diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzahhd" "b/data_all_eng_slimpj/shuffled/split2/finalzzahhd" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzahhd" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\\label{intro}\nInterest in non-Abelian Einstein-Yang-Mills theories was sparked by \nthe discovery of particle-like \\cite{bartnik} and \nnon-Reissner-Nordstr\\\"om (``hairy'') black hole \\cite{bizon}\nsolutions when the gauge group is ${\\mathfrak {su}}(2)$.\nSince then a plethora of hairy black holes, possessing non-trivial\ngeometry and field structure outside the event horizon, have been\nfound (see, for example, \\cite{hairy}), including \ncoloured black holes in ${\\mathfrak {su}}(N)$ \nEinstein-Yang-Mills theories \\cite{klei1,klei2}.\nMany of these objects have as a fundamental requirement for the\nexistence of hair, a non-Abelian gauge field, often coupled to \nother fields, such as a Higgs field \\cite{EM}.\nAs in \\cite{EM}, the existence of gauge hair is not surprising in\nitself, since the gauge field force is long-range.\nHowever, the non-Abelian nature of the field is important \nfor evading the no-hair theorem, for example, for scalar fields\ncoupled to the gauge field.\nThe vast majority of these solutions have been found only numerically,\nwith analytic work in this area at present being limited to \nan extensive study of the ${\\mathfrak {su}}(2)$ case\n\\cite{bfm,smoller,sw}, and some analysis of the field equations\nfor general $N$ \\cite{kunzle,kunz1}.\n\n\\bigskip\nIn this paper we continue this analysis of the coupled \nEinstein-Yang-Mills equations for an ${\\mathfrak {su}}(N)$\ngauge field, and prove analytically the existence of \n``genuine'' hairy black hole solutions for every $N$.\nBy a ``genuine'' ${\\mathfrak {su}}(N)$ black hole, we mean a \nsolution which is not simply the result of embedding a \nsmaller gauge group in ${\\mathfrak {su}}(N)$.\nAs in the ${\\mathfrak {su}}(2)$ case, the solutions are labeled\nby the number of nodes of each of the $N-1$ non-zero functions\nrequired to describe the gauge field.\nWe shall prove that for each integer $n_{1}$, there are an infinite\nnumber of sequences of integers \n$n_{N-1}\\ge n_{N-2}\\ge \\ldots \\ge n_{1}$\ncorresponding to black hole solutions.\nIt is to be expected that in fact {\\em {every}} such sequence\ncorresponds to a black hole solution, but unfortunately we are unable\nto prove this analytically, although we shall present a\nnumerically-based argument for ${\\mathfrak {su}}(3)$ and\nnumerical investigations (such as that done for ${\\mathfrak {su}}(5)$\nin \\cite{klei1}) for higher dimension groups \nwhich indicate that this\nis in fact the case.\nThe method used to prove the main theorem of this paper is \nremarkably simple, drawing only on elementary topological ideas.\n\n\\bigskip\nThe structure of the paper is as follows. \nIn section 2 we review briefly the ${\\mathfrak {su}}(N)$\nEinstein-Yang-Mills field equations and the ansatz and\nnotations we shall employ in the rest of the paper.\nNext we state some elementary properties of these equations,\nincluding results from \\cite{kunz1}.\nThe remainder of the paper has a similar progression of ideas\nas \\cite{bfm}, and we continue by first analyzing the\nbehaviour of solutions to the field equations in the two asymptotic\nregimes, at infinity and close to the event horizon.\nAlthough these forms were discussed in \\cite{kunz1}, we present\nhere a shorter proof.\nSection 5 is devoted to a discussion of the flat space solutions,\nwhich will be important for later propositions. \nThe results here are somewhat weaker than in \\cite{bfm} due\nto the fact that for ${\\mathfrak {su}}(N)$, we have \n$N-1$ variables and therefore have a $2(N-1)$-dimensional phase\nspace rather than a phase plane as in the ${\\mathfrak {su}}(2)$\ncase, so the powerful Poincar\\'e-Bendixson theory no longer applies.\nWe now consider integrating the field equations outward from \nthe event horizon and consider the various possible behaviours\nof the resulting solutions.\nAs in \\cite{bfm}, there are three types of solution: the regular\nblack holes we are seeking, singular solutions (in which the\nlapse function vanishes outside the event horizon), and \noscillating solutions in which the geometry is not asymptotically \nflat.\nThe main results of this paper are in sections 7 and 8.\nAn inductive argument is used to prove the existence of solutions\nfor ${\\mathfrak {su}}(N)$ assuming existence for \n${\\mathfrak {su}}(N-1)$ (since we have rigorous theorems for\n${\\mathfrak {su}}(2)$ \\cite{bfm,smoller}).\nThe argument is presented in detail for ${\\mathfrak {su}}(3)$\nin section 7 and a brief outline of the extension to general\n$N$ in section 8.\nFinally, a summary and our conclusions are presented in section 9.\n\n\n\n\n\\section{Ansatz and field equations}\n\\label{ansatz}\nIn this section we first describe the ansatz we are using and outline\nthe field equations.\n\n\\bigskip\nThe field equations for an ${\\mathfrak {su}}(N)$ \nYang-Mills gauge field coupled to\ngravity have been derived in \\cite{kunzle} for a spherically symmetric\ngeometry.\nWe take the line element, in the usual Schwarzschild co-ordinates,\nto be\n\\begin{equation}\nds^{2}=- S^{2}\\mu \\, dt^{2} + \\mu ^{-1}\\, dr^{2}+\nr^{2} \\, d\\theta ^{2} + r^{2} \\sin ^{2} \\theta \\, d\\phi ^{2},\n\\end{equation}\nwhere the metric functions $\\mu $ and $S$ are functions of $r$ alone\nand \n\\begin{equation}\n\\mu (r) = 1-\\frac {2m(r)}{r}.\n\\end{equation}\nA spherically symmetric ${\\mathfrak {su}}(N)$ \ngauge potential may be written in the\nform \\cite{kunzle}\n\\begin{equation}\n{\\cal {A}}=A\\,dt+B\\, dr + \n\\frac {1}{2} \\left( C-C^{H} \\right) \\, d\\theta \n-\\frac {i}{2} \\left[ \\left( C+C^{H} \\right) \\sin \\theta\n+D\\cos \\theta \\right] \\, d\\phi\n\\end{equation}\nwhere \n$D={\\mbox {Diag}} \\left\\{ k_{1}, k_{2}, \\ldots , k_{N} \\right\\} $\nwith $k_{1}\\ge k_{2} \\ge \\ldots \\ge k_{N}$ integers whose sum is zero.\nIn addition, $C$ is a strictly upper triangular complex matrix \nsuch that $C_{ij}\\neq 0$ only if $k_{i}=k_{j}+2$ and $C^{H}$ its\nHermitian conjugate, and $A$, $B$ are anti-Hermitian matrices that\ncommute with $D$.\nAn irreducible representation of ${\\mathfrak {su}}(N)$ \ncan be constructed by taking\n\\cite{kunzle}\n\\begin{equation}\nD={\\mbox {Diag}}\\left\\{ N-1, N-3, \\ldots , -N+3, -N+1 \\right\\} .\n\\end{equation}\nIn this case $A$, $B$ have trace zero and can be written as\n\\begin{equation}\nA_{jj}=i\\left\\{\n-\\frac {1}{N} \\sum _{k=1}^{j-1} k{\\cal {A}}_{k}\n+\\sum _{k=j}^{N-1} \\left( 1-\\frac {k}{N} \\right) {\\cal {A}}_{k}\n\\right\\}\n\\end{equation}\nfor real functions ${\\cal {A}}_{k}$, and similarly for $B$.\nThe only non-vanishing entries of $C$ are\n\\begin{equation}\nC_{j,j+1}= \\omega _{j} e^{i\\gamma _{j}}\n\\end{equation}\nwhere $\\omega _{j}$ and $\\gamma _{j}$ are real functions.\nAll the functions in this ansatz depend only on $r$.\nNow we make the following simplifying assumption \\cite{kunzle}:\n\\begin{equation}\n{\\cal {A}}_{j}=0, \\qquad\n{\\cal {B}}_{j}+\\gamma _{j}'=0,\n\\qquad\n\\forall j.\n\\end{equation}\nThe remaining gauge freedom can then be used to set $B=0$\n\\cite{brod}, which implies that the $\\gamma _{j}$ are constants, which\nwe choose to be zero for simplicity.\n\n\\bigskip\nThe gauge field equations for the $\\omega _{j}$ then take the form\n\\cite{kunzle,kunz1}(where we have set $\\kappa =2$ in \\cite{kunzle})\n\\begin{equation}\nr^{2} \\mu \\omega _{j}'' + \\left( 2m-2r^{3} p_{\\theta } \\right)\n\\omega _{j}' + \n\\left[ 1-\\omega _{j}^{2} + \\frac {1}{2} \\left(\n\\omega _{j-1}^{2} + \\omega _{j+1}^{2} \\right) \\right] \\omega _{j}\n=0,\n\\label{firsteqn}\n\\end{equation}\nwhere\n\\begin{equation}\np_{\\theta }=\\frac {1}{4r^{4}} \\sum _{j=1}^{N}\n\\left[ \\omega _{j}^{2} -\\omega _{j-1}^{2} -N -1 +2j \n\\right] ^{2}\n\\end{equation}\nand the Einstein equations can be simplified to\n\\begin{equation}\nm'= \\mu G + r^{2} p_{\\theta } ,\n\\qquad\n\\frac {S'}{S} =\\frac {2G}{r}\n\\label{Seqn}\n\\end{equation}\nwhere\n\\begin{equation}\nG= \\sum _{j=1}^{N-1} \\omega _{j}^{'2}.\n\\label{lasteqn}\n\\end{equation}\nNote that $\\omega _{0}\\equiv 0 \\equiv \\omega _{N}$, so that these\nare the usual equations \\cite{bizon} in the ${\\mathfrak {su}}(2)$ \ncase, and have a \nvery similar, but slightly coupled, structure for general $N$.\nThe equations (\\ref{firsteqn}--\\ref{lasteqn}) possess two symmetries\n\\cite{kunz1}.\nFirstly, the substitution $\\omega _{j}\\rightarrow -\\omega _{j}$\nfor any fixed $j$ leaves the equations invariant, exactly as in the\n${\\mathfrak {su}}(2)$ case. \nSecondly, there is an additional symmetry under the transformation\n$j\\rightarrow N-j$ for all $j$.\nWe will not assume that this symmetry is respected by the \nsolutions of the field equations, so that the ${\\mathfrak {su}}(3)$\ncase is not necessarily trivial.\n\n\\bigskip\nWe are concerned in this paper with black holes, which will have a\nregular event horizon at $r=r_{h}$, where $\\mu =0$.\nThe equations in their present form are singular at $r_{h}$, so in\norder to produce a set of equations which are regular as \n$\\mu \\rightarrow 0$, we define a new independent variable $\\tau $ \nby \\cite{bfm}\n\\begin{equation}\n\\frac {dr}{d\\tau }= r {\\sqrt {\\mu }},\n\\end{equation}\ndenote $d\/d\\tau $ by ${\\dot {}}$, and define new dependent variables\n$\\kappa $, $U_{j}$ and $\\Psi $ as follows \\cite{bfm}:\n\\begin{equation}\n\\Psi = {\\sqrt {\\mu }}, \\qquad\nU_{j}=\\Psi \\omega _{j}', \\qquad\n\\kappa = \\frac {1}{2\\Psi } \\left(\n1+\\Psi ^{2} + 2\\mu G - 2r^{2} p_{\\theta } \\right) .\n\\end{equation}\nThen the field equations take the form \\cite{bfm}:\n\\begin{eqnarray}\n{\\dot {r}} & = & r\\Psi \n\\label{fregeqn} \\\\\n{\\dot {\\omega }}_{j} & = & rU_{j} \\\\\n{\\dot {\\Psi }} & = & \n\t\\left( \\kappa - \\Psi \\right) \\Psi - 2\\mu G \\\\\n\\left( S\\Psi \\right) {\\dot {}} & = & \n\tS\\left( \\kappa - \\Psi \\right) \\Psi \\\\\n{\\dot {U}}_{j} & = & \n\t-\\left( \\kappa - \\Psi \\right) U_{j} \n\t-\\frac {1}{r} \\left[ \n\t1-\\omega _{j}^{2} +\\frac {1}{2} \\left(\n\t\\omega _{j+1}^{2} + \\omega _{j-1}^{2} \\right) \\right] \n\t\\omega _{j} \\\\\n{\\dot {\\kappa }} & = & \n\t-\\kappa ^{2} +1 +2\\mu G.\n\\label{lregeqn}\n\\end{eqnarray}\n\n\\bigskip\nThe main thrust of this article is to prove the existence of regular,\nblack hole, solutions of the field equations for general $N$ with \n$N-1$ degrees of freedom. \nIn other words, we begin with a regular event horizon at $r=r_{h}$,\nwhere $\\mu =0$, and integrate outwards with increasing $r$.\nLater we shall classify the possible behaviour of the solutions \nas $r$ increases.\nWe are interested primarily in those solutions which possess the \nphysical properties of a black hole geometry, namely for which\n$\\mu >0$ for all $r>r_{h}$, the $\\omega _{j}$ and their derivatives\nare finite for all $r>r_{h}$ and the spacetime becomes flat in the\nlimit $r\\rightarrow \\infty $.\nWe shall refer to such solutions as\n{\\em {regular black hole solutions}}.\n\n\\section{Elementary results}\nIn this this section we state a few elementary results, which\nwill prove useful in later analysis.\nFirstly, two lemmas which are proved in the\n${\\mathfrak {su}}(N)$ case exactly as in the ${\\mathfrak {su}}(2)$\ncase, see \\cite{bfm}.\n\n\\begin{lemma}\nIf $\\mu (r_{0})<1$ for some $r_{0}$ then $\\mu (r)<1$ for all \n$r\\ge r_{0}$.\n\\end{lemma}\n\n\\begin{lemma}\n\\label{easylem}\nAs long as $0<\\mu <1$ all field variables are regular functions\nof $r$.\n\\end{lemma}\n\nThese two lemmas show that for a regular event horizon at \n$r=r_{h}$, where $\\mu =0$, then $\\mu <1$ for all $r>r_{h}$\nand the field variables are regular functions as long as \n$\\mu >0$.\nThe black hole solutions we seek approach the Schwarzschild geometry\nas $r\\rightarrow \\infty $, so we may\nassume that all $\\omega _{j}$ are bounded for all $r$ (from lemma\n\\ref{easylem} each $\\omega _{j}$ is bounded on every closed interval\n$[r_{h},r_{1}]$, so we are simply assuming that $\\omega _{j}$\nremains bounded as $r\\rightarrow \\infty $).\nIt will be proved in section \\ref{global} that this \nassumption is in fact valid for solutions having $\\mu >0$\nfor all $r$.\nDefine a quantity $M_{j}$ for each $j$ to be the lowest upper bound\non $\\omega _{j}^{2}$, i.e.\n\\begin{equation}\n\\omega _{j}^{2} \\le M_{j}, \\qquad \\forall j=1,2,\\ldots, N-1.\n\\end{equation}\nIt has been shown in \\cite{smoller} that the following result\nis true for $N=2$, and proved in general with the assumption\nthat all $\\omega _{j}$ are bounded via an elegant method in \\cite{kunz1}.\n\n\\begin{theorem}\n\\label{mtheorem}\n$M_{j}\\le j(N-j)$.\n\\end{theorem}\n\nPart of the proof of this theorem in \\cite{kunz1} involves a \nknowledge of where $\\omega _{j}$ may have maxima or minima.\nThe result following is less powerful than in the ${\\mathfrak {su}}(2)$\ncase due to the coupling in the gauge field equation (\\ref{firsteqn}).\n\n\\begin{prop}\n\\label{maxminprop}\nAs long as $\\mu (r)>0$, the function $\\omega _{j}(r)$ cannot have\nmaxima in the regions\n\\begin{equation}\n\\omega _{j}>{\\sqrt {1+\\frac {1}{2}\\left( \n\t\\omega _{j+1}^{2} +\\omega _{j-1}^{2} \\right) }}\n\\qquad {\\mbox {and}}\n\\qquad\n0>\\omega _{j}> -{\\sqrt {1+\\frac {1}{2}\\left( \n\t\\omega _{j+1}^{2} +\\omega _{j-1}^{2} \\right) }}\n\\end{equation}\nor minima in the regions\n\\begin{equation}\n\\omega _{j}<-{\\sqrt {1+\\frac {1}{2}\\left( \n\t\\omega _{j+1}^{2} +\\omega _{j-1}^{2} \\right) }}\n\\qquad {\\mbox {and}}\n\\qquad\n0<\\omega _{j}<{\\sqrt {1+\\frac {1}{2}\\left( \n\t\\omega _{j+1}^{2} +\\omega _{j-1}^{2} \\right) }}.\n\\end{equation}\n\\end{prop}\n\n\\noindent {\\bf {Proof}} \\smallskip \\newline \nWhen $\\omega _{j}'=0$, from (\\ref{firsteqn}),\n\\begin{equation}\n\\mu \\omega _{j}''=-\\frac {1}{r^{2}} \\left[\n1-\\omega _{j}^{2} + \\frac {1}{2} \\left(\n\\omega _{j+1}^{2} +\\omega _{j-1}^{2} \\right) \\right]\n\\omega _{j},\n\\end{equation}\nso that $\\omega _{j}$ will have a maximum if\n\\begin{equation}\n\\omega _{j}>0 \\qquad {\\mbox {and}} \\qquad\n\\omega _{j}^{2}<1+\\frac {1}{2} \\left(\n\\omega _{j+1}^{2} +\\omega _{j-1}^{2} \\right)\n\\end{equation}\nor\n\\begin{equation}\n\\omega _{j}<0 \\qquad {\\mbox {and}} \\qquad\n\\omega _{j}^{2}>1+\\frac {1}{2} \\left(\n\\omega _{j+1}^{2} +\\omega _{j-1}^{2} \\right) .\n\\end{equation}\nSimilarly, $\\omega _{j}$ will have a minimum if\n\\begin{equation}\n\\omega _{j}>0 \\qquad {\\mbox {and}} \\qquad\n\\omega _{j}^{2}>1+\\frac {1}{2} \\left(\n\\omega _{j+1}^{2} +\\omega _{j-1}^{2} \\right) \n\\end{equation}\nor\n\\begin{equation}\n\\omega _{j}<0 \\qquad {\\mbox {and}} \\qquad\n\\omega _{j}^{2}<1+\\frac {1}{2} \\left(\n\\omega _{j+1}^{2} +\\omega _{j-1}^{2} \\right) .\n\\end{equation}\n\\hfill\n$\\square $\n\n\\section{Asymptotic behaviour of solutions}\n\\label{local}\nWe seek solutions to the above field equations representing black\nholes with a regular event horizon at $r=r_{h}$ and finite total\nenergy density. \nIn order to prove the local existence of solutions of the field\nequations with the desired asymptotic behaviour, \nwe shall apply the following\ntheorem \\cite{bfm}.\n\n\\begin{theorem}\n\\label{bfmth}\nConsider a system of differential equations for $n+m$ functions\n${\\mbox {\\boldmath {$a$}}}=(a_{1}, a_{2}, \\ldots , a_{n})$\nand\n${\\mbox {\\boldmath {$b$}}}=(b_{1}, b_{2}, \\ldots , b_{m})$\nof the form\n\\begin{eqnarray}\nx \\frac {da_{i}}{dx} & = & \nx^{p_{i}} f_{i}(x,{\\mbox {\\boldmath {$a$}}},{\\mbox {\\boldmath {$b$}}}),\n\\nonumber \\\\\nx\\frac {db_{i}}{dx} & = &\n-\\lambda _{i} b_{i} + \nx^{q_{i}} g_{i}(x,{\\mbox {\\boldmath {$a$}}},{\\mbox {\\boldmath {$b$}}}),\n\\label{sysde}\n\\end{eqnarray}\nwith constants $\\lambda _{i}>0$ and integers $p_{i},q_{i}\\ge 1$\nand let ${\\cal {C}}$ be an open subset of ${\\mathbb {R}}^{n}$\nsuch that the functions $f_{i}$ and $g_{i}$ are analytic in a\nneighbourhood of $x=0$, \n${\\mbox {\\boldmath {$a$}}}={\\mbox {\\boldmath {$c$}}}$,\n${\\mbox {\\boldmath {$b$}}}={\\mbox {\\boldmath {$0$}}}$, for all\n${\\mbox {\\boldmath {$c$}}}\\in {\\cal {C}}$.\nThen there exists an $n$-parameter family of solutions of the\nsystem (\\ref{sysde}) such that\n\\begin{equation}\na_{i}(x) = c_{i} + O(x^{p_{i}}), \\qquad\nb_{i}(x) = O(x^{q_{i}}),\n\\end{equation}\nwhere $a_{i}(x)$ and $b_{i}(x)$ are defined for \n${\\mbox {\\boldmath {$c$}}}\\in {\\cal {C}}$, and \n$|x|0$ is finite.\n\n\\begin{prop}\n\\label{horprop}\nThere exists an $N$-parameter family of local solutions of\n(\\ref{firsteqn}--\\ref{lasteqn}) near $r=r_{h}$, analytic in $r_{h}$,\n$\\omega _{j,h}$ and $r$ such that\n\\begin{eqnarray}\n\\mu (r_{h}+\\rho ) & = & \\mu '(r_{h})+O(\\rho ), \n\\nonumber \\\\\n\\omega _{j} (r_{h}+\\rho ) & = & \n\t\\omega _{j,h} +\\omega _{j}'(r_{h}) +O(\\rho ^{2})\n\\label{horrels}\n\\end{eqnarray}\nwhere\n$\\mu '(r_{h})$ and $\\omega _{j}'(r_{h})$ are functions of the\n$\\omega _{j,h}$.\n\\end{prop}\n\n\\noindent {\\bf {Proof}} \\smallskip \\newline \nLet $\\rho =r-r_{h}$ be the new independent variable, and define\n\\begin{equation}\nx=r, \\qquad\n\\lambda = \\frac {\\mu }{\\rho }, \\qquad\n\\psi _{j}=\\omega _{j}, \\qquad\n\\xi _{j}=\\frac {\\mu \\omega _{j}'}{\\rho }.\n\\end{equation}\nThen the field equations take the form\n\\begin{eqnarray}\n\\rho \\frac {dx}{d\\rho } & = & \\rho, \\\\\n\\rho \\frac {d\\lambda }{d\\rho } & = & \n\t-\\lambda + \\left[ \\frac {1}{x} - 2xp_{\\theta }\\right]\n\t+\\rho \\frac {\\lambda }{x}\\left[ 1-2G \\right] \n\\nonumber \\\\\n & = & \n\t-\\lambda +\\rho H_{\\lambda }+F_{\\lambda } \\\\\n\\rho \\frac {d\\psi _{j}}{d\\rho } & = & \n\t\\frac {\\rho \\xi _{j}}{\\lambda } \\\\\n\\rho \\frac {d\\xi _{j}}{d\\rho } & = & \n\t-\\xi _{j} +\\rho H_{j} + F_{j} ,\n\\end{eqnarray}\nwhere the $F$'s and $H$'s are polynomials in $x^{-1}$, \n$\\lambda ^{-1}$, and the other variables, and\nthe $F$'s depend only on $x$ and $\\psi _{j}$'s.\nNext define\n\\begin{equation}\n{\\tilde {\\xi }}_{j} = \\xi _{j}-F_{j}, \\qquad\n{\\tilde {\\lambda }} = \\lambda -F_{\\lambda },\n\\end{equation}\nwhose derivatives are given by\n\\begin{eqnarray}\n\\rho \\frac {d{\\tilde {\\xi }}_{j}}{d\\rho } & = & \n\t-{\\tilde {\\xi }}_{j}+\\rho G_{j} \\\\\n\\rho \\frac {d{\\tilde {\\lambda }}}{d\\rho } & = & \n\t-{\\tilde {\\lambda }} + \\rho G_{\\lambda }.\n\\end{eqnarray}\nHere the $G$'s are analytic in $x^{-1}$, $\\lambda ^{-1}$, $\\lambda $,\n$x$, ${\\tilde {\\lambda }}$, $\\psi _{j}$, ${\\tilde {\\xi }}_{j}$.\nApplying theorem \\ref{bfmth}, there exist solutions of the form\n\\begin{equation}\nx=r_{h}+\\rho , \\qquad\n\\psi _{j}=\\omega _{j,h}+O(\\rho ), \\qquad\n{\\tilde {\\lambda }},{\\tilde {\\xi }}_{j}=O(\\rho ),\n\\end{equation}\nwhich gives the behaviour (\\ref{horrels}), together with the\nrequired analyticity.\nFrom the field equations (\\ref{firsteqn}--\\ref{lasteqn}), setting\n$\\mu (r_{h})=0$ gives the following relations:\n\\begin{eqnarray}\n\\mu '(r_{h}) & = & \\frac {1}{2} r_{h}^{2} p_{\\theta }(r_{h}) \\\\\n\\omega _{j}'(r_{h}) & = & -\n\\frac {\\left[ 1-\\omega _{j,h}^{2}+\\frac {1}{2} \\left(\n\\omega _{j-1,h}^{2}+\\omega _{j+1,h}^{2} \\right) \\omega _{j,h}\n\\right] }{r_{h} -r_{h}^{3}p_{\\theta }(r_{h})} \n\\end{eqnarray}\nwhere\n\\begin{equation}\np_{\\theta }(r_{h})=\n\\frac {1}{4r_{h}^{4}} \\sum _{j=1}^{N} \\left[\n\\omega _{j,h}^{2} -\\omega _{j-1,h}^{2} -N-1+2j \\right] ^{2}.