(j)},w_{t-1}^{(j)}\\}_{j=1}^M\\sim\\pi_{t-1}$\n \\STATE Reweight $w_t^{(j)} = w_{t-1}^{(j)}\\pi_t(\\theta_{t-1}^{(j)})\/\\pi_{t-1}(\\theta_{t-1}^{(j)})$. Result: $\\{\\theta_{t-1}^{(j)},w_t^{(j)}\\}_{j=1}^M\\sim\\pi_t$.\n \\IF{particle weights not degenerate (see text)}\n \\STATE $\\{\\theta_{t}^{(j)},w_{t}^{(j)}\\}_{j=1}^M \\leftarrow \\{\\theta_{t-1}^{(j)},w_{t-1}^{(j)}\\}_{j=1}^M$\n \\STATE $t\\rightarrow t+1$.\n \\ELSE\n \\STATE \\label{alg:IBIS_state_resample} Resample: let $\\mathcal{K}=\\{k_1,\\ldots,k_M\\}\\subseteq\\{1,\\ldots,M\\}$ be the resampling indices, then $\\{\\theta_{t-1}^{(k)},1\/M\\}_{k\\in\\mathcal K}\\sim\\pi_t$. Relabel: $k_j \\leftarrow j$, the $j$th resampling index so that $\\{\\theta_{t-1}^{(j)},1\/M\\}_{j=1}^M\\sim\\pi_t$.\n \\STATE Move via $\\pi_t$--invariant MCMC kernel. Result: $\\{\\theta_t^{(j)},1\/M\\}_{j=1}^M\\sim\\pi_t$.\n \\ENDIF\n \\ENDFOR\n \\end{algorithmic}\n\\end{algorithm}\n\n\nChopin's IBIS algorithm is a special case of the resample--move (RM) algorithm of \\cite{gilks2001} and the general algorithm described by \\cite{delmoral2006} (note that the latter method applies beyond MCMC--within--SMC and provides a unifying framework for sampling from sequences of targets). The main difference between RM and IBIS is that, in their presentation of RM, \\cite{gilks2001} use resampling and move steps at each iteration of the sampler. Chopin noticed that at a particular iteration it may be better to just reweight the particles, rather than incur the computational cost and degeneracy induced by a resample--move step. Another related algorithm, a development of simulated annealing \\citep{kirkpatrick1983} due to \\cite{neal2001}, utilises an alternative `likelihood tempered' sequence of targets. The proposed target sequence is $\\pi_t(\\theta)=\\pi(\\theta)\\pi(y_{1:n}|\\theta)^{\\xi_t}$, where $\\{\\xi_t\\}$ is a sequence of real numbers starting at $0$ (the prior) and ending on $1$ (the posterior). Since each move step requires evaluation of the likelihood over all available observations, the main disadvantage of likelihood tempering is computational cost, although for models with sufficient statistics this is not an issue. Further disadvantages of Neal's proposed algorithm are the absence of resampling steps which eventually leads to sample degeneracy; and the lack of interpretability of intermediate target densities.\n\nThe efficiency of an SMC algorithm, such as IBIS, depends on the mixing properties of the associated MCMC kernel. Within SMC there is the advantage of being able to use the current set of particles to help tune an MCMC kernel. For example, the weighted particles can give an estimate of the posterior covariance matrix, which can be used within a random walk proposal. However even in this case, the proposal variance still needs to be appropriately scaled \\citep{roberts2001,sherlock2009}. In the next section the new adaptive SMC procedure will be introduced, the algorithm can learn an appropriate tuning for the MCMC kernel, and can also be used to choose between a set of possible kernels.\n\n\\section{The Adaptive SMC Sampler\\label{sect:newmethod}} \n\nFirst consider the case where the move step in the IBIS algorithm involves one type of MCMC kernel. Let $\\pi_t$ be an \\emph{arbitrary} continuous probability density (the target) and $K_{h,t}$ a $\\pi_t$--invariant MCMC kernel with tuning parameter, $h$. The parameter $h$ is to be chosen to maximise the following utility function,\n\\begin{eqnarray} \\label{eqn:optfun}\n g^{(t)}(h) &=& \\int \\pi_t(\\theta_{t-1})K_{h,t}(\\theta_{t-1},\\theta_t)\\Lambda(\\theta_{t-1},\\theta_t)\\mathrm{d}\\theta_{t-1}\\mathrm{d}\\theta_t,\\\\\n &=& \\mathbb{E}\\left[\\Lambda(\\theta_{t-1},\\theta_t)\\right], \\nonumber\n\\end{eqnarray}\nwhere $\\Lambda(\\theta_{t-1},\\theta_t)>0$ is a measure of mixing of the chain. Most MCMC adaptation criteria can be viewed in this way \\citep{andrieu2008}. Note that for simplicity of presentation, $\\Lambda$ only depends on the current and subsequent state, though the idea readily extends to more complex cost functionals, for example involving multiple transitions of the MCMC chain. The function $g^{(t)}$ is the average performance of the chain with respect to $\\Lambda$, which would normally be some measure of mixing. The ideal choice for $\\Lambda$ would be the integrated autocorrelation time (whence the goal would be to maximise $-g^{(t)}$), but a computationally simpler measure of mixing is the expected square jumping distance (ESJD). Maximising the ESJD is equivalent to minimising the lag-1 autocorrelation; this measure is often used within adaptive MCMC, see for example \\cite{sherlock2009,pasarica2010}. \n\nIn the following it will be assumed that the proposal distribution can depend on quantities calculated from the current set of particles (for example estimates of the posterior variance), but this will be suppressed in the notation.\nThe main idea of ASMC is to use the observed instances of $\\Lambda(\\theta_{t-1},\\theta_t)$ to help choose the best $h$. The tuning parameter will be treated as an auxiliary random variable. At time-step $t$ the aim is to derive a density for the tunings, $\\pi^{(t)}(h)$. If a move step is invoked at this time, a sample of $M$ realisations from $\\pi^{(t)}(h)$, denoted $\\{h_t^{(j)}\\}_{j=1}^M$, will be drawn and `allocated' to particles at random. \n\nWhen moving the $j$th resampled particle, the tuning parameter $h_t^{(j)}$ will be used within the proposal distribution.\nFor notational simplicity this value will be denoted $h$ in the following.\nLet $\\theta_{t-1}^{(j)}$ be the $j$th resampled particle (see step \\ref{alg:IBIS_state_resample} of Algorithm \\ref{alg:IBIS}). In moving this particle, $\\tilde{\\theta}_t^{(j)}$ is drawn from $q_{h}(\\theta_{t-1}^{(j)},\\,\\cdot\\,)$, and accepted with probability \n$\\alpha_h(\\theta_{t-1}^{(j)},\\tilde\\theta_t^{(j)})$, given by (\\ref{eqn:MHacc}). If the proposed particle is accepted then $\\theta_t^{(j)}=\\tilde{\\theta}_t^{(j)}$ otherwise $\\theta_t^{(j)}={\\theta}_{t-1}^{(j)}$.\n\nThe utility function in (\\ref{eqn:optfun}) simplifies to,\n\\begin{equation*}\n g^{(t)}(h) = \\int \\pi_t(\\theta_{t-1})q_h(\\theta_{t-1},\\tilde\\theta_t)\\tilde\\Lambda(\\theta_{t-1},\\tilde\\theta_t)\\mathrm{d}\\theta_{t-1}\\mathrm{d}\\tilde\\theta_t,\n\\end{equation*}\nwhere \n\\begin{equation*}\n \\tilde\\Lambda(\\theta_{t-1}^{(j)},\\tilde\\theta_t^{(j)})= \\alpha_h(\\theta_{t-1}^{(j)},\\tilde\\theta_t^{(j)})\\Lambda(\\theta_{t-1}^{(j)},\\tilde\\theta_t^{(j)}). \n\\end{equation*}\nSince by assumption the resampled particles are approximately drawn from $\\pi_t$ and proposed particles are drawn from $q_h(\\theta_{t-1},\\tilde\\theta_t)$, the quantity $\\tilde\\Lambda(\\theta_{t-1}^{(j)},\\tilde\\theta_t^{(j)})$ can be viewed as an unbiased estimate of $g^{(t)}(h)$.\n\nThe approach in this paper is to use the observed $\\tilde\\Lambda(\\theta_{t-1}^{(j)},\\tilde\\theta_t^{(j)})$ to update the distribution $\\pi^{(t)}(h)$ to a new distribution $\\pi^{(t+1)}(h)$. In particular each $h_t^{(j)}$ will be assigned a weight, $f(\\tilde\\Lambda(\\theta_{t-1}^{(j)},\\tilde\\theta_t^{(j)}))$, for some function $f:\\mathbb{R}^+\\rightarrow\\mathbb{R}^+$. The new density of scalings will be defined,\n\\begin{equation} \\label{eq:pit}\n \\pi^{(t+1)}(h) \\propto \\sum_{i=1}^M f(\\tilde\\Lambda(\\theta_{t-1}^{(j)},\\tilde\\theta_t^{(j)})) R(h-h_t^{(j)}),\n\\end{equation}\nwhere $R(h-h_t^{(j)})$ is a density for $h$ which is centred on $h_t^{(j)}$. Simulating from $\\pi^{(t+1)}(h)$ is achieved by first resampling the $h_t^{(j)}$s with probabilities proportional to their weight and then adding noise to each resampled value; the distribution of this noise is given by $R(\\,\\cdot\\,)$. The motivation for adding noise to the resampled $h$--values is to avoid the distributions $\\pi^{(t)}(h)$ degenerating too quickly to a point-mass on a single value. Similar ideas are used in dynamic SMC methods for dealing with fixed parameters, for example \\cite{west1993,SMCMPliuwest}. In practice the variance of the noise can depend on the variance of $\\pi^{(t)}(h)$ and by analogy to Kernel density estimation should tend to 0 as the number of particles gets large.\n\nIf there is no resampling at step $t$ then set $\\pi^{(t+1)}(h)=\\pi^{(t)}(h)$. The scheme is initiated with an arbitrary distribution $\\pi(h)$.\nThe specific choice of $f$ considered in this paper is a simple linear weighting scheme,\n\\begin{equation*}\n f(\\tilde\\Lambda) = a + \\tilde\\Lambda,\\qquad a\\geq 0.\n\\end{equation*} \nTheoretical justification for this choice is given in the next section.\n\nOne assumption of the proposed approach is that a good choice of $h$ at one time-step will be a good choice at nearby time-steps. Note that this is based on an implicit assumption within SMC that successive targets are similar (see \\cite{chopin2002,delmoral2006} for example). Furthermore, using estimates of posterior variances within the proposal distribution can also help ensure that good values of $h$ at one time-step will be a good choice at nearby time-steps. Some theoretical results concerning this matter will be presented in Section \\ref{sect:theoryASMC}.\n\nTo choose between different types of MCMC kernel is now a relatively straightforward extension of the above. Assume there are $I$ different MCMC kernels, each defined by a proposal distribution $q_{h,i}$, where $i \\in \\{1,\\ldots,I\\}$. The algorithm now learns a set of distributions, $\\pi^{(t)}(h,i)$, for the pair of kernel type and associated tuning parameter. Each particle is assigned a random kernel type and tuning drawn form this distribution, with the pair, $(h_{t-1}^{(j)},i_{t-1}^{(j)})$, associated with $\\theta_{t-1}^{(j)}$. The algorithm proceeds by weighting this pair based on the observed $\\tilde\\Lambda(\\theta_{t-1}^{(j)},\\tilde\\theta_t^{(j)})$ values as before, and updating the distribution, \n\\begin{equation}\\label{eqn:newkernelmixture}\n \\pi^{(t)}(h,i) \\propto \\sum_{j=1}^M f(\\tilde\\Lambda(\\theta_{t-1}^{(j)},\\tilde\\theta_t^{(j)})) R(h-h_{t-1}^{(j)})\\delta_{i_{t-1}^{(j)}}(i).\n\\end{equation}\nwhere $\\delta_{i_{t-1}^{(j)}}(i)$ is a point mass on $i=i_{t-1}^{(j)}$. \nThe method is described in detail below, see Algorithm \\ref{alg:ASMC}. Within the specific implementation described, the sample of pairs, $(h,i)$, from $\\pi^{(t)}(h,i)$ are allocated to particles randomly immediately after the resample--move step at iteration $t$. These pairs are then kept until the next iteration a resample--move step is called.\n\n\\begin{algorithm}\n \\caption{The Adaptive SMC algorithm. Here, $\\pi_0(\\,\\cdot\\,),\\ldots,\\pi_n(\\,\\cdot\\,)$ are an arbitrary sequence of targets; an MCMC kernel is assumed for particle dynamics.}\n \\label{alg:ASMC}\n \\begin{algorithmic}[1]\n \\STATE Initialise from the prior $\\{\\theta_0^{(j)},w_0^{(j)}\\}_{j=1}^M\\sim\\pi_0$.\n \\STATE Draw a selection of pairs of MCMC kernels with associated tuning parameters, $\\{(h_0^{(j)},K_{h,0}^{(j)})\\}_{j=1}^M\\equiv\\{(h_0^{(j)},i_0^{(j)})\\}_{j=1}^M\\sim\\pi(h,i)$, and attach one to each particle arbitrarily.\n \\FOR{$t=1,\\ldots,n$}\n \\STATE Assume current $\\{\\theta_{t-1}^{(j)},w_{t-1}^{(j)}\\}_{j=1}^M\\sim\\pi_{t-1}$\n \\STATE Reweight $w_t^{(j)} = w_{t-1}^{(j)}\\pi_t(\\theta_{t-1}^{(j)})\/\\pi_{t-1}(\\theta_{t-1}^{(j)})$. Result: $\\{\\theta_{t-1}^{(j)},w_t^{(j)}\\}_{j=1}^M\\sim\\pi_t$.\n \\IF{particle weights not degenerate (see text)}\n \\STATE $\\{\\theta_{t}^{(j)},w_{t}^{(j)}\\}_{j=1}^M \\leftarrow \\{\\theta_{t-1}^{(j)},w_{t-1}^{(j)}\\}_{j=1}^M$\n \\STATE $\\{(h_{t}^{(j)},K_{h,t}^{(j)})\\}_{j=1}^M \\leftarrow \\{(h_{t-1}^{(j)},K_{h,t-1}^{(j)})\\}_{j=1}^M$\n \\STATE $t\\rightarrow t+1$.\n \\ELSE\n \\STATE Resample: let $\\mathcal{K}=\\{k_1,\\ldots,k_M\\}\\subseteq\\{1,\\ldots,M\\}$ be the resampling indices, then $\\{\\theta_{t-1}^{(k)},1\/M\\}_{k\\in\\mathcal K}\\sim\\pi_t$. Relabel: $k_j \\leftarrow j$, the $j$th resampling index so that $\\{\\theta_{t-1}^{(j)},1\/M\\}_{j=1}^M\\sim\\pi_t$. DO NOT resample kernels or tuning parameters at this stage.\n \\STATE Move $\\theta_{t-1}^{(j)}$ via the $\\pi_t$--invariant MCMC kernel, $K_{h,t}^{(j)}$, and tuning parameter $h_{t-1}^{(j)}$, denote the proposed new particle as $\\tilde\\theta_t^{(j)}$ and accepted\/rejected particle as $\\theta_t^{(j)}$. Result: $\\{\\theta_t^{(j)},1\/M\\}_{j=1}^M\\sim\\pi_t$.\n \\STATE To obtain $\\{(h_{t}^{(j)},K_{h,t}^{(j)})\\}_{k=1}^M\\equiv\\{(h_t^{(j)},i_t^{(j)})\\}$, sample $M$ times from (\\ref{eqn:newkernelmixture}). Allocate the new selection to particles at random.\n \\ENDIF\n \\ENDFOR\n \\end{algorithmic}\n\\end{algorithm}\n\n\n\\section{Theoretical Results \\label{sect:theoryASMC}}\n\nIn this section the proposed algorithm will be justified by a series of theoretical results; guidance as to how it should best be implemented will also be given. The results presented here apply in the limit as the number of particles, $M\\rightarrow\\infty$. As discussed above, in this limit, the variance of the kernel $R(\\,\\cdot\\,)$ in (\\ref{eq:pit}) tends to 0.\n\nTo simplify the discussion, it will be assumed that tunings are one dimensional (the arguments presented extend readily to the multivariate case). For a slight notational simplification, the criterion $\\Lambda$ will be used, rather than $\\tilde\\Lambda$ (as suggested in algorithm \\ref{alg:ASMC}); this does not affect the validity of any of the arguments, which also hold for $\\tilde\\Lambda$. The section is split into two parts.\n\nFirstly, in section \\ref{sect:onestep}, it is of interest to examine what happens to the distribution of the $h$s after one step of reweighting and resampling; this result will lead to a criterion for the choice of weight function that guarantees MCMC mixing improvement with respect to $\\Lambda$. In section \\ref{sect:manystep}, the sequential improvement of $h$s will be considered over many steps of the ASMC algorithm and with a changing target. General conditions for convergence of ASMC to the optimal kernel and tuning parameter will be provided.\n\n\\subsection{One Step Improvement and Weighting Function\\label{sect:onestep}}\n\nIn this section and in the relevant proofs, it is appropriate to temporarily drop the $t$ superscript, eg $g^{(t)}\\equiv g$, $\\theta_{t-1}\\equiv\\theta$ and $\\theta_t\\equiv\\theta'$. To study the effect of reweighting and resampling on the distribution of the $h$s, suppose that currently $\\{h^{(j)}\\}_{j=1}^M\\stackrel{\\text{iid}}{\\sim}\\pi(h)$, the pdf of a random variable, $H$. The dependence on current and proposed particles means the weight attached to $h^{(j)}$ is random, but also, due to the independence of $h$ with the particles, is an unbiased estimator of the `true' weight, $\\mathbb{E}_{\\Theta,\\Theta'|H}[f(\\Lambda)|H=h^{(j)}]$, where $\\mathbb{E}_{\\Theta,\\Theta'|H}$ denotes the expectation with respect to the joint density of the random variables $\\Theta$ and $\\Theta'$ conditional on $H$. The true \\emph{weighting function} will be denoted,\n\\begin{equation}\\label{eqn:weightfun}\n w(h) = \\mathbb{E}_{\\Theta,\\Theta'|H}[f(\\Lambda)|H=h].\n\\end{equation}\nThe following proposition, which is used repeatedly in subsequent results, shows how reweighting and resampling affects $\\pi(h)$.\n\n\\begin{Propn}\\label{propn:wboot}\n Suppose that currently $\\{h^{(j)}\\}_{j=1}^M\\stackrel{\\text{iid}}{\\sim}\\pi(h)$, the pdf of a random variable, $H$, independent of $\\theta$. Let $w(h)$ be the weighting function defined as in (\\ref{eqn:weightfun}). Then in the limit as $M\\rightarrow\\infty$, the distribution of the reweighted and subsequently resampled $h$s is,\n \\\n \\pi^\\star(h) = \\frac{w(h)\\pi(h)}{\\int w(h)\\pi(h)\\mathrm{d} h}.\n \\\n\\end{Propn}\n\n \\textbf{Proof:} See Appendix \\ref{sect:proofwboot}.\\qed\n\nSince ASMC uses a selection of $h$s, it is appropriate as a starting point to look for conditions under which their \\emph{distribution} is improved. It would be desirable if, over $\\pi^\\star(h)$, the objective function would on average take a higher value, for then the new distribution would on average perform better with respect to $\\Lambda$ than the old. This criterion can be stated in mathematical form: conditions on $f$ are sought for which,\n\\begin{equation*}\n \\int\\pi^\\star(h)g(h)\\mathrm{d} h \\geq \\int\\pi(h)g(h)\\mathrm{d} h.\n\\end{equation*}\n\n\n\\begin{Lem}\\label{lem:improve_criterion}\nAssuming $g$ is $\\pi(h)$--integrable, in the limit as $M\\rightarrow\\infty$,\n\\begin{equation}\\label{eqn:cov_improve}\n \\mathbb{E}_{\\pi^\\star(h)}[g(h)] \\geq \\mathbb{E}_{\\pi(h)}[g(h)]\\iff\\mathrm{cov}_{\\pi(h)}[g(h),w(h)] \\geq 0.\n\\end{equation}\nThat is, provided there is positive correlation between the objective function $g(h)$ and the weighting function, $w(h)$, the new distribution of $h$s will on average perform better (on $g(h)$) with respect to $\\Lambda$ than the old.\n\\end{Lem}\n\\textbf{Proof:} The result is obtained by expanding definitions in $(\\ref{eqn:cov_improve})$:\n\\begin{eqnarray*}\n \\mathbb{E}_{\\pi^\\star(h)}[g(h)] &\\geq& \\mathbb{E}_{\\pi(h)}[g(h)],\\\\\n \\iff\\mathbb{E}_{\\pi(h)}[w(h)g(h)] &\\geq& \\mathbb{E}_{\\pi(h)}[w(h)]\\mathbb{E}_{\\pi(h)}[g(h)],\\\\\n \\iff\\mathrm{cov}_{\\pi(h)}[g(h),w(h)] &\\geq& 0.\n\\end{eqnarray*}\\qed\n\nAlthough this result does not directly yield a general form for $f$, it does give a simple criterion that must be fulfilled by any candidate function. An immediate corollary gives more concrete guidance:\n\n\\begin{Cor}\nA simple linear weighting scheme, $f(\\Lambda) = a + \\Lambda$, where $a\\geq0$, satisfies (\\ref{eqn:cov_improve}). \n\\end{Cor}\n\\textbf{Proof:} This is trivially verified using the linearity property of the covariance.\\qed\n\nA consequence of this lemma is that the ASMC algorithm with linear weights will lead to sequential improvement with respect to $\\Lambda$ under very weak assumptions on the target and initial density for $h$. \nA linear weighting scheme may at first glance seem sub--optimal, and that it should be possible to learn $h$ more quickly using a function $f(\\Lambda)$ that increases at a super--linear rate. The present authors conjecture that such functions will not always guarantee an improvement in the distribution of $h$. For example consider $f(\\Lambda) = \\Lambda^2$, where the weighting function takes the form, $w(h) = g(h)^2 + \\mathbb{V}[\\Lambda|H=h]$. Because of the $\\mathbb{V}[\\Lambda|H=h]$ term, which may be large for values of $h$ where $g(h)$ is small, it is no longer true that $\\mathrm{cov}_{\\pi(h)}[g(h),w(h)]\\geq0$ in general.\n\n\\subsection{Convergence Over a Number of Iterations\\label{sect:manystep}}\n\nThe goal of this section is to provide a theoretical result concerning the ability of ASMC to update the distribution of $h$s with respect to a sequence of targets, $\\pi_1(\\theta_1),\\ldots,\\pi_n(\\theta_n)$. To simplify notation, it will be assumed that a move occurs at each iteration of the algorithm. The result can be extended to the case where moves occur intermittently, providing they incur infinitely often in the limit as the number of data points goes to infinity. \n\nDefine a set of functions, $\\{g^{(t)}(h)\\}_{t=1}^n$,\n\\begin{equation*}\n g^{(t)}(h) = \\int \\pi_t(\\theta_{t-1})K_{h,t}(\\theta_{t-1},\\theta_t)\\Lambda(\\theta_{t-1},\\theta_t)\\mathrm{d}\\theta_{t-1}\\mathrm{d}\\theta_t\\geq0,\n\\end{equation*}\nwhere, for each $t$, $K_{h,t}$ is a $\\pi_t$--invariant MCMC kernel. \n\nFor a linear weighting scheme,\n\\begin{equation*}\n \\pi^{(t)}(h) \\propto \\pi(h)\\prod\\limits_{s=1}^t(a+g^{(s)}(h)).\n\\end{equation*}\nBelow it will be shown that as $t\\rightarrow \\infty$ if the sequence of functions, $\\{g^{(t)}(h)\\}$, converge to a fixed function, $g(h)$, and if $g$ has a unique global maximum, $h_{\\text{opt}}$, then $\\pi^{(t)}(h)$ will converge to a point mass on $h_{\\text{opt}}$.\n\nThe key assumption of this theorem regards the convergence of the functions $\\{g^{(t)}(h)\\}$. This assumption is linked to the idea that a good value of $h$ for the target at time $t$ is required to be a good value at times later on. As mentioned above, the motivation behind SMC is that successive targets should be similar. Moreover, standard Bayesian asymptotic theory shows that as the number of observations, $n$, tends to infinity, the posterior tends in distribution to that of a Gaussian random variable. Thus, providing information from the current parameters about the posterior variance is used appropriately, it should be expected that the sequence of functions, $\\{g^{(t)}(h)\\}$, would also converge. This issue will be explored empirically in the next section.\n\n\\begin{Thm} \\label{thm:convvari}\n Let $\\pi(h)$ be the initial density for the tuning parameter with support $\\mathcal{H}\\subseteq\\mathbb{R}$ and $a>0$. Define, as above, \n \\begin{equation*}\n \\pi^{(t)}(h) \\propto \\pi(h)\\prod\\limits_{s=1}^t(a+g^{(s)}(h)).\n \\end{equation*}\n Suppose there exists a function $g:\\mathcal{H}\\rightarrow\\mathbb{R}_{\\geq0}$ such that\n \\begin{equation*}\n \\sup_{h\\in\\mathcal H}|g^{(t)} - g| \\leq k_gt^{-\\alpha},\\qquad \\alpha\\in(0,1),\\ k_g>0.\n \\end{equation*}\n Furthermore, suppose $g$ has a unique global maximum, $h_{\\text{opt}}$, contained in the interior of $\\mathcal H$ and that $g$ is twice differentiable in an interval containing $h_{\\text{opt}}$. Then as $t\\rightarrow\\infty$, $\\pi^{(t)}(h)$ tends to a Dirac mass centred on the optimal scaling, $h_{\\text{opt}}$.\n\\end{Thm}\n\n\\textbf{Proof:} See Appendix \\ref{sect:proofvari}.\\qed\n\n\n\n\n\n\n\n\\section{Results \\label{sect:resultsASMC}}\n\nThis section is organised as follows. In Section \\ref{sect:convergeh}, the convergence of $h$ to an optimal scaling will be demonstrated empirically using a linear Gaussian model. Then in Section \\ref{sect:bayesian_mixtures} the problem of Bayesian mixture analysis will be introduced. In Sections \\ref{sect:detailsofSMCsims} and \\ref{sect:resultsofsims} the proposed method will be evaluated in simulation studies using the example of Bayesian mixture posteriors as defining the sequence of targets of interest.\n\nFollowing \\cite{sherlock2009}, the expected (Mahalanobis) square jumping distance will be considered as an MCMC performance criterion:\n\\begin{equation*}\n \\Lambda(\\theta_{t-1},\\theta_t) = (\\theta_{t-1}-\\theta_t)^T\\hat\\Sigma_{\\pi_t}^{-1}(\\theta_{t-1}-\\theta_t),\n\\end{equation*}\nwhere $\\theta_{t-1}$ and $\\theta_t$ are two points in the parameter space and $\\hat\\Sigma_{\\pi_t}$ is an empirical estimate of the target covariance obtained from the current set of particles. \n\nTwo different MCMC kernels will be considered within the SMC algorithm, these are defined by the following two proposals:\n\\begin{eqnarray*}\n q_{\\text{rw}}(\\theta_{t-1},\\tilde\\theta_t) &=& \\mathcal{N}(\\theta_{t-1},h^2\\hat\\Sigma_{\\pi_t}),\\\\\n q_{\\text{lw}}(\\theta_{t-1},\\tilde\\theta_t) &=& \\mathcal{N}(\\alpha\\theta_{t-1} + (1-\\alpha)\\bar{\\theta}_t,h^2\\hat\\Sigma_{\\pi_t}), \\qquad h\\in(0,1],\\ \\alpha=\\sqrt{1-h^2},\n\\end{eqnarray*}\nwhere $\\bar{\\theta}_t$ and $\\hat\\Sigma_{\\pi_t}$ are respectively estimates of the target and covariance. The first of these is a \\emph{random--walk} proposal. The second is based upon a method for updating parameter values in \\cite{SMCMPliuwest}, here named the `\\emph{Liu\/West}' proposal. \nThe Liu\/West proposal has mean shrunk towards the mean of the target and the imposed choice of $\\alpha=\\sqrt{1-h^2}$ sets the mean and variance of proposed particles to be the same as that of the current particles. Note that if the target is Gaussian, then this proposal can be shown to be equivalent to a Langevin proposal \\citep{roberts1996}.\n\n\\subsection{Convergence of $h$\\label{sect:convergeh}}\n\nIt is of interest to see an example $g(h)$ and demonstrate convergence of one of the proposed algorithms to the optimal scaling. This will be achieved using a Gaussian target, for which a useable analytic expression for the optimal scaling for the random walk kernel is available. The results in this section are based on 100 observations simulated from a 5--dimensional standard Gaussian density, $y_{1:100}\\stackrel{\\text{iid}}{\\sim}\\mathcal{N}(0,\\mathbb{I}_5)$, where $\\mathbb{I}_5$ is the $5\\times5$ identity matrix. The observation variance was assumed to be known and therefore the probability model, or likelihood, was specified as,\n\\begin{equation*}\n \\pi(y|\\theta) = \\mathcal{N}(y;\\theta,\\mathbb{I}_5).\n\\end{equation*}\nThe prior on the unknown parameter, $\\theta$, the vector of means, was set to $\\mathcal{N}(0,5\\mathbb{I}_5)$. ASMC with a random walk proposal was used to generate $M=2000$ particles from the posterior. Resampling was invoked when the ESS dropped below $M\/2$ and no noise was added to the $h$s after resampling. The initial distribution for $h$ was chosen to be uniform on $(0,10)$. Note that this model admits exact inference via the Kalman Filter.\n\nThe left hand plot in Figure \\ref{fig:gaussiansimresults} shows $g(h)$ for this target. Note in this case that the sequence of functions, $\\{g^{(t)}\\}$, does not change much since each intermediate target is exactly Gaussian and the proposal is scaled by the approximate variance of the target. The optimum scaling, $h_{\\text{opt}}$, was computed using 1-dimensional numerical integration and Theorem 1 of \\cite{sherlock2009}. The right hand plot illustrates several features of the adaptive RWM; the resampling frequency, that the algorithm does indeed converge to the true optimal scaling and the approximate rate of this convergence.\n\n\n\\begin{figure}[htbp]\n \\begin{minipage}{0.5\\textwidth}\n \\includegraphics[width=0.9\\textwidth,height=0.9\\textwidth]{5dimgaussiang_of_h.pdf}\n \\end{minipage}\n \\begin{minipage}{0.5\\textwidth}\n \\includegraphics[width=0.9\\textwidth,height=0.9\\textwidth]{5dimgaussianconvergence.pdf}\n \\end{minipage}\n \\caption{\\label{fig:gaussiansimresults}Left plot: $g(h)$ for a 5--dimensional Gaussian Target, explored with RWM and with ESJD as the optimization criterion. Right hand plot: convergence of $h$ for the same density based on 100 simulated observations; the horizontal line is the approximately optimal scaling, $1.06$.}\n\\end{figure}\n\n\n\\subsection{Bayesian Mixture Analysis \\label{sect:bayesian_mixtures}}\n\nThe ability of the ASMC algorithm to learn MCMC tuning parameters in more complicated scenarios was evaluated using simulated data from mixture likelihoods (for a complete review of this topic, see \\cite{fruhwirth-schnatter}). Let $p_1,\\ldots,p_r>0$ be such that $\\sum_{i=1}^rp_i=1$. Let $\\mathcal{N}(\\,\\cdot\\,;\\mu,v)$ denote the normal density\nfunction with mean $\\mu$ and variance $v$. Let $\\theta=\\{p_{1:r-1},v_{1:r},\\mu_{1:r}\\}$.\n\nThe likelihood function for a single observation, $y_i$,is\n\\begin{equation}\\label{eqn:mixlik}\n \\pi(y_i|\\theta) = \\sum_{j=1}^rp_j\\mathcal{N}(y_i;\\mu_j,v_j).\n\\end{equation}\nThe prior $\\theta$ was multivariate normal, on a transformed space using the generalised logit scale for the weights, log scale for variances, and leaving the means untransformed. The components of $\\theta$ were assumed independent \\emph{a priori}; the priors were $\\log(p_j\/p_r)\\sim\\mathcal{N}(0,1^2)$, $\\log(v_j)\\sim\\mathcal{N}(-1.5,1.3^2)$ and $\\mu_j\\sim\\mathcal{N}(0,0.75^2)$, where $j=1,\\ldots,r-1$ in the case of the weights and $j=1,\\ldots,r$ for the means and variances. The MCMC moves within the SMC algorithm were performed in the transformed space, using the appropriate inverse transformed values to compute the likelihood in (\\ref{eqn:mixlik}).\n\nAn issue with mixture models is that for the above choice of prior, the likelihood and posterior are invariant to permutation of the component labels \\citep{stephens2000}. As a result the posterior distribution has a multiple of $r!$ modes, corresponding to each possible permutation. One way of overcoming this problem is by introducing a constraint on the parameters, such as labelling the components so that $\\mu_1<\\mu_2<\\cdots<\\mu_r$, or so that $v_10$ terms of the scattering formula cannot be computed exactly, but in \\cite{bimonteprecise} it was shown that they can be computed very precisely using an asymptotic small-distance formula, which includes corrections to PFA, based on a recently proposed Derivative Expansion (DE) of the Casimir interaction \\cite{fosco1,bimonte1,fosco2,bimonte2,fosco3}. \n In \\cite{bimonteprecise} it was proved that the semi-analytic approximate formula for the sphere-plate Casimir force, resulting from the combination of the exact $n=0$ term with the approximate expression of the $n>0$ terms, is indeed extremely precise {\\it for all separations}. The formula derived in \\cite{bimonteprecise} is in excellent agreement with the results of the large numerical simulation of \\cite{gert}. In this paper, we extend the construction of \\cite{bimonteprecise} to the sphere-sphere system.\n\nThe paper is organized as follows: in Sec. II we review the PFA for two spheres, the scattering formula\nand display the exact solution for the classical Casimir energy of two Drude spheres discovered in \\cite{bimonteex1}. In Sec. III we compute the contribution of the positive Matsubara modes using the DE. In Sec. IV we display our complete formula for the Casimir energy of two spheres at finite temperature, and use it to compute deviations from PFA. In Sec. IV we also review the recent two-sphere experiment \\cite{garrett} and describe the correct parametrization of corrections to PFA that should be used in the data analysis of experiments with two spheres. In Sec. V we present our conclusions. Finally, in Appendix A we review the DE and in Appendix B we use the DE to compute the leading curvature correction to the force gradient between two spheres. \n\n\n\n\\section{Casimir interaction of two spheres}\n\nWe consider a system composed by two spheres of respective radii $R_1$ and $R_2$ placed in a vacuum and separated by a gap of width $a$ (see Fig. 1). \n\nAs it was explained in the introduction, until recently there were no tools to exactly compute the Casimir force in non-planar geometries, and so one had to resort to the old-fashioned PFA. In the case of two spheres\n\\cite{parse} the PFA formula for the Casimir force is:\n\\begin{equation}\nF^{(\\rm PFA)}(a,R_1,R_2)= 2 \\pi {\\tilde R} \\;{\\cal F}^{(\\rm pp)}(a)\\;,\\label{PFA1}\n\\end{equation} \nwhere ${\\tilde R}$ is the {\\it effective} radius of the two spheres\n\\begin{equation}\n{\\tilde R}=\\frac{R_1 R_2}{R_1+R_2}\\;,\n\\end{equation}\nand ${\\cal F}^{(\\rm pp)}(a)$ is the unit-area Casimir free energy for two plane-parallel slabs respectively made of the same materials as the sphere and plate, whose expression was derived long ago by Lifshitz \\cite{lifs}:\n\\begin{eqnarray}\n&&\n{\\cal F}^{(\\rm pp)}(a,T)=\\frac{k_BT}{2\\pi} \\sum_{n \\ge 0}\\;\\!\\!'\n \\int_{0}^{\\infty}\nk_{\\bot}dk_{\\bot} \\nonumber\n\\\\\n&&\\times\n\\sum\\limits_{\\alpha={\\rm\nTE,TM}}\\ln\\left[1-\n{r_{\\alpha}^2({\\rm i}\\,\\xi_n,k_{\\bot})}e^{-2aq_n}\\right],\\label{PBeq3}\n\\end{eqnarray}\nwhere the prime in the sum over $n$ indicates that the $n=0$ term is taken with a weight 1\/2, $T$ is the temperature of the plates, $k_{\\bot}$ is the in-plane momentum, $r_{\\alpha}({\\rm i}\\,\\xi_n,k_{\\bot})$ denotes the Fresnel reflection coefficient for polarization $\\alpha={\\rm TE, TM}$ of a thick slab, evaluated for the imaginary frequency $\\omega={\\rm i}\\, \\xi_n$ and \n$q_n=\\sqrt{\\xi_n^2\/c^2+k_{\\bot}^2}$. The PFA for the force gradient $F' \\equiv \\partial F\/\\partial a$ (here and in what follows a prime shall denote a derivative with respect to the separation) easily follows from Eq. (\\ref{PFA1}):\n\\begin{equation}\n{F'}^{(\\rm PFA)}(a,R_1,R_2)=- 2 \\pi {\\tilde R}\\; {F}^{(\\rm pp)}(a)\\;,\\label{PFA2}\n\\end{equation}\nwhere ${F}^{(\\rm pp)}(a)=-\\partial {\\cal F}^{(\\rm pp)}\/\\partial a$ is the unit-area Casimir force for two parallel slabs. The PFA force and force-gradient for a sphere of radius $R$ opposite a plane, is recovered from Eqs. (\\ref{PFA1}) and (\\ref{PFA2}), respectively, by taking the radius of one of the two spheres to infinity, i.e. substituting ${\\tilde R}$ by $R$. It is important to remark that within the PFA, both $F$ and $F'$ depend on the radii of the two spheres only via the effective radius ${\\tilde R}$.\n \nFor a proper interpretation of current precision Casimir experiments it has become important to estimate curvature corrections beyond PFA. This has been impossible until recently, when \nan exact scattering formula providing the Casimir energy of two compact dielectric bodies has been worked out \\cite{sca1,sca2,kenneth}. The general structure of the scattering formula is:\n\\begin{equation}\n{\\cal F}=k_B T \\sum_{n \\ge 0}\\;\\!\\!' \\;{\\rm Tr} \\ln[1-\\hat{M}(\\rm {i} \\xi_n)]\\;,\\label{freeen}\n\\end{equation} \nwhere the prime sign in the sum indicates again that the $n=0$ term is taken with weight 1\/2. The trace ${\\rm Tr}$ in this equation is over both spherical multipoles indices $(l,m)$ and polarization indices $\\alpha={\\rm TE,TM}$:\n\\begin{equation}\n{\\rm Tr}= \\sum_{m=-\\infty}^{\\infty} \\sum_{l= |m|}^{\\infty} {\\rm tr}\\;,\n\\end{equation} \nwhere ${\\rm tr}$ denotes the trace over $\\alpha$. The matrix elements $M_{lm\\alpha,m'l'\\alpha'}$ of ${\\hat M}$ shall not be reported here for brevity. Their explicit expressions can be found for example in Refs.\\cite{rahi,teo2}. We just recall that the matrix $M_{lm\\alpha,m'l'\\alpha'}$ involves a product of the T-matrices for the two bodies (i.e. the Mie scattering coefficients in the case of two spheres), both evaluated for the imaginary Matsubara frequencies ${\\rm i} \\xi_n$, intertwined with translation matrices that serve to convert the mutipole basis relative to either body into the multipole basis relative to the other body (see Refs.\\cite{rahi,teo2} for details). The expressions for the Casimir force $F=- {\\cal F}'$ and its gradient $F'$ are obtained by taking derivatives of Eq. (\\ref{freeen}) with respect to the separation $a$. Using the scattering formula it has been possible to prove eventually that the PFA formula is indeed asymptotically exact for small separation in the sphere-plate and cylinder-plate geometries \\cite{bordag}. \n\nHaving at our disposal the exact representation Eq. (\\ref{freeen}), it is natural to ask whether it can be used efficiently to accurately compute the Casimir force in concrete experimental situations. Unfortunately, this is not easy at all. Consider as an example the geometry of a sphere of radius $R$ at a minimum distance $a$ from a plate. The problem is that to obtain a precise estimate of the Casimir force for experimentally relevant sphere-plate separations (typical aspect ratios $a\/R \\sim 10^{-3}$) it is necessary to include a huge number of multipoles in the computation. Previous\nworks \\cite{sca2,antoine1,antoine2,bimonteprecise,gert} found that the multipole order $l_{\\rm max}$ for which convergence is achieved scales as $l_{\\rm max} \\sim R\/a$. To date, the largest numerical simulation of the sphere-plate scattering formula reached up to $l_{\\rm max}=2 \\times 10^4$ \\cite{gert}, which allowed the authors of \\cite{gert} to probe the Casimir force in the experimentally relevant region $a\/R \\sim 10^{-3}$. Managing such a large number of multipoles on a computer is not easy at all, and sophisticated algorithms are needed to handle the problem. \n\n\\begin{figure}\n\\includegraphics[width=.9\\columnwidth]{setup\n\\caption{\\label{setup} The sphere-sphere Casimir setup. The sphere-sphere geometry is characterized by the {\\it effective} radius ${\\tilde R}=R_1 R_2\/(R_1+R_2)$ and the dimensionless parameter $u={\\tilde R}^2\/R_1 R_2$}\n\\end{figure}\n\nAt this point we turn to our objective of deriving a simple and very accurate formula for the Casimir force between two spheres. To do this, we go back to the general scattering formula Eq. (\\ref{freeen}). As we see, it has the form of a sum of terms ${\\cal F}=\\sum'_{n \\ge 0}{\\cal F}_{n}$ over the Matsubara frequencies $\\xi_n,\\;n=0,1,\\dots$. It is convenient for our purposes to separate the first term ${\\cal F}_{n=0}$ of the series from the the remaining terms with $n>0$. We accordingly decompose Eq. (\\ref{freeen}) as:\n\\begin{equation}\n{\\cal F}={\\cal F}_{n=0}+{\\cal F}_{n>0}\\;,\n\\end{equation}\nwhere we set ${\\cal F}_{n>0}=\\sum_{n > 0}{\\cal F}_{n}$.