\n\\end{equation}\n\\hfill\n$\\square $\n\n\\section{Flat space solutions}\n\\label{flat}\nThe behaviour of the gauge field equations in the flat space limit\nwill be useful later in section \\ref{global} when we come to examine the \nproperties of regular solutions.\nWe are not concerned here with the global existence of flat space \nsolutions but only the local properties pertinent to the curved space\nproblem.\n\n\\bigskip\nThe analysis here is similar to that in \\cite{bfm}, but note that\nfor general $N$ there will be $N-1$ coupled gauge field equations.\nThe powerful Poincar\\'e-Bendixson theory of autonomous systems \nis not applicable when $N>2$, and so we will be able to derive \nonly correspondingly weaker results, and cannot draw a phase portrait.\nFortunately the theorems later require only a knowledge of the \nnature of the critical points and the local behaviour close to these\npoints, which is the subject of this section.\n\n\\bigskip\nIn flat space the gauge field equations reduce to:\n\\begin{equation}\nr^{2} \\omega _{j}'' +\\left[ 1-\\omega _{j}^{2} +\\frac {1}{2}\n\\left( \\omega _{j+1}^{2} +\\omega _{j-1}^{2} \\right) \\right] \n\\omega _{j} =0.\n\\end{equation}\nThese equations can be made autonomous by changing variables to\n$\\tau =\\log r$, and denoting $d\/d\\tau $ by ${\\dot {}}$:\n\\begin{equation}\n{\\ddot {\\omega }}_{j}-{\\dot {\\omega }}_{j}+\n\\left[ 1-\\omega _{j}^{2} +\\frac {1}{2}\n\\left( \\omega _{j+1}^{2} +\\omega _{j-1}^{2} \\right) \\right] \n\\omega _{j} =0.\n\\end{equation}\nThis system of $N-1$ coupled equations has critical points when\n\\begin{equation}\n\\left[ 1-\\omega _{j}^{2} +\\frac {1}{2}\n\\left( \\omega _{j+1}^{2} +\\omega _{j-1}^{2} \\right) \\right] \n\\omega _{j} =0.\n\\qquad\n\\forall j=1,2,\\ldots ,N-1.\n\\end{equation}\nIf $\\omega _{j}\\neq 0$ for all $j$ then \n\\begin{equation}\n\\omega _{j}={\\sqrt {j(N-j)}}.\n\\end{equation}\nThere is also a critical point when $\\omega _{j}=0$ for all $j$.\n\n\\bigskip\nThe other critical points can be described as follows.\nSuppose that $\\omega _{i}=0$ and $\\omega _{i+k}=0$ but that\n$\\omega _{i+m}\\neq 0$ for $m=1,\\ldots ,k-1$ (where $k\\ge 2$).\nThe critical point is then described by the equations\n\\begin{eqnarray}\n0 & = & \\omega _{i} \\nonumber \\\\\n0 & = & 1-\\omega _{i+1}^{2}+\\frac {1}{2} \\left( \\omega _{i}^{2}\n+\\omega _{i+2}^{2} \\right) \n\\nonumber \n\\\\\n & \\vdots & \n\\nonumber\n\\\\\n0 & = & 1-\\omega _{i+k-1}^{2} +\\frac {1}{2} \\left( \\omega _{i+k-2}^{2}\n+\\omega _{i+k}^{2} \\right)\n\\nonumber\n\\\\\n0 & = & \\omega _{i+k}\n\\end{eqnarray}\nwhich have the solution\n\\begin{equation}\n\\omega _{i+m}={\\sqrt {m(k-m)}} \\qquad \n{\\mbox {for $m=1,\\ldots , k-1$}}.\n\\end{equation}\nIn other words, if we have a run of $k$ non-zero $\\omega $'s with\nzero $\\omega _{i}$ at each end, then the solution for the non-zero\n$\\omega $'s is the same as the $N=k$ case with no zero $\\omega $'s.\nWe can also put together a run of as many zero $\\omega $'s as we like\nin the critical point.\n\n\\bigskip\nIn order to classify the critical points, we linearize the field\nequations.\nLet\n\\begin{equation}\n\\omega _{j}(\\tau )=\\omega _{j}^{(0)}+\\epsilon _{j}(\\tau )\n\\end{equation}\nwhere $\\omega _{j}^{(0)}$ is the value of $\\omega _{j}$ at\nthe critical point and $\\epsilon _{j}$ is a small perturbation.\nThe equation for $\\epsilon _{j}$ is, to first order,\n\\begin{eqnarray}\n0 & = & \n{\\ddot {\\epsilon }}_{j} -{\\dot {\\epsilon }}_{j} \n+\\epsilon _{j}\\left[ 1-\\omega _{j}^{(0)2} +\\frac {1}{2}\n\\left( \\omega _{j+1}^{(0)2} +\\omega _{j-1}^{(0)2} \\right) \\right] \n\\nonumber\n\\\\ \n & & \n+\\omega _{j}^{(0)} \\left[ -2\\epsilon _{j}\\omega _{j}^{(0)}\n+\\epsilon _{j+1} \\omega _{j+1}^{(0)} +\\epsilon _{j-1}\n\\omega _{j-1}^{(0)} \\right] .\n\\end{eqnarray}\nWe shall take the two cases we need to consider in turn.\n\n\\bigskip\nFirstly, the case where $\\omega _{j}^{(0)}=0$, when the equation\nfor $\\epsilon _{j}$ reduces to\n\\begin{equation}\n{\\ddot {\\epsilon }}_{j}-{\\dot {\\epsilon }}_{j}+\n\\epsilon _{j} \\left[ 1+\\frac {1}{2} \\left(\n\\omega _{j+1}^{(0)2} +\\omega _{j-1}^{(0)2} \\right) \\right] =0.\n\\end{equation}\nIn order to find the nature of the critical point, let\n\\begin{equation}\n\\epsilon _{j}=e^{\\lambda \\tau },\n\\end{equation}\nthen $\\lambda $ satisfies the equation\n\\begin{equation}\n\\lambda ^{2}-\\lambda +\\left[ \n1+\\frac {1}{2} \\left(\n\\omega _{j+1}^{(0)2} +\\omega _{j-1}^{(0)2} \\right) \\right] =0.\n\\end{equation}\nThis implies that\n\\begin{equation}\n\\lambda = \\frac {1}{2} \\pm \\frac {1}{2} i\\alpha,\n\\end{equation}\nwhere $\\alpha $ is the positive real number given by\n\\begin{equation}\n\\alpha ^{2} = 3+2\\omega _{j+1}^{(0)2} +2\\omega _{j-1}^{(0)2}.\n\\end{equation}\nWe conclude that, in the $(\\epsilon _{j}, {\\dot {\\epsilon }}_{j})$\nplane, we have an unstable focal point, as found in the ${\\mathfrak {su}}(2)$ \ncase in \\cite{bfm}.\n\n\\bigskip\nSecondly, we consider the situation in which there is at least\none $\\omega _{j}^{(0)}$ which is non-zero.\nSuppose that $\\omega _{i}^{(0)}=0$, and $\\omega _{i+k}^{(0)}=0$,\nbut $\\omega _{i+m}^{(0)}\\neq 0$ for $m=1,\\ldots ,k-1$, where we \ninclude the case that $i=0$ and $k=N$.\nThen we have a series of coupled perturbation equations:\n\\begin{eqnarray}\n0 & = & \n{\\ddot {\\epsilon }}_{i+1} - {\\dot {\\epsilon }}_{i+1}\n-2\\epsilon _{i+1} \\omega _{i+1}^{(0)2} \n+\\epsilon _{i+2} \\omega _{i+2}^{(0)} \\omega _{i+1}^{(0)} \n\\nonumber \\\\\n0 & = & \n{\\ddot {\\epsilon }}_{i+2} - {\\dot {\\epsilon }}_{i+2}\n-2\\epsilon _{i+2} \\omega _{i+2}^{(0)2} \n+\\epsilon _{i+3} \\omega _{i+3}^{(0)} \\omega _{i+2}^{(0)}\n+\\epsilon _{i+1} \\omega _{i+1}^{(0)} \\omega _{i+2}^{(0)}\n\\nonumber \\\\\n & \\vdots & \n\\nonumber \\\\\n0 & = & \n{\\ddot {\\epsilon }}_{i+k-1} - {\\dot {\\epsilon }}_{i+k-1}\n-2\\epsilon _{i+k-1} \\omega _{i+k-1}^{(0)2} \n+\\epsilon _{i+k-2} \\omega _{i+k-2}^{(0)} \\omega _{i+k-1}^{(0)} .\n\\end{eqnarray}\nDefine a vector ${\\mbox {\\boldmath {$\\epsilon $}}}$ by\n\\begin{equation}\n{\\mbox {\\boldmath {$\\epsilon $}}} = \\left( \n\\epsilon _{i+1}, \\ldots , \\epsilon _{i+k-1} \\right) ^{T}\n\\end{equation}\nand consider solutions of the form \n\\begin{equation}\n{\\mbox {\\boldmath {$\\epsilon $}}} = e^{\\lambda \\tau }\n{\\mbox {\\boldmath {$q$}}}\n\\end{equation}\nwhere ${\\mbox {\\boldmath {$q$}}}$ is a constant vector.\nThen $\\lambda ^{2}-\\lambda $ are eigenvalues of the matrix\n${\\cal {M}}_{k-1}$ given by (\\ref{matrix}).\nAs discussed in section \\ref{local}, the matrix ${\\cal {M}}_{k-1}$ \nhas positive real eigenvalues ${\\cal {E}}_{j}$.\nThen\n\\begin{equation}\n\\lambda = \\frac {1\\pm {\\sqrt {1+4{\\cal {E}}_{j}}}}{2}\n=j+1, -j\n\\end{equation}\nwill have positive and negative real values and there is a \nsaddle point.\nAgain this is in direct analogy with the \n${\\mathfrak {su}}(2)$ case \\cite{bfm}.\n\n\n\\section{Global behaviour of the solutions}\n\\label{global}\nIn this section we investigate the behaviour of solutions of\nthe regular field equations (\\ref{fregeqn}--\\ref{lregeqn}) as \nfunctions of $\\tau $.\nFrom section \\ref{local}, we know that, given any starting values\n$\\omega _{j,h}$ for the gauge field functions, then there is \na local solution of the field equations in a neighbourhood of \nthe regular event horizon at $r=r_{h}$, $\\tau =0$, which is analytic\nin $\\tau $ and the initial parameters.\nFurthermore, from lemma \\ref{easylem} as long as $0<\\Psi <1$ the\nsolutions remain regular.\nTherefore, as we integrate out in $\\tau $ from the event horizon there\nare only three possibilities:\n\\begin{enumerate}\n\\item\nThere is a $\\tau _{0}>0$ such that $\\Psi (\\tau _{0})=0$.\n\\item\nFor all $\\tau >0$, we have $\\Psi (\\tau )>0$ and $r(\\tau )$ remains\nbounded as $\\tau \\rightarrow \\infty $.\n\\item\nFor all $\\tau >0$, we have $\\Psi (\\tau )>0$ and \n$r(\\tau )\\rightarrow \\infty $ as $\\tau \\rightarrow \\infty $.\n\\end{enumerate}\nWe refer to solutions of the first type as {\\em {singular}} solutions.\nThose of type 2 are known as {\\em {oscillating}} solutions in\n\\cite{bfm} and we retain their terminology.\nIt will be shown below (proposition \\ref{rmaxprop}), that\nthere is a maximum value $r_{max}$ of $r$ for\noscillating solutions.\nFinally, we denote by $S_{\\infty }$ solutions of the\nfirst type for which $r(\\tau _{0})0$ for all $\\tau $ and \n$\\lim _{\\tau \\rightarrow \\infty }r(\\tau )=\\infty $ then\nall the $\\omega _{j}$ remain bounded as $r\\rightarrow \\infty $.\n\\end{lemma}\n\n\\noindent {\\bf {Proof}} \\smallskip \\newline \nFrom the local existence propositions \\ref{horprop} and \\ref{infprop}\nand lemma \\ref{easylem}, each $\\omega _{j}$ is an analytic\nfunction of $r$ as long as $\\mu >0$.\nIntroduce a new variable $x=r^{-1}$, then each $\\omega _{j}$\nis an analytic function of $x$ in a neighbourhood of $x=0$,\nexcept possibly at $x=0$, and can be written as a Laurent series:\n\\begin{equation}\n\\omega _{j}(x)=\\sum _{n=-\\infty }^{\\infty } a_{n}^{j}x^{n}.\n\\end{equation}\nIn order for $\\mu >0$ for all $r$, it must be the case that\n$2mr^{-1}=2mx<1$, although $m$ itself need not necessarily \nremain bounded.\nHence $2mx$ is bounded in a neighbourhood of $x=0$, and in \nparticular is analytic at $x=0$.\nTherefore we can write\n\\begin{equation}\n2mx=\\sum _{n=0}^{\\infty }b_{n}x^{n}.\n\\end{equation}\nThus\n\\begin{equation}\nm=\\frac {1}{2} \\sum _{n=0}^{\\infty }b_{n}x^{n-1},\n\\qquad\n\\frac {dm}{dr}=-x^{2}\\frac {dm}{dx}\n=-\\frac {1}{2} \\sum _{n=0}^{\\infty }b_{n}(n-1)x^{n},\n\\end{equation}\nso that $dm\/dr$ is analytic in a neighbourhood of $x=0$.\nFrom the field equations,\n\\begin{equation}\n\\frac {dm}{dr}=\\mu G +r^{2}p_{\\theta }\n\\end{equation}\nwhere\n\\begin{equation}\nG=\\sum _{j=1}^{N-1} \\left( \\frac {d\\omega _{j}}{dr} \\right) ^{2}\n\\qquad\nr^{2}p_{\\theta }=\\frac {x^{2}}{4}\n\\sum _{j=1}^{N} \\left[ \n\\omega _{j}^{2}-\\omega _{j-1}^{2}-N-1+2j \\right] ^{2}.\n\\end{equation}\nSince both $G$ and $r^{2}p_{\\theta }$ are positive, they must\neach be analytic in a neighbourhood of $x=0$.\nConsider $G$ first, using\n\\begin{equation}\n\\frac {d\\omega _{j}}{dr}=-x^{2}\\frac {d\\omega _{j}}{dx}\n=-\\sum _{n=-\\infty }^{\\infty }a_{n}^{j} nx^{n+1}\n\\end{equation}\nand since $\\left( \\frac {d\\omega _{j}}{dr}\\right) ^{2}$\nmust be analytic near $x=0$, it must be the case that\n\\begin{equation}\na_{n}^{j}=0 \\qquad \\forall n<-1.\n\\end{equation}\nThen\n\\begin{equation}\n\\omega _{j}^{2}=\\frac {a_{-1}^{j2}}{x^{2}}+\n\\sum _{n=-1}^{\\infty }c_{n}^{j}x^{n}\n\\end{equation}\nfor constants $c_{n}^{j}$ and hence\n\\begin{equation}\n\\omega _{j}^{2}-\\omega _{j-1}^{2}-N-1+2j\n=\\frac {1}{x^{2}}\\left[ a_{-1}^{j2}-a_{-1}^{(j-1)2} \\right]\n+\\sum _{n=-1}^{\\infty }d_{n}^{j}x^{n}.\n\\end{equation}\nSince $r^{2}p_{\\theta }$ is analytic, \n\\begin{equation}\na_{-1}^{j}=\\pm a_{-1}^{j-1}\n\\qquad\n\\forall j.\n\\end{equation}\nBut $\\omega _{0}\\equiv 0$, so $a_{-1}^{0}=0$ and thus $a_{-1}^{j}=0$\nfor all $j$.\nWe conclude that $\\omega _{j}$ is analytic at $x=0$, and therefore\nfinite as $x=0$.\nTherefore $\\omega _{j}$ is bounded for all $r\\in [r_{h},\\infty )$.\n\\hfill\n$\\square $\n\n\\begin{prop}\n\\label{flatprop}\nIf $\\Psi (\\tau )>0$ for all $\\tau $ and \n$\\lim _{\\tau \\rightarrow \\infty }r(\\tau )=\\infty $ then\nthe solution tends to one of the flat space critical points, with \nthe exception of the origin.\n\\end{prop}\n\nThe proof of this proposition closely follows \nProposition 14 of Ref. 7, \nthe main\nthrust of which is the following lemma.\n\n\\begin{lemma}\n\\label{flatlem}\nIf $\\Psi (\\tau )>0$ for all $\\tau $ and \n$\\lim _{\\tau \\rightarrow \\infty }r(\\tau )=\\infty $ then\n\\begin{equation}\n\\lim _{\\tau \\rightarrow \\infty } \\Psi (\\tau )=1.\n\\end{equation}\n\\end{lemma}\n\n\\noindent {\\bf {Proof}} \\smallskip \\newline \nThere are two cases to consider.\n\\begin{enumerate}\n\\item\nThere is at least one $\\omega _{j}$ which has zeros for arbitrarily\nlarge $r$.\n\\item\nAll $\\omega _{j}$ have only a finite number of zeros.\n\\end{enumerate}\n\n\\bigskip\n\\noindent\n{\\em {Case 1}}\n\n\\smallskip\n\\noindent\nThe proof in this situation follows exactly that of \nProposition 14 of Ref. 7 (equations (74--76)) \nsince all the $\\omega _{j}$'s are \nbounded (lemma \\ref{boundlem}).\n\n\\bigskip\n\\noindent\n{\\em {Case 2}}\n\n\\smallskip\n\\noindent\nIn this case there is a $T_{1}>0$ such that for all $\\tau >T_{1}$\nno $\\omega _{j}$ has a zero.\nHowever, since the equations governing the $\\omega _{j}$ are\ncoupled, the $\\omega _{j}$ do not necessarily have to be\nmonotonic, provided proposition \\ref{maxminprop} is satisfied.\n\n\\bigskip\nSuppose, initially, that there is a $T_{2}>T_{1}$ such that for all\n$\\tau >T_{2}$ every $\\omega _{j}$ is monotonic.\nIn this situation each $\\omega _{j}$ has a limit and so \n$\\lim _{\\tau \\rightarrow \\infty }U_{j}=0$ for all $j$.\nThen the proof of Proposition 14 of Ref. 7 carries over directly\nto show that $m(\\tau )$ is bounded and \n$\\lim _{\\tau \\rightarrow \\infty } \\Psi (\\tau )=1$.\nThe proof proceeds as follows.\nFirstly, in analogy to equation (74) of Ref. 7, integrating the\nfield equations (\\ref{fregeqn}--\\ref{lregeqn}) gives,\nfor each $j$,\n\\begin{equation}\n|\\Psi U_{j}(\\tau _{2})-\\Psi U_{j}(\\tau _{1})|\n\\le \\frac {c_{j}}{r(\\tau _{1})}\n\\end{equation}\nfor all $\\tau _{2}>\\tau _{1}>T_{2}$, and some constant $c_{j}$,\nsince all the $\\omega _{j}$ are bounded.\nFix $\\tau _{1}$ for the moment, then for all $\\tau _{3}>\\tau _{1}$,\n\\begin{equation}\n\\int _{\\tau _{1}}^{\\tau _{3}} r\\Psi U_{j}^{2} \\, d\\tau '\n\\le c_{j}'\\int _{\\tau _{1}}^{\\tau _{3}} {\\dot {\\omega }}_{j} \\, \nd\\tau ' \\le\nc_{j}''\n\\end{equation}\nfor constants $c_{j}'$ and $c_{j}''$, since $\\omega _{j}$ has a limit.\nHence\n\\begin{equation}\nm(\\tau _{3})-m(\\tau _{1})\\le \\frac {1}{2} \\sum _{j=1}^{N-1}\nc_{j}''\n\\end{equation}\nwhich is bounded.\n\n\n\\bigskip\nNext turn to the other extreme situation, where $\\omega _{j}^{2}$\nhas minima for arbitrarily large $r$, for every $j$.\nIf $\\omega _{j}^{2}$ has a minimum at $r=r_{0}$, then by\nproposition \\ref{maxminprop},\n\\begin{equation}\n\\omega _{j}^{2}(r_{0}) \\ge 1+\\frac{1}{2} \\left(\n\\omega _{j+1}^{2} (r_{0}) +\\omega _{j-1}^{2} (r_{0}) \\right).\n\\end{equation}\nDefine $N_{j}$ to be the greatest lower bound of the set of\nminimum values of $\\omega _{j}^{2}$ for $\\tau >T_{1}$,\nwhere $N_{j}$ can be zero (although $\\omega _{j}^{2}$ cannot). \nThen we have\n\\begin{equation}\nN_{j}\\ge 1+\\frac {1}{2}\\left( N_{j+1}+N_{j-1} \\right) .\n\\label{nineq}\n\\end{equation} \nThese inequalities may be solved to give\n\\begin{equation}\nN_{j}\\ge j(N-j) \\qquad \\forall j.\n\\end{equation}\nHowever, since $\\omega _{j}^{2}\\le j(N-j)$ for all $j$ from \ntheorem \\ref{mtheorem}, it follows that \n$\\omega _{j}^{2}(\\tau )=j(N-j)$ for sufficiently large $\\tau $\nand all $j$. \nTherefore $U_{j}(\\tau )$ is zero for sufficiently large $\\tau $,\nhence $m(\\tau )$ is bounded as $\\tau \\rightarrow \\infty $ and\n$\\lim _{\\tau \\rightarrow \\infty } \\Psi (\\tau )=1$.\n\n\\bigskip\nThe remaining intermediate case is where $\\omega _{j}$ and\n$\\omega _{j+l}$, where $l>1$, are monotonic for all \nsufficiently large $\\tau $,\nwhilst $\\omega _{j+i}^{2}$, $i=1,\\ldots ,l-1$ have minima for\narbitrarily large $\\tau $.\nWe include the possibility that $j=0$, to cover the case where\nthere is only one $\\omega _{i}$ which has a limit.\nDefine $L_{1}$ and $L_{2}$ by\n\\begin{equation}\n\\omega _{j}^{2} \\rightarrow L_{1}, \\qquad\n\\omega _{j+l}^{2} \\rightarrow L_{2} \n\\end{equation}\nas $\\tau \\rightarrow \\infty $.\nGiven $\\epsilon >0$, there is a $T_{1}$ such that\n\\begin{equation}\n\\left| \\omega _{j}^{2}-L_{1} \\right| <\\epsilon ,\n\\qquad\n\\left| \\omega _{j+l}^{2}-L_{2} \\right| <\\epsilon\n\\qquad \n\\forall \\tau > T_{1}.\n\\end{equation} \nIf $\\omega _{j+1}^{2}$ has a minimum at $\\tau _{0}>T_{1}$, then\n\\begin{eqnarray}\n\\omega _{j+1}^{2}(\\tau _{0}) & \\ge & \n1+\\frac {1}{2} \\left( \\omega _{j+2}^{2}(\\tau _{0}) +\n\\omega _{j}^{2}(\\tau _{0}) \\right) \n\\nonumber\n\\\\\n & \\ge & 1+\\frac {1}{2} \\left(\n\\omega _{j+2}^{2}(\\tau _{0}) +L_{1} -\\epsilon \\right) .\n\\end{eqnarray}\nWith $N_{j+1}$ as before, then\n\\begin{equation}\nN_{j+1} \\ge 1+\\frac {1}{2} N_{j+2} +K_{1}\n\\end{equation}\nwhere $K_{1}=\\frac {1}{2} \\left( L_{1}-\\epsilon \\right)$.\nSimilarly,\n\\begin{equation}\nN_{j+l-1}\\ge 1+\\frac {1}{2} N_{j+l-2}+K_{2}\n\\end{equation}\nwhere $K_{2}=\\frac {1}{2} \\left( L_{2}-\\epsilon \\right) $.\nThe remaining inequalities for $N_{j+2},\\ldots ,N_{j+l-2}$\nare exactly as before (\\ref{nineq}).\nThe previous method can be repeated to give\n\\begin{equation}\nN_{j+i}\\ge i(l-i)+K_{1}+K_{2} \\qquad i=1,\\ldots ,l-1.\n\\end{equation}\n\n\\bigskip\nNow define ${\\tilde {N}}_{i}$ to be the lowest upper bound of the \nset of maximum values of $\\omega _{i}^{2}$ (which exists since\n$\\omega _{i}^{2}$ is bounded).\nIf $\\omega _{j+1}^{2}$ has a maximum at $\\tau _{0}>T_{1}$, then\n\\begin{eqnarray}\n\\omega _{j+1}^{2}(\\tau _{0}) & \\ge & \n1+\\frac {1}{2} \\left( \\omega _{j+2}^{2}(\\tau _{0})+\n\\omega _{j}^{2}(\\tau _{0}) \\right)\n\\nonumber\n\\\\\n& \\ge & 1+\\frac {1}{2} \\left(\n\\omega _{j+2}^{2} +L_{1}+\\epsilon \\right) .\n\\end{eqnarray}\nThe analysis now proceeds exactly as in the previous paragraph\nand yields\n\\begin{equation}\n{\\tilde {N}}_{j+i}\\le i(l-i)+{\\tilde {K}}_{1}+{\\tilde {K}}_{2}\n\\qquad\ni=1,\\ldots ,l-1\n\\end{equation}\nwhere ${\\tilde {K}}_{i}=\\frac {1}{2} \\left( L_{i}+\\epsilon \\right)$ for\n$i=1,2$.\nIn addition, by definition it must be the case that\n\\begin{equation}\n{\\tilde {N}}_{j+i}\\ge N_{j+i}\n\\end{equation}\nwhich means that\n\\begin{equation}\ni(l-i)+\\frac {1}{2} \\left( L_{1}+L_{2} \\right) -\\epsilon\n\\le N_{j+i} \\le {\\tilde {N}}_{j+i} \\le\ni(l-i)+\\frac {1}{2} \\left( L_{1}+L_{2} \\right) +\\epsilon\n\\end{equation}\nfor all $\\epsilon >0$, whence ${\\tilde {N}}_{j+i}=N_{j+i}$ for\nall $i$ and\n\\begin{equation}\n\\omega _{j+i}^{2} =i(l-i)+\\frac {1}{2} \\left( L_{1}+L_{2} \\right)\n\\end{equation}\nfor sufficiently large $\\tau $.\n\n\\bigskip\nIn conclusion, then, we have shown that the intermediate case has\nsome $\\omega _{j}$'s which are monotonic for sufficiently \nlarge $\\tau $, with the remaining $\\omega _{j}$'s being\nconstant for sufficiently large $\\tau $. \nHence in this case also $m$ is bounded as $\\tau \\rightarrow \\infty $\nand $\\lim _{\\tau \\rightarrow \\infty }\\Psi (\\tau )=1$.\n\\hfill\n$\\square $\n\n\\bigskip\n\\noindent\n{\\bf {Proof of Proposition \\ref{flatprop}}}\n\\smallskip\n\\newline\nIn order to show that the geometry becomes flat as \n$\\tau \\rightarrow \\infty $, it remains to show that $S\\rightarrow 1$\nas $\\tau \\rightarrow \\infty $.\nFrom the proof of lemma \\ref{flatlem}, for $\\omega _{j}$\nhaving a limit as $\\tau \\rightarrow \\infty $, we have\n\\begin{equation}\nrU_{i}={\\dot {\\omega }}_{i}\\rightarrow 0 \n\\end{equation}\nas $\\tau \\rightarrow \\infty $.