\nConsider first ${\\cal F}_{n=0}$. This term represents a {\\it classical} contribution to the Casimir energy, which provides the dominant contribution to the full Casimir energy ${\\cal F}$ of the system in the limit of large separations $a \\gg \\lambda_T$. \n\nIn Ref. \\cite{bimonteex1}, it was shown that this classical term can be evaluated {\\it exactly} in the following two cases. The first one is that of a scalar field obeying Dirichlet (D) boundary conditions (bc) on the surfaces of two spheres of arbitrary radii, including the sphere-plate geometry as a special case. The second case is that of a scalar field obeying so-called Drude bc on the surfaces of a sphere opposite a plate \\footnote{The sphere-plate solution with Drude bc presented in \\cite{bimonteex1} does not extend to the sphere-sphere geometry.}. The latter bc is identical to D bc, apart from the fact that in the Drude case charge monopoles (corresponding to index $l=0$) are excluded from the scattering formula. Both sets of bc can be used to describe ohmic conductors, depending on the electric configuration of the system. Drude bc describe isolated conductors, whose total charge is fixed, while D bc describe conductors whose voltages are fixed. The latter is the experimentally important situation, since in all Casimir experiments (including the experiment \\cite{garrett}) one plate is grounded, while the other is connected with a voltage generator which serves to apply a bias potential, to compensate for unavoidable potential differences between the plates, resulting from differences in the respective work functions. The classical sphere-sphere Casimir energies implied by D and Drude bc are undistinuishable for separations much smaller than the spheres radii, while the two models lead to distinct asymptotic behaviors in the limit of large separations $a \\gg (R_1,R_2)$, since ${\\cal F}^{\\rm (D)}_{\\rm n=0} \\sim -k_B T R_1 R_2\/a^2$, while ${\\cal F}_{\\rm n=0}^{(\\rm Dr)} \\sim -k_B T R_1^3 R_2^3\/a^6$. \n\nFor two metallic spheres obeying D bc the exact solution worked out in \\cite{bimonteex1} is: \n\\begin{equation}\n{\\cal F}_{n=0}^{(\\rm ex)}=\\frac{k_B T}{2} \\sum_{l=0}^{\\infty} (2 l +1) \\ln[1-Z^{2 l+1}] \\; ,\\label{enerDr}\n\\end{equation}\nwhere $Z$ is:\n\\begin{equation}\nZ=[1+x+x^2 u\/2+\\sqrt{(x+x^2 u\/2)\\,(2+x+x^2 u\/2)}]^{-1}\\;.\\label{Zdef}\n\\end{equation}\nIn the above Equation, $x=a\/\\tilde{R}$, and $u=\\tilde{R}^2\/R_1 R_2$. The parameter $u$ depends only on the ratio between $R_1$ and $R_2$ and takes values in the interval $[0,1\/4]$, the upper bound $u=1\/4$ corresponding to two equal spheres $R_1=R_2$, while the lower bound $u=0$ is approached as either of the two radii becomes infinite. The special case of a sphere of radius $R$ opposite a plate is thus recovered by taking $u=0$ into Eq. (\\ref{Zdef}), and setting $\\tilde{R}=R$. The range of the variable $Z$ in Eq. (\\ref{Zdef}) is the interval $[0,1]$, the upper (lower) bound $Z=1$ ($Z=0$) corresponding to the limit of vanishing (infinite) separation $x\\rightarrow 0$ ($x\\rightarrow \\infty$). \nThe corresponding expression for the Casimir force $F_{n=0}^{(\\rm ex)}= - {{\\cal F}'}_{n=0}^{(\\rm ex)}$ and its derivative ${F'}_{n=0}^{(\\rm ex)}$ are easily obtained by deriving Eq. (\\ref{enerDr}) with respect to $a$. As we see from the expression of $Z$ in Eq. (\\ref{Zdef}), the exact classical energy ${\\cal F}_{n=0}^{(\\rm ex)}$ depends not only on the effective radius ${\\tilde R}$, but also on the ratio among the radii via the variable $u$. This feature marks an important difference with respect to the PFA formula.\n\nIn \\cite{bimonteex1} the small distance expansion of ${\\cal F}_{n=0}^{(\\rm ex)}$ for the sphere-plate system was worked out, by setting $Z=\\exp(-\\mu)$ and then taking the small-$\\mu$ asymptotic expansion of the series on the r.h.s. of Eq. (\\ref{enerDr}). Using the formulae of Ref. \\cite{bimonteex1}, it is easy to verify that for small separations the sphere-sphere force gradient has the expansion:\n\\begin{equation}\n {F'}_{n=0}^{(\\rm ex)}=k_B T\\frac{ \\zeta(3) {\\tilde R}}{4\\, a^3} \\left(1+\\frac{1}{12 \\zeta(3)} \\frac{a}{\\tilde R} +o(a\/{\\tilde R}) \\right)\\;,\\label{DEnzero}\n\\end{equation}\nwhere $\\zeta(x)$ is Riemann zeta function.\nThe leading term coincides with the PFA Eq. (\\ref{PFA2}), since from Lifshitz formula we find ${F}^{(\\rm pp)}_{n=0}=-k_B T \\zeta(3)\/(8 \\pi a^3)$ for two Drude-metal plates, while the next term provides the leading correction to PFA. Interestingly, like the PFA, also the latter correction is independent of the parameter $u$. The correction to PFA in Eq. (\\ref{DEnzero}) is consistent with the DE (see Eq. (\\ref{DEformula})).\n\nAt this point we need consider the contribution ${\\cal F}_{n>0}$ of the positive Matsubara modes $n>0$ to the free energy. Unfortunately, differently from the classical term ${\\cal F}_{n=0}$, the quantity ${\\cal F}_{n>0}$ cannot be computed exactly. Of course ${\\cal F}_{n>0}$ be computed numerically, using the scattering formula truncated to a finite multipole order $l_{\\rm max}$. As we explained earlier, such a computation is however very challenging, because the multipole order $l_{\\rm max}$ that is necessary\nis very large for experimentally relevant values of the radii and separation. Below we obtain a very precise and simple analytical formula for ${\\cal F}_{n>0}$, by using the so-called Derivative Expansion \n \\cite{fosco1,bimonte1,fosco2,bimonte2}.\n\n\\section{Derivative Expansion of ${\\cal F}_{n>0}$}\\\n\nThe DE \\cite{fosco1,bimonte1,fosco2,bimonte2,fosco3} is an analytical technique to compute curvature corrections to proximity forces beween two surfaces of small slope. For the benefit of the reader, we provide in Appendix A a short review to the DE and a guide to the relevant References.\n\nIn this Section we use the DE to estimate the contribution ${\\cal F}_{n>0}$ of the non-zero Matsubara modes to the Casimir energy. The DE is particularly well suited to this task, as we now explain. By it very nature, the DE is expected to be very precise in situations in which the slope of the surfaces is small within the interaction region. A little reasoning shows that this condition is met in the problem at hand for all separations $a$ between the spheres, just provided (as it always is the case in current experiments) that the radii of the spheres are both large compared to the thermal length $\\lambda_T$ ($\\lambda_T=1.2\\;\\mu$m at room temperature). \n\nIt is a well known fact \\cite{parse,book2} that the Casimir interaction between two surfaces, with a characteristic radius of curvature $R$, is localized inside a disk of radius $\\rho \\sim \\sqrt{a R}$ around the point of either surface which is closest to the other. \nThus, one is led to expect that the DE is applicable in general only for separations $a$ such that $\\rho\/R=\\sqrt{a\/R} \\ll1$. This condition is usually well satisfied in Casimir experiments, for which typically ${a\/R} < 0.01$. A closer inspection reveals however that the DE of ${\\cal F}_{n>0}$ is in fact valid for {\\it all} separations, provided that $R_1$ and $R_2$ are both larger than $\\lambda_T$. The point to consider is that {\\it positive} Matsubara modes of (imaginary) frequency $\\xi_n$ can only propagate across a distance\nof order $\\ell_n=c\/\\xi_n =\\lambda_T\/n \\le \\lambda_T$. Because of this constraint, the true size of the interaction region is actually not $\\sqrt{a R}$, but instead $\\rho=\\min (\\sqrt{a R},\\sqrt{\\lambda_T R})$. This implies that the DE for ${\\cal F}_{n>0}$ is actually valid for separations such that $\\rho\/R=\\min (\\sqrt{a\/R},\\sqrt{\\lambda_T\/R}) \\ll 1$. The latter condition is clearly satisfied for all separations, provided that $R_1$ and $R_2$ are both much larger than $\\lambda_T$. \n\nWe have thus established that the DE is a valid method for ${\\cal F}_{n>0}$. In Appendix B it is shown that the DE leads to a simple general formula Eq. (\\ref{DEformula}) for the force gradient between two spheres. The formula for the DE expansion of ${\\cal F}_{n>0}$ can obtained by making into Eq. (\\ref{DEformula}) the appropriate substitutions: \n\\begin{equation}\n{ F}'_{n>0}= -2 \\pi {\\tilde R} {F}^{(\\rm pp)}_{n>0}(a) \\left[1- \\left(\\tilde{\\theta}(a)+u \\,\\kappa(a) \\right) \\frac{a}{\\tilde R} \\right]\\;, \\label{DEformula1}\n\\end{equation}\nwhere the coefficients $\\tilde \\theta(a)$ and $\\kappa(a)$ are (see Eqs. (\\ref{thetacoe}) and (\\ref{kappacoe}))\n\\begin{eqnarray}\n{\\tilde \\theta}&=& \\frac{{{\\cal F}}^{(\\rm pp)}_{n>0}(a) -2 \\alpha_{n>0}(a)}{a {F}^{(\\rm pp)}_{n>0}(a)}\\;,\\label{thetacoe1}\\\\\n\\kappa(a) &=& 1-2 \\frac{{ {\\cal F}}^{(\\rm pp)}_{n>0}(a)}{ a { F}^{(\\rm pp)}_{n>0}(a)} \\;.\\label{kappacoe1}\n\\end{eqnarray}\nIn the above Equations, $ {\\cal F}^{(\\rm pp)}_{n>0}$ denotes the contribution of the $n>0$ modes to Lifshitz formula Eq. (\\ref{PBeq3}), and ${F}^{(\\rm pp)}_{n>0}=-\\partial {\\cal F}^{(\\rm pp)}_{n>0}\/\\partial a$ is the corresponding (unit area) force. \n\nAn important ingredient of Eqs. (\\ref{DEformula1}-\\ref{kappacoe1}) is the coefficient $\\alpha_{n>0}(a)$ which enters into the expression of $\\tilde{\\theta}$ in Eq. (\\ref{thetacoe1}). As it is explained in Appendix A, this coefficient can be extracted from the Green function \n${\\tilde G}^{(2)}(k;a)$ of the second-order perturbative expansion of ${\\cal F}_{n>0}$ for a small amplitude deformation of one of the plates around the plane-parallel geometry. The computation of this coefficient for gold plates at room temperature can be carried out following the procedure described in \\cite{bimonte2}, and we address the interested reader to that Reference for details. The coefficients $\\tilde \\theta$ and $\\kappa$ for perfect conductors (PC) in the limit of zero temperature are both independent of the separation. Their values can be determined using the formulae listed in \\cite{bimonte1}: \n\\begin{equation}\n\\tilde \\theta^{(\\rm PC)}_{T=0}=\\frac{20}{3 \\pi^2}-\\frac{1}{9}=0.564\\;,\\;\\;\\;\\;\\;\\;\\kappa^{(\\rm PC)}_{T=0}=\\frac{1}{3}\\;.\n\\end{equation} For gold surfaces at room temperature, both coefficients depend on the separation (as well as on the temperature and on the material lengths characterizing the optical properties of gold, in particular the plasma length). In Table \\ref{tab.1} we list the values of $\\tilde \\theta(a)$ and $\\kappa(a)$ for gold at room temperature, that we computed using tabulated \\cite{palik} optical data \\footnote{The weighted Kramers-Kronig dispersion relations \\cite{bimonteKK} was used to compute precisely $\\epsilon({\\rm i} \\xi_n)$ starting from the real-frequency optical data given by Palik. }, for several values of the separation in the range from 100 nm to 2 micron. \nUsing these values of $\\tilde \\theta(a)$ and $\\kappa(a)$ together with Eq. (\\ref{DEformula}), it is easily possible to compute ${ F}'_{n>0}$ for any combinations of sphere radii. \n\nIn view of later applications, it is important to note that while in the Proximity Approximation ${ F}'_{n>0}$ depends only on the effective radius ${\\tilde R}$ (see Eq. (\\ref{PFA2})), the more accurate expression of ${ F}'_{n>0}$ in Eq. (\\ref{DEformula1}), which includes curvature corrections to PFA, depends also on the parameter $u$, i.e. on the ratio of the radii of the two spheres.\n\\begin{widetext}\n\\label{tab.1}\n\\begin{center}\n\\begin{table*}\n\\begin{tabular}{ccccccccccccc} \\hline\n$a (\\mu m)$\\;\\; &0.10& 0.15 & 0.2 \\;\\;& 0.25\\;\\;\\;& 0.3\\;\\; &0.35 \\;\\;& 0.4 \\;\\; & 0.45 & 0.5\\;\\;& 0.55 & 0.6\\;\\; & 0.65 \\\\ \\hline \\hline\n${\\tilde \\theta}$\\;\\;& 0.456 &0.4715 & 0.470\\;\\; &0.463 & 0.454 \\;\\;&0.4445 & 0.435 \\;&0.425 & 0.415 &0.4055 & 0.396& 0.387 \\\\ \\hline\n${\\kappa}$\\;\\;&0.245 &0.270 &\\; 0.289\\;\\; & 0.305 & 0.319 \\;\\;&0.331 & 0.342\\;\\;&0.353 & 0.362\\;\\;&0.371 & 0.380\\;\\; &0.389 \\;\\; \\\\ \\hline \\\\\n \\hline\n$a (\\mu m)$\\; &0.70&0.75 & 0.8 & 0.85 & 0.9& 0.95& 1 &1.2&1.4 &1.6 & 1.8 & 2 \\\\ \\hline \\hline\n${\\tilde \\theta}$ \\;\\; &0.379 &0.370 &0.362 &0.3545 &0.347&0.3395&0.332 &0.306&0.282&0.261& 0.242 &0.225 \\\\ \\hline\n${\\kappa}\\;\\;$&0.397&\\;0.405 \\;&0.413 &0.421&0.429&0.437& 0.444\\;\\;&0.474&0.502&0.529&0.554&0.578 \\\\ \\hline \\hline\n\\end{tabular}\n\\caption{Values of the coefficients ${\\tilde \\theta}$ and $\\kappa$ for Au at room temperature.}\n\\end{table*}\n\\end{center}\n\\end{widetext}\n\n\n\n\\section{Beyond-PFA corrections.}\n\nCombining Eq. (\\ref{enerDr}) with Eq. (\\ref{DEformula1}) we obtain the following formula for the force gradient between two gold spheres:\n\\begin{equation}\nF'= {F'}_{n=0}^{(\\rm ex)} -2 \\pi {\\tilde R} {F}^{(\\rm pp)}_{n>0}(a) \\left[1- \\left(\\tilde{\\theta}(a)+u\\, \\kappa(a)\\right) \\frac{a}{\\tilde R} \\right]\\;,\\label{approx}\n\\end{equation}\nwhich constitutes the main result of the present work. A nice feature of the formula above is that, by construction, it is {\\it exact in both limits} $a\/{\\tilde R}\\rightarrow 0$ and $a\/\\lambda_T\\rightarrow \\infty$. This is so because, on one hand, Eq. (\\ref{approx}) is exact for $a\/{\\tilde R}\\rightarrow 0$, since in this limit the DE, which we used to compute the contribution of the positive Matsubara modes, is asymptotically exact. On the other hand, Eq. (\\ref{approx}) is exact also for $a\/\\lambda_T \\rightarrow \\infty$, because for separations larger than the thermal length $a \\gg \\lambda_T$ the relative contribution of the positive Matsubara modes vanishes exponentially fast, and then Eq. (\\ref{approx}) reduces to the exact $n=0$ mode. Comparison with high precision numerical computations of the sphere-plate scattering formula in \\cite{bimonteprecise} revealed that Eq. (\\ref{approx}) is in fact very accurate also for {\\it all intermediate} separations. For a gold sphere as small as 8 micron, the maximum error made by Eq. (\\ref{approx}) was only of 0.1$\\%$, and this was for the large aspect ratio $a\/R=0.12$. The error is expected to be far smaller in the conditions of the experimet \\cite{garrett}, which used spheres with radii larger than 29.8 micron, and probed distances corresponding to aspect ratios smaller than 0.017.\n\nFor later use, it is useful to work out the small distance limit of our formula for the sphere-sphere force gradient. This can be easily done subsituting $ {F'}_{n=0}^{(\\rm ex)} $ on the r.h.s. of Eq. (\\ref{approx}) by its small-distance expansion Eq. (\\ref{DEnzero}). After simple algebraic manipulations, one finds:\n\\begin{equation}\nF'=-2 \\pi {\\tilde R} {F}^{(\\rm pp)}(a) \\left[1- \\left(\\hat{\\theta}(a)+u\\, \\hat{\\kappa}(a)\\right) \\frac{a}{\\tilde R}+ o(a\/{\\tilde R}) \\right]\\;.\\label{DE2s}\n\\end{equation}\n The coefficients $\\hat{\\theta}(a)$ and $\\hat{\\kappa}(a)$ are \n\n\\begin{eqnarray}\n\\hat{\\theta}(a)&=&\\frac{ {F}^{(\\rm pp)}_{n>0}}{{F}^{(\\rm pp)}} \\; \\tilde{\\theta}-\\frac{{F}^{(\\rm pp)}_{n=0}}{{F}^{(\\rm pp)}} \\,\\frac{1}{12\\, \\zeta(3)}\\;,\\nonumber \\\\\n\\hat{\\kappa}(a)&=&\\frac{{F}^{(\\rm pp)}_{n>0}}{{F}^{(\\rm pp)}}\\;{\\tilde \\kappa}\\;.\\\\\n\\end{eqnarray}\n\nIn Table \\ref{tab.2}, we provide the values of the coefficients ${\\hat \\theta}$ and ${\\hat \\kappa}$ for gold at room temperature. We note that the coefficient ${\\hat \\theta}$ introduced here coincides with the opposite of the coefficient ${\\hat \\theta}_1$ of \\cite{bimonte2}. \n\\begin{widetext}\n\\label{tab.2}\n\\begin{center}\n\\begin{table*}\n\\begin{tabular}{ccccccccccccccc} \\hline\n$a (\\mu m)$\\;\\;&0.05 &\\;0.10 & 0.15 & 0.2 \\;\\;& 0.25\\;\\;\\;& 0.3\\;\\; &0.35 \\;\\;& 0.4 & 0.45 & 0.5 & 0.55 & 0.6 \\\\ \\hline \\hline\n${\\hat \\theta}$\\;\\;& 0.378 &\\;\\;0.439 &\\; 0.449\\;\\; &\\;0.443\\;\\; &\\; 0.432 \\;\\;&0.419 & \\;0.405 \\;&\\;0.392&\\;\\; 0.378 &\\;\\;0.365 & \\;\\;0.352 &\\;\\;0.340 \\\\ \\hline\n${\\hat \\kappa}$\\;\\;&0.209 &\\;\\;0.237 &\\; 0.259\\;\\; & 0.275 &\\; 0.288 \\;\\;&0.298 &\\; 0.306\\;\\;&\\;0.313 &\\;\\;\\;0.320 &\\;\\;0.325 &\\;\\;0.330 &\\;\\; 0.334 \\\\\n \\hline \\hline\n\\end{tabular}\n\\caption{Values of the coefficients ${\\hat \\theta}$ and ${\\hat \\kappa}$ for Au at room temperature.}\n\\end{table*}\n\\end{center}\n\\end{widetext}\nIn order to show the deviations of the force gradient from PFA predicted by our formula, in Fig. \\ref{deviations} we plot the quantity $({\\tilde R}\/a)(F'\/F'_{\\rm PFA}-1)$ for a system of two identical spheres of radius $R_1=R_2=30\\;\\mu$m (lower solid line) and $R_1=R_2=100\\;\\mu$m (lower dashed line).\n\\begin{figure}\n\\includegraphics[width=.9\\columnwidth]{deviations\n\\caption{\\label{deviations} Beyond-PFA corrections for the gradient of the Casimir force between two gold spheres at room temperature are shown as a function of the separation for two identical spheres of radius $R=30\\;\\mu$m (lower solid line) and $R=100\\;\\mu$m (lower dashed line). The upper pair of lines is for a sphere-plate system with sphere radius $R=30\\;\\mu$m (upper solid line) and $R=100\\;\\mu$m (upper dashed line). }\n\\end{figure}\nThe upper solid and dashed lines in Fig. \\ref{deviations} refer to a sphere-plate system, for a sfere of radius $R=30\\;\\mu$m (solid line) and $R=100\\;\\mu$m (dashed line). Fig. \\ref{deviations} demonstrates that deviations from PFA of the force gradient are practically independent of the effective radius $\\tilde R$, for fixed value of $u$. However deviations from PFA depend significantly on the ratio among the radii of the spheres via the parameter $u$. \n\\begin{figure}\n\\includegraphics[width=.9\\columnwidth]{deviations2\n\\caption{\\label{deviations2} Beyond-PFA corrections for the gradient of the Casimir force between two gold spheres at room temperature are shown as a function of the parameter $u={\\tilde R}^2\/ (R_1 R_2)$ (for constant ${\\tilde R}$ and $a$). Solid lines are for ${\\tilde R}=30\\;\\mu$m, dashed lines for ${\\tilde R}=100\\;\\mu$m. The four pairs of solid and dashed lines from top to bottom correspond to the four separations $a=1\\mu$m, 800 nm, 400 nm and 100 nm respectively. The extreme values $u=0$ and $u=1\/4$ correspond, respectively, to a sphere-plate configuration and to two spheres of equal radii.}\n\\end{figure}\nThis is clearly seen from Fig. \\ref{deviations2}, where the quantity $({\\tilde R}\/a)(F'\/F'_{\\rm PFA}-1)$ is displayed versus the parameter $u$ (for constant ${\\tilde R}$ and $a$). In Fig. \\ref{deviations2} solid lines are for ${\\tilde R}=30\\;\\mu$m, while dashed lines are for ${\\tilde R}=100\\;\\mu$m. The four pair of solid and dashed lines from top to bottom correspond to the four separations $a=1\\mu$m, 800 nm, 400 nm and 100 nm respectively. We recall that the extreme values $u=0$ and $u=1\/4$ correspond, respectively, to a sphere-plate and to two spheres of equal radii. Fig. \\ref{deviations2} shows that the $u$-dependence of the deviations from PFA is linear for the considered separations.\n\n\nFor small values of $a\/{\\tilde R}$, the quantity displayed in Fig. \\ref{deviations} and in Fig. \\ref{deviations2} can be basically identified with the parameter $\\beta'$ that was introduced in the sphere-plate experiment \\cite{ricardo} as a measure of the deviation of the data from PFA. In \\cite{ricardo} starting from the force gradient $F'$, an effective pressure $P^{(\\rm eff)}(a,R)$ was defined as:\n\\begin{equation}\nP^{(\\rm eff)}(a,R)\\equiv-\\frac{F'}{2 \\pi R} \\;.\n\\end{equation}\nIf the PFA Eq. (\\ref{PFA2}) were exact, $P^{(\\rm eff)}(a,R)=F^{(\\rm pp)}(a)$. However, the PFA is not exact, and so the authors of \\cite{ricardo} parametrized deviations from PFA by a coefficient $\\beta'(a)$ such that:\n\\begin{equation}\nP^{(\\rm eff)}=F^{(\\rm pp)}(a)\\left(1+\\beta' \\frac{a}{R}+o(a\/R)\\right)\\;.\\label{ricardo}\n\\end{equation} \n It is clear from Eq. (\\ref{ricardo}) that, up to higher order corrections, $\\beta'(a)$ coincides with the quantity $({\\tilde R}\/a)(F'\/F'_{\\rm PFA}-1)$.\nThe parameter $\\beta'$ was determined in \\cite{ricardo} by measuring the effective pressure $P^{(\\rm eff)}$ for certain fixed sphere plate-separations using spheres of different radii, and then fitting $P^{(\\rm eff)}$ versus $1\/R$ with a straight line. \nThe experiment \\cite{ricardo} placed a bound $|\\beta'|<0.4$ at 95 \\% CL in the separation range from 150 to 300 nm. This bound is in substantial agreement with the theoretical prediction (see the upper curves of Fig. \\ref{deviations}). \n\nThe authors of \\cite{garrett} used the same procedure to study deviations of their data from PFA. In particular, they assumed that the measured forces can be parametrized as:\n\\begin{equation}\n\\frac{F'}{\\tilde R}=-2 \\pi F^{(\\rm pp)}(a)\\left(1+\\beta' \\frac{a}{\\tilde R}+o(a\/{\\tilde R})\\right)\\;,\\label{garrett}\n\\end{equation} \ni.e. by a function of the same form as that of the sphere-plate system, apart from the substitution of $R$ by the effective radius ${\\tilde R}$ of the two-sphere system. Importantly, they assumed that $\\beta'$ is {\\it independent of the radii of the spheres}. Based on this assumption, the authors of \\cite{garrett} tried to determine $\\beta'(a)$ by doing a linear fit of $F'\/{\\tilde R}$ versus $1\/{\\tilde R}$ using for that purpose 12 measurements of $F'$. Three sets of data were taken in sphere-plate setups (as in the experiment \\cite{ricardo}), using three different spheres of radii $R=$ 40.7 $\\mu$m, 36.1 $\\mu$m and 34.2 $\\mu$m, while the remaining nine data sets were taken with nine different sphere-sphere setups, corresponding to the different combinations of each of three spheres with radii $R_1=$ 34.2 $\\mu$m, 36.1 $\\mu$m and 40.7 $\\mu$m, with each of the three spheres of radii $R_2=$ 29.8 $\\mu$m, 38.0 $\\mu$m and 46.9 $\\mu$m. The measurements and the corresponding fits were repeated for 26 values of the separation, in the interval from 40 nm to $300$ nm. It was found that $\\beta'=-6 \\pm 27$ was within the 2$\\sigma$ confidence interval of their calculated $\\beta'$ for all considered separations.\n\nThe procedure used in \\cite{garrett} to determine $\\beta'$ is not entirely correct, however, because the parametrization in Eq. (\\ref{garrett}) misses the dependence of $\\beta'$ on the parameter $u$. Indeed, by comparing Eq.(\\ref{garrett}) with the small-distance expansion of the sphere-sphere force Eq. (\\ref{DE2s}) one finds that $\\beta'$ has the expression\n\\begin{equation}\n\\beta'=- \\left(\\hat{\\theta}(a)+u\\, \\hat{\\kappa}(a)\\right)\\;.\\label{betass}\n\\end{equation}\nThis formula shows that in the two-sphere case, contrary to the assumption made in \\cite{garrett}, $\\beta'$ does depend on the radii of the spheres via the parameter $u$. Of course, this dependence disappears in the sphere-plate case, for which $u=0$. The linear dependence on $u$ of deviations from PFA is clearly visible from Fig. \\ref{deviations2}. Since in \\cite{garrett} the 12 combinations of radii used to determine $\\beta'$ correspond to values of $u$ that vary from zero (for the three sphere-plate setups) to 0.2498 (corresponding to the sphere-sphere setup with $R_1=36.1\\;\\mu$m and $R_2= 38\\;\\mu$m), the dependence of $\\beta'$ on $u$ should be considered in the data analysis. Substituing Eq. (\\ref{betass}) into Eq. (\\ref{garrett}), we find that in a two-sphere system $F'\/{\\tilde R}$ has the expression:\n\\begin{equation}\n\\frac{F'}{\\tilde R}=-2 \\pi F^{(\\rm pp)}(a)\\left(1- \\frac{a\\, \\hat{\\theta}}{\\tilde R}-\\frac{a\\,\\hat{\\kappa}}{R_1+R_2}+o(a\/{\\tilde R})\\right)\\;,\\label{garrett2}\n\\end{equation}\nwhere in the second term between the brackets we used the relation $u\/{\\tilde R}=1\/(R_1+R_2)$. This formula shows that the correct procedure to determine the coefficients $\\hat \\theta$ and $\\hat \\kappa$, is to make a joint 2-dimensional linear fit of $F'\/{\\tilde R}$ versus $1\/{\\tilde R}$ and $1\/(R_1+R_2)$. \n\nThe present sensitivity of the experiment \\cite{garrett} is not yet sufficient to detect the small deviations from PFA predicted by Eq. (\\ref{garrett2}). We estimate that an increase in the sensitivity by over one order of magnitude would be necessary for that purpose. It is hoped that future improvements of the apparatus will achieve this goal.\n\n\\section{Conclusions}\n\nMotivated by the recent experiment in \\cite{garrett}, we have performed a precise computation of the gradient of the Casimir force between two gold spheres at room temperature. Our computation provides an accurate estimate of beyond PFA corrections for this system. The semi-analytic formula for the Casimir force that we construct is valid for all separations and can be easily used to interpret future experiments in both the sphere-plate and sphere-sphere configurations. We have also described the correct parametrization of the corrections to PFA that should be used to carry out the data analysis in experiments using the sphere-sphere geometry. \n\nIn our computations we modelled the god plates as ohmic conductors (connected to charge reservoirs). In recent years it has been argued by some researchers \\cite{book2} that a better agreement with Casimir experiments is obtained if metallic bodies are modelled as dissipationless plasmas. The main change introduced by this model is in the classical $n=0$ Matsubara term for TE polarization, which is zero within the Drude prescription, but different from zero in the plasma model. A detailed comparison between the Drude and plasma models for the sphere-plate configuration, based on a large scale numerical simulation of the scattering formula, has been reported in \\cite{gert}, where it was shown that the deviations from PFA engendered by the plasma prescription have the same qualitative behavior as the Drude model, but are slightly larger in magnitude and show a more pronounced dependence on the aspect ration $a\/R$. We plan to study the plasma prescription for the sphere-sphere case in a forthcoming work \\cite{bimontenext}.\n\n\\acknowledgments\n\nThe author thanks T. Emig, N. Graham, M. Kruger, R. L. Jaffe and M. Kardar for valuable discussions. \n\n\\section*{APPENDIX}\n\n\\subsection{The DE expansion}\n\nFor the convenience of the reader, in this Appendix we briefly review general properties of the DE that are useful for the present work. The DE \\cite{fosco1,bimonte1,fosco2,bimonte2,fosco3} is an asymptotic expansion that allows to compute curvature corrections of any sufficiently {\\it local} functional ${\\hat {\\cal F}}$ that describes the interaction between two (non-intersecting) surfaces $\\Sigma_1$ and $\\Sigma_2$. The idea behind the DE is intuitive. One considers that the two surfaces can be described by {\\it smooth} height profiles $z=H_1(x,y)$ and $z=H_2(x,y)$, where $(x,y)$ are cartesian coordinates spanning some reference plane $\\Sigma$ and $z$ is a coordinate perpendicular to $\\Sigma$ (see Fig. 1). Since the two surfaces are non-intersecting, it can always be assumed that $H_2(x,y) < H_1(x,y)$. At this point one considers that for surfaces of small slopes $| \\nabla H_i| \\ll 1,\\;i=1,2$ \\footnote{In fact it is sufficient that the small slope condition is satisfied only in the relevant interaction area around the point of closest approach between the two surfaces} it should be possible to expand ${\\hat {\\cal F}}[H_1,H_2]$ in powers of {\\it derivatives} of increasing order of the height profiles, at least up to some order. It is rather easy to convince oneself that for a functional ${\\hat {\\cal F}}[H_1,H_2]$ that is invariant under simultaneous rotations and translations of $H_1$ and $H_2$ in the reference plane $\\Sigma$ (like the Casimir force between two plates made of a homogeneous and isotropic material) the most general expression of the DE valid to second order in the slopes of the surfaces is of the form: \n\\begin{eqnarray}\n{\\hat {\\cal F}}[H_1,H_2]=\\int_{\\Sigma} d^2 x \\left[{\\hat {\\cal F}}^{(\\rm pp)}(H)\\right. &+&\\left. \\;\\alpha_1(H) (\\nabla H_1)^2 \\right. \\nonumber \\\\\n+ \\alpha_2(H) (\\nabla H_2)^2 \\!\\!&+& \\!\\!\\alpha_{\\times}(H) \\nabla H_1 \\cdot \\nabla H_2 \\nonumber\\\\\n+ \\alpha_{-}(H) \\nabla H_1 \\!\\!& \\times &\\! \\!\\nabla H_2 \\left. \\right]+ \\rho^{(2)}\\;,\\label{derexp}\n\\end{eqnarray}\nwhere we set $H=H_1-H_2$ and $\\rho^{(2)}$ is a remainder that becomes negligible as the local radii of curvature of the surfaces go to infinity for fixed minimum surface-surface distance $a$. \nNote that invariance of ${\\hat {\\cal F}}$ under translations of $\\Sigma$ in the $z$ direction implies that ${\\hat {\\cal F}}^{(\\rm pp)}$ and the $\\alpha$'s can depend only on the height difference $H$ an not on the individual heights $H_1$ and $H_2$.\nIt is evident that the quantity ${\\hat {\\cal F}}^{(\\rm pp)}(a)$ in Eq. (\\ref{derexp}) provides the (unit-area) interaction of two plane-parallel surfaces at distance $a$, and thus the first term on the r.h.s of Eq. (\\ref{derexp}) reproduces the Derjaguin Approximation (DA) \\footnote{The DA approximation is sometimes referred to as the \"exact\" PFA.} for the functional ${\\hat {\\cal F}}$: \n\\begin{equation}\n {\\hat {\\cal F}}^{(\\rm DA)} = \\int_{\\Sigma} d^2 x \\;{\\hat {\\cal F}}^{(\\rm pp)}(H)\\;,\n\\end{equation}\nThe integrals on the r.h.s. of Eq. (\\ref{derexp}) that are proportional to $\\alpha$'s represent curvature corrections beyond the DA, and thus we see that the DE provides a systematic way to improve the old-fashioned DA. \nArbitrariness in the choice of the reference plane $\\Sigma$ further constraints the three coefficients $\\alpha$ in Eq. (\\ref{derexp}) \\cite{bimonte1}. In particular, invariance of ${\\hat {\\cal F}}$ with respect to tilting of $\\Sigma$ (for details, see \\cite{bimonte1}) implies:\n\\begin{eqnarray}\n&&2(\\alpha_1(H)+\\alpha_2(H)+\\alpha_{\\times}(H))+H \\frac{d {\\hat {\\cal F}}^{(\\rm pp)} }{d H}-{\\hat {\\cal F}}^{(\\rm pp)}=0\\;,\\nonumber\\\\\n&&\\alpha_{-}(H)=0\\;.\\label{alpharel}\n\\end{eqnarray}\nThe above relations show that, to second order in the gradient expansion, the two-surface problem actually reduces to the simpler problem of a single curved surface opposite a plane, since $\\alpha_1$ and $\\alpha_2$ can be determined in that case, and then $\\alpha_{\\times}$ follows from the first of Eqs. (\\ref{alpharel}). We now make the simplifying assumption that the field(s) that mediate the interaction obeys the same boundary conditions on $\\Sigma_1$ and $\\Sigma_2$. Then \n\\begin{equation}\\alpha_1(H)=\\alpha_2(H)\\equiv\\alpha(H)\\;.\\label{onemat}\n\\end{equation}\nTaking advantage of Eqs. (\\ref{alpharel}) and (\\ref{onemat}) the DE can then be recast in the form:\n$$\n{\\hat {\\cal F}}[H_1,H_2]= {\\hat {\\cal F}}^{(\\rm DA)} + \\int_{\\Sigma} d^2 x \\; \\frac{}{} \\alpha(H) (\\nabla H)^2 \n$$\n\\begin{equation}\n+ \\frac{1}{2} \\int_{\\Sigma} d^2 x \\left( {\\hat {\\cal F}}^{(\\rm pp)}-H \\frac{d {\\hat {\\cal F}}^{(\\rm pp)} }{d H} \\right)\\nabla H_1 \\cdot \\nabla H_2 + \\rho^{(2)}\\;.\\label{derexp2}\n\\end{equation}\nWe thus see that to second order in the slope the interaction ${\\hat {\\cal F}}$ is fully determined by knowledge of the (unit area) interaction $ {\\hat {\\cal F}}^{(\\rm pp)}(a)$ of two parallel plates and by the single coefficient $\\alpha(H)$. \n\nThe latter coefficient can be determined by comparing the DE Eq. (\\ref{derexp2}) to a {\\it perturbative} expansion of the functional ${\\hat {\\cal F}}[H,0]$ around flat plates $H=a + h(x,y)$ to second order in the deformation $h(x,y)$. Note that the latter perturbation requires a deformation of small amplitude $h(x,y)\/a \\ll 1$, while the DE relies on the condition that the slope of the surface be small. To second order in $h$ the perturbative expansion of ${\\hat {\\cal F}}$ reads:\n$$\n{\\hat {\\cal F}}[a+h({\\bf x})]=A {\\hat {\\cal F}}^{(\\rm pp)}(a)+\\mu(a) {\\tilde h}({\\bf 0})\n$$\n\\begin{equation}\n+\\int \\frac{d^2{\\bf k}}{(2 \\pi)^2}\\;{\\tilde G}^{(2)}(k;a)|{\\tilde h}({\\bf k})|^2+{\\tilde \\rho}^{(2)}[h]\\;,\n\\end{equation}\nwhere $A$ is the surface area, ${\\bf k}$ is the in-plane wave-vector, ${\\tilde h}({\\bf k})$ is the Fourier transform of $h(x)$, and ${\\tilde \\rho}^{(2)}[h]$ refers to higher order corrections. The function $\\alpha(H)$ can now be determined if the kernel ${\\tilde G}^{(2)}(k;a)$ can be expanded to second order in $k$. Indeed, matching the expansion\n\\begin{equation}\n{\\tilde G}^{(2)}(k;a)= \\gamma(a)+\\delta(a) k^2+o(k^2)\\;,\\label{green1}\n\\end{equation}\nto Eq. (\\ref{derexp2}) one finds:\n\\begin{equation}\n {\\hat {\\cal F}}^{(\\rm pp)}\\!\\frac{}{}'(a)=\\mu(a)\\;,\\;\\;{\\hat {\\cal F}}^{(\\rm pp)}\\!\\frac{}{}''(a)=2 \\gamma(a)\\;,\\;\\;\\;\\alpha(a)=\\delta(a)\\;,\\label{green2}\n\\end{equation}\nwhere a prime denotes a derivative with respect to $a$. The above Equation shows that a {\\it necessary} condition for existence of the second order DE is existence of the Taylor expansion of the perturbative kernel ${\\tilde G}^{(2)}(k;a)$ to second order in the in-plane momentum. Indeed, it can be shown that the DE can be formally recovered by an (infinite) resummation of the perturbative series for small in-plane momenta \\cite{fosco3}.\n\nWhenever applicable, the DE has been successfully used to compute curvature corrections beyond the PA in various problems involving interactions among gently curved surfaces. In the context of Casimir physics, it was used in \\cite{fosco1} to compute curvature corrections to the zero temperature Casimir energy for a scalar field obeying Dirichlet (D) boundary conditions (bc) in the sphere-plate and cylinder-plate geometries. The zero temperature Casimir problem for the em field with perfect conductor (PC) bc, as well as a scalar field obeying Neumann bc, or mixed DN bc (i.e. D bc on one surface and N on the other), was studied in \\cite{bimonte1} for two spheres and for two inclined cylinders. The curvature corrections obtained in the latter work for the em field with PC bc in the sphere-plate and sphere-sphere geometries were subsequently confirmed in \\cite{bordagteo,teo2} by working out a rigorous small-distance expansion of the scattering formula. The experimentally important case of the Casimir interaction between gold sphere and plate at finite temperature was instead studied in \\cite{bimonte2}. Even in this case, the results obtained by the DE were later shown to be in agreement with the small-distance expansion of the scattering formula \\cite{teogold}. Curvature corrections obtained by the DE have also been found to be in agreement \nwith the small distance expansion of the rare exact Casimir energies in non planar geometries that have been discovered so far, i.e. in the cases of two Drude or D spheres in the classical limit \\cite{bimonteex1}, and for two three-spheres with D or PC bc in four euclidean dimensions \\cite{euclidean}. \nThe DE has been also used to study curvature effects in the Casimir-Polder interaction of a particle with a gently curved surface \\cite{CPbimonte1,CPbimonte2}, \nand to estimate the shifts of the rotational levels of a diatomic molecule due to its van der Waals interaction with a curved dielectric surface \\cite{CPbimonte3}. In a non Casimir context, the DE hase been also used to compute curvature corrections to the scattering amplitude for an em wave impinging on a curved surface \\cite{bimonteref} and to the electrostatic interaction among two curved plates \\cite{foscoel}.