\nIn case 1 of the proof of lemma \\ref{flatlem}, the proof\nof \\cite{bfm} carries straight over to show that \n$U_{i}\\rightarrow 0$ as $\\tau \\rightarrow \\infty $ also in this\nsituation.\nNow\n\\begin{equation}\n\\frac {{\\dot {S}}}{S}=\\Psi G = \n\\frac {1}{\\Psi } \\sum _{i=1}^{N-1} U_{i}^{2} \\le\n\\frac {C}{\\Psi r^{2}}\n\\end{equation}\nfor some constant $C$ and sufficiently large $\\tau $.\nTherefore\n\\begin{equation}\n\\frac {S'}{S}=\\frac {{\\dot {S}}}{S} \\frac {1}{r\\Psi } \\le\n\\frac {C}{\\mu r^{3}}\n\\le \\frac {{\\tilde {C}}}{r^{3}}\n\\end{equation}\nfor some constant ${\\tilde {C}}$ and sufficiently large $\\tau $,\nas $\\mu \\rightarrow 1$ as $\\tau \\rightarrow \\infty $.\nThis means that $S$ has a finite limit as $\\tau \\rightarrow \\infty $.\n\n\\bigskip\nThe field equations only involve $\\frac {S'}{S}$ rather than just $S$\nitself. \nTherefore $S$ is defined only up to a multiplicative constant,\nand without loss of generality we may therefore take the\nfinite limit of $S$ as $\\tau \\rightarrow \\infty $ to be 1,\nso that the spacetime is asymptotically flat.\n\n\\bigskip\nLet $\\delta (\\tau )=2\\Psi - \\kappa -1$ be a small perturbation,\nthen the equation for $\\omega _{j}$ reads\n\\begin{equation}\n{\\ddot {\\omega }}_{j}+\\left[ 1 -\\omega _{j}^{2} \n+\\frac {1}{2} \\left( \\omega _{j+1}^{2} +\\omega _{j-1}^{2} \\right)\n\\right] \\omega _{j} = \\left( 1+ \\delta (\\tau ) \\right)\n{\\dot {\\omega }}_{j}\n\\end{equation}\nas $\\tau \\rightarrow \\infty $.\nSince $\\delta $ is small, it does not alter the position or nature\nof the critical points as compared with the exactly \nflat space case. \nLemma \\ref{flatlem} showed that each \n${\\dot {\\omega }}_{j},{\\ddot {\\omega }}_{j}\\rightarrow 0$ \nas $\\tau \\rightarrow \\infty $.\nHence $\\omega _{j}$ must approach one of the critical points \nwhose nature was elucidated in section \\ref{flat}.\nThe flat space analysis showed that, if $\\omega _{j}=0$,\nthen there is an unstable focal point in the \n$(\\omega _{j},{\\dot {\\omega }}_{j})$ plane.\nHence our solution cannot approach this value, unless \n$\\omega _{j}\\equiv 0$.\nThe solution has to tend to one of the saddle points along a\nstable direction, and, in direct analogy with the \n${\\mathfrak {su}}(2)$ case \\cite{ershov}, \nthere are no solutions where $\\omega _{j}\\rightarrow 0$\nas $\\tau \\rightarrow \\infty $.\nThe solution must therefore be a member of the family found in the\nlocal existence theorem \\ref{infprop}.\n\\hfill\n$\\square $\n\n\\bigskip\nThe power of proposition \\ref{flatprop} lies in that, if we can prove\nthe existence of solutions for which $\\Psi >0$ for all $\\tau >0$\nand $r\\rightarrow \\infty $, then these solutions are automatically \nthe regular black holes (and, if $r_{h}\\rightarrow 0$, \nsoliton solutions \\cite{bartnik})\nwe seek. \nWe close this section by determining the asymptotic behaviour of\nsolutions of type 2.\n\n\\begin{prop}\n\\label{rmaxprop}\nIf $\\Psi (\\tau )>0$ for all $\\tau >0$ and $r(\\tau )$ remains \nbounded, then $\\Psi \\rightarrow 0$ and $\\kappa \\rightarrow 1$\nas $\\tau \\rightarrow \\infty $.\nIn addition, there is at least one $j$ for which\n$\\omega _{j} \\rightarrow 0$ as\n$\\tau \\rightarrow \\infty $, and this $\\omega _{j}$ has\ninfinitely many zeros.\n\\end{prop}\n\n\\noindent {\\bf {Proof}} \\smallskip \\newline \nSince $r$ is monotonic increasing and bounded, it has a limit $r_{0}$.\nIn addition, $m$ is monotonic increasing and bounded since\nthe positivity of $\\mu $ implies that\n\\begin{equation}\nm<\\frac {r}{2} \\le \\frac {r_{0}}{2}.\n\\end{equation}\nAs $\\tau \\rightarrow \\infty $, ${\\dot {r}}\\rightarrow 0$\nand hence $\\Psi \\rightarrow 0$ from (\\ref{fregeqn}).\nConsider the quantity\n\\begin{equation}\nE=-\\frac {r^{2}}{4}\\left( 1+\\Psi ^{2} -2\\kappa \\Psi \\right)\n=-\\frac {r^{2}}{2} \\left( -\\mu G+r^{2} p_{\\theta } \\right).\n\\end{equation}\nThen\n\\begin{eqnarray}\n{\\dot {E}} & = & 2r^{2}\\mu G \\Psi -r^{2}\\kappa \\mu G\n\\nonumber \n\\\\\n& < &\n r^{2}\\mu G (1-\\kappa) \n\\nonumber\n\\\\\n& < & 0,\n\\end{eqnarray}\nfor sufficiently large $\\tau $ for which $\\Psi < 1\/2$, and since\n$\\kappa >1$ \\cite[Lemma 10]{bfm}.\nHence \n\\begin{equation}\nE\\rightarrow -\\frac {r_{0}^{2}}{4}\n\\label{elim}\n\\end{equation}\nas $\\tau \\rightarrow \\infty $, since $E$ is monotonically decreasing\nfor sufficiently large $\\tau $.\nThen, following \\cite[Proposition 15]{bfm}, we have \n$\\kappa \\rightarrow 1$ as $\\tau \\rightarrow \\infty $.\n\n\\bigskip\nLet $\\delta (\\tau )=\\kappa -2\\Psi -1$ be small, then the\nequation for $\\omega _{j}$ reads\n\\begin{equation}\n{\\ddot {\\omega }}_{j} = {\\dot {\\omega }}_{j} \n\\left( -1-\\delta \\right) - \\omega _{j} \\left[\n1-\\omega _{j}^{2} +\\frac {1}{2} \\left(\n\\omega _{j+1}^{2} +\\omega _{j-1}^{2} \\right) \\right] .\n\\end{equation}\nReplacing $\\tau $ by $-\\tau $ yields the equation\n\\begin{equation}\n{\\ddot {\\omega }}_{j} = {\\dot {\\omega }}_{j} \n\\left( 1+\\delta \\right) - \\omega _{j} \\left[\n1-\\omega _{j}^{2} +\\frac {1}{2} \\left(\n\\omega _{j+1}^{2} +\\omega _{j-1}^{2} \\right) \\right] .\n\\end{equation}\nwhich is the equation for flat space solutions, so that $\\omega _{j}$\nmust approach one of the critical points.\nThe fact that $E\\neq 0$ means that $p_{\\theta }$ cannot tend to zero\nas $\\tau \\rightarrow \\infty $ because $\\mu G$ does vanish in the limit.\nHence at least one of the $\\omega _{j}$'s must be zero as\n$\\tau \\rightarrow \\infty $.\nSince $\\delta $ is small it alters neither the position nor\nthe characteristics of the critical points, and hence this \n$\\omega _{j}$ will go into the focus at zero. \nTherefore it oscillates infinitely many times\n before it hits zero.\n\\hfill\n$\\square $\n\n\\bigskip\nWe may determine the value of $r_{0}$ as follows.\nSince $\\kappa $ is finite and we have (\\ref{elim}), it follows\nthat $r^{2}p_{\\theta }\\rightarrow 1\/2$ as\n$\\tau \\rightarrow \\infty $.\nHence\n\\begin{equation}\nr_{0}^{2}=\\frac {1}{2} \\lim _{\\tau \\rightarrow \\infty }\n\\sum _{j=1}^{N}\n\\left( \\omega _{j}^{2}-\\omega _{j-1}^{2} -N-1+2j \\right) ^{2}.\n\\end{equation}\nThe maximum possible value of $r_{0}^{2}$ is when all\n$\\omega _{j}\\rightarrow 0$ as $\\tau \\rightarrow \\infty $.\nIn this case:\n\\begin{equation}\nr_{0,max}^{2} =\\frac {1}{2} \\sum _{j=1}^{N} \\left(\n-N-1+2j \\right) ^{2} = \\frac {1}{6} N(N-1)(N+1).\n\\end{equation}\nThe minimum possible value of $r_{0}^{2}$ is when only one\n$\\omega _{j}\\rightarrow 0$ as $\\tau \\rightarrow \\infty $, when\n\\begin{equation}\nr_{0}^{2} =j^{2}(N-j)^{2}\n\\end{equation}\nwhich has a minimum when $j=1$ or $N-1$, and\n\\begin{equation}\nr_{0}^{2} =(N-1)^{2}.\n\\end{equation}\nWith this value of $r_{0}$, we follow \\cite{bfm} and denote\nby $S_{\\infty }$ singular solutions for which $\\mu $ vanishes\noutside the event horizon when $r0$\nfor all $\\tau $ and $r(\\tau )$ remains bounded as \n$\\tau \\rightarrow \\infty $, has $r\\rightarrow r_{0}$ as\n$\\tau \\rightarrow \\infty $, where\n\\begin{equation}\n(N-1)^{2} \\le r_{0}^{2} \\le \n\\frac {1}{6} N(N+1)(N-1).\n\\end{equation}\nFor $N=3$, therefore, $r_{0}=2$. \nFixing $r_{h}>2$ will therefore rule out the possibility of such\nsolutions.\nThis is analogous to the value $r_{h}>1$ which rules out\nsuch solutions in the ${\\mathfrak {su}}(2)$ case.\nAt the end of this section we shall return to the existence proof\nfor $r_{h}\\le 2$.\n\n\\bigskip\nFor every pair of starting values $(\\omega _{1,h},\\omega _{2,h})$\nthere are then just two possibilities:\n\\begin{enumerate}\n\\item\n$\\mu (\\tau )>0$ for all $\\tau $ and we have a regular black hole\nsolution;\n\\item\nthere is a $\\tau _{0}>0$ such that $\\mu (\\tau _{0})=0$\nand we have a singular solution.\n\\end{enumerate}\nDefine a new variable $R$ by\n\\begin{equation}\nR=\\frac {r-r_{h}}{r}\n\\end{equation}\nand $R_{m}$ by the maximum value of $R$ for each solution, that is,\nin case one $R_{m}=1$ (corresponding to $r\\rightarrow \\infty $), and\nin case two $R_{m}=R(\\tau _{0})$.\nFor each solution we define $n_{i}$ to be the number of zeros of\nthe function $\\omega _{i}$ between $\\tau =0$ and $\\tau =\\infty $\nin case 1 and $\\tau =\\tau _{0}$ in case 2, where $n_{i}$ can be\ninfinite.\nDefining new variables $N_{i}$ by\n\\begin{equation}\nN_{i}=\\frac {n_{i}}{1+n_{i}}\n\\end{equation}\nwe allow the possibility that $N_{i}=1$.\n\n\\bigskip\nEach pair of non-zero starting values $(\\omega _{1,h},\\omega _{2,h})$\ncan then be mapped to the three quantities\n$(R_{m},N_{1},N_{2})$ for the corresponding solution, giving\nrise to a map from ${\\mathbb {R}}^{2}$ \n($(\\omega _{1,h},\\omega _{2,h})$ space) to \n${\\mathbb {B}}^{2}\\times [0,1]$ ($(R_{m},N_{1},N_{2})$ space),\nwhere ${\\mathbb {B}}$ is the discrete set\n$\\{ 0, 1\/2, 2\/3, 3\/4, \\ldots , 1\\} $.\nCall this map $f$.\nFigures 1--3 sketch the nature of $(\\omega _{1,h},\\omega _{2,h})$\nspace, ${\\mathbb {B}}\\times [0,1]$ and ${\\mathbb {B}}^{2}$\nrespectively.\nNote that this map will not be 1-1 for $N=3$, although\nit was conjectured in \\cite{bfm} that for $N=2$ the map is 1-1.\n\n\\bigskip\nNote that since we know the nature of the solution space \nwhen one of the $\\omega _{i,h}$ vanishes (since in this case \n$\\omega _{i}\\equiv 0$), we do not need to extend the map $f$\nto the co-ordinate axes in $(\\omega _{1,h},\\omega _{2,h})$\nspace.\nWith this notation, we are ready to prove proposition\n\\ref{firstprop}, once we have the following lemma.\n\n\\begin{lemma}\n\\label{fcont}\nSuppose that for starting values\n$({\\bar {\\omega }}_{1,h},{\\bar {\\omega }}_{2,h})$\nthere is a singular solution with $\\mu ({\\bar {\\tau }}_{0})=0$\nand the gauge field functions having node structure \n$({\\bar {n}}_{1},{\\bar {n}}_{2})$.\nThen the map $f$ is continuous at \n$({\\bar {\\omega }}_{1,h},{\\bar {\\omega }}_{2,h})$.\n\\end{lemma}\n\n\\noindent {\\bf {Proof}} \\smallskip \\newline \nFrom proposition \\ref{horprop}, the field variables are \ncontinuous\nin $\\tau $ and the starting parameters.\nThus all starting values \n$(\\omega _{1,h},\\omega _{2,h})$ in a sufficiently small \nneighbourhood of \n$({\\bar {\\omega }}_{1,h},{\\bar {\\omega }}_{2,h})$\nwill give rise to a singular solution with $\\mu (\\tau _{0})=0$\nwhere $\\tau _{0}$ is close to ${\\bar {\\tau }}_{0}$,\nthe values of the field variables at $\\tau _{0}$ will be\nclose to those at ${\\bar {\\tau }}_{0}$ in the original solution\nand the node structure will be $({\\bar {n}}_{1},{\\bar {n}}_{2})$\nsince the gauge field functions cannot have double zeros\n(proposition \\ref{maxminprop}).\nIn other words, $f$ is continuous at\n$({\\bar {\\omega }}_{1,h},{\\bar {\\omega }}_{2,h})$.\n\\hfill\n$\\square $\n\n\\bigskip\n\\noindent\n{\\bf {Proof of Proposition \\ref{firstprop}}}\n\\smallskip\n\\newline\nConsider the open subset of the $(\\omega _{1,h},\\omega _{2,h})$\nplane given by\n\\begin{equation}\nD=\\{ (\\omega _{1,h},\\omega _{2,h}) : \n0<\\omega _{1,h}<\\omega _{2,h} \\} .\n\\end{equation}\nThe subset $D'=\\{ (\\omega _{1,h},\\omega _{2,h}):\n0<\\omega _{2,h}<\\omega _{1,h} \\} $\ncan be treated similarly.\nThe symmetries of the field equations (\\ref{firsteqn}--\\ref{lasteqn})\nmean that it is sufficient to consider only the first\nquadrant of the $(\\omega _{1,h},\\omega _{2,h})$ plane.\nFrom \\cite{bfm} we know that along the line\n$\\omega _{1,h}=\\omega _{2,h}$ there are singular solutions with\nnode structure $(n_{1},n_{1})$ for all $n_{1}=1,2,\\ldots $.\nTherefore there are neighbourhoods of $D$ corresponding\nto singular solutions with node structure $(n_{1}, n_{1})$\nfor all $n_{1}=1,2,\\ldots $.\nHence $f(D)$ contains points corresponding to $(R_{m},N_{1},N_{1})$\nfor all $N_{1}\\in {\\mathbb {B}}$.\nTherefore $f(D)$ is not a connected set.\nTherefore $f$ cannot be continuous everywhere on $D$ because $D$ is\nconnected.\nHence we conclude that there exists at least one \n$(\\omega _{1,h},\\omega _{2,h})\\in D$ corresponding to a black hole\nsolution because $f$ is continuous for all \n$(\\omega _{1,h},\\omega _{2,h})$ corresponding to singular solutions.\n\\hfill\n$\\square $\n\n\\bigskip\nThis approach allows us to see quite simply how the transversality\nproperty conjectured in \\cite{bfm} arises in the ${\\mathfrak {su}}(2)$\ncase.\nAlong the line of values of $\\omega _{h}$, singular solutions\nhaving different numbers of nodes must be separated by at least\none regular solution. \nFor ${\\mathfrak {su}}(2)$, it is known that values of\n$\\omega _{h}$ sufficiently close to $1$ correspond to singular\nsolutions with $\\omega $ having one node,\nwhilst sufficiently small $\\omega _{h}$ correspond to solutions\nhaving as many nodes as we like.\nIn addition, if we have a regular solution with $n$ nodes,\nthen all singular solutions with $\\omega _{h}$\nsufficiently close have either $n$ or $n+1$ nodes.\nStarting with the singular solutions with one node, and decreasing\n$\\omega _{h}$, we first hit a regular solution with one node\n(there being only the trivial solution with no nodes).\nThen there may be more singular solutions with one node, or two nodes.\nIn the former case there must be another regular solution with\none node, in the latter case there will next be a regular solution\nwith two nodes.\nEither way, there must be a regular solution with two nodes before\nwe can move on to singular solutions with $n>2$ nodes.\nThe process continues for all $n$.\n\n\\bigskip\nHaving shown that there exist genuine ${\\mathfrak {su}}(3)$\nblack holes, the main result of this section is the following\ntheorem, which we prove at the moment for the case $r_{h}>2$,\nreturning to the case $r_{h}\\le 2$ at the end of the section.\n\n\\begin{theorem}\n\\label{exist}\nGiven ${\\bar {n}}_{1}=0,1,\\ldots $, then there exist\nregular black hole solutions of the ${\\mathfrak {su}}(3)$\nEinstein-Yang-Mills equations with $\\omega _{1}(r)$ having\n${\\bar {n}}_{1}$ nodes and $\\omega _{2}(r)$ having $n_{2}$ nodes,\nfor infinitely many $n_{2}\\ge {\\bar {n}}_{1}$.\n\\end{theorem}\n\nA similar result holds with the roles of $\\omega _{1}$ and\n$\\omega _{2}$ reversed. \nNote that this result is slightly weaker than the corresponding\ntheorem for ${\\mathfrak {su}}(2)$, since we cannot guarantee that\nevery combination of $(n_{1},n_{2})$ is the node structure of\nthe gauge fields for some black hole solution, only that an infinite\nnumber of such combinations does occur for each $n_{1}$.\nAt the end of this section we shall give an argument, based on\na numerical analysis, that in fact black holes exist for all\n$(n_{1},n_{2})$, although we are not able to prove this\nanalytically.\nThe proof of theorem \\ref{exist} will proceed via a series of lemmas.\n\n\\begin{lemma}\n\\label{weaklem}\nSuppose that the starting parameters $(0,{\\bar {\\omega }}_{2,h})$\ncorrespond to a charged regular black hole solution in which\n$\\omega _{2}(r)$ has ${\\bar {n}}_{2}$ zeros.\nThen, given $n_{0}$, for all sufficiently small $\\omega _{1,h}$\nand $\\omega _{2,h}$ sufficiently close to ${\\bar {\\omega }}_{2,h}$,\nthe solutions are regular or singular solutions with \n$n_{1}\\ge n_{0}$ and $n_{2}\\ge {\\bar {n}}_{2}$.\n\\end{lemma}\n\n\\noindent {\\bf {Proof}} \\smallskip \\newline \nBy continuity, all field variables will remain close to the original\ncharged solution until $r\\gg 1$ and the geometry is approximately\nflat.\nAt this point $\\omega _{1}$ will be very small and $\\omega _{2}$\nwill be close to $1$.\nThe equations for $\\epsilon _{1}=\\omega _{1}$ and \n$\\epsilon _{2}=\\omega _{2}-1$ as functions of $\\tau $ are, in \nthis regime, to first order,\n\\begin{eqnarray}\n0 & = & {\\ddot {\\epsilon }}_{1}-{\\dot {\\epsilon }}_{1}\n+\\frac {3}{2}\\epsilon _{1} \n\\nonumber\n\\\\\n0 & = & {\\ddot {\\epsilon }}_{2} -{\\dot {\\epsilon }}_{2}\n-2\\epsilon _{2}.\n\\end{eqnarray}\nThis corresponds to a focus in the \n$(\\omega _{1},{\\dot {\\omega }}_{1})$ plane, and hence with \n$\\omega _{1,h}$ sufficiently small, $\\epsilon _{1}$ will have\nat least $n_{0}$ zeros.\n\\hfill\n$\\square $\n\n\\begin{lemma}\n\\label{reglem}\nIf $({\\bar {\\omega }}_{1,h},{\\bar {\\omega }}_{2,h})$\nleads to a regular black hole solution with $\\omega _{1}$ having\n${\\bar {n}}_{1}$ nodes and $\\omega _{2}$ having ${\\bar {n}}_{2}$\nnodes, then all $(\\omega _{1,h},\\omega _{2,h})$ sufficiently \nclose to\n$({\\bar {\\omega }}_{1,h},{\\bar {\\omega }}_{2,h})$\nlead to regular or singular solutions with $\\omega _{1}$ having\nat least ${\\bar {n}}_{1}$ zeros and $\\omega _{2}$ having\nat least ${\\bar {n}}_{2}$ zeros.\n\\end{lemma}\n\n\\noindent {\\bf {Proof}} \\smallskip \\newline \nSince the solutions are continuous in the starting parameters and\n$r$, for $(\\omega _{1,h},\\omega _{2,h})$ sufficiently close to\n$({\\bar {\\omega }}_{1,h},{\\bar {\\omega }}_{2,h})$,\nthe gauge function $\\omega _{1}$ will have ${\\bar {n}}_{1}$\nzeros and $\\omega _{2}$ will have ${\\bar {n}}_{2}$ zeros for \n$rn_{0}$.\nThen there is a neighbourhood \n${\\tilde {D}}_{0}\\subset D_{0}, {\\tilde {D}}_{0}\\neq D_{0}$\nfor which all solutions are regular or singular solutions with\n$n_{1}\\ge {\\tilde {n}}_{0}$ and $n_{2}\\ge {\\bar {n}}_{2}$.\nIn ${\\tilde {D}}_{0}$ there must be at least one singular solution\nhaving $n_{1}\\ge {\\tilde {n}}_{0}$ and $n_{2}\\ge {\\bar {n}}_{2}$,\nby lemma \\ref{reglem}, corresponding to starting values\n$({\\tilde {\\omega }}_{1,h},{\\tilde {\\omega }}_{2,h})$.\nConsider a curve joining \n$({\\tilde {\\omega }}_{1,h},{\\tilde {\\omega }}_{2,h})$\nand $({\\hat {\\omega }}_{1,h},{\\hat {\\omega }}_{2,h})$\nand lying in $D_{0}$.\nThen there must be at least one regular solution along this curve.\nLet $(\\omega _{1,h}^{R},\\omega _{2,h}^{R})$\nbe the regular solution closest to \n$({\\hat {\\omega }}_{1,h},{\\hat {\\omega }}_{2,h})$,\nhaving $n_{1}=n_{1}^{R}$ and $n_{2}=n_{2}^{R}$.\nSince $(\\omega _{1,h}^{R},\\omega _{2,h}^{R})\\in D_{0}$, it follows \nthat $n_{1}^{R}\\ge n_{0}$ and $n_{2}^{R}\\ge {\\bar {n}}_{2}$.\nAlso, from lemma \\ref{reglem}, \n$n_{1}^{R}\\le n_{0}$ and $n_{2}^{R}\\le {\\bar {n}}_{2}$,\nsince sufficiently close to \n$(\\omega _{1,h}^{R},\\omega _{2,h}^{R})$ there are singular solutions\nhaving $n_{1}=n_{0}$ and $n_{2}={\\bar {n}}_{2}$.\nTherefore $n_{1}^{R}=n_{0}$ and $n_{2}^{R}={\\bar {n}}_{2}$.\n\\hfill\n$\\square $\n\n\\begin{lemma}\n\\label{existlem}\nGiven ${\\bar {n}}_{2}$ and $n_{0}>{\\bar {n}}_{2}$, then there exist\nregular and singular solutions with $n_{2}={\\bar {n}}_{2}$ and\n$n_{1}\\ge n_{0}$.\n\\end{lemma}\n\n\\noindent {\\bf {Proof}} \\smallskip \\newline \nConsider the point $(0,{\\bar {\\omega }}_{2,h})$ which corresponds\nto the charged regular solution with $n_{2}={\\bar {n}}_{2}$.\nThen all $(\\omega _{1,h},\\omega _{2,h})$ sufficiently close to\n$(0,{\\bar {\\omega }}_{2,h})$ are regular or singular solutions\nwith $n_{1}\\ge n_{0}$ and $n_{2}\\ge {\\bar {n}}_{2}$.\nLet all such $(\\omega _{1,h},\\omega _{2,h})$ form a neighbourhood\n$D_{0}$ of $(0,{\\bar {\\omega }}_{2,h})$.\nFrom \\cite{bfm}, there exist $(0,{\\hat {\\omega }}_{2,h})\\in D_{0}$\ncorresponding to singular solutions having $\\omega _{1}\\equiv 0$\nand $n_{2}={\\bar {n}}_{2}$.