\n \n\n\\subsection{Computing the leading curvature correction to the force gradient. }\n\n\nThe small-slope approximation of the interaction energy ${\\hat {\\cal F}}$ provided by Eq. (\\ref{derexp2}) still involves a surface integral over $\\Sigma$ of functions depending on the height profiles of the surfaces. As such, \nEq. (\\ref{derexp2}) is not very convenient for a practical use. A better route is to expand Eq. (\\ref{derexp2}) in powers of the small parameter $a\/R$, where $R$ is the characteristic radius of curvature of the surfaces. The leading order of this expansion will reproduce the standard PFA, while in the next order it shall provide us with the desired curvature correction beyond the PFA. \nWe shall carry out this expansion not directly for the energy ${\\hat {\\cal F}}$, but rather for the gradient of the force ${\\hat F}'=-{\\hat {\\cal F}}''$, which is the quantity that was measured in the experiment \\cite{garrett}. Moreover, we shall restrict attention to the sphere-sphere system, which is again the geometry used in \\cite{garrett}.\n\nAccording to Eq. (\\ref{derexp2}), the formula for the force gradient ${\\hat F}'$ can be split as\n\\begin{equation}\n{\\hat F}'={\\hat { F'}}^{(\\rm DA)}+I_2+I_3\\;,\n\\end{equation}\nwhere\n\\begin{equation}\n{\\hat { F'}}^{(\\rm DA)}=\\int_{\\Sigma} d^2 x \\;{\\hat { F'}}^{(\\rm pp)}(H)\n\\end{equation}\nand we set\n\\begin{eqnarray}\n&I_2& =\\frac{1}{2} \\int_{\\Sigma} d^2 x\\; (H {\\hat {\\cal F}}^{(\\rm pp)}\\!\\frac{}{}'')'\\;\\nabla H_1 \\cdot \\nabla H_2\\;,\n\\nonumber \\\\\n&I_3&=- \\int_{\\Sigma} d^2 x \\; \\alpha''(H) \\; (\\nabla H)^2\\;.\n\\end{eqnarray}\nWe consider proximity forces that decay rapidly wih the distance, like the Casimir force. For forces of this nature, the interaction among the surfaces is localized within a small area, typically of radius $\\rho \\sim \\sqrt{a {\\tilde R}}$, around the point of closest approach. Under such circumstances, it is legitimate to take the Taylor expansion of the height profiles $H_1(x,y)$ and $H_2(x,y)$ of the two spheres around their tips, that we imagine placed at $x=y=0$. Since the position of the reference plane $\\Sigma$ in Fig. 1 is immaterial, we are free to take for $\\Sigma$ the tangent plane to the sphere of radius $R_2$, passing through the sphere tip. Then: \n\\begin{eqnarray}\nH_1(x,y) &=& a+\\frac{r^2}{2R_1}+\\frac{r^4}{8 R_1^3}+\\dots\\;,\\nonumber \\\\\nH_2(x,y) &=&-\\frac{r^2}{2R_2}-\\frac{r^4}{8 R_2^3}+\\dots\\;,\n\\end{eqnarray}\nwhere $r^2=x^2+y^2$. To evaluate the integrals $I_j$ it is convenient to introduce polar coordinates $(r, \\theta)$ in the $(x,y)$ plane, and then substitute $r$ by the dimensionless quantity $\\xi=r^2\/a {\\tilde R}$. An essential property of the integrals $I_j$ is that they involve derivatives of certain functions (i.e. ${\\hat {\\cal F}}^{(\\rm pp)}$ and $\\alpha$) of the height difference $H$ with respect to the separation $a$. These derivatives can be converted into derivatives with respect to $\\xi$, using the identity\n\\begin{eqnarray}\nU'&=&U_{ ,\\xi} \\left(H_{,\\xi}\\right)^{-1} \\nonumber \\\\\n&=&2\\frac{U_{ ,\\xi} }{a} \\left[1-\\frac{\\xi}{2} \\left(\\frac {a{\\tilde R}^2}{R_1^3}+\\frac{a{\\tilde R}^2}{R_2^3} \\right) +o(a\/{\\tilde R})\\right]\\;,\\label{dera}\n\\end{eqnarray}\n(comas denote derivatives) which holds for any function $U$ of $H$. We are ready now to take the small-distance expansion of ${\\hat F}'$. We start from ${\\hat { F'}}^{(\\rm DA)}$. \nUsing Eq. (\\ref{dera}), and omitting corrections of order $o(a\/{\\tilde R})$ we find \n$$\n{\\hat { F'}}^{(\\rm DA)}\\!\\!\\!=2\\pi {\\tilde R} \\!\\!\\int_0^{\\infty} \\!\\!\\!\\!d \\xi \\,{\\hat {F}}^{(\\rm DA)}_{,\\xi}\\!\\left[1-\\frac{\\xi}{2} \\left(\\!\\frac{a{\\tilde R}^2}{R_1^3}+\\frac{a{\\tilde R}^2}{R_2^3}\\! \\right)\\right] \n$$\n$$\n=-2 \\pi {\\tilde R} {\\hat F}^{(\\rm pp)}(a)-a\\, \\pi \\left(\\frac{{\\tilde R}^3}{R_1^3}+\\frac{{\\tilde R}^3}{R_2^3} \\right)\n \\int_0^{\\infty} d \\xi \\;{\\hat {\\cal F}}^{(\\rm pp)}\\! \\frac{}{}'\n$$\n\\begin{equation}\n=-2 \\pi {\\tilde R} {\\hat F}^{(\\rm pp)}(a)+ 2\\pi \\left(\\frac{{\\tilde R}^3}{R_1^3}+\\frac{{\\tilde R}^3}{R_2^3} \\right) {\\hat {\\cal F}}^{(\\rm pp)}(a)\\;.\\label{first}\n\\end{equation}\nThe first term on the last line of Eq. (\\ref{first}) coincides with the standard PFA for the force gradient (see Eq. (\\ref{PFA2})), while its second term represents a curvature correction. \n\nBy following an analogous procedure for $I_2$, and again omitting higher order terms, we obtain:\n\\begin{eqnarray}\n&I_2 & = - \\pi\\frac{ a {\\tilde R}^2}{R_1 R_2} \\int_0^{\\infty} d \\xi\\,(H {\\hat {\\cal F}}^{(\\rm pp)}\\!\\frac{}{}'')_{,\\xi}\\,\\xi\n\\nonumber \\\\\n&=&\n \\pi \\frac{a {\\tilde R}^2}{R_1 R_2} \\int_0^{\\infty} d \\xi\\,(H {\\hat {\\cal F}}^{(\\rm pp)}\\!\\frac{}{}'')=\\frac{2\\pi {{\\tilde R}^2}}{R_1 R_2} \\int_0^{\\infty} d \\xi\\,H ({\\hat {\\cal F}}^{(\\rm pp)}\\!\\frac{}{}')_{,\\xi}\n\\nonumber \\\\\n&=& \\frac{2\\pi {{\\tilde R}^2}}{R_1 R_2} a {\\hat F}^{(\\rm pp)}(a)-\\frac{2\\pi {{\\tilde R}^2}}{R_1 R_2} \\int_0^{\\infty} d \\xi\\, {\\hat {\\cal F}}^{(\\rm pp)}_{,\\xi} \\nonumber \\\\\n&=& \\frac{2\\pi {{\\tilde R}^2}}{R_1 R_2} \\left[a {\\hat F}^{(\\rm pp)}(a) +{\\hat {\\cal F}}^{(\\rm pp)}(a)\\right]\\;.\\label{I2}\n\\end{eqnarray}\nFinally, for $I_3$ we obtain:\n\\begin{equation}\nI_3 = - 4 \\pi \\int_0^{\\infty} d \\xi\\, \\alpha_{,\\xi \\xi} \\,\\xi =-4 \\pi \\alpha(a)\\;.\\label{I3}\n\\end{equation}\nUpon combining Eqs. (\\ref{first}-\\ref{I3}), after simple algebraic transformations, we obtain the following small-distance expansion of the force gradient, correct up to terms of order $o(a\/{\\tilde R})$:\n\\begin{eqnarray}\n{\\hat F}' &=&-2 \\pi {\\tilde R} {\\hat F}^{(\\rm pp)}(a)+ 2 \\pi \\left[{\\hat {\\cal F}}^{(\\rm pp)}(a) -2 \\alpha(a)\\right]\n\\nonumber \\\\\n&+& 2 \\pi u \\left[ a {\\hat F}^{(\\rm pp)}(a)-2 {\\hat {\\cal F}}^{(\\rm pp)}(a) \\right]\\;\n\\nonumber \\\\\n&\\equiv& -2 \\pi {\\tilde R} {\\hat F}^{(\\rm pp)}(a) \\left[1- \\left(\\tilde{\\theta}(a)+u \\kappa(a) \\right) \\frac{a}{\\tilde R} \\right]\\;, \\label{DEformula}\n\\end{eqnarray}\nwhere the coefficients $\\tilde \\theta(a)$ and $\\kappa(a)$ are\n\\begin{eqnarray}\n{\\tilde \\theta} &=& \\frac{{\\hat {\\cal F}}^{(\\rm pp)}(a) -2 \\alpha(a)}{a {\\hat F}^{(\\rm pp)}(a)}\\;,\\label{thetacoe}\n\\\\\n\\kappa(a) &=& 1-2 \\frac{{\\hat {\\cal F}}^{(\\rm pp)}(a)}{ a {\\hat F}^{(\\rm pp)}(a)} \\;.\\label{kappacoe}\n\\end{eqnarray}\n \n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nLike symplectic manifolds, contact manifolds have no local invariants. Darboux's theorem tells us that locally all contact structures are the same. Moreover Gray stability tells us that there is no deformation theory. Nonetheless there are contact manifolds which are not contactomorphic. \nSometimes one can distinguish these different contact structure via the Chern classes of the underlying symplectic vector bundle defined by the contact distribution. However this is insufficient. In ~\\cite{Gi1} Giroux shows that the contact structures $$\\xi_{n} = ker(\\alpha_{n} =cos(n\\theta )dx + sin(n\\theta)dy)$$ are pairwise noncontactomorphic. Moreover $c_{1}(\\xi_{n}) = 0$ for all $n$.\n\n\nCalculations of this type have been entirely dependent on the specific geometric conditions of the example. However, we now have contact homology, a small piece of the symplectic field theory ~\\cite{EGH} of Elisahberg, Givental, and Hofer. Using this powerful tool Ustilovsky ~\\cite{Ust} was able to find infinitely many exotic contact structures on odd dimensional spheres all in the same homotopy class of almost complex structures. Similarly Otto Van Koert, in his thesis ~\\cite{OVK}, made a similar calculation for a larger class of Brieskorn manifolds using the Morse-Bott form of the theory. It should be mentioned that it came to the author's attention upon completion of this work that Miguel Abreu, using different methods, has, indepentently, computed a general formula for contact homology for toric contact manifolds with $c_{1}(\\xi)=0.$ \n\nThe ideas in this paper were originally motivated by examples related to a question of Lerman about contact structures on various $S^{1}$-bundles over $\\mathbb{CP}^1 \\times \\mathbb{CP}^1$ ~\\cite{Ler1}. We are now able to distinguish these structures essentially using an extension of a theorem of Bourgeois for $S^1$-bundles over symplectic manifolds which admit perfect Morse functions ~\\cite{Bourg1}. \n\\begin{thm}[Bourgeois]\nLet $(M, \\omega)$ be a symplectic manifold with $[\\omega] \\in H_2 (M, \\mathbb{Z})$ satisfying $c_1(TM) =\\tau[\\omega]$ for some $\\tau \\in \\mathbb{R}.$ Assume that $M$ admits a perfect Morse function. Let $V$ be a Boothby-Wang fibration over $M$ with its natural contact structure. Then contact homology $HC^{cylindrical}_{\u2217} (V, \\xi)$ is the homology of the chain complex generated by infinitely many copies of $H_{*}(M, \\mathbb{R})$, with degree shifts $2 c k - 2$, $k \\in \\mathbb{N},$ where $c$ is the first Chern class of $T(M)$ evaluated on a particular homology class. The differential is then given in terms of the Gromov-Witten potential of $M.$\n\\end{thm}\n\nThis theorem exploits the fact that in the case of $S^{1}$-bundles the differential in contact homology is especially simple since there is essentially one type of orbit for each multiplicity (ie., simple orbits can be paramatrized so that their periods are all $1$). Perfection of the Morse function kills the Morse-Smale-Witten part of the differential in Morse-Bott contact homology, so the chain complex reduces to the homology of $M.$ The grading in contact homology comes from the fact that the index calculations may be made via integration of $c_{1}(TM)$ over certain spherical two dimensional homology classes. In the case of simply connected reduced spaces the cohomology ring of the base has a particularly nice form in terms of $2$ dimensional cohomology classes, obtained from the moment map as a Morse function. Moreover all the two dimensional homology of the base in these cases is generated by spheres. This fact makes Maslov\/Conley-Zehnder indices easier to compute without the need to find a stable trivialization of the contact distribution. In this way we are able to extend the theorem of Bourgeois. For homogeneous spaces the above theorem works ``out of the box'', but we must allow for non-monotone bases, ie., $c_{1}(\\xi) \\neq 0$. To generalize the idea to general toric contact manifolds it is necessary to consider symplectic bases (of the circle bundle) which are orbifolds.\\footnote{It would be interesting to determine conditions under which the Reeb vector field could be perturbed so that the quotient of the Reeb action becomes an honest manifold when the contact manifold is simply connected. There are known counter-examples, even in the toric case. It would be interesting to find SFT obstructions to this.} The main new idea of this paper is to formalize this process for index computation from ~\\cite{Bourg1} and extend the result to an orbifold base, ie., the case where Reeb action is only locally free. Moreover we can extend this to orbifolds bases which also admit a Hamiltonian action, since over $\\mathbb{C}$ their cohomology ring is still a polynomial ring in $H^{2}$ with spherical representatives of the ``diagonal'' homology classes. \n\\begin{thm} \\label{thm:main}\n Let $M$ be a contact manifold, which is an $S^1$ bundle over the symplectic orbifold $\\mathcal{Z},$ where $\\mathcal{Z}$ admits a strongly Hamiltonian action of a compact Lie group. Suppose that the curvature form, $d \\alpha$ of $M$ as a circle bundle over $\\mathcal{Z}$ is given by $$\\sum w_{j}\\pi^{*}c_{j},$$ where $c_{j}$ are the Chern classes associated to the Hamiltonian action. Assume that the $c_{j}$ generate $H^{*}(\\mathcal{Z};\\mathbb{C})$ as variables in a truncated polynomial ring. Let $\\tilde{w_{j}}$ be the coeffecient of $c_{j}$ in $c_{1}(T(\\mathcal{Z})).$ Assume further regularity of the $\\bar{\\partial}_{J}$-operator, and that $\\sum_{j} \\tilde{w_{j}} >1.$ Then contact homology is generated by copies of the homology of the critical submanifolds for any of the Morse-Bott functions given by the components of the moment map, with degree shifts given by twice-integer multiples of the sums of the $\\tilde{w_{j}}$ plus the dimension of the stratum, $S$, of $\\mathcal{Z}$ in which the given Reeb orbit is projected under $\\pi.$ \n\\end{thm}\nAs corollaries we obtain contact homology for both toric and homogeneous contact manifolds. The reader should also beware that such calculations are only formal without some sort of transversality of the $\\bar{\\partial}_{J}$ operator. Though several such statements and proofs have been published there are still certain persistent gaps. There are many current developments around this issue and the underlying analysis. Without such a result there is no way to know if the counts that we make are actually correct. In section $3$ we give an alternate argument for transversality for homogeneous contact manifolds and for toric contact manifolds using only elementary tools from algebraic geometry, this is possible only because the almost complex structures involved are integrable in these cases.\n\n\\begin{ack}\n I would like to thank Charles Boyer for his patience, interest, and assistance in my work. I would also like to thank Alexandru Buium for many useful conversations about the dark arts of algebraic geometry.\n\\end{ack}\n\n\\section{Preliminary Notions}\nIn this section we give some basic definitions, theorems, and ideas in order to fix notation and perspective. \nFirst of all we define a contact structure:\n\\begin{defn}\n A \\textbf{contact structure} on a manifold $M$ of dimension $2n-1$ is a \\textbf{maximally non-integrable} $2n-2$-plane distribution, $\\xi \\subset T(M).$\nIn other words $\\xi$ is a field of $2n-2$-planes which is the kernel of some $1$-form $\\alpha$ which satisfies $$\\alpha \\wedge d\\alpha^{n-1} \\neq 0.$$ Such an $\\alpha$ is called a \\textbf{contact 1-form}. \n\\end{defn}\nNotice that given such a $1$-form $\\alpha$, $f\\alpha$ is also a contact $1$-form for $\\xi$ whenever $f$ is smooth and non-vanishing. In the following, a contact manifold has dimension $2n-1$, hence the symplectization has dimension $2n$ and our symplectic bases all have dimension $2n-2.$ Unless otherwise stated we assume $M$ compact without boundary. \nGiven a choice of contact $1$-form, $\\alpha$ we define the \\emph{Reeb vector field} of\n $\\alpha$ as the unique vector field $R_{\\alpha}$ satisfying $$i_{R_{\\alpha}} d \\alpha = 0$$ and $$i_{R_{\\alpha}} \\alpha = 1.$$ \n A contact form $\\alpha$ is called \\emph{quasi-regular} if, in flow-box coordinates with respect to the flow of the Reeb vector field, orbits intersect each flow box a finite number of times. If that number can be taken to be $1$ we call the $\\alpha$ \\emph{regular}. Note that if the manifold is compact this makes all Reeb orbits periodic\\footnote{By Poincar\\'{e} recurrence, for example.} of the same period. In contrast, the standard contact form on $\\mathbb{R}^{2n-1}$ is regular. We call a contact manifold (quasi)regular if there is a contact form $\\alpha$ for $\\xi$ such that $\\alpha$ is (quasi)regular. Given $(M, \\alpha)$, we denote by $V$ the \\emph{symplectization} of $M$: $$V := (M \\times \\mathbb{R}, \\omega = d(e^t \\alpha)).$$ A contact structure defines a symplectic vector bundle with transverse symplectic form $d\\alpha.$ We can then choose an almost complex structure $J_{0}$ on $\\xi.$ We extend this to a complex structure $J$ on $V$ by $$J|_{\\xi}= J_{0}$$ and $$J\\frac{\\partial}{\\partial t} = R_{\\alpha}$$ where $t$ is the variable in the $\\mathbb{R}$-direction. Note that we also get a metric, as usual, on $V$ compatible with $J$ given by $$g(v, w) = \\omega(v, Jw).$$\n\nWe have the important result from ~\\cite{BG00b}, originally proved in the regular case in ~\\cite{BW58}:\n\\begin{thm}[Orbifold Boothby-Wang]\n Let $(M, \\xi)$ be a quasiregular contact manifold. Then $M$ is a principal $S^1$-orbibundle over a symplectic orbifold $(\\mathcal{Z}, \\omega)$ with connection $1$-form $\\alpha$ whose curvature satisfies $$d\\alpha = \\pi^*\\omega.$$ If $\\alpha$ is \\textbf{regular} then $\\mathcal{Z}$ is a manifold, and $M$ is the total space of a principal $S^1$-bundle. \n\\end{thm}\n\nThis enables us to study the nature of $M$ via the cohomology of $\\mathcal{Z}.$ As we will see, in nice enough cases, the cohomology of $\\mathcal{Z}$ along with the bundle data of the Boothby-Wang fibration will determine the contact homology as well. Notice that if we really want to study the quasi-regular case via the base we are forced to consider symplectic orbifolds, a complete study of contact geometry with symmetries necessarily must include orbifolds.\n\nThe first important examples here are the homogeneous contact manifolds. These are contact manifolds which have a transitive action of a Lie group via contactomorphisms. In all of these examples we can apply the theorem of Bourgeois \\emph{almost} directly\\footnote{We say almost, since these manifolds are not always monotone. ie., that $c_{1}$ is a multiple of the symplectic form.}. Next we have the toric contact manifolds. The toric situation is not quite as nice as with the homogeneous manifolds, but we will still be able to make a similar statement and read off contact homology from the Boothby-Wang data.\nBefore we describe the contact situation in detail, we describe the corresponding symplectic objects over which we will construct our $S^1$ bundles. We will have to work with symplectic orbifolds as in some of our examples the action of the Reeb vector field is only locally free.\n\\subsection{Toric Symplectic and Toric Contact Manifolds}\n\n\\begin{defn}\n Let $M$ be a symplectic orbifold. Suppose that the Lie group $G$ acts on $M$ by (orbifold) symplectomorphisms. The action is called \\textbf{Hamiltonian} is for each $\\tau \\in \\mathfrak{g}$ the associated vector field $X_{\\tau}$ on $M$ satisfies $$i_{X_{\\tau}}\\omega = dH$$ for some $H \\in C^{\\infty}(M , \\mathbb{R}).$\n\\end{defn} \nWe will actually be interested in something stronger. We will assume that our actions are what some authors call \\emph{strongly} Hamiltonian, ie., that there exists a \\emph{moment map} $$\\mu : M \\rightarrow \\mathfrak{g}^*$$ such that \n $$d\\langle \\mu(p),\\tau \\rangle=-i_{X_{\\tau}} \\omega.$$\n\nThis result ~\\cite{LerTol} will be extremely important for us when we determine how to compute contact homology.\n\\begin{thm}(Lerman-Tolman)\n Let $(M, \\omega)$ be a symplectic orbifold. Let $G$ be a compact Lie group acting effectively, and such that the action is strongly Hamiltonian with moment map $\\mu$. Then each component of the moment map is a Morse-Bott function whose critical submanifolds are all even dimensional with even Morse-Bott index. \n\\end{thm}\n\n\nWhen the group $G$ is a torus of maximal possible dimension we have a name for these manifolds:\n\\begin{defn} Suppose that a symplectic orbifold $(M, \\omega)$ of dimension $2n$ admits an effective Hamiltonian action of a torus $T^n$ of dimension $n.$ Then we call $(M, \\omega)$ a \\textbf{toric} symplectic orbifold.\n \\end{defn}\n\nNow we consider a compact Lie group $G$ acting via contactomorphisms on the contact manifold $(M^{2n-1}, \\xi).$ By averaging over $G$ we can always obtain a $G$-invariant contact form for $\\xi.$ We find it convenient to limit all of our actions. First given $\\zeta \\in \\mathfrak{g}$, let $X_{\\zeta}$ denote the fundamental vector field associated to $\\zeta$ defined via the exponential map. The following definition was introduced in ~\\cite{BG00}. \n\n\\begin{defn} Let $G$ be a Lie group which acts on the contact manifold $(M, \\xi).$ The action is said to be of \\textbf{Reeb type} if there is a contact $1$-form $\\alpha$ for $\\xi$ and an element $\\zeta \\in \\mathfrak{g}$, such that $X^{\\zeta}=R_{\\alpha}.$ \n\\end{defn}\n\n This is important since as long as such an action is proper, we know that our contact manifold is an $S^1$-bundle. The proof of the following can be found in ~\\cite{BG} \n\n\\begin{prop}\n If an action of a torus $T^k$ is of Reeb type then there is a quasi-regular contact structure whose $1$-form satisfies, $ker(\\alpha)=\\xi.$ Moreover, then $M$ is the total space of a $S^1$ bundle over a symplectic orbifold which admits a Hamiltonian action of a torus $T^{k-1}.$ \n\\end{prop}\n\n\nWe now need the definition:\n\\begin{defn}\n A \\textbf{toric contact manifold} is a co-oriented contact manifold $(M^{2n-1}, \\xi)$ with an effective action of a torus, $T^{n}$ of maximal dimension $n$ and a moment map\\footnote{The contact moment map can be defined in terms of the symplectization, $V$, or intrinsically in terms of the annhilator of $\\xi$ in $TM^*$. For more information about this see ~\\cite{Ler2} or ~\\cite{BG}} into the dual of the Lie algebra of the torus. \n\\end{defn}\n\nNotice now that by constraining to actions of Reeb type we get that all of our toric contact manifolds are circle bundles over symplectic orbifolds. Even more is true, the bases are toric. So we limit our study strictly to those tori of Reeb type.\\footnote{Actually, our homology calculations are potentially false otherwise, since in dimension $3$, there are toric manifolds with over-twisted contact structures. These always have vanishing contact homology (when contact homology is well-defined).} An action being of Reeb type is quite a strong condition, but by the classification theorem of Lerman ~\\cite{Ler2} we see that it includes many cases of interest. We only include his classification for $2n-1 >3.$\n\\begin{thm}\n Let $(M, \\xi)$ be a toric contact manifold of dimension greater than $3.$ \n\\begin{enumerate}\n \\item[i.] If the torus action is free, then $M$ is a principal torus bundle over a sphere.\n\\item[ii.] If the action is not free then the \\textbf{moment cone} is a \\textbf{good rational polyhedral cone}. \n\\end{enumerate}\n\n\\end{thm}\nWe are interested mostly in the second case and since we have the following proposition in~\\cite{BG} it is completely reasonable for us to restrict ourselves to actions of Reeb type.\n\\begin{prop}\n Let $M$ be a contact toric manifold of dimension greater than $3$. Then the action of the torus is of Reeb type if and only if it is not free and the moment cone contains no non-zero linear subspace. \n\\end{prop}\n\n\\subsection{Cohomology Rings of Reduced Spaces}\nIn this section we follow ~\\cite{GlSt}. The thing that really makes our calculation possible in its simple form is the wonderful structure of the cohomology rings of symplectically reduced spaces, ie, they are all truncated polynomial rings in the Chern classes. Moreover in the simply connected case, we know that all of $H_{2}$ can be represented by spheres. Even better, we can always relate all of these homology and cohomology classes to the moment map. \n\nFirst let's work out what we get in general. Let $(M, \\omega)$ be a symplectic manifold of dimension $2n.$ Let $G$ be a compact connected Lie group of dimension d which acts via (strongly) Hamiltonian symplectomorphisms, $\\mathfrak{g} = Lie(G).$ Let $$\\mu: M \\rightarrow \\mathfrak{g^{*}}$$ denote the corresponding moment map. Let $\\tau$ be a regular value of $\\mu$, let $$X_{\\tau}=\\mu^{-1}(\\tau).$$ Then $X_{\\tau}\/G$ is a symplectic orbifold of dimension $2(n-d)$. Set $\\mathcal{Z} = X_{\\tau}\/G$. Let $c_{1}, \\ldots, c_{n}$ be the Chern classes of the fibration $M \\rightarrow \\mathcal{Z}.$\n\n\\begin{thm}\n If the $c_{1}, \\ldots, c_{n}$ generate $H^{*}(\\mathcal{Z} ; \\mathbb{C})$ then\n$$H^{*}(\\mathcal{Z}; \\mathbb{C}) \\simeq \\mathbb{C}[x_{1}, \\ldots, x_{n}]\/ann(v_{top})$$\nwhere $ann(v_{top})$ is the annhilator of the highest order homogeneous part of the symplectic volume of $\\mathcal{Z}.$\n\\end{thm}\n\nAgain the following results are in ~\\cite{GlSt}\n\\begin{prop}\n Let $\\mathcal{Z}$ be a toric orbifold. Then the Chern classes as above generate $H^{*}(\\mathcal{Z};\\mathbb{C}).$ \n\\end{prop}\nTo apply this to all homogeneous contact manifolds we need not only the case of flag manifolds but also of \\emph{generalized} flag manifolds. These are quotients of a complex semi-simple Lie group $G$ by a \\emph{parabolic} subgroup $P.$ These include the flag manifolds. We extend the result from ~\\cite{GlSt} about flag manifolds to $G\/P.$ For more about generalized flag manifolds see ~\\cite{BE} and ~\\cite{BGG}\n\\begin{prop}\n Let $G\/P$ be a generalized flag manifold. Then the cohomology is generated by the Chern classes as above. \n\\end{prop}\n\\begin{proof}\n Since $P$ is parabolic, it contains a Borel subgroup. Each Schubert cell in $G\/P$ lifts to one in $G\/B.$ This gives an injective map $$H^{*}(G\/P ; \\mathbb{C}) \\rightarrow H^{*}(G\/B;\\mathbb{C}).$$ Thus we need only to see that the Chern classes generate $H^{*}(G\/B; \\mathbb{C})$ which is known from ~\\cite{Bor1}.\n\\end{proof}\n\n\n\\subsection{The Maslov Index and the Conley-Zehnder Index}\nThe Maslov index associates to each path of symplectic matrices a rational number. This particular definition originally appeared in Salamon and Robbins. This index determines the grading for the chain complex in contact homology. The Maslov index should be thought of as analagous to the Morse index for a Morse function. The analogy isn't perfect, since the actual Morse theory we consider should give information about the loop space of the contact manifold. Also note that our action functional has an infinite dimensional kernel. \nIt should be noted that we will describe three indices in the following. Two of them will be called the Maslov index. This is unfortunate, but it will always be clear which Maslov index we will use at any particular time.\n\\begin{remark}Historically, the Maslov index arose as an invariant of loops of Lagrangian subspaces in the Grassmanian of Lagrangian subspaces of a symplectic vector space V. In this setting the Maslov index is the intersection number of a path of Lagrangian subspaces with a certain algebraic variety called the Maslov cycle.\nThis is of course related to our Maslov index of a path of symplectic matrices, since we can consider a path of Lagrangian subspaces given by the path of graphs of the desired path of symplectic matrices. For more information on this see ~\\cite{McDSal}, and ~\\cite{SalRob}.\n\\end{remark}\n\\begin{remark}For a symplectic vector bundle, $E$, over a Riemann surface, $\\Sigma$ there is symplectic definition of the first Chern number $\\langle c_{1}(E),\\Sigma \\rangle$. It turns out that this Chern number is the loop Maslov index of a certain loop of symplectic matrices, obtained from local trivializations of $\\Sigma$ decomposed along a curve $\\gamma \\subset \\Sigma$. This Chern number agrees with the usual definition, considering $E$ as a complex vector bundle, and can be obtained via a curvature calculation. \n\\end{remark}\nLet $\\Phi(t)$, $t \\in[0, T]$ be a path of symplectic matrices starting at the identity such that $det(I - \\Phi(T)) \\neq 0$\\footnote{This is the \\emph{non-degeneracy} assumption. In the context of the Reeb vector field, this condition implies that all closed orbits are isolated}. We call a number $t \\in [0,T]$, a \\emph{crossing} if $det(\\Phi(t) - I)=0.$ A crossing is called \\emph{regular} if the \\emph{crossing form} (defined below) is non-degenerate. One can always homotope a path of symplectic matrices to one with regular crossings, which, as we will see below, does not change the Maslov index. \n\nFor each crossing we define the \\emph{crossing form} \n$$\\Gamma(t)v = \\omega(v, D \\dot{\\Phi}(t)).$$ \nWhere $\\omega$ is the standard symplectic form on $\\mathbb{R}^{2n}.$\n\\begin{defn}\n The Conley-Zehnder of the path $\\Phi{t}$ under the above assumptions is given by:\n$$\\mu_{CZ}(\\Phi) = \\frac{1}{2}sign(\\Gamma(0)) + \\sum_{t\\neq 0\\,, \\,t\\, a\\,\\, crossing}sign(\\Gamma(t))$$\n\\end{defn}\nThe Conley-Zehnder index satisfies the following axioms: \n\\begin{itemize}\n\\item [Homotopy:] $\\mu_{CZ}$ is invariant under homotopies which fix endpoints.\n\\item [Naturality:] $\\mu_{CZ}$ is invariant under conjugation by paths in $Sp(n, \\mathbb{R}).$\n\\item [Loop:] For any path, $\\psi$ in $Sp(n, \\mathbb{R}),$ \nand a loop $\\phi$, $$\\mu_{CZ}(\\psi \\cdot \\phi) = \\mu_{CZ}(\\psi) + \\mu_{l}(\\phi).$$ Where $\\mu_{l}$ is the Maslov index for loops of symplectic matrices.\n\\item [Direct Sum:] If $n = n' + n''$ and $\\psi_{1}$ is a path in $Sp(n', \\mathbb{R})$ \nand $\\psi_{2}$ is a path in $Sp(n'', \\mathbb{R})$ \nthen for the path $\\psi_{1} \\oplus \\psi_{2} \\in Sp(n', \\mathbb{R}) \\bigoplus Sp(n'', \\mathbb{R}),$ we have \n$$\\mu(\\psi_{1} \\oplus \\psi_{2}) = \\mu(\\psi_{1}) + \\mu(\\psi_{2}).$$\n\\item [Zero:] If a path has no eigenvalues on $S^1$, \nthen its Conley-Zehnder index is 0.\n\\item [Signature:] Let $S$ be symmetric and nondegenerate with $$||S|| < 2\\pi.$$ Let $\\psi(t) = exp(JSt)$, then $$\\mu_{CZ}(\\psi) = \\frac{1}{2}sign(S).$$ \n\\end{itemize}\nThe Conley-Zehnder index is still insufficient for our purposes since we need the assumption that at time $T=1$ the symplectic matrix has no eigenvalue equal to 1.\nWe introduce yet another index for arbitrary paths from ~\\cite{SalRob}. We will call this index the Maslov index and denote it $\\mu.$ \n\nFor this new index we simply add half of the signature of the crossing form at the terminal time of the path to the formula for the Conley-Zehnder index.\n$$ \\mu(\\Phi(t)) = \\frac{1}{2}sign(\\Gamma(0)) + \\sum_{t\\neq 0\\,, \\,t\\, a\\,\\, crossing}sign(\\Gamma(t)) + \\frac{1}{2}sign(\\Gamma(T))$$\nThis Maslov index satisfies the same axioms as $\\mu_{CZ}$ as well as the new property of catenation. This means that the index of the catenation of paths is the sum of the indices.\n\n\\subsection{Indices for homotopically trivial closed Reeb orbits}\nLet $\\gamma$ be a closed orbit of a Reeb vector field. Choose a symplectic trivialization of this orbit in $M,$ ie., take a map $u : D \\rightarrow M$ from a disk into $M,$ with the property that the boundary of the image of $u$ is $\\gamma$ and a bundle isomorphism between $u^* \\xi$ and standard symplectic $\\mathbb{R}^{2n}$, $(\\mathbb{R}^{2n} , \\omega_{0}).$ Now we look at the Poincare time $T$ return map of the associated flow (with respect to this trivialization, choosing a framing), where $T$ is the period of $\\gamma.$ If the linearized flow has no eigenvalue equal to $1$, we define the Conley-Zehnder index of $\\gamma$ to be the Conley-Zehnder index of the path of matrices given by the linearized Reeb flow. If there are eigenvalues equal to $1$ we calculate the Maslov index of the path of matrices coming from the flow (in an appropriate symplectic trivialization.) Note that when there is no eigenvalue equal to $1$, the two indices agree. \n\nThe Conley-Zehnder and Maslov indices depends on the choice of spanning disk or Riemann surface used in the symplectic trivialization. Different choices of disks will change the index by twice the first Chern class\\footnote{This is the reason that so often in the literature on contact homology authors insist that $c_{1}(\\xi) = 0$. This index defines the grading of contact homology so if this Chern class is non-zero we must be careful to keep track of which disks we use to cap orbits.} of $\\xi$. Intuitively, given a periodic orbit of the Reeb vector field, this index reveals how many times nearby orbits ``wrap around'' the given orbit. \n\n\n\n\\section{Morse-Bott Contact Homology}\nOur invariants come from a sort of infinite dimensional Morse theory on the loop space of the symplectization $V$ of $M$. In this context, the analogue of gradient trajectories is given by pseudo-holomorphic images of punctured Riemann surfaces. For us a $J$-holomorphic curve is a $C^{\\infty}$ function from a Riemann surface into $V$ whose total derivative is almost-complex linear with respect to $J$ on $V$, and the standard complex structure on the Riemann surface. \n\nWe want to count so-called \\emph{rigid} curves between critical points of a Morse functional. To do this we study certain moduli spaces of \\emph{stable} maps of punctured Riemann surfaces into $V.$ As we will see these curves are \\emph{asymptotically cylindrical} over closed Reeb orbits. The number of punctures corresponds to the number of orbits. We denote such moduli spaces by $$\\mathcal{M}_{0,J}^A(\\gamma_{1}, \\ldots ,\\gamma_{s};\\gamma_{s+1}, \\ldots , \\gamma_{s+k},V).$$ This reads ``the set of genus $0$ ie., \\emph{rational} maps into $V$ with $s$ positive punctures and $k$ negative punctures representing the $2$ dimensional homology class $A$. When no confusion can arise we will drop the ''$V$`` from the notation. Also, when it is understood that there is only one positive puncture, we omit the semicolon. \nThe trouble is that the relevant moduli spaces of curves have, at times, quite an elusive geometric structure. We try to capture their structure as zero sets of a section of a certain infinite dimensional vector bundle over the space of maps from $\\Sigma$ into $V.$ The relevant section is a Fredholm operator with a nice index given in terms of the dynamics of the Reeb vector field of $\\alpha.$ The problem is that one cannot directly apply the implicit function theorem to determine the dimension of the moduli space since the linearized operator could have a nonzero cokernel. Therefore, until we can prove that the operator is transverse to the zero section of the bundle we cannot conclude any kind of manifold structure or dimension formula from the Fredholm index of this operator. \n\nIn the cases at hand we have an integrable almost complex structure so we can try to use Dolbeault cohomology of an appropriate complex. As it turns out, this works under suitable positivity conditions, so we can determine transversality from bounds on spherical chern numbers in $M.$ \n\nContact homology is the homology of the differential graded algebra generated by periodic orbits of the Reeb vector field and graded by the reduced Conley-Zehnder index equal to $\\mu_{CZ} +n-3.$ Whenever there are no Reeb orbits of reduced Conley-Zehnder index equal to $0$, $1$, or $-1$, we can actually use a simpler chain complex. We then consider the graded \\emph{vector space} generated by the closed Reeb orbits, in this case we call the homology \\emph{cylindrical contact homology}.\\footnote{There is an alternative to this called \\emph{linearized contact homology} which ''chops off`` unwanted legs in pairs of pants via an augmentation. This homology is well-defined even when cylindrical homology is not. Whenever cylindrical homology \\emph{is} well-defined these two invariants are the same.} The reader should be wary that when the first Chern class of the contact distribution\\footnote{We consider $\\xi$ as a symplectic vector bundle, defining a first Chern class.} is non-zero, then in order to define a reasonable grading we must keep track of all choices made, spanning surfaces for Reeb orbits, for example. \nFor cylindrical contact homology the differential is given as follows:\n $$ d\\gamma=\\sum_{\\gamma^{'}} n_{\\gamma , \\gamma^{'}} \\gamma^{'}.$$ Where $$n_{\\gamma, \\gamma^{'}} = 0$$\\\\*\nwhenever $$dim(\\mathcal{M}_{0,J}^{A}(\\gamma, \\gamma^{'}, V))\/\\mathbb{R} \\neq 0$$\notherwise it is equal to the signed count of these curves.\\footnote{Recall that we have an action of $\\mathbb{R}$ on the compact moduli space $\\mathcal{M}_{0}^{A}(\\gamma, \\gamma^{'}, V, J)$ by translation in $t$, thus we make a count when the \\emph{quotient} is $0$-dimensional.}\nThe following is from ~\\cite{EGH}.\n\\begin{thm}\n Suppose that there are no Reeb orbits of Conley-Zehnder index $0, -1, 1.$ Then $d\\circ d =0$, and cylindrical contact homology, $CH_{*}(M, \\xi)$ is defined to be the homology of the above chain complex. Moreover it is independent of all choices made, including $J$ and $\\alpha$, in particular it is an invariant of the \\textbf{contact structure} $\\xi.