\nBy continuity, all $(\\omega _{1,h},\\omega _{2,h})$ sufficiently close to\n$(0,{\\hat {\\omega }}_{2,h})$ correspond to singular solutions having\n$n_{1}\\ge n_{0}$ and $n_{2}={\\bar {n}}_{2}$.\nHence, by lemma \\ref{singlem}, there are also in this neighbourhood\nregular black hole solutions having $n_{1}\\ge n_{0}$ and\n$n_{2}={\\bar {n}}_{2}$.\n\\hfill\n$\\square $\n\n\\bigskip\n\\noindent \n{\\bf {Proof of Theorem \\ref{exist}}} \n\\smallskip \n\\newline \nFix ${\\bar {n}}_{2}$.\nThen we have regular solutions with node structure \n$({\\bar {n}}_{2}, {\\bar {n}}_{2})$.\nNow let $n_{0}={\\bar {n}}_{2}+1$, then from lemma \\ref{existlem}\nthere are regular solutions having $n_{2}={\\bar {n}}_{2}$ and\n$n_{1}\\ge n_{0}$.\nLet $n_{0}'$ be the smallest such $n_{1}$.\nNow set $n_{0}=n_{0}'+1$ and repeat the process.\n\\hfill\n$\\square $\n\n\\bigskip\nIn order to guarantee the existence of black hole solutions having\nnode structure $({\\bar {n}}_{1},{\\bar {n}}_{2})$ for every pair of\nintegers $({\\bar {n}}_{1},{\\bar {n}}_{2})$, we would require the\nfollowing lemma in addition to lemma \\ref{existlem}.\n\n\\begin{lemma}\n\\label{tightlem}\nSuppose $({\\bar {\\omega }}_{1,h},{\\bar {\\omega }}_{2,h})$\ncorresponds to a regular black hole solution in which\n$\\omega _{1}$ has ${\\bar {n}}_{1}$ nodes and\n$\\omega _{2}$ has ${\\bar {n}}_{2}$ nodes.\nThen, for $(\\omega _{1,h},\\omega _{2,h})$ sufficiently close\nto $({\\bar {\\omega }}_{1,h},{\\bar {\\omega }}_{2,h})$,\nsolutions for which $\\omega _{2}$ still has ${\\bar {n}}_{2}$\nnodes are such that $\\omega _{1}$ has either ${\\bar {n}}_{1}$\nor ${\\bar {n}}_{1}+1$ nodes.\n\\end{lemma}\n\nThe argument we now give for lemma \\ref{tightlem} is not an analytic\nproof because we require a numerical analysis.\n\n\\bigskip\n\\noindent {\\bf {Proof}} \\smallskip \\newline \nBy continuity, we know that for $(\\omega _{1,h},\\omega _{2,h})$\nsufficiently close to \n$({\\bar {\\omega }}_{1,h},{\\bar {\\omega }}_{2,h})$,\nthe gauge function $\\omega _{1}$ has ${\\bar {n}}_{1}$ zeros\nand $\\omega _{2}$ has ${\\bar {n}}_{2}$ zeros for $rr_{1}$, then, $\\omega _{2}$ will\nbe of one sign.\nTherefore we may consider the new dependent variable\n\\begin{equation}\n\\psi = \\frac {\\omega _{1}}{\\omega _{2}}\n\\end{equation}\nwhich will have the same number of zeros as $\\omega _{1} (r)$\nfor $r>r_{1}$.\nThen the equation satisfied by $\\psi $ is\n\\begin{equation}\nr^{2} \\mu \\psi '' +\\left( 2m-2r^{3}p_{\\theta } \\right)\n\\psi ' +2r^{2}\\mu \\frac {\\omega _{2}'}{\\omega _{2}} \n\\psi '\n+\\frac {3}{2} \\omega _{2}^{2} \\psi \\left( 1-\\psi ^{2} \\right) =0.\n\\label{psieqn}\n\\end{equation}\nOn some interval, $r_{1}1$.\nIn other words, for $\\epsilon _{1}>0$ initially, $\\epsilon _{1}>0$\nalways and $\\omega _{1}$ will have no additional zeros for $r>r_{1}$.\nHence we need only consider the case where $\\epsilon _{1}<0$\ninitially.\nUnfortunately the non-linear perturbation equations cannot be\nintegrated analytically and so we present a numerical argument.\nFor initial $\\epsilon _{1},\\epsilon _{2}$ sufficiently small,\nthe values of $\\epsilon _{1}$ and $\\epsilon _{2}$ will remain close\nto those on the unstable manifold for all $\\tau <\\tau _{1}$ for\nsome $\\tau _{1}$, where $\\tau _{1}$ can be taken to be as large\nas we like by making the initial perturbations sufficiently small.\nNumerical integration of the non-linear equations with the \ninitial point on the unstable manifold and $\\epsilon _{1}<0$,\nshows that $\\epsilon _{2}$ increases monotonically and \n$\\epsilon _{1}$ decreases monotonically with just one zero, and \ncuts through $\\epsilon _{1}=-2$ (corresponding to $\\psi =-1$),\nfrom whence $\\psi $ must be monotonically decreasing\n(see figures 4 and 5).\nHence in this case $\\omega _{1}$ has ${\\bar {n}}_{1}+1$ zeros.\n\\hfill\n$\\square $ \n\n\\bigskip\nLemma \\ref{tightlem} allows us to prove that every combination\nof integers $(n_{1},n_{2})$ must correspond to a black hole\nsolution.\nIn the proof of theorem \\ref{exist}, we begin with regular\nsolutions with node structure $({\\bar {n}}_{2},{\\bar {n}}_{2})$\nand then from lemma \\ref{tightlem} there are either regular \nor singular solutions with node structure\n$({\\bar {n}}_{2}+1,{\\bar {n}}_{2})$.\nUsing lemma \\ref{singlem}, there are then regular solutions\nhaving this node structure and the proof of theorem \\ref{exist}\nfollows as before. \n\n\\bigskip\nSo far we have proved the existence of infinitely many \n${\\mathfrak {su}}(3)$ black holes only for $r_{h}>2$.\nThe remainder of this section will be spent proving the result\nfor $r_{h}\\le 2$.\nIn this case, the points at which the map $f$ is not continuous\n(which must still exist by the argument used in proving proposition\n\\ref{firstprop}) do not necessarily correspond to regular black hole\nsolutions: they could also be oscillating solutions.\nA couple of lemmas concerning oscillating solutions are required\nbefore the proofs of proposition \\ref{firstprop} and theorem\n\\ref{exist} can be extended.\n\n\\begin{lemma}\n\\label{osclem}\nIf $({\\bar {\\omega }}_{1,h},{\\bar {\\omega }}_{2,h})$\nleads to an oscillating solution with \n$\\omega _{1}(\\tau )\\rightarrow 0$ and \n$\\omega _{2}(\\tau )\\rightarrow 1$ as \n$\\tau \\rightarrow \\infty $, and\n$\\omega _{2}$ has ${\\bar {n}}_{2}$ zeros, then all \n$(\\omega _{1,h},\\omega _{2,h})$ sufficiently close to\n$({\\bar {\\omega }}_{1,h},{\\bar {\\omega }}_{2,h})$\nlead to one of the following types of solution:\n\\begin{enumerate}\n\\item\nan oscillating solution in which $\\omega _{1}(\\tau )\\rightarrow 0$ and \n$\\omega _{2}(\\tau )\\rightarrow 1$ as \n$\\tau \\rightarrow \\infty $, with $\\omega _{2}$ having at least\n${\\bar {n}}_{2}$ zeros;\n\\item\ngiven $n_{0}$, either a regular or a singular solution with\n$n_{1}\\ge n_{0}$ and $n_{2}\\ge {\\bar {n}}_{2}$.\n\\item\nan $S_{\\infty }$ solution.\n\\end{enumerate}\n\\end{lemma}\n\n\\noindent {\\bf {Proof}} \\smallskip \\newline \nSince in the original solution $\\omega _{1}$ has infinitely many\nzeros and the solutions are analytic in $\\tau $ and the starting\nvalues, there is some $\\tau _{1}$ such that \nall solutions with $(\\omega _{1,h},\\omega _{2,h})$ \nsufficiently close to $({\\bar {\\omega }}_{1,h},{\\bar {\\omega }}_{2,h})$\nmust have at least $n_{0}$ zeros of $\\omega _{1}$ and \n${\\bar {n}}_{2}$ zeros of $\\omega _{2}$ for $\\tau <\\tau _{1}$,\nand all variables will be close to their asymptotic values.\n\\hfill\n$\\square $\n\n\\begin{lemma}\nThere exists $({\\bar {\\omega }}_{1,h},{\\bar {\\omega }}_{2,h})\\in D$\ncorresponding to an oscillating solution.\n\\end{lemma}\n\n\\noindent {\\bf {Proof}} \\smallskip \\newline \nFrom \\cite{bfm}, we know that there are points on the line\n$\\omega _{1,h}=\\omega _{2,h}$ corresponding to $S_{\\infty }$\nsolutions.\nBy continuity, there is a neighbourhood $D_{0}$ of the line\n$\\omega _{1,h}=\\omega _{2,h}$ corresponding to $S_{\\infty }$\nsolutions.\nHowever, lemma \\ref{reglem} still applies in this case, \nso that the regular solutions on the line \n$\\omega _{1,h}=\\omega _{2,h}$ will have singular solutions in $D$\nsufficiently close to them for which $r(\\tau _{0})>2$.\nTherefore, by lemma \\ref{osclem}, there must be points in $D$\nwhich correspond to oscillating solutions.\n\\hfill\n$\\square $\n\n\\bigskip\nThe proofs of proposition \\ref{firstprop} and theorem \\ref{exist}\nare now exactly the same as for $r_{h}>2$. \nSince oscillating solutions\nhave an infinite number of zeros for at least one of the gauge field\nfunctions, and using lemma \\ref{osclem},\nregular solutions are still needed to separate \nsingular solutions having different node structures.\n\n\\section{${\\mathfrak {su}}(N)$ black holes}\n\\label{sun}\nThe detailed discussion of the previous section now enables us to\nproceed quite swiftly to the analogues of proposition \\ref{firstprop}\nand theorem \\ref{exist} for general $N$. \nThe method of proof will be by induction, which was the basic idea used \nin the previous section, where we proved existence for \n${\\mathfrak {su}}(3)$ exploiting known results about \n${\\mathfrak {su}}(2)$.\n\n\\bigskip\nThe first step is to illustrate how solutions which are from \n${\\mathfrak {su}}(n)$, where $n3$, there are additional embeddings which arise from setting\nat least one of $\\omega _{2}, \\ldots , \\omega _{N-2}\\equiv 0$,\nso that the gauge field equations decouple into two or more\nsets of coupled components, the sets being coupled to each other \nonly through the metric. Solutions of this form have been found\nnumerically in \\cite{klei1}.\nWe illustrate how the existence of black holes of this type may\nbe proved by considering the simplest case, which arises when\n$N=4$.\nIn this case, we have three non-zero gauge field functions, \n$\\omega _{1},\\omega _{2}, \\omega _{3}$.\nIf we set $\\omega _{2}\\equiv 0$, then the field equations take\nthe form\n\\begin{eqnarray}\nr^{2}\\mu \\omega _{1}'' & = & \n-\\left( 2m-2r^{3}p_{\\theta } \\right) \\omega _{1}'\n-\\left( 1-\\omega _{1}^{2} \\right) \\omega _{1} \n\\label{omone}\n\\\\\nr^{2}\\mu \\omega _{3}'' & = & \n-\\left( 2m-2r^{3}p_{\\theta } \\right) \\omega _{3}'\n-\\left( 1-\\omega _{3}^{2} \\right) \\omega _{3} \n\\label{omtwo}\n\\\\\nm' & = & \\left( \\mu G +r^{2} p_{\\theta }\\right)\n\\\\\n\\frac {S'}{S} & = & \\frac {2G}{r}\n\\end{eqnarray}\nwhere\n\\begin{equation}\nG=\\omega _{1}^{'2} + \\omega _{3}^{'2},\n\\qquad\np_{\\theta } = \\frac {1}{2r^{4}} \\left( \n\\left[ \\omega _{1}^{2}-1 \\right] ^{2}+ \n\\left[ \\omega _{3}^{2}-1 \\right] ^{2}+ 8 \\right) .\n\\end{equation}\nThese equations look very much like two uncoupled ${\\mathfrak {su}}(2)$\ndegrees of freedom, however, the two $\\omega $'s are (albeit weakly)\ncoupled.\nThe fact that we have two gauge field functions here means that we\ncan use the methods of section \\ref{su3} to prove the existence of\nregular black hole solutions.\nIn fact, all the results of that section carry directly over to this\nsituation on replacing $\\omega _{2}$ there by $\\omega _{3}$ here.\nThere is one exception. \nThe equations (\\ref{omone},\\ref{omtwo}) are slightly different from\nthose in section \\ref{su3} and this enables us to strengthen\nlemma \\ref{reglem}.\n\n\\begin{lemma}\nIf $({\\bar {\\omega }}_{1,h},{\\bar {\\omega }}_{3,h})$\nleads to a regular black hole solution with $\\omega _{1}$ having\n${\\bar {n}}_{1}$ nodes and $\\omega _{3}$ having ${\\bar {n}}_{3}$\nnodes, then all $(\\omega _{1,h},\\omega _{3,h})$ sufficiently \nclose to\n$({\\bar {\\omega }}_{1,h},{\\bar {\\omega }}_{3,h})$\nlead to regular or singular solutions with $\\omega _{1}$ having\neither ${\\bar {n}}_{1}$ or ${\\bar {n}}_{1}+1$ zeros and \n$\\omega _{3}$ having either ${\\bar {n}}_{3}$ or ${\\bar {n}}_{3}+1$\nzeros.\n\\end{lemma}\n\n\\noindent {\\bf {Proof}} \\smallskip \\newline \nSince the solutions are continuous in the starting parameters and $r$,\nfor $(\\omega _{1,h},\\omega _{3,h})$ sufficiently close to\n$({\\bar {\\omega }}_{1,h},{\\bar {\\omega }}_{3,h})$,\nthere is some $r_{1}$ such that $\\omega _{1}$ will have ${\\bar {n}}_{1}$\nzeros and $\\omega _{3}$ will have ${\\bar {n}}_{3}$ zeros for\n$rr_{1}$ such that all field variables\nare close to their asymptotic values for $r\\in [r_{1},r_{2}]$.\nThe flat space field equations (see section \\ref{flat}) decouple\ncompletely since $\\omega _{2}\\equiv 0$, and so the phase plane\nanalysis of \\cite{bfm} is valid in this case.\nThe phase plane analysis enables us to be more precise about\nthe number of zeros of the perturbed gauge field functions, \nunlike the more general case where the phase space had more than\ntwo coupled dimensions.\nTherefore the conclusions of \\cite[proposition 22]{bfm} can \nbe applied directly here and each gauge field function\ncan have at most one zero for $r>r_{2}$.\n\\hfill\n$\\square $\n\n\\bigskip\nWith this more powerful lemma, the analysis of section \\ref{su3}\nreaches the conclusion that there are regular black hole\nsolutions of the required form for each $r_{h}$ and node structure\n$(n_{1},n_{3})$.\n\n\\bigskip\nFor general $N$, similar embeddings of the form of two or more\n${\\mathfrak {su}}(n)$ ($n\\frac {1}{6}N(N+1)(N-1)$ (in which\ncase no oscillating solutions exist), and \n$r_{h}^{2}\\le \\frac {1}{6} N(N+1)(N-1)$.\nNote that, as in the ${\\mathfrak {su}}(3)$ case, we are not able to\nprove analytically that every sequence of integers \n$(n_{1},n_{2},\\ldots ,n_{N-1})$ corresponds to a black hole solution\nwhose gauge functions have this node structure, although it might\nreasonably be expected that this is indeed the case.\n\n\n\n\n\\section{Results and conclusions}\nIn this paper we have proved the existence of a vast number of\nhairy black holes in ${\\mathfrak {su}}(N)$ Einstein-Yang-Mills\ntheories.\nThe solutions are described by N-1 parameters,\ncorresponding to the number of nodes of the gauge field\nfunctions.\nThe result of \\cite{brod} tells us that each of these solutions\nwill have a topological instability, similar to the flat-space\n``sphaleron'', as was the case for ${\\mathfrak {su}}(2)$\nblack holes.\nThis instability does\nnot necessarily diminish the physical importance of these objects,\nas they may have an important role in processes such as cosmological\nparticle creation \\cite{gibbons}.\n\n\\bigskip\nWe now briefly mention another important reason for studying\n${\\mathfrak {su}}(N)$ black holes, namely the behaviour\nof the solutions as $N\\rightarrow \\infty $.\nBlack holes in ${\\mathfrak {su}}(\\infty )$\nEinstein-Yang-Mills theory would possess an infinite amount\nof hair, requiring an infinite number of parameters to\ndescribe their geometry.\nThis might be analogous to the ``W-hair'' found in \nnon-critical string theory \\cite{ellis}, and have \ndrastic consequences for the Hawking radiation,\ninformation loss and quantum decoherence processes\nassociated with black holes.\nThe existence of infinite amounts of hair might also \nrender such objects stable.\nIt is already known that the ${\\mathfrak {su}}(\\infty )$\nLie algebra is simply that of the diffeomorphisms of the sphere\n\\cite{floratos}.\nThis means that the limit as $N\\rightarrow \\infty $ cannot\nbe taken smoothly, in particular the results of the present paper\nare only valid when $N$ is finite.\nWe hope to return to this matter in a subsequent publication.\n\n\n\n\\section*{Acknowledgments}\nThe work of N.E.M. is supported by a P.P.A.R.C. advanced fellowship\nand that of E.W. is supported by a fellowship at Oriel College, Oxford.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nExploding stars have been noted for millennia, and observed (in a scientific\nsense) for somewhat over a century. \nIt wasn't until the middle of the 20$^{th}$ century that a distinction\ncould be made\nbetween the supernovae, the novae, and the eruptive phenomena seen\nin cataclysmic variables (the dwarf novae).\n\\cite{K63} was the first to suggest that the\nnovae were the consequence of explosive hydrogen burning on the surface\nof a degenerate dwarf. It is now well accepted that\nthe novae are manifestations of runaway thermonuclear reactions\non the\nsurface of a white dwarf (WD) accreting hydrogen in a close binary system\n(e.g., \\citealt{S71}).\nThe novae are highly dynamic phenomena, with\ntimescales ranging from seconds to millennia, occurring in complex systems\ninvolving two stars and mass transfer.\n\nThe primary driver of the evolution of the observational characteristics\nof a nova is \nthe temporal decrease of the optical depth in an expanding atmosphere.\nThe novae are marked by an extraordinary spectral evolution \\citep{Wil91,Wil92}.\nIn the initial phases one often\nsees an optically thick, expanding pseudo-photosphere. In some cases\none sees the growth and then disappearance of inverse P~Cygni\nabsorption lines from the cool,\nhigh velocity ejecta. As the pseudo-photosphere becomes optically thin, emission\nlines of the Hydrogen Balmer series strengthen, accompanied by either a\nspectrum dominated by permitted lines of Fe~II, or of helium and nitrogen.\nThe emission line profiles and line ratios evolve as the optical depth\nof the ejecta decreases, and the nova transitions from the permitted to the \nnebular phase \\citep{Wil91}.\n\nBeyond this template, in detail the novae exhibit a panoply\nof individual behaviors.\n\\cite{PG57} and \\cite{McL60}\ndescribed the evolution of novae as they were known at the time. \n\\cite{Wil92} discussed the formation of the lines, and\ndivided novae into the \\ion{Fe}{2} and He-N classes, based on \nwhich emission lines dominated (aside from the ubiquitous\nBalmer lines of hydrogen).\nNovae are also categorized as recurrent and classical novae, with the former\nhaving more than one recorded outburst. Over a long enough baseline, it is\nlikely that all novae are recurrent (e.g., \\citealt{For78}).\n\n\\cite{BE08} present a recent set of reviews of the nova phenomenon.\n\nThere exist well-sampled photometric records for many novae, such as those \npresented by \\cite{SSH10}. They classify the photometric light curves, from \nplates amassed over the past century, into 7 distinct photometric classes.\nOn the other hand, spectroscopic observations of novae have rarely been \npursued far past maximum because most novae fade rapidly, and time on the\nlarge telescopes required for spectroscopy is precious.\nThe most comprehensive past work was the Tololo Nova atlas\n\\citep{Wil94} of 13 novae followed spectroscopically over a\n5 year interval.\nThe availability of the SMARTS\\footnote{The Small and Medium Aperture\nTelescope System, directed by Charles Bailyn, is an ever-evolving\npartnership that has overseen operations of 4 small telescopes at\nCerro Tololo Interamerican Observatory since 2003.}\ntelescope facilities \\citep{S10} makes\npossible routine synoptic monitoring programs, \nboth photometric and spectroscopic,\nof time-variable sources.\nIt is timely, therefore, to undertake a comprehensive, systematic,\nhigh-cadence study of the\nspectrophotometric evolution of the galactic novae.\n\nThis atlas collects photometry and spectra of the novae we have observed\nwith SMARTS.\nMost of the novae in the atlas are recent novae, discovered since 2003. Most\nare in the southern hemisphere. The observing cadences are irregular; we have\nconcentrated on He-N and recurrent novae, novae in the LMC, and\nnovae that otherwise show unusual characteristics.\n\nOur purpose here is to introduce this atlas.\nOur scientific aim is to facilitate a detailed comparison of the\nvarious characteristics of the novae. These data can be used by and of\nthemselves to study individual objects, \nfor systematic studies to further define the phenomenon, \nand for correlative studies with other comprehensive data sets,\nsuch as the {\\it Swift} Nova Working Group's survey of X-ray and UV\nobservations of recent novae \\citep{Swift11}.\nOur aim in making the atlas public is to make the data accessible to \nthe community.\nWe are focusing on certain novae, \nand on particular aspects of the nova phenomenon (\\S\\ref{sec-disc}),\nbut simply cannot do justice to the full dataset.\n\n\\section{Observations and Data Analysis}\n\n\\subsection{Low Dispersion Spectroscopy}\\label{sec-lds}\nThe spectra reported here have been obtained with the venerable RC \nspectrograph\\footnote{\\url{http:\/\/www.ctio.noao.edu\/spectrographs\/60spec\/60spec.html}}\non the SMARTS\/CTIO 1.5m telescope.\nObservations are queue-scheduled, and are taken by\ndedicated service observers.\n\nThe detector is a Loral 1K CCD. We use a variety of spectroscopic modes, with\nmost of the spectra having been obtained with one of the standard modes \nshown in Table~\\ref{tbl-spsetups}.\n\nWe use slit widths of 1 and 0.8 arcsec in the low and the higher resolution\n47\/II modes, respectively.\nThe slit is oriented E-W and is not rotated during the night.\n\nWe routinely obtain 3 spectra of each target in order to filter for cosmic \nrays. We combine the 3 images and extract the spectrum by fitting a Gaussian \nin the spatial direction at each pixel. Wavelength calibration is accomplished \nby fitting a 3$^{rd}$ to 6$^{th}$ order polynomial to the\nTh-Ar or Ne calibration lamp \nline positions. We observe a spectrophotometric standard star, generally \nLTT~4364 \\citep{Ham92,Ham94} or Feige~110 \\citep{Oke90,Ham92,Ham94},\non most nights to determine the counts-to-flux conversion.\nBecause of slit losses, possible changes in transparency and seeing \nduring the night, and parallactic losses due to the fixed slit orientation,\nthe flux calibration is imprecise. We generally recover the correct \nspectral shape, except at the shortest wavelengths ($<$3800\\AA)\nwhere apparent changes in the slope of the continuum \nare likely attributable to airmass-dependent parallactic slit losses.\nWe have the capability to use simultaneous or contemporaneous photometry\nto recalibrate the spectra.\n\nThere are some quality control issues that have not been fully dealt with,\nespecially when we are near the sensitivity\nlimits of the telescope. These include\nobservations of an incorrect star, obviously incorrect flux calibrations,\nor spectra indistinguishable from noise. We are going through the data\nas time permits to address these issues.\n\n\\subsection{High Dispersion Spectroscopy}\n\nWe have a small number of high resolution spectra of some of the\nbrighter novae near maximum.\nThese were obtained with the Bench-Mounted \nEchelle\\footnote{\\url{http:\/\/www.ctio.noao.edu\/noao\/content\/fiber-echelle-spectrograph}}, and\ncurrently with the Chiron echelle\nspectrograph\\footnote{\\url{http:\/\/www.ctio.noao.edu\/noao\/content\/chiron}}. \nThese data will be incorporated into the atlas at a later time.\n\n\\subsection{Photometry}\n\nMost of the photometry was obtained using the \nANDICAM\\footnote{\\url{http:\/\/www.astronomy.ohio-state.edu\/ANDICAM\/detectors.html}}\ndual-channel imager on the 1.3m telescope.\nObservations are queue-scheduled, with dedicated service observers.\n\nThe ANDICAM optical channel is a 2048$^2$ pixel Fairchild 447 CCD.\nIt is read out with 2x2 binning, which yields a 0.369 arcsec\/pixel\nplate scale. The field of view is roughly 6x6 arcmin, but until\nrecently there has been \nsignificant unusable area on the east and south sides on the chip.\nThe finding charts in the atlas\nshow examples of ANDICAM images. We normally obtain single\nimages, since the fraction of pixels marred by cosmic rays\nand other events is small.\nExposure times range from 1 second to about 2 minutes. We use the standard \nJohnson-Kron-Cousins $B$, $V$,\n$R_C$, and $I_C$ filters (the $U$ filter has been unavailable since 2005, but\nwe have extensive $U$~band for some of the earlier novae, particularly\nV475~Sct and V5114~Sgr).\n\nThe ANDICAM IR channel is a Rockwell 1024$^2$ HgCdTe ``Hawaii'' Array.\nIt is read out in 4 quadrants with 2x2 binning, which yields a\n0.274 arcsec\/pixel plate scale and a 2.4 arcmin field of view.\nThe observations are dithered using an internal mirror. In most cases\nwe use 3 dither positions, with integration times from 4 seconds (the minimum\nintegration time) to about 45 seconds. We use the CIT\/CTIO $J$, $H$, and $K_s$\nfilter set.\n\nThe optical and IR channels are observed simultaneously using a dichroic beam\nsplitter.\nThe observing cadence varies from nightly for new novae to $\\sim$annual\nmonitoring for the oldest novae in our list.\n\nWe perform aperture photometry on the target and between 1 and 25\ncomparison stars in the field. The aperture radius R is either 5 or 7\npixels, depending on field crowding and sky brightness.\nThe background is the median\nvalue in an annulus of inner radius 2R and outer radius 2R+20 pixels centered\non the extraction aperture.\nInstrumental magnitudes are recorded for each star. There are cases where the\nfading remnant becomes blended with nearby stars (within $\\sim$1.5~arcsec).\nTo date we have not accounted for such blending.\nEventually we plan to employ PSF-fitting techniques\nin these crowded regions.\n\nOn most photometric nights an observation of a \\cite{Landolt92} standard field\nis taken. On those nights \nwe determine the zero-point correction and determine the magnitudes of the\ncomparison stars. We adopt the mean magnitudes for each comparison star.\nThese are\ngenerally reproducible to better than 0.02~mag; variable stars are identified\nthrough their scatter around the mean, and are not used in the \ndifferential photometry.\nWith only a single observation of a standard star field each night,\nwe assume the nominal atmospheric extinction law and zero color correction. \nUsing differential photometry, we can\nrecover the apparent magnitude of a target with \na typical uncertainty of $<$0.03 mag at 20$^{th}$ magnitude.\n\nWhile we could do the same with the IR channel images, we find it simpler to\nuse the catalogued 2MASS magnitudes of the standard stars. We implicitly\nassume that the 2MASS comparison stars are non-variable, and that the color\nterms in the photometric solution are negligible.\n\nIn addition, some higher cadence data have been obtained with the SMARTS 0.9m\nand 1.0m telescopes.\nThe 0.9m detector is a 2048x2046 CCD with a 0.401~arcsec\/pixel plate scale.\nOn the 1.0m, we used the 512x512 Apogee camera that was employed prior to\ninstallation of the 4K camera.\nWe perform the differential photometry\nin a manner identical to that for the ANDICAM, and merge the data sets.\nThese data are not yet fully incorporated into the atlas.\n\n\\section{Setup of the Atlas}\n\nThe atlas is on-line at\n\\url{http:\/\/www.astro.sunysb.edu\/fwalter\/SMARTS\/NovaAtlas\/}.\n\nThe atlas consists of a main page for each nova, giving finding charts (in\nboth $V$ and $K$ bands), coordinates, and links to the spectra and photometry.\nThe spectra are available as images, and may be downloaded in ascii (text)\nformat. The photometry page shows plots of the light curve and colors,\nand permits one to download the data in ascii format.\nNote that the plots on the photometry page only show data with formal\nuncertainties $<$0.5~mag, while all measured magnitudes and uncertainties\nare included in the ascii listings.\nThere is a link to a page of references for other observations of the novae.\n\n\\section{The Novae}\n\nAs of 1 July 2012 the atlas includes data on 64 novae. \nOf these, 29 are still bright enough (V$<$18) to reach spectroscopically\nwith the 1.5m\/RC spectrograph. Most are still detectable photometrically\nwith the 1.3m\/ANDICAM imager. Only 5 are no longer on our photometric\ntarget list because\nthey are too faint or too confused with brighter companions.\n\nThe spatial distribution of these novae is shown in\nFigure~\\ref{fig_spdist} in both celestial and galactic\ncoordinates.\n\nLists of our targets and particulars \non the number and observing date distribution of the \nobservations are in Tables~\\ref{tbl-obs} (novae from before 2012);\n\\ref{tbl-obs_new} (novae discovered in 2012), and \\ref{tbl-obs_LMC} (novae in\nthe LMC). The reference time is ideally the time of peak brightness, but this\nis often not well known. In general, T$_0$ is the time is discovery.\nIn the case of T Pyx, which rises very slowly, T$_o$ is the time of peak\nbrightness as estimated from our photometry. For novae that were discovered well\npast peak, including N Sgr 2012b and XMMU J115113.3-623730, T$_0$ is a guess.\nAll the dates in the Tables are referenced to T$_0$.\nThe tabulated $V$ is the last observed $V$ magnitude;\nin most cases this is\nthe brightness in June 2012.\nThe Tables are current as of 1 July 2012. \n\n\n\\normalsize\n\\subsection{Observing Statistics}\n\nAs of 1 July 2012 the full atlas contains 64 novae.\nWe have between 1 and 368 spectra for the novae, with a median of 28\nspectra per nova. The number of photometric points varies between 1 and 265,\nwith a median of 35,\nfor 53 novae. Since some of the observations were taken through thick clouds,\nnot all observations have the best possible S\/N.\n\nThe photometric and spectral coverage is generally non-uniform in time. \nIn addition to annual gaps due to the Sun, there is spotty coverage during the\naustral winter when the weather becomes worse. We do not have unlimited \nobserving time, so we concentrate on those novae that tickle our astronomical\nfancy - the He-N novae, and those showing unusual characteristics. We \ndo not attempt spectroscopy of targets fainter than $V\\sim$18, because\nthe 1.5m telescope has limited grasp. \n\nWe generally do not make great efforts to obtain photometry from day 0, because\namateur astronomers do such a good job. In many cases data available from the\nAAVSO can fill in the first few weeks, while the nova is bright (we do have\nbright limits near $V=8$ and $K=6$). Our forte is the ability\nto a) follow the evolution to quiescence, and b) to do so in the 7 photometric\nbands from $B$ through $K_s$. \nIn one case we were on the nova 1.1 days after discovery,\nbut the median delay is 15 days.\n\nWe try to start the spectroscopic monitoring sooner, because this is a unique\ncapability of SMARTS. The first spectrum is obtained with a median delay of\n8.0 days from discovery, but we have observed 1 nova within 0.6~days of\ndiscovery, 9 within 2 days, and 14 within 3 days. \n\nWe have multi-epoch photometry of 52 novae over timespans of up to\n3173 days (8.7 years),\nand multi-epoch spectroscopy of 63 novae over timespans of up to\n3156 days (8.6 years). These durations will increase with time so long as\nSMARTS continues operating, and the targets are sufficiently bright. The median\nobservation durations of 1317 and 360 days, respectively, for the photometry\nand spectroscopy, are limited mostly by target brightness.\nThe median time between observations is skewed by the growing number of old,\nfaint targets that are now observed with a cadence of 1-2 observations per\nyear, so that the median time between spectra is 8.4 days, and is 27 days\nfor photometric observations.\n\n\\subsection{The Example of V574 Pup}\n\nWe illustrate possible uses of the atlas with the example of V574 Pup,\nan \\ion{Fe}{2} nova for which we have good coverage.\nAside from near-IR observations \\citep{N10}\nand analysis of the super-soft X-ray source \\citep{Swift11},\nthere has been little discussion of this bright nova.\n\nThe main atlas page (Figure~\\ref{fv574main})\npresents finding charts in $V$ and $K$, along with the coordinates, time of\ndiscovery, and links to the spectral and photometric data and references.\n\nThe photometry consists of observations taken on 100 days with the 1.3m\nANDICAM imager, starting on day 32 and\nrunning through day 2723 (5 May 2012). Most of these sets include all 7 ANDICAM\nbands, $BVRIJHK_s$. This light curve is shown in Figure~\\ref{v574_lc}.\nIt is possible to fill in the first 30 days with data from other sources,\nsuch as \\cite{S05}, or by using data from the AAVSO (www.aavso.org).\n\nWe supplemented these with data taken on 20 days using\na temporary small CCD on the SMARTS 1.0m telescope. These were opportunistic\nobservations enabled by the unavailability of the wide-field 4k camera.\nWe use the $\\sim$2 hour long sequences to search for short\nperiodicities. (Similar data exist for very few of the novae in the\natlas.) Three long sequences in the $B$~band,\non days 87, 195, and 196, showed sinusoidal-like modulations.\nRemoving a linear trend from the data on day 87 and normalizing to the mean\nmagnitudes, we find a likely period of 0.0472 days (68 minutes; see\nFigure~\\ref{fv574_var}) from a shortest-string analysis \\citep{D83}.\nHowever, we cannot exclude some aliases. This is shorter than the minimum\norbital period for CVs, and may be half an orbital period (ellipsoidal\nvariability is a possible explanation).\nThe amplitude of the best-fit sinusoid decreased from 0.02 to 0.007~mag\nfrom day 87 to days 195-196.\n\nWe obtained 107 spectra before the target became too faint for the 1.5m\ntelescope.\nWe illustrate two types of investigations\nthat can be supported by high cadence spectral observations.\n\n\\begin{enumerate}\n\\item Figure~\\ref{fv574pc} shows the evolution of the P~Cygni line profiles\nas the wind evolves against the backdrop of the optically-thick\npseudo-photosphere. With daily spectra, it is clear that the absorption\nvelocities are not constant, but rather are accelerating. Through day 14\nthe velocities can be described as a quadratic function of time. Hence the\nacceleration is linear in time. It is hard to see how this can result\nfrom decreasing optical depth effects in an envelope with a monotonic\nvelocity law increasing outwards. \\cite{SNS11} show how similar structures\nseen in T~Pyx can be explained as an outward-moving recombination front in\nan envelope with a linear velocity law.\n\n\n\\item Figure~\\ref{fv574fe} shows the time-evolution of a series of lines of\ndiffering temperatures as the nova evolves through the nebular and coronal\nphases. The [\\ion{Fe}{10}] $\\lambda$6375\\AA\\ line requires high excitation, and \nits presence correlates well with the super-soft (SSS) X-ray emitting\nphase \\citep{Swift11}. V574~Pup was in its SSS phase from before\nday 180 through day 1118; it ended before day 1312. We can use data such as\nthese to explore how well optical lines are diagnostic of the SSS phase.\n\n\\end{enumerate}\n\n\n\\subsection{Notes on the Novae}\n\nNotes here are not meant to be complete or definitive in any sense. They are \nmeant to highlight past or ongoing work on select novae, or to note some\nparticularly interesting cases. We have made no attempt to provide complete\nreferences here; they are in the on-line atlas.\nFor the convenience of the reader, we have collected in\nTable~\\ref{tbl-meas} various basic measurements. These are:\n\\begin{itemize}\n\\item Spectroscopic class. This is a phenomenological classification\nbased on the appearance of the spectrum\nin the first few spectra after the emission lines appear.\nPhysically, this is likely an\nindicator of the optical depth of the envelope.\nOf the 63 classifiable targets, most (47\/63, or 75\\%) are \\ion{Fe}{2} type;\n15 (24\\%) are or may be He-N, and one is a possible symbiotic nova.\nIn one case we cannot tell because our first spectrum was obtained nearly \n2 years after peak.\n\nWe append a ``w'' in those cases where there is a clear P~Cygni absorption\nin the Balmer lines (and sometimes in \\ion{Fe}{2})\nindicative of an optically-thick wind.\nHalf (32 of the 64 novae) show such P~Cyg absorption.\nWe caution that the absence of P~Cyg absorption may\nbe caused by the cadence of the observations.\n\n\\item Photometric class. We examined the $V$ band light\ncurves for the first 500 days and categorized them by eye into one or more\nof the 7 classes defined by \\cite{SSH10}. In many cases we have very little data\nduring the first 3 months, and do not attempt to categorize these. In some cases\nwe had a hard time shoe-horning the lightcurve into one class, and have given\nmultiple classes. For example, N LMC 2005 maintained a fairly flat light\ncurve for about 50 days (class F), then exhibited a cusp (class C). It also\nformed dust (class D), though the dip is not particularly pronounced. The\npresence of dust is indicated by the increase in the $H$ and $K$ fluxes\nas the optical fades.\n\nIn many cases a significant brightening in $K$, suggestive of dust formation,\nis not accompanied by an optical dip, suggesting an asphericity in the dust.\n\nIn some cases there is significant color evolution between the optical\nand near-IR. We will quantify this later.\n\n\\item The FWHM of the H$\\alpha$ emission line. We measure the first grating 47\nspectrum (3.1\\AA\\ resolution) that does not show P~Cyg wind absorption,\nand report\nthe day on which that spectrum was obtained. Uncertainties are of order 2\\%.\nNote that the FWHM can change significantly with time in the \\ion{Fe}{2} novae.\nFor the He-N novae we measure the FWHM of the broad base, ignoring the narrow\ncentral emission component. In some cases there is a faint but broader\ncomponent visible early on. The measurement of the FWZI of this component\nwould be more representative of the maximum expansion velocity.\nWe do not tabulate this because of incompleteness, and because of the\ndifficulty defining the continuum level in some cases.\n\\end{itemize}\n\nWe have not estimated the times for\nthe light to decay by 2 and 3 magnitudes at $V$\n(t$_2$ and t$_3$, respectively) in\nany systematic manner, because it is only in rare cases that we have \nsufficiently dense photometric sampling early enough to make a good estimate.\nWe discuss these in the notes on individual novae. \n\nWe note that the estimates of t$_2$ and t$_3$ can be highly uncertain,\nespecially for fast novae. The reported discovery times are often past\nthe peak. The discovery magnitudes are often visual estimates, or \nunfiltered CCD magnitudes, necessitating a color correction to $V$.\nA full analysis of the light curves,\nincorporating other published literature, AAVSO data (which are much denser\nnear peak), and data from other\nsources, is beyond the scope of this paper.\n\n\n\\subsubsection{N Aql 2005 = V1663 Aql}\nThis is a standard \\ion{Fe}{2} nova. On day 50 there was prominent\n$\\lambda$4640\\AA\\ Bowen blend emission.\nThe auroral [\\ion{O}{3}] lines were strong by day 85. \nOur last spectrum, on day 414, is dominated by\nH$\\alpha$, [\\ion{O}{3}] 4959\/5007, [\\ion{N}{2}] 5755, [\\ion{Fe}{7}] 6087,\n[\\ion{O}{1}] 6300, and [\\ion{Ar}{3}] 7136.\n\n\\subsubsection{N Car 2008 = V679 Car}\nThis \\ion{Fe}{2} nova never seemed to develop a coronal phase. \nWe have limited photometric coverage.\n\n\\subsubsection{N Car 2012 = V834 Car}\nThis recent \\ion{Fe}{2} nova exhibited a strong wind through day 36.\nEvolution of the light curve has been uneventful. There was some jitter of\n$\\pm$0.5 mag from a smooth trend from days 12-40. We estimate\nt$_2$ and t$_3$ to be 20 and 38 days, respectively, with uncertainties of order\n$\\pm$3 days for t$_2$ and $\\pm$1 day for t$_3$.\n\n\\subsubsection{N Cen 2005 = V1047 Cen}\nWe have no photometry, and only two spectra, of this \\ion{Fe}{2} nova.\n\n\\subsubsection{N Cen 2007 = V1065 Cen}\nThis dusty \\ion{Fe}{2} nova was analyzed by \\cite{Hel10}, using SMARTS spectra\nthrough day 719. The atlas includes additional photometry, from days\n944 though 1850.