$ \n\\end{thm}\n\nThis differential counts \\emph{rigid} \\footnote{Rigid, in this context means that these curves live in $0$-dimensional moduli spaces} $J$-holomporphic curves in the symplectization of $M$ asymptotically over $\\gamma$ at $\\infty$ and over $\\gamma^{'}$ at $- \\infty.$ \n\n\\begin{remark}\n Contact homology and in particular, the Floer-like cylindrical contact homology, are just the tip of the iceburg of the larger \\textbf{symplectic field theory} which contains many more subtle invariants.\n\\end{remark}\n\nTo use this complex, however, we really need the condition that the periodic Reeb orbits are isolated. Since we are working exclusively with $S^1$-bundles, all Reeb orbits are periodic so are never isolated. It turns out there there is a Morse-Bott version of all of this theory. \n\nTo make this work we actually work with Morse theory on the base. If we work in an orbifold we have different orbifold strata given by the various orbit types of the original action. Recall that we assume all actions are strongly Hamiltonian.\nWe now introduce some terminology.\n\\begin{defn}\n The \\textbf{action functional} is the map $$\\mathcal{A} : C^{\\infty}(S^1, M) \\rightarrow \\mathbb{R}$$ given by $$\\gamma \\mapsto \\int_{\\gamma} \\alpha,$$ where $\\alpha$ is a contact 1-form for $(M, \\xi).$\n \\end{defn}\nNote that critical points of $\\mathcal{A}$ are periodic orbits of the Reeb vector field. \n\\begin{defn} Let $(M, \\xi)$ be a contact manifold with contact form $\\alpha.$ The \\textbf{action spectrum}, $$\\sigma(\\alpha) = \\{r \\in \\mathbb{R} | r = \\mathcal{A}(\\gamma)\\}$$ for $\\gamma$ a periodic orbit of the Reeb vector field.\n \n\\end{defn}\n\\begin{defn} Let $T \\in \\sigma(\\alpha).$ Let $$N_{T} = \\{ p \\in M | \\phi_{p}^{T} = p\\},$$ $$S_{T} = N_{T}\/S^{1},$$ where $S^1$ acts on $M$ via the Reeb flow. Then $S_{T}$ is called the \\textbf{orbit space} for period $T$.\n \\end{defn}\nHere the orbit spaces are precisely the orbifold strata when $M$ is considered as an $S^1$ orbibundle.\n\nFor our Morse-Bott set-up we assume that our contact form is of Morse-Bott type, ie. \n\\begin{defn}\nA contact form is said to be of Morse-Bott type if\n\\begin{enumerate}\n \\item[i.] The action spectrum:\n$$\\sigma(\\alpha) := \\{r \\in \\mathbb{R} : \\mathcal{A}(\\gamma) = r,\\, for\\, some\\, periodic\\, Reeb\\, orbit\\, \\gamma.\\}$$ is discrete.\n\\item[ii.] The sets $N_{T}$ are closed submanifolds of M, such that the rank of $d\\alpha|_{N_{T}}$ is locally constant and $$T_{p}(N_{T}) = ker(d\\phi_{T} - I).$$\n\\end{enumerate}\n\\end{defn}\n\\begin{remark}These conditions are the Morse-Bott analogues for the functional on the loop space of $M.$\n\\end{remark}\n\n \nNotice that in the case of $S^1$ bundles this is always satisfied. The key observation, as we soon shall see, is that we can relate $J$-holomorphic curves to Morse theory on the symplectic base. Since we consider only quasi-regular contact manifolds here, we can always approximate the contact structure by one with a dense open set of periodic orbits of period 1, say, and a finite collection of strata of orbits of smaller period, each such stratum has even dimension and has codimension at least 2 (other than the dense set of regular points, of course). \n\n\n\\subsection{Moduli}\nBefore we describe the Morse-Bott chain complex we need to describe the moduli spaces of pseudoholomorphic curves with which we will be working. So, as before let $(M, \\xi)$ be a contact manifold, $V$ its symplectization, and $J$ an almost complex structure on $V$ compatible with the transverse symplectic structure on $M.$ The curves that we're interested in are $J$-holomorphic maps from punctured $S^2$'s into the the symplectization of our contact manifold. Such curves are \\emph{asymptotically cylindrical} over closed Reeb orbits.\n\\begin{defn}\nLet $\\Sigma$ be a Riemann surface with a set of punctures $Z = \\{z_{1}, \\ldots, z_{k}\\}$ A map $$u = (a(s,t), f(s,t)): \\Sigma \\setminus Z \\rightarrow V$$ is called \\textbf{asymptotically cylindrical} over the set of Reeb orbits $\\gamma_{1}, \\ldots, \\gamma_{k}$ if for each $z_{j} \\in Z$ there are cylindrical coordinates centered at $z_{j}$ such that \n$$lim_{s \\rightarrow \\infty} f(s, t) = \\gamma (T t)$$ \nand\n$$lim_{s \\rightarrow \\infty} \\frac{a(s, t)}{s} = T$$\n\\end{defn} \n\nHere are some precise statements from ~\\cite{BEHWZ}:\n\\begin{prop}Suppose that $\\alpha$ is of Morse,\\footnote{Here we mean that for each periodic Reeb orbit, the time 1 return map has no eigenvalue equal to $1$.} or Morse-Bott type. Let $$u = (a, f ) : \\mathbb{R}^{+} \\times \\mathbb{R}\/\\mathbb{Z} \\rightarrow (\\mathbb{R} \\times M, J)$$ be a $J$-holomorphic curve of finite energy.\\footnote{For the most general definitions of energy see ~\\cite{BEHWZ}} Suppose that the image of $u$ is unbounded in $\\mathbb{R} \\times M$. Then there exist a number $T \\neq 0$ and a periodic orbit $\\gamma$ of $R_{\\alpha}$ of period $|T|$ such that\n\n$$lim_{s \\rightarrow \\infty} f(s, t) = \\gamma (T t)$$ \nand\n$$lim_{s \\rightarrow \\infty} \\frac{a(s, t)}{s} = T.$$\n\n\\end{prop}\n\nThis immediately implies \n\\begin{prop}\nLet $(S,j)$ be a closed Riemann surface and let $$Z=\\{z_1,\\ldots , z_k \\}\\subset S$$ be a set of punctures. Every $J$-holomorphic curve $$F = (a,f):(S \\setminus Z)\\rightarrow \\mathbb{R} \\times M$$ of finite energy and without removable singularities is asymptotically cylindrical near each puncture $z_i$ over a periodic orbit $\\gamma_i$ of $R_{\\alpha}$.\n\\end{prop}\n\nThese propositions are extremely important to us because they show that it is reasonable to define gradient trajectories between Reeb orbits to be $J$-holomorphic curves in the symplectization. \nWe have even more, ie., a Gromov compactness theorem, which says that although the moduli spaces are not necessarily compact, we can compactify them by adding stable curves of height $2$. Even better, via a gluing construction ~\\cite{Bourg1} we show that the boundary of the moduli space is equal to set of height $2$ curves. \n\nAs above we consider the moduli space of generalized $J$-holomorphic curves from an $s$-times punctured Riemann surface into $V$ asymptotically cylindrical over the orbit spaces $S_1, \\ldots, S_{s}$ representing the class $A$:\n$$\\mathcal{M}^{A}_{0,J}(S_{1}; \\ldots, S_{s}, V).$$\n\nIn this notation the first orbit space is from the \\emph{positive} puncture, all others are negative. These moduli spaces are the analogues of the gradient trajectories of Morse theory. We only count them when they come in zero dimensional families. Thus we need a dimension formula for these spaces. \n\\begin{prop}\n The virtual dimension for the moduli space of generalized genus $0$ $J$-holomorphic asymptotic over the orbit spaces $S_{T_{0}}, \\ldots, S_{T_{1}},\\ldots, S_{T_{s}}$ (with 1 positive, and $s$ negative punctures) representing $A$ is equal to\n$$(n-3)(1-s) + \\mu(S_{T_{0}}) + \\frac{1}{2}dim(S_{T_{0}}) - \\sum_{i=0}^s (\\mu(S_{T_{i}}) + \\frac{1}{2} dim (S_{T_{i}})) + 2c_{1}(\\xi, \\Sigma),$$ where $\\Sigma$ is a Riemann surface used to define the symplectic trivialization and homology class $A.$ \n\\end{prop}\n\nIn cylindrical contact homology, since we only are keeping track of cylinders, we take $s=1$ and this formula reduces to $$ \\mu(S_{T_{+}}) + \\frac{1}{2}dim(S_{T_{+}}) + \\mu(S_{T_{-}}) + \\frac{1}{2} dim (S_{T^{-}}) + 2c_{1}(\\xi, \\Sigma).$$\nOf course if $\\xi$ has a regular structure this boils down to $$ \\mu(S_{T^{+}}) - \\mu(S_{T^{-}}) + 2n-2 + 2c_{1}(\\xi, \\Sigma).$$\nFor a proof of this formula see ~\\cite{Bourg1}. Bourgeois' proof is of interest as traditionally these kinds of results come from a spectral flow analysis. Bourgeois, however, makes interesting use of the Riemann-Roch theorem.\n \nWe want to understand the structure of the moduli space since our Morse-like chain complex uses these curves to construct the differential. The reader should be aware that the formula for the dimension of the moduli space above is really a \\emph{virtual} dimension until some sort of transversality is achieved for some $\\bar{\\partial}_{J}$-operator. This formula is obtained via Fredholm analysis on the space of $C^{\\infty}$ maps from $S^2 \\setminus \\{z_1, z_2, \\ldots, z_{j}\\}$ into $V.$ The $\\bar{\\partial}_{J}$ turns out to be a Fredholm section of a certain infinite dimensional bundle over this space whose kernel is precisely the set of $J$-holomorphic curves. The Fredholm index $\\bar{\\partial}_{J}$ is the dimension formula above. The trouble is that a priori, we cannot rule out a non-zero cokernal, hence our dimension formula could be \\emph{under counting} the relevant curves. There have been many attempts at transversality proofs, and it seems as though the new \\emph{polyfold} theory of Hofer, Zehnder and Wysocki should solve the problem. There are also proofs using virtual cycle techniques, (cf. ~\\cite{Bourg1}) however even here it seems that there may be potential gaps. Therefore we show how, in some cases, we can justify the validity of our curve counts through more elementary geometric considerations.\n\nIn the cases that we are considering in this paper, the almost complex structure will be integrable, thus we can use algebro-geometric techniques to find conditions for regularity of $J$.\\footnote{The word \\emph{regularity} is over used. Here we mean that for this $J$, the $\\bar{\\partial}_{J}$ operator is surjective, as a section in a suitable infinite dimensional vector bundle.} \n\nNow let us describe the relationship between moduli spaces of stable curves in a symplectic orbifold and the moduli space of curves into the symplectization of its Boothby-Wang manifold. Notice that the symplectization $W$ is just the associated line (orbi)bundle to the principle $S^1$-(orbi)bundle\\footnote{Of course, in the situations we are dealing with in this paper, $M$ is a manifold even if it is the total space of an orbibundle.}, $M$, with the zero section removed. Given as many marked points as punctures we actually get a fibration, here curves upstairs are sections of $L$ with zeroes of order $k$ and poles of order $l$ once we fix the phase of a section we actually get unique curves. This is described for the case of \\emph{regular} contact structures in ~\\cite{EGH}. For $S^{1}$-bundles over $\\mathbb{CP}^1$ with isolated cyclic singularities Rossi extended this result in ~\\cite{PR}, we'll actually need an extension of this to higher dimension. The point here is that we want to coordinate our curve counts upstairs with the ``Gromov-Witten'' curve count downstairs. In the case where the base is an orbifold we must make sure that we can get an appropriate curve in the sense of Gromov-Witten theory on orbifolds. It should be noted that in symplectizations all moduli spaces come with an $\\mathbb{R}$-action by translation. Whenever we talk about $0$-dimensional moduli spaces, we really mean that we are considering $1$-dimensional moduli spaces quotiented out by the $\\mathbb{R}$-action giving $0$-dimensional manifolds\\footnote{Actually these moduli spaces in general are branched orbifolds with corners, however in the cases that we consider in this paper they really are manifolds.}. \n\n\nThe following lemma comes from ~\\cite{CR}.\n\\begin{lemma}\n Suppose that $u$ is a $J$-holomorphic curve into the symplectic orbifold $\\mathcal{Z}$, then either $u$ is completely contained in the orbifold singular locus or it intersects it in only finitely many points.\n\\end{lemma}\nWe use this to prove:\n\\begin{lemma}\nLet $u : \\Sigma \\rightarrow \\mathcal{Z}$ be a non-constant $J$-holomorphic map between a Riemann surface and a symplectic orbifold. Then there is a unique orbifold structure on $\\Sigma$ and a unique germ of a $C^{\\infty}$-lift $\\tilde{u}$ of $u$ such that $u$ is an orbicurve.\n\\end{lemma}\n\\begin{proof}\nFirst let us assume that the marked points are all mapped into the singular locus, since otherwise the curve only intersects the singular locus in a finite number of points. Now $u_{z_{i}}$ corresponds to a closed Reeb orbit of non-generic period, ie., a curve in $S_{T_{k}}$, say. Take an element from the moduli space of curves into $W$ asymptotically cylindrical over $S_{T_{k}}$ in some slot. We need only to take a local uniformizing chart equivariant with respect to $\\mathbb{Z}_{T_{k}}.$\n\\end{proof}\n\nFrom this we actually get a fibration.\n\\begin{prop}\n There is a fibration $$pr :\\mathcal{M}_{0, J}(S_{T_{1}}; S_{T_{2}}, \\ldots , S_{T_{k}}) \\rightarrow \\mathcal{M}_{0, k}(a_{1}, \\ldots, a_{k}).$$ \n\\end{prop}\n\n\nWith this understanding of the moduli space, assuming now that $J$ is integrable we see that that the linearized Cauchy-Riemann operator is the $\\bar{\\partial}$ operator of the right Dolbeault complex on $\\mathcal{Z}$. This leads to the following criterion for regularity of $\\bar{\\partial}_{J}$ at each $u$ adapted to the $S^1$ bundle case from ~\\cite{McDSal2}. First we note that $u: \\mathbb{CP}^1 \\rightarrow V$, we can look at $u^*T(V),$ whose characteristic classes look like those of $u^{*}\\xi.$ Over $\\mathbb{CP}^{1}$, $u^*\\xi$ splits as a sum of line bundles:\n$$u^*\\xi = \\bigoplus_{j} L_{j}.$$ \nThis splitting doesn't necessarily work for general orbifolds as ambient spaces (unless the sphere doesn't intersect the orbifold singular locus at all, but for toric orbifolds this splitting principle always works ~\\cite{Gmn97}. Of course, in the case of a regular contact form, this is just the classical Groethendieck splitting principle. \n\\begin{thm}[Regularity criterion for genus $0$ moduli spaces when $J$ is integrable]\nSuppose that $J$ is integrable. Suppose that $$\\langle c_{1}(L_{j}), A \\rangle \\geq -2 + s -t$$ for every $A \\in H_{2}(\\mathcal{Z})$ which is represented by a 2-sphere. Then the linearized Cauchy-Riemann operator is surjective and the genus $0$ moduli space of curves with $s$ positive punctures and $t$ negative punctures is a smooth manifold of dimension given by the Fredholm index. In the case that $\\mathcal{Z}$ is an orbifold, we require that $$c_{1}(L_{j}) \\geq \\sum_{\\alpha}(1-\\frac{1}{m_{\\alpha}})c_{1}(\\mathcal{O}(D_{\\alpha})) -2-s-t,$$ where $D_{\\alpha}$ are branch divisors.\n\\end{thm}\n\\begin{proof}\nLet $z_{1}, \\ldots, z_{s}, \\ldots z_{s +t}$ be distinct points on $S^2.$ Consider the divisor $$D= k_{1}z_{1} + \\ldots + k_{s}z_{s} - k_{s+1}z_{s+1} - \\ldots - k_{s+t}z_{s+t}.$$ Then the Cauchy-Riemann operator is just the $\\bar{\\partial}$-operator of the Dolbeault complex for the line orbibundle $L_{j} \\otimes \\mathcal{O}(D).$\n$$ \\bar{\\partial} : \\Omega^{0}(\\mathbb{CP}^1, L_{j} \\otimes \\mathcal{O}(D)) \\rightarrow \\Omega^{0,1}(\\mathbb{CP}^1, L_{j} \\otimes \\mathcal{O}(D)).$$\nThe cokernel of $\\bar{\\partial}$ is just the $(0,1)$ cohomology of that complex. But we have the following isomorphisms: \n$$H_{\\bar{\\partial}}^{0,1}(\\mathbb{CP}^1, L_{j} \\otimes \\mathcal{O}(D)) \\simeq H_{\\bar{\\partial}}^{1,0}(\\mathbb{CP}^1, (L_{j} \\otimes \\mathcal{O}(D))^*)^* \\simeq H_{\\bar{\\partial}}^{0,1}(\\mathbb{CP}^1, (L_{j} \\otimes \\mathcal{O}(D))^* \\otimes K).$$\nFor the last group to be $0$, we must have $$c_{1}(L_{j} \\otimes \\mathcal{O}(D))^* \\otimes K) <0.$$ This happens whenever $$c_{1}(L_{j}) > -2 - deg(D).$$ \nIn the quasi-regular case we check this in the orbifold sense. So we consider the orbifold first Chern class :\n$$c_{1}^{orb}= c_{1}(L_{j}) - \\sum_{\\alpha}(1-\\frac{1}{m_{\\alpha}})c_{1}(\\mathcal{O}(D_{\\alpha}))$$ where the sum at the end is non-zero only in the presence of branch divisors, $D_{\\alpha}.$ Here we must bound this below by $-2 -deg(D).$ \n\\end{proof}\n\\begin{thm}\n$$\\langle c_{1}(L), A \\rangle \\geq -2 + s -t$$ whenever $L$ is a line (orbi)bundle obtained by the Boothby-Wang fibration whose total space is either of the following:\n\\begin{enumerate}\n\\item[i.] a homogeneous contact manifold\n\\item[ii.] a toric Fano contact manifold.\n\\end{enumerate}\n\\end{thm}\n\\begin{proof}\nThe proof of (i) is nearly the same Proposition 7.4.3 in ~\\cite{McDSal2} with $u^{*}TM$ replaced with $u^*(\\xi).$ Note that in case (i) we are always dealing with manifolds at each level rather than with orbifolds.\nFor (ii), for a splitting of $$u^*(\\xi) = \\bigoplus_j L_{j}$$ we get sections and positivity of Chern classes via the Fano condition. Here the splitting takes place in the toric orbifold, then all the lines are pulled back to the sphere. \n\\end{proof} \n\n\\begin{cor}\n For homogeneous and toric contact manifolds the dimensions of all genus $0$ moduli spaces are given by the Fredholm index as predicted.\n\\end{cor}\nWe would like now to set up the Morse-Bott chain complex. This was originally done in ~\\cite{Bourg1} and discussed for circle bundles in ~\\cite{EGH}. We have already discussed some of the basic setup, now, much like the case in Morse theory we would like to relate the Morse-Bott case to the generic case. The idea is to perturb our contact structure so it's periodic orbits are in $1$-$1$ correspondence with the critical points of some Morse-function. In our case we would like to use our moment maps to get a perfect Morse or Morse-Bott function $f.$\nSo first for appropriate $\\epsilon$ and our Morse or Morse-Bott function$f$ we take the new contact form\n$$ \\alpha_{f} = (1 + \\epsilon f) \\alpha.$$\nThen critical points of $f$ correspond to periodic orbits of $\\alpha_{f}.$\nIf $f$ is a perfect Morse function as in the toric case we take this as our contact form. Else, suppose $f$ is Morse-Bott, then we choose Morse functions on each critical submanifold. Note that in the case of a Hamiltonian action of a compact Lie group all such submanifolds have even index and even dimension. By $G$-invariance of the moment map this restricts to all orbit spaces. \nIn this case the periodic orbits are in $1$-$1$ correspondence with critical points of Morse functions on the critical submanifolds of $f.