\n\n\\subsubsection{N Cen 2009 = V1213 Cen}\nThis \\ion{Fe}{2} nova became a bright super-soft X-ray source. The coronal phase\nextended from about days 300 to 1000, roughly coinciding with the SSS phase\n\\citep{Swift11}, with strong lines of\n[\\ion{Fe}{10}], [\\ion{Fe}{11}], and [\\ion{Fe}{14}].\nIn quiescence the remnant is blended with two other objects of\ncomparable brightness.\n\n\\subsubsection{PNV J13410800-5815470 = N Cen 2012}\nThis recent \\ion{Fe}{2} nova exhibited wind absorption through day 25. \nt$_2$ is about 16$\\pm$1 days; t$_3$ occurs about day 34.\nThe 2 mag brightening in $K$ starting about day 35,\nwith a contemporaneous drop in the $B$ and $V$~band brightness, \nsuggests dust formation.\nThe strong emission in the \\ion{Ca}{2} near-IR triplet on\nday 11 had disappeared by day 74.\n\n\\subsubsection{PNV J14250600-5845360 = N Cen 2012b}\nThe $K$~band brightness increased by 2 magnitudes between days 18 and 32,\nsuggesting dust formation, but no drop is seen at optical magnitudes.\nThe smooth $V$ light curve yields t$_2$ and t$_3$ of 12.3 and 19.8 days, with \nuncertainties $<$1 day.\nThe spectral development is similar to N Cen 2012.\nThe strong emission in the \\ion{Ca}{2} near-IR triplet on\nday 16 had disappeared by day 61.\n\n\\subsubsection{N Cir 2003 = DE Cir}\nThis fast nova was discovered by \\cite{L03} in the glare of the setting Sun.\nSpectra obtained on days 11 and 12, at high air mass, show this was a He-N\nnova. We did not obtain any photometry until after it reappeared from behind the\nSun. Since then it has been in quiescence at V$\\sim$17, with a variance of\n$\\pm$0.4~mag. The strongest line in the quiescent spectrum is\nHe~II~$\\lambda$4686\\AA.\n\n\\subsubsection{N Cru 2003 = DZ Cru}\nThis is another nova that was discovered in the west in the dusk twilight.\nDespite the discussion about the ``peculiar'' early spectrum \\citep{iauc8185},\nour spectra show this was an \\ion{Fe}{2} nova discovered before maximum, as\nconcluded by \\cite{Rus08}.\n\n\\subsubsection{N Dor 1937 = YY Dor}\nThis is the second recurrent nova discovered in the LMC \\citep{lil04}. \nIt is a fast (t$_2$,t$_3$=4.0, 10.9 days, respectively)\nHe-N nova with the broad tripartite Balmer lines seen in many\nfast recurrent novae.\nAn analysis is in preparation.\n\n\\subsubsection{N Eri 2009 = KT Eri}\nKT Eri is a fast He-N nova with a bright quiescent counterpart.\n\\cite{Hou10} reported a spectacular pre-maximum light curve from\nthe SMEI instrument. The light curve shows two plateaus (Figure~\\ref{kteri-lc}), \nmuch like those seen in U~Sco, prior to dropping to quiescence.\n\\cite{Jur12} find a 737 day period in the quiescent source\nfrom archival plate material. \\cite{Hun11} claim a 56.7 day period \nduring the second plateau. In quiescence, after day 650, \nthere are hints of a period near 55 days,\nand a possibly 4.2 day spectroscopic period (Walter et al. in preparation). \nDue to its brightness and RA (there is less competition for time towards the\ngalactic anti-center), we have excellent spectral time coverage.\nFigure~\\ref{kteri-trsp} shows the time-evolution of KT~Eri in the blue, over\n790 days, from 83 low dispersion blue spectra.\n\nKT~Eri is located in a sparse field; there is only a single comparison star\navailable in the small IR channel field of view.\n\n\n\\subsubsection{N Lup 2011 = PR Lup}\nThe light curve of this slow \\ion{Fe}{2} nova showed\na second maximum about day 3.5. There was little appreciable decay\nduring the first 2 months.\nWind absorption was evident through day 58.\nAs of day 300 it has not entered the coronal phase. \n\n\\subsubsection{N Mus 2008 = QY Mus}\nWe picked up this fairly late, but a combination of the\nspectroscopy and photometry span the time that dust formed.\nIt seems to be a standard \\ion{Fe}{2} nova that bypassed the coronal phase \nand is now in the nebular phase.\n\n\\subsubsection{N Nor 2005 = V382 Nor}\nThis appears to be a standard \\ion{Fe}{2} nova.\n\n\\subsubsection{N Nor 2007 = V390 Nor}\nWe have good spectroscopic coverage of this \\ion{Fe}{2} nova for 4 months,\nbut no photometry. During this time it did not evolve any hot lines.\n\n\\subsubsection{RS Oph}\nThis is the prototypical long period, wind-driven recurrent nova. \nThe emission lines are narrow.\nWe are continuing observations to characterize it well into quiescence. \n\n\\subsubsection{N Oph 2003 = V2573 Oph}\nBased on two spectra, this appears to be a standard\n\\ion{Fe}{2} nova.\n\n\\subsubsection{N Oph 2004 = V2574 Oph}\nOur photometric coverage consists of two observations, 4 and 8 years\nafter the eruption. Neither was obtained on a photometric night, so the\ndata are not yet photometrically calibrated.\nWe have good spectroscopic coverage showing the transition\nfrom an optically thick pseudophotosphere to the permitted line spectrum in\nthis \\ion{Fe}{2} nova.\n\n\\subsubsection{N Oph 2006 = V2575 Oph}\nThis is a standard \\ion{Fe}{2} nova.\n\n\\subsubsection{N Oph 2006b = V2576 Oph}\nMost photometric observations are in $R$ only. It is a standard\n\\ion{Fe}{2} nova.\n\n\\subsubsection{N Oph 2007 = V2615 Oph}\nThis is a normal \\ion{Fe}{2} nova. Photometric coverage begins after 1.5 years.\n\n\\subsubsection{N Oph 2008 = V2670 Oph}\nThis is an \\ion{Fe}{2} nova.\n\n\\subsubsection{N Oph 2008b = V2671 Oph}\nThis \\ion{Fe}{2} nova faded rapidly, and was undetectable,\nexcept at $R$, after 1 year.\n\n\\subsubsection{N Oph 2009 = V2672 Oph}\n\\cite{Mun11} reported on this very fast nova. t$_2$ and t$_3$ passed before our\nfirst photometry; \\cite{Mun11} quote values of 2.3 and 4.2 days, respectively.\nThe spectra and spectral evolution are similar to those\nof U~Sco. We followed this nova spectroscopically through day 31.6, and did not\nsee the deceleration reported by \\cite{Mun11}.\nThis has the broadest H$\\alpha$ line, at FWZI$\\sim$11,000~km\/s, of all the novae\nin the atlas, exceeding that of U~Sco by about 20\\%.\n\n\\subsubsection{PNV J17260708-2551454 = N Oph 2012}\nThis recent slow \\ion{Fe}{2} nova showed no significant decline in\nbrightness from days 10 through 90. Then dust formed, with a drop in the\n$B$ and $V$ brightness by over 5 magnitudes.\nThe lines are narrow: FWHM(H$\\alpha$)$\\sim$950~km\/s.\nThere is little spectral evolution through day 90, with \npersistent P~Cygni line profiles.\n\n\\subsubsection{PNV J17395600-2447420 = N Oph 2012b}\nThis is an \\ion{Fe}{2} nova with an H$\\alpha$ FWHM about 3000~km\/s.\nThrough the first 45 days the brightness drops monotonically in $B$ through $K$.\n\n\\subsubsection{N Pup 2004 = V574 Pup}\nSee \\S4.2. Our first photometry was on day 30, suggesting t$_3<27$ days. \n\\cite{Swift11} quote t$_2$=13 days.\n\n\\subsubsection{N Pup 2007 = V597 Pup}\nThis slowly developing, broad-lined nova developed a coronal phase after\n3-4 months.\n\n\\subsubsection{N Pup 2007b = V598 Pup}\nThis X-ray-discovered nova was in its nebular phase when reported by\n\\cite{RSE07}. The quiescent counterpart appears fairly bright.\n\n\\subsubsection{T Pyx}\nThis well-known recurrent novae is, as has been pointed out by many authors,\nvery different from the fast He-N recurrent novae. Its light curve and spectral\ndevelopment mimic those of slow classical novae. \nThe rate of the photometric decay remained unchanged by the turn-on\/turn-off\nof the SSS X-ray emission (Figure~\\ref{tpyx-sss}).\nThe slope of the photometric decay decreased about day 300.\nThe H$\\alpha$ line width shown in Table~\\ref{tbl-meas} refers to the\nwidth of the broad base after the line profile stabilized; it was about half\nthat during the first 40 days post-peak.\n\n\\cite{Eva12} include some of our near-IR photometry in an analysis of the\nheating of the dust already present in the system.\n\nAs T Pyx remains bright, we are continuing our monitoring.\n\n\\subsubsection{U Sco}\nThis is the prototypical short orbital period recurrent nova.\nSome spectral analysis in included in \\cite{Max12}.\n\n\\subsubsection{N Sco 2004b = V1187 Sco}\nThis well-observed \\ion{Fe}{2} nova became a super-soft X-ray source.\n\n\\subsubsection{N Sco 2005 = V1188 Sco}\nWe have only limited coverage of this \\ion{Fe}{2} nova.\n\n\\subsubsection{N Sco 2007a = V1280 Sco}\nThis extraordinarily slow nova has remained bright ($V$ generally\nbetween 10 and 11) for nearly 2000 days. \\cite{N12} present a lot of data\nfor this nova; our monitoring provides finer temporal coverage, which\nshow absorption events with durations $<$60 days that may be due to dust\nformation in small mass ejection events. The latest spectra, at an age of 5.3\nyears, still show evidence for wind absorption.\n\n\\subsubsection{N Sco 2008 = V1309 Sco}\nThis is a very narrow-lined system, and likely a symbiotic nova\nor a merger \\citep{Mas10}. The spectrum is dominated by narrow Balmer line\nemission.\n\n\\subsubsection{N Sco 2010 No.~2 = V1311 Sco}\nThis nova faded rapidly. It is likely a He-N class nova. On day 9\npossible [Ne III] $\\lambda$3869 is seen, and the $\\lambda$4640 Bowen blend\nis in emission.\n\n\\subsubsection{N Sco 2011 = V1312 Sco}\nThis \\ion{Fe}{2} nova developed coronal line emission.\n\n\\subsubsection{N Sco 2011 No.~2 = V1313 Sco}\nThis nova exhibited very strong \\ion{He}{1} and H-Paschen line emission.\nStrong \\ion{He}{2}\n$\\lambda$4686 emission appeared between days 16 and 19.\nThe Balmer lines have a narrow central core atop a broad base, as in the He-N\nand recurrent novae, but there is also likely \\ion{Fe}{2} multiplet 42\nemission at\n$\\lambda$5169\\AA\\ (other multiplet 42 lines are overwhelmed by \\ion{He}{1}\nemission lines), and wind absorption at least through day 3. This seems to be\na hybrid nova.\nt$_2$ lies between 5.8 and 8.5 days, and t$_3$ between 13 and 18 days,\ndepending on whether the peak was on\n2011 Sep 6.37 or 2011 Sep 7.51 \\citep{Sea11}.\nThe continuum is red. This may be a symbiotic nova.\n\n\\subsubsection{N Sct 2003 = V475 Sct}\nThis was the first nova that we concentrated on. Results appear in \n\\cite{Str06}. This fairly narrow-lined \\ion{Fe}{2} nova may have formed dust;\nno coronal phase was seen. We have $U$-band photometry for the first 100 days,\n\n\\subsubsection{N Sct 2005 = V476 Sct}\nWe have very limited observations of this \\ion{Fe}{2} nova.\n\n\\subsubsection{N Sct 2005b = V477 Sct}\nThis is probably an \\ion{Fe}{2} nova; we have very poor coverage.\n\n\\subsubsection{N Sct 2009 = V496 Sct}\nThis \\ion{Fe}{2} nova likely formed dust. As it is fairly bright, we have good\nspectral coverage for nearly 2 years.\n\n\\subsubsection{N Sgr 2002c = V4743 Sgr}\nThis is the first nova we started observing. We picked it up about 200 days\nafter discovery. There is currently no photometry in the atlas.\n\n\\subsubsection{N Sgr 2003 = V4745 Sgr}\nThis was the first nova to explode during the SMARTS era.\nThere are two prominent P~Cygni absorption line systems,\ninitially at -780 and -1740\nkm\/s, visible from days 12 through 66; they disappear by day 71.\nThere is currently no photometry in the atlas.\n\n\\subsubsection{N Sgr 2004 = V5114 Sgr}\nData for this \\ion{Fe}{2} nova have been analyzed and published by \\cite{E06}.\n\\cite{E06} quote t$_2$ and t$_3$ values of 11 and 21 days; for a peak\n$V$=8.38 we find a marginally slower nova, with t$_2$ and t$_3$ of 14 and\n25 days, respectively.\nWe have $U$ band photometry for the first 180 days.\n\n\\subsubsection{N Sgr 2006 = V5117 Sgr}\nThis is a standard \\ion{Fe}{2} nova. We estimate t$_2$ and t$_3$ are about\n16$\\pm$1 and 42 days, respectively, with uncertainties of perhaps a week in t$_3$.\n\n\\subsubsection{N Sgr 2007 = V5558 Sgr}\nThis very slow nova resembles V723~Cas. We started the photometry at about\nday 100; the nova remained at $7.00$, $1 \\leq k \\leq d$, can be used to define a family of criteria for optimal experimental design, concave for $\\tau\\leq 1\/k$, for which an equivalence theorem can be formulated.\n\n\n\n\\subsection{Quadratic entropy and learning}\\label{S:learning}\nIn a series of papers \\citep{rao1982a, rao1982b, rao1984convexity, rao2010quadratic} C.R. Rao and co-workers have introduced a quadratic entropy which is a generalised version of the $k=2$ functional of this section but with a general kernel $K(x_1,x_2)$ in $\\mathds{R}^d$:\n\\begin{equation}\\label{quadratic-entropy}\n Q_R = \\int \\int K (x_1,x_2) \\mu(\\dd x_1)\\mu(\\dd x_2)\\,.\n\\end{equation}\nFor the discrete version\n$$\nQ_R = \\sum_{i=1}^N \\sum_{j=1}^N K(x_{i},x_{j})\\,p_i\\,p_j,\n$$\nRao and co-workers developed a version of the Analysis of Variance (ANOVA), which they called Anaysis of Quadratic Entropy (ANOQE), or Analysis of Diversity (ANODIV).\nThe Gini coefficient, also used in the continuous and discrete form is a special case with $d=1$ and $K(x_1,x_2) = |x_1-x_2|$.\n\nAs pointed in \\citep[Chap.~3]{rao1984convexity}, a necessary and sufficient condition for the functional $Q_R$ to be concave is\n\\begin{equation}\n\\label{semi}\n\\int \\int K(x_1,x_2) \\nu(\\dd x_1)\\nu(\\dd x_2) \\leq 0\n\\end{equation}\nfor all measures $\\nu$ with $\\int \\nu(\\dd x)=0$. The discrete version of this is\n$$\n\\sum_{i=1}^N \\sum_{j=1}^N K (x_i,x_j)\\, q_i\\, q_j \\leq 0\n$$\nfor any choice of real numbers $q_1,\\ldots,q_N$ such that $\\sum_{i=1}^N q_i=0$.\n\\citet{schilling2012bernstein} discuss\nthe general problem of finding for what class of continuous functions $B(\\cdot)$ of $\\|x_1 - x_2\\|^2$ does the kernel\n$\nK(x_1, x_2) = B\\left (\\|x_1 - x_2\\|^2\\right)\n$\nsatisfy \\eqref{semi}: the solution is that $B(\\cdot)$ must be a so-called Bernstein function. We do not develop these ideas here, but note that $B(\\lambda) = \\lambda^{\\alpha}$ is a Bernstein function for all $0 < \\alpha \\leq 1$.\nThis is the reason that, above,\nwe can claim concavity for $k=1$ and all $0 < \\delta \\leq 2$ in \\eqref{dV}.\n\n\\cite{hainy2014learning}\ndiscuss the link to embedding and review some basic results related to Bayesian learning. One asks\nwhat is the class of functionals $\\psi$ on a distribution $\\mu(\\theta)$ of a parameter in the Bayesian statistical learning\nsuch that for all $\\mu(\\theta)$ and all sampling distributions $\\pi(x | \\theta)$ one expects to learn, in the preposterior sense:\n$\\psi(\\mu(\\theta)) \\leq \\Ex_\\nu \\psi(\\pi(\\theta|X))$, with $X\\sim \\nu$.\nThe condition is that $\\psi$ is convex, a result which has a history but is usually attributed to\n\\cite{degroot1962uncertainty}.\nThis learning is enough to justify calling such a functional a generalised information functional, or a general learning functional. Shannon information falls in this class, and earlier versions of the result were for Shannon information. It follows that wherever, in this paper, we have a concave functional then its negative is a learning functional.\n\n\\section{Functionals based on squared volume}\n\\label{S:squared volume}\n\n\nIn the rest of the paper we focus our attention on the functional\n$$\n\\mu\\in\\SM \\longrightarrow \\psi_k(\\mu)=\\phi_{[k],2,1}(\\mu)=\\Ex\\{ \\SV_k^2(x_1,\\ldots,x_{k+1}) \\}\\,,\n$$\nwhich corresponds to the mean squared volume of simplices of dimension $k$ formed by $k+1$ independent samples from $\\mu$. For instance,\n\\begin{equation}\\label{psi2}\n \\psi_2(\\mu) = \\int\\int\\int \\SV_2^2(x_1,x_2,x_3)\\, \\mu(\\dd x_1)\\, \\mu(\\dd x_2)\\, \\mu(\\dd x_3)\\,,\n\\end{equation}\nwith $\\SV_2(x_1,x_2,x_3)$ the area of the triangle formed by the three points with coordinates $x_1$, $x_2$ and $x_3$ in $\\mathds{R}^d$, $d\\geq 2$. Functionals $\\phi_{[k],\\delta,\\tau}(\\mu)$ for $\\delta \\neq 2$ will be considered in another paper, including the case of negative $\\delta$ and $\\tau$ in connection with space-filling design for computer experiments.\n\nTheorem~\\ref{Prop:1} of Section~\\ref{S:main-theorem} indicates how $\\psi_k(\\mu)$ can be expressed as a function of $V_\\mu$, the covariance matrix of $\\mu$, and shows that $\\phi_{[k],2,1\/k}(\\cdot)$ satisfies properties (a), (b) and (c) of Section~\\ref{S:intro}. The special case of $k=d$\nwas known to Wilks (1932, 1960)\\nocite{wilks1932, wilks1960} in his introduction of generalised variance, see also \\cite{van1965note}. The connection with U-statistics is exploited in Section~\\ref{S:empirical}, where an unbiased minimum-variance estimator of $\\psi_k(\\mu)$ based on a sample $x_1,\\ldots,x_n$ is expressed in terms of the empirical covariance matrix of the sample.\n\n\\subsection{Expected squared $k$-simplex volume}\\label{S:main-theorem}\n\n\\begin{theorem}\\label{Prop:1} Let the $x_i$ be i.i.d.\\ with the probability measure $\\mu\\in\\SM$. Then, for any $k\\in\\{1,\\ldots,d\\}$, we have\n\\begin{eqnarray}\n\\psi_k(\\mu) &=& \\frac{k+1}{k!} \\, \\sum_{i_10$ for $p>0$ and any $q>0$. The family of functionals \\eqref{Kiefer-phi_p} is therefore unable to detect the true dimensionality of the data. On the other hand, $\\psi_k(\\mu)=0$ for all $k>q$ when rank $V_\\mu=q$.\n\n\\subsection{Empirical version and unbiased estimates}\\label{S:empirical}\n\nLet $x_1,\\ldots,x_n$ be a sample of $n$ vectors of $\\mathds{R}^d$, i.i.d.\\ with the measure $\\mu$. This sample can be used to obtain an empirical estimate $({\\widehat\\psi}_1)_n$ of $\\psi_k(\\mu)$, through the consideration of the ${n \\choose k+1}$ $k$-dimensional simplices that can be constructed with the $x_i$. Below we show how a much simpler (and still unbiased) estimation of $\\psi_k(\\mu)$ can be obtained through the empirical variance-covariance matrix of the sample. See also \\cite{wilks1960, Wilks1962}.\n\nDenote\n\\begin{eqnarray*}\n\\widehat x_n &=& \\frac1n\\, \\sum_{i=1}^n x_i \\,, \\\\\n\\widehat V_n &=& \\frac{1}{n-1}\\, \\sum_{i=1}^n (x_i-\\widehat x_n)(x_i-\\widehat x_n)\\TT = \\frac{1}{n(n-1)}\\, \\sum_{i1$, this property would not hold if $\\widehat V_n$ were replaced by another unbiased estimator of $V_\\mu$.\n\n\\vsp\nThe value of $({\\widehat\\psi}_k)_n$ only depend on $\\widehat V_n$, with $\\Ex\\{({\\widehat\\psi}_k)_n\\}=\\psi_k(V_\\mu)$, but its variance depends on the distribution itself.\nAssume $\\Ex\\{\\SV_k^4(x_1,\\ldots,x_{k+1})\\}<\\infty$.\nFrom \\cite[Lemma A, p.~183]{Serfling80}, the variance of $({\\widehat\\psi}_k)_n$ satisfies\n$$\n\\var[({\\widehat\\psi}_k)_n]=\\frac{(k+1)^2}{n}\\, \\omega + O(n^{-2}) \\,,\n$$\nwhere $\\omega= \\var[h(x)]$, with $h(x)=\\Ex\\{\\SV_k^2(x_1,x_2,\\ldots,x_{k+1})|x_1=x\\}$. Obviously, $\\Ex[h(x)]=\\psi_k(\\mu)$ and calculations similar to those in the proof of Theorem~\\ref{Prop:1} give\n\\begin{eqnarray}\n&& \\hspace{0.5cm} \\omega = \\frac{1}{(k!)^2}\\, \\sum_{I,J} \\det[\\{V_\\mu\\}_{I\\times I}] \\, \\det[\\{V_\\mu\\}_{J\\times J}] \\label{zeta} \\\\\n&& \\times\\, \\left[\\Ex\\left\\{ (E_\\mu-x)_I\\TT \\{V_\\mu\\}_{I\\times I}^{-1} (E_\\mu-x)_I (E_\\mu-x)_J\\TT \\{V_\\mu\\}_{J\\times J}^{-1} (E_\\mu-x)_J \\right\\} - k^2\\right]\\,, \\nonumber\n\\end{eqnarray}\nwhere $I$ and $J$ respectively denote two sets of indices $i_10$ and $F_{\\psi_k}(\\mu_k^*;\\nu) \\leq 0$ for all $\\nu\\in\\SM$, that is\n\\begin{equation}\\label{CNS-nu}\n\\tr\\left\\{\\nabla_{\\Psi_k}[V_{\\mu_k^*}]\\,\\frac{\\dd V_{(1-\\ma)\\mu_k^*+\\ma\\nu}}{\\dd\\ma}\\bigg|_{\\ma=0} \\right\\} \\leq 0 \\,, \\ \\forall\\nu\\in\\SM\\,.\n\\end{equation}\nWe obtain the following.\n\n\\begin{theorem}\\label{th:equivTh} The probability measure $\\mu_k^*$ such that $\\psi_k(\\mu_k^*)>0$ is $\\psi_k$-optimal, that is, maximises $\\psi_k(\\mu)$ with respect to $\\mu\\in\\SM$, $k\\in\\{1,\\ldots,d\\}$, if and only if\n\\begin{equation}\\label{CNS}\n \\max_{x\\in\\SX} (x-E_{\\mu_k^*})\\TT \\frac{\\nabla_{\\Psi_k}[V_{\\mu_k^*}]}{\\Psi_k(V_{\\mu_k^*})}(x-E_{\\mu_k^*}) \\leq k \\,.\n\\end{equation}\nMoreover,\n\\begin{equation}\\label{support-mu*}\n(x-E_{\\mu_k^*})\\TT \\frac{\\nabla_{\\Psi_k}[V_{\\mu_k^*}]}{\\Psi_k(V_{\\mu_k^*})}(x-E_{\\mu_k^*}) = k\n\\end{equation}\nfor all $x$ in the support of $\\mu_k^*$.\n\\end{theorem}\n\n\n\\begin{proof}\nFirst note that the Newton equations \\eqref{Newton} and the recurrence \\eqref{nabla-E_k} for $\\nabla_{\\mathcal{E}_k}[\\cdot]$ imply that $\\tr(V\\nabla_{\\Psi_k}[V])=k\\Psi_k(V)$ for all $k=1,\\ldots,d$.\n\nThe condition \\eqref{CNS} is sufficient. Indeed, suppose that $\\mu_k^*$ such that $\\psi_k(\\mu_k^*)>0$ satisfies \\eqref{CNS}. We obtain\n$$\n\\int (x-E_{\\mu_k^*})\\TT \\nabla_{\\Psi_k}[V_{\\mu_k^*}](x-E_{\\mu_k^*})\\, \\nu(\\dd x) \\leq \\tr\\left\\{V_{\\mu_k^*}\\nabla_{\\Psi_k}[V_{\\mu_k^*}] \\right\\}\n$$\nfor any $\\nu\\in\\SM$, which gives \\eqref{CNS-nu} when we use \\eqref{derivativeV}. The condition is also necessary since \\eqref{CNS-nu} must be true in particular for $\\delta_x$, the delta measure at any $x\\in\\SX$, which gives \\eqref{CNS}. The property \\eqref{support-mu*} on the support of $\\mu_k^*$ follows from the observation that $\\int (x-E_{\\mu_k^*})\\TT \\nabla_{\\Psi_k}[V_{\\mu_k^*}](x-E_{\\mu_k^*})\\, \\mu_k^*(\\dd x) = \\tr\\left\\{V_{\\mu_k^*}\\nabla_{\\Psi_k}[V_{\\mu_k^*}] \\right\\}$.\n\\end{proof}\n\nNote that for $k0$ implies that $\\psi_{k-1}(\\mu)>0$, $k=2,\\ldots,d$.\n\n\\begin{remark} As a natural extension of the concept of potential in case of order-two interactions ($k=1$), we call\n$P_{k,\\mu}(x)= \\psi_k(\\mu,\\ldots,\\mu,\\delta_x)$ the potential of $\\mu$ at $x$, where\n$$\n\\psi_k(\\mu_1,\\ldots,\\mu_{k+1})= \\int \\ldots \\int \\SV_k^2(x_1,\\ldots,x_{k+1})\\, \\mu_1(\\dd x_1) \\ldots \\mu_{k+1}(\\dd x_{k+1}) \\,.\n$$\nThis yields $F_{\\psi_k}(\\mu;\\nu) = (k+1)\\, [\\psi_k(\\mu,\\ldots,\\mu,\\nu)-\\psi_k(\\mu)]$, where $\\mu$ appears $k$ times in $\\psi_k(\\mu,\\ldots,\\mu,\\nu)$. Therefore, Theorem~\\ref{th:equivTh} states that $\\mu_k^*$ with $\\psi_k(\\mu_k^*)>0$ is $\\psi_k$-optimal if and only if $\\psi_k(\\mu_k^*,\\ldots,\\mu_k^*,\\nu) \\leq \\psi_k(\\mu_k^*)$ for any $\\nu\\in\\SM$, or equivalently $P_{k,\\mu_k^*}(x) \\leq \\psi_k(\\mu_k^*)$ for all $x\\in\\SX$.\n\nIt can be shown that for any measure $\\mu\\in\\SM$, $\\min_{x\\in\\SX} P_{k,\\mu}(x)$ is reached for $x=E_\\mu$, which extends the result of \\citet{wilks1960} about the minimum property of the internal scatter.\n\\end{remark}\n\n\\begin{remark} \\label{R:Radoslav}\nConsider Kiefer's $\\Phi_p$-class of orthogonally invariant criteria and their associated functional $\\varphi_p(\\cdot)$, see \\eqref{Kiefer-phi_p}.\nFrom a result in \\citep{Harman2004-MODA}, if a measure $\\mu_p$ optimal for some $\\varphi_p(\\cdot)$ with $p\\in(-\\infty,1]$ is such that $V_{\\mu_p}$ is proportional to the identity matrix $I_d$, then $\\mu_p$ is simultaneously optimal for all orthogonally invariant criteria. A measure $\\mu_p$ having this property is therefore $\\psi_k$-optimal for all $k=1,\\ldots,d$.\n\\end{remark}\n\n\\begin{remark}\\label{R:otherET} Using \\eqref{EkEd-k}, when $V$ is nonsingular we obtain the property\n$$\n\\Psi_k(V)=\\frac{(k+1)(d-k)!}{(d-k+1)k!}\\, \\det(V)\\, \\Psi_{d-k}(V^{-1})\n$$\nwhich implies that maximising $\\Psi_k(V)$ is equivalent to maximising $\\log\\det(V)+\\log\\Psi_{d-k}(V^{-1})$. Therefore, Theorem~\\ref{th:equivTh} implies that $\\mu_k^*$ with nonsingular covariance matrix $V_{\\mu_k^*}$ maximises $\\psi_k(\\mu)$ if and only if\n$$\n \\max_{x\\in\\SX} (x-E_{\\mu_k^*})\\TT \\left[V_{\\mu_k^*}^{-1}- V_{\\mu_k^*}^{-1}\\,\\frac{\\nabla_{\\Psi_{d-k}}[V_{\\mu_k^*}^{-1}]}{\\Psi_{d-k}(V_{\\mu_k^*}^{-1})}\\,V_{\\mu_k^*}^{-1} \\right](x-E_{\\mu_k^*}) \\leq d- k \\,,\n$$\nwith equality for $x$ in the support of $\\mu_k^*$. When $k$ is large (and $d-k$ is small), one may thus check the optimality of $\\mu_k^*$ without using the complicated expressions of $\\Psi_k(V)$ and $\\nabla_{\\Psi_k}[V]$.\n\\end{remark}\n\n\\subsubsection{A duality property}\n\nThe characterisation of maximum-diversity measures can also be approached from the point of view of duality theory.\n\nWhen $k=1$, the determination of a $\\psi_1$-optimal measure $\\mu_1^*$ is equivalent to the dual problem of constructing the minimum-volume ball $\\SB_d^*$ containing $\\SX$. If this ball has radius $\\rho$, then $\\psi_1(\\mu_1^*)=2\\rho^2$, and the support points of $\\mu_1^*$ are the points of contact between $\\SX$ and $\\SB_d^*$; see \\cite[Th.~6]{Bjorck56}. Moreover, there exists an optimal measure with no more than $d+1$ points.\n\nThe determination of an optimal measure $\\mu_d^*$ is also dual to a simple geometrical problem: it corresponds to the determination of the minimum-volume ellipsoid $\\SE_d^*$ containing $\\SX$. This is equivalent to a $D$-optimal design problem in $\\mathds{R}^{d+1}$ for the estimation of $\\beta=(\\beta_0,\\beta_1\\TT)\\TT$, $\\beta_1\\in\\mathds{R}^d$, in the linear regression model with intercept $\\beta_0+\\beta_1\\TT x$, $x\\in\\SX$, see \\cite{Titterington75}. Indeed, denote\n$$\nW_\\mu= \\int_\\SX (1\\ \\ x\\TT)\\TT (1\\ \\ x\\TT) \\, \\mu(\\dd x)\\,.\n$$\nThen $\\SE_{d+1}^*=\\{z\\in\\mathds{R}^{d+1}: z\\TT W^{-1}_{\\mu_d^*} z \\leq d+1\\}$, with $\\mu_d^*$ maximising $\\det(W_\\mu)$, is the minimum-volume ellipsoid centered at the origin and containing the set $\\{z\\in\\mathds{R}^{d+1}: z=(1\\ \\ x\\TT)\\TT,\\ x\\in\\SX \\}$. Moreover, $\\SE_d^*$ corresponds to the intersection between $\\SE_{d+1}^*$ and the hyperplane $\\{z\\}_1=1$; see, e.g., \\cite{ShorB92}. This gives $\\psi_d(\\mu_d^*)=(d+1)\/d!\\, \\det(W_{\\mu_d^*})$. The support points of $\\mu_d^*$ are the points of contact between $\\SX$ and $\\SE_d^*$, there exists an optimal measure with no more than $d(d+3)\/2+1$ points, see \\cite{Titterington75}.\n\nThe property below generalises this duality property to any $k\\in\\{1,\\ldots,d\\}$.\n\\begin{theorem}\\label{Th:duality}\n$$\n\\max_{\\mu\\in\\SM} \\Psi_k^{1\/k}(V_\\mu) = \\min_{M,c:\\ \\SX\\subset\\SE(M,c)} \\frac{1}{\\phi_k^\\infty(M)} \\,,\n$$\nwhere $\\SE(M,c)$ denotes the ellipsoid $\\SE(M,c) = \\{x\\in\\mathds{R}^d: (x-c)\\TT M(x-c)\\leq 1 \\}$ and $\\phi_k^\\infty(M)$ is the polar function\n\\begin{equation}\\label{polar-phi}\n \\phi_k^\\infty(M) = \\inf_{V\\succeq 0:\\ \\tr(MV)=1} \\frac{1}{\\Psi_k^{1\/k}(V)} \\,.\n\\end{equation}\n\\end{theorem}\n\nThe proof is given in Appendix. The polar function $\\phi_k^\\infty(\\cdot)$ possesses the properties of what is called an information function in \\citep[Chap.~5]{Pukelsheim93}; in particular, it is concave on the set of symmetric non-negative definite matrices. This duality property has the following consequence.\n\n\\begin{corollary} The determination of a covariance matrix $V_k^*$ that maximises $\\Psi_k(V_\\mu)$ with respect to $\\mu\\in\\SM$ is equivalent to the determination of an ellipsoid $\\SE(M_k^*,c_k^*)$ containing $\\SX$, minimum in the sense that $M_k^*$ maximizes $\\phi_k^\\infty(M)$. The points of contact between $\\SE(M_k^*,c_k^*)$ and $\\SX$ form the support of $\\mu_k^*$.\n\\end{corollary}\n\nFor any $V\\succeq 0$, denote by $M_*(V)$ the matrix\n\\begin{equation}\\label{M_*}\n M_*(V) = \\frac{\\nabla_{\\Psi_k}[V]}{k\\,\\Psi_k(V)]} = \\frac1k\\, \\nabla_{\\log\\Psi_k}[V]\\,.\n\\end{equation}\nNote that $M_*(V)\\succeq 0$, see \\cite[Lemma~7.5]{Pukelsheim93}, and that\n$$\n\\tr[VM_*(V)]=1 \\,,\n$$\nsee the proof of Theorem~\\ref{th:equivTh}.\nThe matrix $V\\succeq 0$ maximises $\\Psi_k(V)$ under the constraint $\\tr(MV)=1$ for some $M\\succeq 0$ if and only if $V[M_*(V)-M]=0$. Therefore, if $M$ is such that there exists $V_*=V_*(M)\\succeq 0$ such that $M=M_*[V_*(M)]$, then $\\phi_k^\\infty(M)=\\Psi_k^{-1\/k}[V_*(M)]$. When $k1$, since $\\psi_k(\\mu_1^*)=0$, $k>1$.\n\n\\paragraph{Example 3}\n\nTake $\\SX=\\SB_d(\\0b,\\rho)$, the closed ball of $\\mathds{R}^d$ centered at the origin $\\0b$ with radius $\\rho$. Let $\\mu_0$ be the uniform measure on the sphere\n$\\SS_d(\\0b,\\rho)$ (the boundary of $\\SB_d(\\0b,\\rho)$). Then, $V_{\\mu_0}$ is proportional to the identity matrix $I_d$, and $\\tr[V_{\\mu_0}]=\\rho^2$ implies that\n$V_{\\mu_0}=\\rho^2 I_d\/d$. Take $k=d$. We have $E_{\\mu_0}=0$ and\n$$\n\\max_{x\\in\\SX} (x-E_{\\mu_0})\\TT \\nabla_{\\Psi_d}[V_{\\mu_0}](x-E_{\\mu_0}) = \\frac{(d+1) \\rho^{2d}}{d^{d-1}d!} =\\tr\\{V_{\\mu_0}\\nabla_{\\Psi_d}[V_{\\mu_0}]\\} \\,,\n$$\nso that $\\mu_0$ is $\\psi_d$-optimal from \\eqref{CNS}.\n\nLet $\\mu_d$ be the measure that allocates mass $1\/(d+1)$ at each vertex of a $d$ regular simplex having its $d+1$ vertices on $\\SS_d(\\0b,\\rho)$, with squared volume $\\rho^{2d} (d+1)^{d+1}\/[d^d (d!)^2]$. We also have $V_{\\mu_d}=\\rho^2 I_d\/d$, so that $\\mu_d$ is $\\psi_d$-optimal too. In view of Remark~\\ref{R:Radoslav}, $\\mu_0$ and $\\mu_d$ are $\\psi_k$-optimal for all $k$ in $\\{1,\\ldots,d\\}$.\n\nLet now $\\mu_k$ be the measure that allocates mass $1\/(k+1)$ at each vertex of a $k$ regular simplex $\\SP_k$, centered at the origin, with its vertices on $\\SS_d(\\0b,\\rho)$. The squared volume of $\\SP_k$ equals $\\rho^{2k}\\, (k+1)^{k+1}\/[k^k (k!)^2]$.\nWithout any loss of generality, we can choose the orientation of the space so that $V_{\\mu_k}$ is diagonal, with its first $k$ diagonal elements equal to $\\rho^2\/k$ and the other elements equal to zero. Note that $\\psi_{k'}(\\mu_k)=0$ for $k'>k$. Direct calculations based on \\eqref{Psi_kE_k} give\n$$\n\\psi_k(\\mu_k)= \\frac{k+1}{k!}\\, \\frac{\\rho^{2k}}{k^k} \\leq \\psi_k(\\mu_0) = \\frac{k+1}{k!}\\, {d \\choose k}\\, \\frac{\\rho^{2k}}{d^k} \\,,\n$$\nwith equality for $k=1$ and $k=d$, the inequality being strict otherwise.\n\n\n\\subsection{Optimal design in regression models}\\label{S:design}\n\nIn this section we consider the case when $V=M^{-1}(\\xi)$, where $M(\\xi)$ is the information matrix\n$$\nM(\\xi) = \\int_{\\ST} f(t)f\\TT(t)\\, \\xi(\\dd t)\n$$\nin a regression model $Y_j=\\mt\\TT f(t_j)+\\mve_j$ with parameters $\\mt\\in\\mathds{R}^d$, for a design measure $\\xi\\in\\Xi$. Here $\\Xi$ denotes the set of probability measures on a set $\\ST$ such that $\\{f(t): t\\in\\ST\\}$ is compact, and $M^{-1}(\\xi)$ is the (asymptotic) covariance matrix of an estimator $\\mth$ of $\\mt$ when the design variables $t$ are distributed according to $\\xi$. The value $\\psi_k(\\mu)$ of Theorem~\\ref{Prop:1} defines a measure of dispersion for $\\mth$, that depends on $\\xi$ through $V_\\mu=M^{-1}(\\xi)$. The design problem we consider consists in choosing $\\xi$ that minimises this dispersion, as measured by $\\Psi_k[M^{-1}(\\xi)]$, or equivalently that maximises $\\Psi_k^{-1}[M^{-1}(\\xi)]$.\n\n\\subsubsection{Properties}\nIt is customary in optimal design theory to maximise a concave and Loewner-increasing function of $M(\\xi)$, see \\cite[Chap.~5]{Pukelsheim93} for desirable properties of optimal design criteria. Here we have the following.\n\n\\begin{theorem}\\label{Th:design} The functions $M \\longrightarrow \\Psi_k^{-1\/k}(M^{-1})$, $k=1,\\ldots,d$, are Loew\\-ner-increasing, concave and differentiable on the set $\\mathbb{M}^+$ of $d\\times d$ symmetric positive-definite matrices. The functions $\\Psi_k(\\cdot)$ are also orthogonally invariant.\n\\end{theorem}\n\n\\begin{proof}\nThe property \\eqref{EkEd-k} yields\n\\begin{equation}\\label{design-criterion}\n \\Psi_k^{-1\/k}(M^{-1}) = \\left(\\frac{k+1}{k!}\\right)^{-1\/k}\\, \\frac{\\det^{1\/k}(M)}{\\mathcal{E}_{d-k}^{1\/k}(M)}\n\\end{equation}\nwhich is a concave function of $M$, see Eq.~(10) of \\cite[p.~116]{MarcusM64}. Since $\\Psi_k(\\cdot)$ is Loewner-increasing, see \\cite{Lopez-FR-D98-MODA}, the function $M \\longrightarrow \\Psi_k^{-1\/k}(M^{-1})$ is Loewner-increasing too. Its orthogonal invariance follows from the fact that it is defined in terms of the eigenvalues of $M$.\n\\end{proof}\n\nNote that Theorems~\\ref{Prop:1} and \\ref{Th:design} imply that the functions\n$M \\longrightarrow - \\log \\Psi_k(M)$\nand $M \\longrightarrow \\log \\Psi_k(M^{-1})$ are convex for all $k=1,\\ldots,d$, a question which was left open in \\citep{Lopez-FR-D98-MODA}.\n\nAs a consequence of Theorem~\\ref{Th:design}, we can derive a necessary and sufficient condition for a design measure $\\xi_k^*$ to maximise $\\Psi_k^{-1\/k}[M^{-1}(\\xi)]$ with respect to $\\xi\\in\\Xi$, for $k=1,\\ldots,d$.\n\n\\begin{theorem}\\label{th:equivTh2} The design measure $\\xi_k^*$ such that $M(\\xi_k^*)\\in \\mathbb{M}^+$ maximises $\\tilde\\psi_k(\\xi)=\\Psi_k^{-1\/k}[M^{-1}(\\xi)]$ with respect to $\\xi\\in\\Xi$ if and only if\n\\begin{equation}\\label{CNS-design-a}\n \\max_{t\\in\\ST} f\\TT(t)M^{-1}(\\xi_k^*)\\, \\frac{\\nabla_{\\Psi_k}[M^{-1}(\\xi_k^*)]}{\\Psi_k[M^{-1}(\\xi_k^*)]}\\,M^{-1}(\\xi_k^*)f(t) \\leq k\n\\end{equation}\nor, equivalently,\n\\begin{equation}\\label{CNS-design}\n \\max_{t\\in\\ST} \\left\\{ f\\TT(t)M^{-1}(\\xi_k^*)f(t) - f\\TT(t)\\frac{\\nabla_{\\Psi_{d-k}}[M(\\xi_k^*)]}{\\Psi_{d-k}[M(\\xi_k^*)]}f(t) \\right\\} \\leq d - k \\,.\n\\end{equation}\nMoreover, there is equality in \\eqref{CNS-design-a} and \\eqref{CNS-design} for all $t$ in the support of $\\xi_k^*$.\n\\end{theorem}\n\n\\begin{proof}\nFrom \\eqref{design-criterion}, the maximisation of $\\tilde\\psi_k(\\xi)$ is equivalent to the maximisation of\n$\\tilde\\phi_k(\\xi)=\\log\\det[M(\\xi)]-\\log\\Psi_{d-k}[M(\\xi)]$.\nThe proof is similar to that of Theorem~\\ref{th:equivTh} and is based on the following expressions for the directional derivatives of these two functionals\nat $\\xi$ in the direction $\\nu\\in\\Xi$,\n$$\nF_{\\tilde\\psi_k}(\\xi;\\nu) = \\tr\\left( \\frac1k\\, M^{-1}(\\xi)\\, \\frac{\\nabla_{\\Psi_k}[M^{-1}(\\xi)]}{\\Psi_k[M^{-1}(\\xi)]}\\,M^{-1}(\\xi)\\, [M(\\nu)-M(\\xi)] \\right) $$\nand\n$$\nF_{\\tilde\\phi_k}(\\xi;\\nu) = \\tr\\left( \\left\\{M^{-1}(\\xi)-\\frac{\\nabla_{\\Psi_{d-k}}[M(\\xi)]}{\\Psi_{d-k}[M(\\xi)]} \\right\\}[M(\\nu)-M(\\xi)] \\right) \\,,\n$$\nand on the property $\\tr\\{M\\nabla_{\\Psi_{j}}[M]\\}=j\\, \\Psi_{j}(M)$.\n\\end{proof}\n\nIn particular, consider the following special cases for $k$ (note that $\\Psi_0(M)=\\mathcal{E}_0(M)=1$ for any $M$).\n\\begin{eqnarray*}\n&&k=d: \\hspace{1.2cm} \\tilde\\psi_d(\\xi) = \\log\\det[M(\\xi)] \\,, \\\\\n&&k=d-1: \\hspace{0.5cm} \\tilde\\psi_{d-1}(\\xi) = \\log\\det[M(\\xi)] - \\log\\tr[M(\\xi)] - \\log 2 \\,,\\\\\n&&k=d-2: \\hspace{0.5cm} \\tilde\\psi_{d-2}(\\xi) = \\log\\det[M(\\xi)] \\\\\n&& \\hspace{4cm} - \\log\\left\\{\\tr^2[M(\\xi)]-\\tr[M^2(\\xi)]\\right\\} - \\log(3\/4)\\,. \\\\\n\\end{eqnarray*}\nThe necessary and sufficient condition \\eqref{CNS-design} then takes the following form:\n\\begin{eqnarray*}\n&&\\hspace{-0.5cm} k=d: \\hspace{1cm} \\max_{t\\in\\ST} f\\TT(t)M^{-1}(\\xi_k^*)f(t) \\leq d \\,,\\\\\n&&\\hspace{-0.5cm} k=d-1: \\hspace{0.3cm} \\max_{t\\in\\ST} \\left\\{ f\\TT(t)M^{-1}(\\xi_k^*)f(t) - \\frac{f\\TT(t)f(t)}{\\tr[M(\\xi_{k}^*)]} \\right\\} \\leq d -1\\,, \\\\\n&&\\hspace{-0.5cm} k=d-2: \\hspace{0.3cm} \\max_{t\\in\\ST} \\left\\{ f\\TT(t)M^{-1}(\\xi_k^*)f(t) \\right. \\\\\n&& \\hspace{3cm} \\left. - 2\\,\\frac{\\tr[M(\\xi_{k}^*)]f\\TT(t)f(t)-f\\TT(t)M(\\xi_{k}^*)f(t)}{\\tr^2[M(\\xi_{k}^*)]-\\tr[M^2(\\xi_{k}^*)]} \\right\\} \\leq d -2 \\,.\\\\\n\\end{eqnarray*}\nAlso, for $k=1$ condition \\eqref{CNS-design-a} gives\n$$\n\\max_{t\\in\\ST} f\\TT(t) \\frac{M^{-2}(\\xi_1^*)}{\\tr[M^{-1}(\\xi_1^*)]} f(t) \\leq 1\n$$\n(which corresponds to $A$-optimal design), and for $k=2$\n$$\n\\max_{t\\in\\ST} \\frac{\\tr[M^{-1}(\\xi_2^*)] f\\TT(t)M^{-2}(\\xi_2^*)f(t) - f\\TT(t)M^{-3}(\\xi_2^*)f(t)}{\\tr^2[M^{-1}(\\xi_2^*)]-\\tr[M^{-2}(\\xi_2^*)]} \\leq 1 \\,.\n$$\n\nFinally, note that a duality theorem, in the spirit of Theorem~\\ref{Th:duality}, can be formulated for the maximisation of $\\Psi_k^{-1\/k}[M^{-1}(\\xi)]$; see \\cite[Th.~7.12]{Pukelsheim93} for the general form a such duality properties in optimal experimental design.\n\n\\subsubsection{Examples}\n\\paragraph{Example 4} For the linear regression model on $\\mt_0+\\mt_1\\,x$ on $[-1,1]$, the optimal design for $\\tilde\\psi_k(\\cdot)$ with $k=d=2$ or $k=1$ is\n$$\n\\xi_k^*=\\left\\{\n\\begin{array}{cc}\n-1 & 1 \\\\\n1\/2 & 1\/2\n\\end{array} \\right\\}\\,,\n$$\nwhere the first line corresponds to support points and the second indicates their respective weights.\n\n\\paragraph{Example 5} For linear regression with the quadratic polynomial model $\\mt_0+\\mt_1\\,t+\\mt_2\\,t^2$ on $[-1,1]$, the optimal designs for $\\tilde\\psi_k(\\cdot)$ have the form\n$$\n\\xi_k^*=\\left\\{\n\\begin{array}{ccc}\n-1 & 0 & 1 \\\\\nw_k & 1-2w_k & w_k\n\\end{array} \\right\\}\\,,\n$$\nwith $w_3 = 1\/3$, $w_2 = (\\sqrt{33}-1)\/16 \\simeq 0.2965352$ and $w_1 = 1\/4$. Define the efficiency $\\mathrm{Eff}_k(\\xi)$ of a design $\\xi$ as\n$$\n\\mathrm{Eff}_k(\\xi) = \\frac{\\tilde\\psi_k(\\xi)}{\\tilde\\psi_k(\\xi_k^*)} \\,.\n$$\nTable~\\ref{Tab:Ex5} gives the efficiencies $\\mathrm{Eff}_k(\\xi_j^*)$ for $j,k=1,\\ldots,d=3$. The design $\\xi_2^*$, optimal for $\\tilde\\psi_2(\\cdot)$, appears to make a good compromise between $A$-optimality (which corresponds to $\\tilde\\psi_1(\\cdot)$) and $D$-optimality (which corresponds to $\\tilde\\psi_3(\\cdot)$).\n\n\\begin{table}[t]\n\\centering\n\\caption{\\small Efficiencies $\\mathrm{Eff}_k(\\xi_j^*)$ for $j,k=1,\\ldots,d$ in Example 5.}\n{\\small \\begin{tabular}{lccc\n\\hline\n & $\\mathrm{Eff}_1$ & $\\mathrm{Eff}_2$ & $\\mathrm{Eff}_3$ \\\\\n\\hline\n$\\xi_1^*$ & 1 & 0.9770 & 0.9449 \\\\\n$\\xi_2^*$ & 0.9654 & 1 & 0.9886 \\\\\n$\\xi_3^*$ & 0.8889 & 0.9848 & 1 \\\\\n \\hline\n\\end{tabular}}\n\n\\label{Tab:Ex5}\n\\end{table}\n\n\\paragraph{Example 6} For linear regression with the cubic polynomial model $\\mt_0+\\mt_1\\,t+\\mt_2\\,t^2+\\mt_3\\,t^3$ on $[-1,1]$, the optimal designs for $\\tilde\\psi_k(\\cdot)$ have the form\n$$\n\\xi_k^*=\\left\\{\n\\begin{array}{cccc}\n-1 & -z_k & z_k & 1 \\\\\nw_k & 1\/2-w_k & 1\/2-w_k & w_k\n\\end{array} \\right\\}\\,,\n$$\nwhere\n$$\n \\begin{array}{ll}\n z_4 = 1\/\\sqrt{5} \\simeq 0.4472136\\,, & w_4 =0.25\\,, \\\\\n z_3 \\simeq 0.4350486\\,, & w_3 \\simeq 0.2149859\\,, \\\\\n z_2 \\simeq 0.4240013\\,, & w_2 \\simeq 0.1730987\\,, \\\\\n z_1 = \\sqrt{3\\sqrt{7}-6}\/3 \\simeq 0.4639509\\,, & w_1 = (4-\\sqrt{7})\/9 \\simeq 0.1504721\\,,\n \\end{array}\n$$\nwith $z_3$ satisfying the equation $2z^6-3z^5-45z^4+6z^3-4z^2-15z+3=0$ and \\\\\n$$\nw_3=\n\\,{\\frac {5\\,{z}^{6}+5\\,{z}^{4}+5\\,{z}^{2}+1-\n\\sqrt {{z}^{12}+2\\,{z\n}^{10}+3\\,{z}^{8}+60\\,{z}^{6}+59\\,{z}^{4}+58\\,{z}^{2}+73}}\n{12({z}^{6}+{\nz}^{4}+{z}^{2}-3)}}\\,,\n$$\nwith $z=z_3$. For $k=d-2=2$, the numbers $z_2$ and $w_2$ are too difficult to express analytically. Table~\\ref{Tab:Ex6} gives the efficiencies $\\mathrm{Eff}_k(\\xi_j^*)$ for $j,k=1,\\ldots,d$. Here again the design $\\xi_2^*$ appears to make a good compromise: it maximises the minimum efficiency $\\min_k \\mathrm{Eff}_f(\\cdot)$ among the designs considered.\n\n\\begin{table}[t]\n\\centering\n\\caption{\\small Efficiencies $\\mathrm{Eff}_k(\\xi_j^*)$ for $j,k=1,\\ldots,d$ in Example 6.