$\nWe want to relate the Conley-Zehnder indices of the generic form with those of the original. Since we have a $1$-$1$ correspondence between critical points and orbits, we will think of the chain complex associated to $\\alpha_{f}$ as critical points of $f.$ The index is the grading so we'll write the Conley-Zehnder index of the orbit corresponding to $p$, $\\mu_{CZ}(\\gamma_{p})$ as $|p|.$ In the Morse case we have\n$$ |p| = \\mu(S_{T_{k}}) -\\frac{1}{2}dim(S_{T_{k}}) + ind_{p} f.$$\\\\* \nIn the case of a Morse-Bott function, we just proceed as in ordinary Morse theory to get a new Morse function on each critical submanifold, but now in this formula we use the Morse-Bott indices. \nNow we can define the differential for Morse-Bott contact homology.\n$$dp = \\partial p + \\sum_{q} n_{pq} q.$$\nWhere $\\partial p$ is just the Morse-Smale-Witten boundary operator, $n_{pq}$ is similar to the coeffecient in the generic case, and $ind_{p}$ is the Morse index for critical points. Bourgeois proves, in his thesis ~\\cite{Bourg1}, that this homology computes contact homology. For us, the particular form of the differential does not matter much since it will vanish for index reasons. \n\n\\begin{thm}(Bourgeois)\n When the homology defined above exists it is isomorphic to the standard contact homology for non-degenerate contact forms. \n\\end{thm}\n\\section{Morse-Bott Contact Homology in the Homogeneous and Toric Cases}\nNow we apply the results from the previous section. Let us first set some notation. Suppose first that $(M, \\xi)$ is compact, simply connected, and admits a strongly Hamiltonian action of a Lie group as discussed in the introduction which is of Reeb type. Then we know that there is a quasi-regular contact form $\\alpha$ for $(M, \\xi)$ equivariant with respect to the action.\nAs above, let $S_{T_{k}}$ denote the stratum in $\\mathcal{Z} = M\/(S^1)$ corresponding to Reeb orbits of period $T_{k}.$ Let $\\Gamma_j$ denote the local uniformizing group for the stratum $S_{T_{k}}.$ Recall that each stratum is a K\\\"{a}hler sub-orbifold of $\\mathcal{Z}.$ \nIn what follows assume that $H^{*}(\\mathcal{Z}; \\mathbb{C})$ is a truncated polynomial ring generated by elements in $H^{2}(\\mathcal{Z}; \\mathbb{C})$, ie., the Chern classes coming from the symplectic reduction defining $\\mathcal{Z}$ as a symplectically reduced orbifold. Let us write such a basis of $H^2(\\mathcal{Z};\\mathbb{C})$ as $\\{c_{1}, \\ldots, c_{k} \\}.$ Now choose $1$ forms $\\tilde{c_{j}}$ representing the $c_{j}$'s Now we just consider circle bundles over $\\mathcal{Z}$ by choosing connection $1$-forms $\\alpha$ with curvature $$ d \\alpha = \\sum_{j}\\pi^{*} w_{j} \\tilde{c_{j}}.$$ Notice that for $\\mathcal{Z}$ a toric orbifold, this construction yields all possible toric contact structures of Reeb type. Note that we implicitly choose a symplectic form $\\omega = \\sum w_{i} \\tilde{c_{i}}$ on $\\mathcal{Z}$ during this process. Then $$c_{1}(T(\\mathcal{Z})) = \\sum \\tilde{w_{i}} \\tilde{c_i},$$ where $\\tilde{w_i}$ is obtained via the spectral sequence for the Boothby-Wang fibration.\n\n\\begin{remark}\nIn the case of contact reduction in $\\mathbb{C}^n$ by a circle (where the action is of Reeb type) the coeffecients of $|z_{j}|^2$ in the (circle) moment map can be chosen to be the $\\tilde{w_{j}}$'s. \n\\end{remark} \n\nNow we choose elements of $H_{2}(\\mathcal{Z}; \\mathbb{Z})$, $A_{1}, \\ldots, A_{n}$, with $$\\langle \\tilde{c_{i}},A_{i} \\rangle =1.$$ This is possible because the cohomology is a truncated polynomial ring generated by the $c_{j},$ all elements having even degree. Now let $$A= \\sum_{j} A_{j}.$$ Then for any K\\\"{a}hler suborbifold $ i :S \\hookrightarrow \\mathcal{Z},$ $$\\sum_{i}\\langle i^{*}\\tilde{c_{i}},A \\rangle$$ is nonzero. Thus we can also do this for each $S_{T_j}$ by pulling the Chern classes back along the inclusion maps, then choosing homology classes in each stratum as above in terms of $i_{j}^{*} \\tilde{c_{i}},$ where $i_{j}: S_{T_{j}} \\rightarrow \\mathcal{Z}$ is the inclusion, and $\\{c_{i} \\}$ are the Chern classes generating $H^{*}(\\mathcal(Z); \\mathbb{C}).$ Call the corresponding homology class $A_{S_{T_j}}.$ The purpose here is to find a nice diagonally embedded sphere with which to make our calculations. Now let's use this set-up to do some index calculations. \nFirst we must find suitable trivializations and capping disks for Reeb orbits. The idea here is to find two trivializations for each Reeb orbit, then use the loop property of the Maslov index to calculate the index via integration of $c_{1}(T(\\mathcal{Z}))$ over the sphere obtained by gluing the two disks (from the symplectic trivializations) along their boundaries. The author first encountered this idea in ~\\cite{Bourg1} and ~\\cite{EGH}, however this was only for the regular\\footnote{Regular in the sense of foliation theory.} case. So let $\\gamma_{S_{T_j}}$ be a Reeb orbit of period $T_{j}$, living, of course, in the stratum $S_{T_j}.$ We now pull back $\\xi$ via the inclusion map over $S_{T_j},$ $i_{j}.$ For the first disk we just cap off a tubular neighborhood of the Reeb orbit given by the product framing for $M$. In this framing the Maslov index is $0$, since the return map is always the constant path in $Sp(2n-2, \\mathbb{R})$ given by the identity. Now we need another disk to glue along the Reeb orbit to get a sphere. In order to do this consider a holomorphic sphere, ie., a map $u:S^2 \\rightarrow S_{T_j}$ passing through $p \\in S_{T_{j}}$ such that $[u] =A_{S_{T_j}}.$ This is always possible since the moment map is invariant and since we assume $\\mathcal{Z}$ is simply connected, the Hurewicz homomorphism is surjective. Now consider a holomorphic (orbi)section of $L$ over our sphere with a zero of order equal to the multiplicity of $\\gamma$ and no pole. Such a section exists since we are talking about line (orbi)bundles over $\\mathbb{CP}^1.$ With this set-up we prove:\n\\begin{lemma}\n Let $M$ be an $S^1$-bundle over a symplectic orbifold admitting a Hamiltonian action of a compact Lie group, such that its cohomology is generated by the Chern classes associated to the action. Then the Maslov index of a Reeb orbit in the stratum $S_{T_{j}}$ of multiplicity $m$ is equal to $$\\frac{2m}{|\\Gamma_{j}|}\\int_{A_{S_{T_j}}} i^{*}c_{1}(T(S_{T_{j}})),$$ moreover this number is an integer.\n\\end{lemma}\n\\begin{proof}\nBy the loop property of the Maslov index, the Maslov index of the Reeb orbit is twice the Maslov index of the path of change of coordinate maps between the two disks glued along $\\gamma.$ Since the disk was obtained via an (orbi)section over a sphere representing $A_{S_{T_j}}$, we get $$\\mu(\\gamma) = 2 \\langle c_{1}(\\xi) , \\sigma(u) \\rangle =2 \\langle c_{1}(T(\\mathcal{Z}) , A_{S_{T_{j}}} \\rangle.$$ This is exactly $c_{1}^{orb}(T(S_{T_{j}}))$ evaluated on $A$. Therefore the index of an orbit of multiplicity $m$ is $$2m \\int_{A_{S_{T_j}}} c_{1}^{orb}(T(S_{T_{j}})).$$ Now going back to the work of Satake ~\\cite{Sat57} to compute the integral of an orbifold characteristic class over a homology class, we take intersections with all orbifold strata and divide out by the orders of the local uniformizing groups and sum:$$2k \\int_{A_{S_{T_j}}} c_{1}^{orb}(T(S_{T_{j}}))=2m \\sum_{j}\\frac{1}{|\\Gamma_{j}|}\\int_{A_{S_{T_j}} \\cap \\Sigma_{j}} c_{1}(T(S_{T_{j}}))$$ where $\\Gamma_{j}$ is a local uniformizing group in the orbifold stratum $\\Sigma_{j}=S_{T_j}.$ Now, since each such spherical class is completely contained in $S_{T_{j}},$ we can just compute the integral $$\\frac{2}{|\\Gamma_{j}|}\\int_{A_{S_{t_j}}} c_{1}(T(S_{T_{j}}))|_{S_{T_{j}}} = \\frac{2}{|\\Gamma_{j}|}\\int_{A_{S_{T_j}}} i^{*}c_{1}(T(S_{T_{j}}))$$ for simple orbits, multiplying by $m$ for $m$-multiple orbits. Note however that, although we may compute the integral on $\\mathcal{Z}$, this integral is equal to one which takes place as the evaluation of an integral form on the contact manifold, hence we always get an integer. \n\\end{proof}\n\\begin{remark}\nThe idea above is that $A_{S_{T_j}}$ is a ``sufficiently diagonal'' sphere in $S_{T_k}.$ This ensures that we pick up as much information as possible about the line bundle as possible during the integration. One should also note that in general $c_{1}(\\xi) \\neq 0$ so this grading scheme for contact homology is computed with respect to a \\emph{particular} choice of capping surface for each Reeb orbit. When comparing contact manifolds which are $S^1$-orbibundles over the same base, care must be taken to make the same choices each time, so that the weights are realized via the Chern classes of each \\emph{specific toric structure}. \n\\end{remark}\n\\begin{remark}\nThe reader may wonder what role branch divisors play in the index calculation above. This is encoded in summing over the strata and dividing by the orders of local uniformizers.\n\\end{remark}\nWe want to use these calculations to compute cylindrical contact homology, however this is not well defined unless we can exclude Reeb orbits of degree $0$, $1$, $-1$. To ensure this we must assume that for all $k$ $$2 (\\sum_i i^*c_{i}\\tilde{w_{i}}) -\\frac{1}{2} dim(S_{T_{k}})>0.$$ For this it is sufficient to assume that $$\\sum_{i}\\tilde{w_{i}} >1.$$ We take this as a standing assumption in the following. \n\nNow we notice that there are no rigid $J$-holomorpic cylinders other than the trivial ones. This follows from a simple index computation and comparison with the dimension formula for the relevant moduli spaces. This means that the contact homology is given ompletely by the Morse-Smale-Witten complex of the moment map with degree shifts given by the Maslov indices. The discussion above yields theorem ~\\ref{thm:main}]. We obtain the following corollaries.\n \n\n\n\\begin{cor}\n Let $(M, \\xi)$ be a simply connected compact homogeneous contact manifold. Then $CH_{*}(M)$ is generated by copies of $H_{*}(\\mathcal{Z})$ with degree shifts given by $$2m \\int_{A} c_{1}(T(\\mathcal{Z})) = 2m \\sum_{i} \\tilde{w_{i}} -2$$.\n\\end{cor}\n\\begin{proof}In this case $M$ is an $S^{1}$-bundle over a generalized flag manifold, (recall that in this case there is a \\emph{regular} contact $1$-form, $\\alpha$ for $\\xi$). The cohomology of the base is a polynomial ring as per the discussion earlier, and all the relevant homology classes are spherical. By the regularity theorem for integrable $J$ the dimension of the moduli space is the one predicted by the Fredholm index. Moreover by the index calculation above, and the dimension formula for the moduli spaces, there are no rigid $J$-holomorphic curves connecting orbit spaces. This contact homology is given compeltely in terms of the Morse-Smale-Witten differential, which vanishes since the moment map deterines a perfect Morse function, thus we get a generator for each critical point of the norm squared of the moment map in degree given by the Maslov indices as calculated in the previous discussion.\n\\end{proof}\n\\begin{cor} Let $(M, \\xi)$ be a simply connected compact toric Fano contact manifold with a quasiregular contact form $\\alpha$. Then $CH_{*}(M)$ is generated by copies of $H_{*}(\\mathcal{Z})$ with degree shifts given by the Maslov indices plus the dimension of the stratum containing the particular Reeb orbit as a point. \nIf $\\xi$ has a regular contact form $\\alpha$ then the degree shifts are given by $$2m\\sum_{j}\\tilde{w_{j}}-2 ,$$ where the $\\tilde{w_{j}}$ are defined as above. \n\\end{cor}\n\\begin{proof}The Fano condition gives transversality of the $\\bar{\\partial}_{J}$-operator via the Dolbeault complex. If we assume transversality we can drop the Fano assumption. Again, our cohomology ring is a truncated polynomial ring generated by all possible Chern classes, with spherical second homology because of simple connectivity. The indices are given by the even multiples of the sum of the weights. Again there are no non-trivial $J$-curves. So the homology is that given by the Morse-Smale-Witten complex (whose differential again vanishes by perfection of the Morse function) with the degree shifts given by the Maslov indices as calculated above. \n\\end{proof}\n\n\\section{Examples} \n\\subsection{Wang-Ziller Manifolds}\nNow let's specialize to Wang-Ziller manifolds. \nThese are toric manifolds either obtained from reduction in $\\mathbb{C}^2 \\times \\mathbb{C}^2$ via the moment map $\\mu(z,w) = k|w|^2 - l|z|^2.$ This manifold is also a homogeneous contact manifold. This is how we get transversality. Note that as a toric manifold, this manifold is non-Fano, so we really need to use the homogeneity to get transversaltiy via the Dolbeault complex. \nWe can also see this manifold as a Boothby-Wang manifold. Consider $\\mathcal{Z}= \\mathbb{C}\\mathbb{P}^1 \\times \\mathbb{C}\\mathbb{P}^1,$ and we take the standard symplectic form on each summand and multiply each piece by relatively prime integers $k$ and $l$. \nWe take $P$ to be the circle bundle with a connection form $\\alpha$ satisfying $d \\alpha = \\pi^{*}(k c_{1} + l c_{2}),$ where the $c_{j}$ are actually the generators of the second cohomology of each sphere. Then $$c_{1}(\\xi)=(2k - 2l)\\beta,$$ for $\\beta$ a generator of $H_2(S^2 \\times S^3 ; \\mathbb{Z}),$ and $$c_{1}(T\\mathcal{Z}) = (2kc_{1} + 2lc_{2}),$$ here $\\mathcal{Z}$ is topologically $\\mathbb{CP}^1 \\times \\mathbb{CP}^1$ with the toric structure obtained by with symplectic form determined by $k,l.$ \n$\\mathcal{Z}$ admits a perfect Morse function, and the Maslov indices in this case for orbits of multiplicity $m$ are given by $ 4m(k+l) $. Thus the grading of contact homology is given by\n $$|p| = 4m(k+l) - 2 + d$$\nwhere $m\\in \\mathbb{Z} \\setminus\\{0\\}$, and $d$ ranges over all possible degrees of homology classes in $\\mathcal{Z},$ in this case $d=0,2,4.$b\nThis gives infinitely many distinct contact structure on $S^2 \\times S^3$ since for each choice of relatively prime $k$ and $l$, we get generators of contact homology in minimal dimension $4(k+l) -2.$ Of course, for all pairs such that $k-l = c$ we get a single first Chern class for the contact bundle ~\\cite{WZ}. Choosing now all pairs with $k-l =c,$ we get infinitely many distinct contact structures in the same first Chern class. In ~\\cite{Ler1} Lerman showed that these contact structures are all pairwise non-equivalent as toric contact structures, but he asked whether or not they were pairwise contactomorphic. This answers that question in the negative. Via the above construction we get contact structures $\\xi_{k,l}$ on $S^{2} \\times S^{3}.$\n\\begin{cor}\n Fix $c \\in \\mathbb{Z},$ choose $k,l$ such that $gcd(k,l) =1,$ and $k-l =c$ then the contact structures $\\xi_{k,l}$ are pairwise non-contactomorphic all within the same first Chern class of $4$-plane distribution. \n\\end{cor}\n\n\\begin{remark}\n This example suggests a Kunneth-type formula for the \\textbf{join} ~\\cite{BG07} construction for quasiregular contact manifolds provided each summand has suitable contact homology. Suppose that $(\\mathcal{Z}_1, \\omega_{1})$ and $(\\mathcal{Z}_2 , \\omega_{2})$ are both simply connected symplectic orbifolds which are reduced spaces so that their cohomology rings are polynomials in the Chern classes. Then we can build circle bundles over their product with curvature forms given as integer linear combinations of the $\\omega_{j}.$ By choosing appropriate spheres ``diagonally'' embedded into the product we can evaluate the first Chern class of this bundle in order to get the Maslov indices as above. Assuming transversality of the $\\bar{\\partial}_{J}$-operator this always computes contact homology. \n\\end{remark}\n\\begin{remark}\n One could do a similar computation with Boothby-Wang spaces over $\\mathbb{CP}^2 \\# \\overline{\\mathbb{CP}^2}.$\n\\end{remark}\n\n\\subsection{Circle bundles over weighted projective spaces.} \nIn these examples first off, note that weighted projective spaces are toric Fano, (even in the orbifold sense.)\\footnote{These orbifolds generally have branch divisors.} Note that the cohomology ring is then just the standard one, and we just need to find the right spherical classes. Of course we just pullback $k$-multiples of the standard symplectic form to define our line bundles. Now to compute contact homology of the bundle we integrate $c_{1}(T(\\mathcal{Z}))$ over the class of a line. The base admits a perfect Morse function, so all we need to do is keep track of the strata. Integrating $i^{*}_{j} c_{1}(T(\\mathcal{Z}))$ over spheres representing the K\\\"{a}hler class for each stratum. So the grading of contact homology for an orbit of multiplicity $m$ will be $$2m \\sum_{j}( \\langle i_{j}^{*}c_{1}^{orb}(T\\mathbb{CP}^n), [S^2_{j}]\\rangle + \\frac{1}{2} dim S_{T_{j}})+ d+n-3=(2km\\sum_{j}\\frac{1}{|\\Gamma_{j}|})+ \\frac{1}{2}dimS_{T_{j}} + d+n-3,$$ where the class $[S^{2}_{j}]$ is the class of a line in each stratum and $d$ corresponds to the possible degree of a homology class on $\\mathbb{CP}^{n}$, hence is an even number between $0$ and $2n$ and $S \\in \\mathbb{Z}^{+}.$ The dimension of the moduli space for genus $0$ and $1$ positive and $1$ negative puncture is then never $1.$ Notice that $c_{1}(\\xi) = 0$ in this case. So again we see that these contact manifolds are distinguished by the bundle and orbifold data. \n\\begin{remark}\n One should be able to simplify the above formula when working with branch divisors. We choose to stick with our earlier notation, in which any information about such branch divisors is encoded in the calculation.\n\\end{remark}\n\n\n \n\\newcommand{\\etalchar}[1]{$^{#1}$}\n\\def$'${$'$}\n\\providecommand{\\bysame}{\\leavevmode\\hbox to3em{\\hrulefill}\\thinspace}\n\\providecommand{\\MR}{\\relax\\ifhmode\\unskip\\space\\fi MR }\n\\providecommand{\\MRhref}[2]{%\n \\href{http:\/\/www.ams.org\/mathscinet-getitem?mr=#1}{#2}\n}\n\\providecommand{\\href}[2]{#2}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}