}\n{\\small \\begin{tabular}{lcccc\n\\hline\n & $\\mathrm{Eff}_1$ & $\\mathrm{Eff}_2$ & $\\mathrm{Eff}_3$ & $\\mathrm{Eff}_4$ \\\\\n\\hline\n$\\xi_1^*$ & 1 & 0.9785 & 0.9478 & 0.9166 \\\\\n$\\xi_2^*$ & 0.9694 & 1 & 0.9804 & 0.9499 \\\\\n$\\xi_3^*$ & 0.9180 & 0.9753 & 1 & 0.9897 \\\\\n$\\xi_4^*$ & 0.8527 & 0.9213 & 0.9872 & 1 \\\\\n \\hline\n\\end{tabular}}\n\\label{Tab:Ex6}\n\\end{table}\n\n\\section*{Appendix}\n\n\\paragraph{Shift-invariance and positive homogeneity}\n\nDenote by $\\SM$ the set of probability measures defined on the Borel subsets of $\\SX$, a compact subset of $\\mathds{R}^d$. For any $\\mu\\in\\SM$, any $\\mt\\in \\mathds{R}^d$ and any $\\ml\\in\\mathds{R}^+$, respectively denote by $T_{-\\mt}[\\mu]$ and $H_{\\ml^{-1}}[\\mu]$ the measures defined by:\n$$\n\\mbox{ for any } \\mu\\mbox{-measurable } \\SA\\subseteq\\SX\\,, \\ T_{-\\mt}[\\mu](\\SA+\\mt)=\\mu(\\SA)\\,, \\ H_{\\ml^{-1}}[\\mu](\\ml\\SA)=\\mu(\\SA) \\,,\n$$\nwhere $\\SA+\\mt=\\{x+\\mt: x\\in\\SA\\}$ and $\\ml\\SA=\\{\\ml\\,x: x\\in\\SA\\}$.\nThe shift-invariance of $\\phi(\\cdot)$ then means that $\\phi(T_{-\\mt}[\\mu])=\\phi(\\mu)$ for any $\\mu\\in\\SM$ and any $\\mt\\in\\mathds{R}^d$,\npositive homogeneity of degree $q$ means that $\\phi(H_{\\ml^{-1}}[\\mu])=\\ml^q\\, \\phi(\\mu)$ for any $\\mu\\in\\SM$ and any $\\ml\\in\\mathds{R}^+$.\n\\fin\n\n\\paragraph{The variance is the only concave central moment}\n\nFor $q\\neq 2$, the $q$-th central moment\n$\\Delta_q(\\mu)=\\int |x-E_\\mu|^q\\,\\mu(\\dd x)$\nis shift-invariant and homogeneous of degree $q$, but it is not concave on $\\SM$. Indeed, consider for instance the two-point probability measures\n$$\n\\mu_1=\\left\\{ \\begin{array}{cc}\n0 & 1 \\\\\n1\/2 & 1\/2\n\\end{array}\\right\\}\n\\mbox{ and }\n\\mu_2=\\left\\{ \\begin{array}{cc}\n0 & 101 \\\\\nw & 1-w\n\\end{array}\\right\\}\\,,\n$$\nwhere the first line denotes the support points and the second one their respective weights. Then, for\n$$\nw=1-\\frac{1}{404}\\, \\frac{201^{q-1}-202q+405}{201^{q-1}-101q+102}\n$$\none has\n$\\mp^2 \\Delta_q[(1-\\ma)\\mu_1+\\ma\\mu_2]\/\\mp\\ma^2\\big|_{\\ma=0} \\geq 0$ for all $q \\geq 1.84$, the equality being obtained at $q=2$ only. Counterexamples are easily constructed for values of $q$ smaller than 1.84.\n\\fin\n\n\\paragraph{Proof of Lemma~\\ref{L:1}}\nWe have\n$$\n\\Ex \\left \\{ \\det\\left [\\sum_{i=1}^{k+1} z_i z_i\\TT \\right] \\right\\} = (k+1)!\\, \\det\\left[\n \\begin{array}{cc}\n \\Ex(x_1 x_1\\TT) & E_\\mu \\\\\n E_\\mu\\TT & 1 \\\\\n \\end{array}\n\\right] = (k+1)! \\det[V_\\mu]\\,,\n$$\nsee for instance \\cite[Theorem~1]{Pa98}.\n\\fin\n\n\\paragraph{Proof of Lemma~\\ref{L:2}}\nTake any vector $z$ of the same dimension as $x$. Then $z\\TT V_\\mu z=\\var_\\mu(z\\TT x)$, which is a concave functional of $\\mu$, see Section~\\ref{S:intro}. This implies that\n$z\\TT V_{(1-\\ma)\\mu_1+\\ma\\mu_2} z = \\var_{(1-\\ma)\\mu_1+\\ma\\mu_2}(z\\TT x) \\geq (1-\\ma)\\var_{\\mu_1}(z\\TT x)+\\ma \\var_{\\mu_2}(z\\TT x) = (1-\\ma) z\\TT V_{\\mu_1}z +\\ma z\\TT V_{\\mu_2}z$,\nfor any $\\mu_1$, $\\mu_2$ in $\\SM$ and any $\\ma\\in(0,1)$ (see Section~\\ref{S:intro} for the concavity of $\\var_\\mu$).\nSince $z$ is arbitrary, this implies \\eqref{conc0}.\n\\fin\n\n\n\\paragraph{Proof of Theorem~\\ref{Th:empirical}}\nThe estimate \\eqref{widehat-psi} forms a U-statistics for the estimation of $\\psi_k(\\mu)$ and is thus unbiased and has minimum variance, see, e.g., \\cite[Chap.~5]{Serfling80}.\nWe only need to show that it can be written as \\eqref{widehat-psi-b}.\n\nWe can write\n\\begin{eqnarray*}\n({\\widehat\\psi}_k)_n &=& {n \\choose k+1}^{-1} \\\\\n&& \\times \\sum_{j_10,\\ \\ma \\geq 0} \\left\\{ - \\log \\Psi_k^{1\/k}\\left[\\sum_{x\\in\\SX} \\ma_x\\,(x-c^*)(x-c^*)\\TT\\right] + \\mg - \\log(\\mg) -1 \\right\\} \\,, \\\\\n&& = \\min_{\\ma \\geq 0} - \\log \\Psi_k^{1\/k}\\left[\\sum_{x\\in\\SX} \\ma_x\\,(x-c^*)(x-c^*)\\TT\\right] = - \\log\\Psi_k^{1\/k}(V_k^*) \\,,\n\\end{eqnarray*}\nwhere we have denoted $\\mg=\\sum_{x\\in\\SX} \\beta_x$ and $\\ma_x=\\beta_x\/\\mg$ for all $x$. Therefore\n$T^* \\leq - \\log\\Psi_k^{1\/k}(V_k^*)$, that is,\n$\\log\\left[\\min_{M,c:\\ \\SX\\subset\\SE(M,c)} 1\/\\phi_k^\\infty(M)\\right] \\geq \\log\\Psi_k^{1\/k}(V_k^*)$.\n\\fin\n\n\n\n\\section*{Acknowledgments}\nThe work of the first author was partly supported by the ANR project 2011-IS01-001-01 DESIRE (DESIgns for spatial Random fiElds).\n\n\\bibliographystyle{elsart-harv}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nAn in-coincidence experiment measures simultaneously the outgoing\nmomenta of multiple products of a microscopic reaction\n\\cite{schmidt2002coltrims}. It is an instrument that can study the\ncorrelations in reactions involving multiple particles. In double\nionization, for example, a single photon ionizes, simultaneously, two\nelectrons and the outgoing momenta of both particles are captured\n\\cite{akoury2007simplest}. The reaction probes the correlation between\ntwo electrons in, for example, a chemical bound at the moment of\nphoton impact. The outgoing wave of the two electrons is described by\na 6D correlated wave and results in a cross section that depends on\nfour angles, the directions of the first and the second electron.\n\nFree-electron lasers, and similar experiments around the world, are\nexpected to generate a wealth of this high-dimensional scattering\ndata. This will result in high-dimensional forward and inverse wave\nproblems that need to be solved to interpret the data.\n\nThe experimental cross section are often smooth functions as a\nfunction of the angles. Similarly, some parts of the scattering\nsolution, such as single-ionization, only probes a limited subspace\nof the possible full solution space. The scattering solution can then\nbe described by a low-rank wave function, a product of one-particle\nbound states with scattering waves in the other coordinates.\n\nThis paper introduces a low-rank representation for the\nscattering solutions, not only for the single ionization but also for\ndouble and triple ionization waves that appear in breakup reaction.\n\nWe also propose and analyze an alternating direction algorithm that\ndirectly solves for the low-rank components that describe the\nsolution. This reduces a large-scale linear system to smaller,\nlow-dimensional, scattering problems that are solved in a iterative\nsequence. The proposed method can be generalized to high-dimensional\nscattering problems where a low-rank tensor decomposition is used to\nrepresent the full scattering wave function.\n\nEfficient low-rank tensor representations are used in quantum physics\nfor quite some time already \\cite{huckle2013computations,\n murg2010simulating}. They are also used in the applied mathematics\nliterature to approximate high-dimension problems, for a review see\n\\cite{grasedyck2013literature, hackbusch2012tensor, kolda2009tensor}.\nMethods such ALS \\cite{holtz2012alternating}, DMRG\n\\cite{oseledets2011dmrg}, and AMEn \\cite{dolgov2014alternating} use in\nalternating directions, a small linear system to determine the\nlow-rank components of a tensor decomposition. These innovations have\nnot found their application in computational scattering theory.\n\nTo calculate cross sections, from first principles, we start\nfrom a multi-particle Schr\\\"odinger equation. The equation is\nreformulated into a driven Schr\\\"odinger equation with an unknown scattering\nwave function and a right hand side that describes the\nexcitation, for example, a dipole operator working on the initial\nstate.\n\nSince the asymptotic behaviour of a scattering function for multiple\ncharged particles is in many cases unknown, absorbing boundary\nconditions \\cite{ECS,PML} are used. Here, an artificial layer is\nadded to the numerical domain that dampens outgoing waves. The\noutgoing wave boundary conditions are then replaced with\nhomogeneous Dirichlet boundary conditions at the end of the artificial\nlayer. These boundary do not require any knowledge about the\nasymptotic behaviour, which becomes very complicated for these\nmultiple charged particles.\n\nThe resulting equation is discretized on a grid and results in a\nlarge, sparse indefinite linear system. It is typically solved by a\npreconditioned Krylov subspace method \\cite{cools2016fast}. However,\nthe preconditioning techniques for indefinite systems are not as\nefficient as preconditioners for symmetric and positive definite\nsystem. And solving the resulting equation is still a\ncomputationally expensive task, often requiring a distributed\ncalculation on a supercomputer.\n\nTo compare the resulting theoretical cross sections with experimental\ndata, a further postprocessing step is necessary. The cross section is\nthe farfield map and this is calculated through integrals of the\nscattering wave function, which is the solution of the linear system,\nand a Greens function \\cite{mccurdy2004solving}.\n\nThe main result of the paper is that we show that scattering waves\nthat describe multiple ionization can be represented by a low-rank\ntensor. We first show this for a 2D wave and then generalize the\nresults to 3D waves. The methodology can be generalized to higher\ndimension.\n\nThe outline of the paper is a follows. In section~\\ref{sec:stateofart} we\nreview the methodology that solves a forward scattering problem. It\nresults in a driven Schr\\\"odinger equation with absorbing boundary\nconditions. From the solution we can extract the cross section using\nan integral. In section~\\ref{sec:lowrank} we illustrate, in 2D, that a\nsolution can be approximated by a truncated low-rank approximation. We\nalso show that these low-rank components can be calculated directly\nwith an iterative method. In section~\\ref{sec:3dhelmholtz} we show that\nthis methodology generalizes to 3D and higher dimensional problems.\nWe use a truncated tensor decomposition and determine the components \nwith a similar iterative method.\nA discussion of some numerical results and a comparison of the different\npresented versions of the method is given in section~\\ref{sec:NumericalResults}.\nIn the final section, Sec.~\\ref{sec:discussion}, we summarize some\nconclusions and discuss some possible extensions of the presented method.\n\n\n\n\\section{State of the art} \\label{sec:stateofart}\nThis section summarize the methodology that solves a forward\nbreak-up problems with charged particles. The methodology is developed\nin a series of papers \\cite{rescigno2000numerical,mccurdy2004solving}\nand applied to solve the impact-ionization problem\n\\cite{rescigno1999collisional} and double ionization of molecules\n\\cite{vanroose2005complete, vanroose2006double}. These methods are\nbeing extended to treat, for example, water\n\\cite{streeter2018dissociation}.\n\nThe helium atom, He, is the simplest system with double ionization\n\\cite{briggs2000differential}. It has two electrons with coordinates\n$\\mathbf{r}_1 \\in \\mathbb{R}^3$ and $\\mathbf{r}_2 \\in \\mathbb{R}^3$\nrelative to the nucleus positioned at $\\mathbf{0}$. The driven\nSchr\\\"odinger equation for $u(\\mathbf{r}_1,\\mathbf{r}_2) \\in C^2$ then\nreads\n\\begin{equation}\\label{DrivenSchrodinger}\n\\left( {- \\frac{1}{2} \\Delta_{\\mathbf{r}_1} - \\frac{1}{2} \\Delta_{\\mathbf{r}_2} } { - \\frac{1}{\\|\\mathbf{r}_1\\|}- \\frac{1}{\\|\\mathbf{r}_2\\|}} + {\\frac{1}{\\|\\mathbf{r}_1-\\mathbf{r}_2\\|} }- { E} \\right) u(\\mathbf{r}_1,\\mathbf{r}_2) = \\mathcal{\\mu} \\phi_0(\\mathbf{r}_1,\\mathbf{r}_2) \\quad \\forall \\mathbf{r}_1, \\mathbf{r}_2 \\in \\mathbb{R}^3,\n\\end{equation}\nwhere the right hand side is\nthe dipole operator $\\mathcal{\\mu}$ working on the ground state $\\phi_0$, the\neigenstate with the lowest energy $\\lambda_0$. The operators\n$-\\frac{1}{2} \\Delta_{\\mathbf{r}_1}$ and\n$-\\frac{1}{2}\\Delta_{\\mathbf{r}_2}$ are the Laplacian operators for\nthe first and second electron and model the kinetic energy. The\nnuclear attraction is $-1\/\\|\\mathbf{r}_1\\|$ and $-1\/\\|\\mathbf{r}_2\\|$\nand the electron-electron repulsion is\n$1\/\\|\\mathbf{r}_1-\\mathbf{r}_2\\|$.\n\nThe total energy $E = h\\nu + \\lambda_0$ is the energy\ndeposited in the system by the photon, $h\\nu$, and the energy $\\lambda_0$ of\nthe ground state. If the $E>0$, both electrons can\nescape simultaneously from the system. The solution\n$u(\\mathbf{r}_1,\\mathbf{r}_2)$ then represents a 6D wave emerging from\nthe nucleus.\n\nThe equation can be interpreted as a Helmholtz equation with a\nspace-dependent wave number, $k^2(\\mathbf{r}_1,\\mathbf{r}_2)$,\n\\begin{equation}\\label{eq:helmholtz}\n \\left( - \\Delta_{6D} - { k^2(\\mathbf{r}_1,\\mathbf{r}_2)} \\right)\n u(\\mathbf{r}_1,\\mathbf{r}_2) = { f(\\mathbf{r}_1,\\mathbf{r}_2)} \\quad \\forall (\\mathbf{r}_1,\n \\mathbf{r}_2) \\in \\mathbb{R}^6.\n\\end{equation}\n\nIn this paper we prefer to write this Helmholtz equation as\n\\begin{equation}\n \\left( - \\Delta_{6D} - k_0^2 (1 + \\chi(\\mathbf{r}_1,\\mathbf{r}_2) ) \\right) u(\\mathbf{r}_1,\\mathbf{r}_2) = f(\\mathbf{r}_1,\\mathbf{r}_2) \\quad \\forall (\\mathbf{r}_1, \\mathbf{r}_2) \\in \\mathbb{R}^6,\n\\end{equation}\nwhere $k_0^2$ is a constant wave number, in this case related to the\ntotal energy $E$, and a space-dependent function $\\chi: \\mathbb{R}^6\n\\rightarrow \\mathbb{R}$, that goes to zero if $\\|\\mathbf{r}_1\\|\n\\rightarrow \\infty $ or $\\|\\mathbf{r}_1\\| \\rightarrow \\infty $ that\nrepresents all the potentials.\n\n\\subsection{Expansion in spherical waves and absorbing boundary conditions}\nFor small atomic and molecular system, where spherical symmetry is relevant, the\nsystem is typically written in spherical coordinates and expanded in\nspherical harmonics. With $\\mathbf{r}_1(\\rho_1, \\theta_1,\n\\varphi_1)$ and $\\mathbf{r}_2(\\rho_2, \\theta_2, \\varphi_2)$ we can\nwrite\n\\begin{equation}\\label{eq:partialwaves}\n u(\\mathbf{r}_1,\\mathbf{r}_2) = \\sum_{l_1}^\\infty \\sum_{m_1 = -l_1}^{l_1} \\sum_{l_2=0}^{\\infty} \\sum_{m_2=-l_2}^{l_2} u_{l_1m_1,l_2m_2}(\\rho_1,\\rho_2) Y_{l_1m_1}(\\theta_1,\\varphi_1) Y_{l_2m_2}(\\theta_2,\\varphi_2),\n\\end{equation}\nwhere $Y_{l_1m_1}(\\theta_1, \\varphi_1)$ and $Y_{l_2m_2}(\\theta_2,\n\\varphi_2)$ are spherical Harmonics, the eigenfunctions of the angular\npart of a 3D Laplacian in spherical coordinates. In practice the sum in equation~\\eqref{eq:partialwaves} is\ntruncated. The expansion is then a low-rank, truncated, tensor decomposition\nof a 6D tensor describing the solution.\n\nFor each $l_1$, $m_1$, $l_2$ and $m_2$ combination, the radial function\n$u_{l_1m_1,l_2m_2}(\\rho_1,\\rho_2)$ describes an outgoing wave that\ndepends on the distances $\\rho_1$ and $\\rho_2$ of the two electrons to\nthe nucleus.\n\nA coupled equation that simultaneously solves for all the\n$u_{l_1m_1l_2m_2}(\\rho_1,\\rho_2)$'s is found by inserting the\ntruncated sum in \\eqref{DrivenSchrodinger}, multiplying with $Y_{l_1m_1}^*(\\theta_1,\n\\varphi_1)$ and $Y_{l_2m_2}^*(\\theta_2,\\varphi_2)$ and integrating\nover all the angular coordinates,\n\\begin{equation}\\label{eq:coupledpartialwaves}\n \\begin{aligned}\\left(-\\frac{1}{2}\\frac{d^2}{d \\rho_1^2} + \\frac{l_1(l_1+1)}{2\\rho_1^2 }\n -\\frac{1}{2}\\frac{d^2}{d \\rho_2^2} + \\frac{l_2(l_2+1)}{2\\rho_2^2 } + V_{l_1m_1l_2m_2}\n(\\rho_1,\\rho_2) -E \\right) u_{l_1m_1l_2m_2}(\\rho_1,\\rho_2) \\\\ + \\sum_{l^\\prime_1m^\\prime_1l^\\prime_2m^\\prime_2} V_{l_1m_1l_2m_2, l^\\prime_1m^\\prime_1l^\\prime_2m^\\prime_2}(\\rho_1,\\rho_2) u_{l^\\prime_1m^\\prime_1l^\\prime_2m^\\prime_2}(\\rho_1,\\rho_2) = f_{l_1m_1,l_2m_2}(\\rho_1,\\rho_2) \\quad \\forall l_1m_1 l_1m_2. \\quad \\forall \\rho_1,\\rho_2 \\in [0,\\infty[\n\\end{aligned}\n\\end{equation}\nwith boundary conditions $u(\\rho_1=0,\\rho_2)=0$ for all $\\rho_2 \\ge 0$ and $u(\\rho_1,\\rho_2=0)=0$ for all $\\rho_1 \\ge 0$.\n\nThe equation \\eqref{eq:coupledpartialwaves} is typically discretized\non a spectral elements quadrature grid \\cite{rescigno2000numerical}.\n\nTo reflect the physics, where electrons are emitted from the system,\noutgoing wave boundary conditions need to be applied at the outer\nboundaries. There are many ways to implement outgoing wave boundary\nconditions. Exterior complex scaling (ECS) \\cite{ECS} for\nexample, is frequently used in the computational atomic and molecular\nphysics literature. In the computational electromagnetic scattering a\nperfectly matched layers (PML) \\cite{PML} is used, which can also be\ninterpreted as a complex scaled grid. \\cite{chew19943d}\n\n\\subsection{Calculation of the amplitudes}\nTo correctly predict the probabilities of the arriving particles at\nthe detector, we need the amplitudes of the solution far away from the\nmolecule. These are related to the asymptotic amplitudes of the\nwave functions.\n\nLet us go back to the formulation with Helmholtz equation, as given in\n\\eqref{eq:helmholtz}. Suppose that we have solved the following\nHelmholtz equation with absorbing boundary conditions, in any\nrepresentation,\n\\begin{equation} \\label{Helmholtz}\n \\left(- \\Delta -k_0^2 \\left( 1+ \\chi(\\mathbf{x}) \\right) \\right) u_{\\text{sc}}(\\mathbf{x}) = f(\\mathbf{x}), \\quad \\forall \\mathbf{x} \\in [-L,L]^d,\n\\end{equation}\nwhere $f$ is only non-zero on the real part of the grid $[-L,L]^d\n\\subset \\mathbb{R}^d$. Similarly, $\\chi(\\mathbf{x})$ is only non-zero\non the box $[-L,L]^d$.\n\nThe calculation of the asymptotic amplitudes requires the solution\n$u_{\\text{sc}}(\\mathbf{x})$ for an $\\mathbf{x}$ outside of the box\n$[-L, L]^d$. To that end, we reorganize equation~\\eqref{Helmholtz},\nafter we have solved it, as follows\n\\begin{equation} \\label{eq:reorganized}\n \\left( - \\Delta -k_0^2 \\right) u_{\\text{sc}} = f + k_0^2 \\, \\chi \\, u_{\\text{sc}}.\n\\end{equation}\nThe right hand side of \\eqref{eq:reorganized} is now only non-zero on\n$[-L,L]^d$, since both $f$ and $\\chi$ are only non-zero\nthere. Furthermore, since we have solved \\eqref{Helmholtz} we also\nknow $u_\\text{sc}$ on $[-L,L]^d$. So the full right hand side of\n\\eqref{eq:reorganized} is known. The remaining left hand side of\n\\eqref{eq:reorganized} is now a Helmholtz equation with a constant\nwave number $k_0^2$. For this equation the Greens function is known\nanalytically. \n\\begin{equation}\n \\begin{aligned}\n u_{\\text{sc}}(\\mathbf{x}) &= \\int G(\\mathbf{x},\\mathbf{y}) \\left( f(\\mathbf{y}) + k_0^2 \\,\\chi(\\mathbf{y}) u_\\text{sc}(\\mathbf{y})\\right) d\\mathbf{y} \\\\\n &= \\int\\limits_{[-L,L]^d} G(\\mathbf{x},\\mathbf{y}) \\left( f(\\mathbf{y}) + k_0^2\\, \\chi(\\mathbf{y}) u_\\text{sc} (\\mathbf{y}) \\right) d\\mathbf{y} \\quad \\forall x \\in \\mathbb{R}^d,\n\\end{aligned}\n\\end{equation}\nwhere $f$ and $\\chi$ are limited to $[-L,L]^d$ thus we can truncate the\nintegral to the box $[-L,L]^d$.\n\nThis methodology was successfully applied to calculate challenging\nbreak up problems, for example \\cite{rescigno1999collisional}.\n\n\\subsection{Single ionization versus double ionization}\nLet us discuss the qualitative behaviour of the solution for single\nand double ionization. To illustrate the behaviour, we truncate the\npartial wave expansion, \\eqref{eq:partialwaves}, to the first\nterm. This is known as a $s$-wave expansion. The 6D wave function is\nthen approximated as\n\\begin{equation}\n u(\\mathbf{r}_1,\\mathbf{r}_2) \\approx u(\\rho_1,\\rho_2) Y_{00}(\\theta_1,\\varphi_1) Y_{00}(\\theta_2,\\varphi_2).\n\\end{equation}\nThe radial wave, $u(\\rho_1,\\rho_2)$, then fits a 2D Helmholtz equation\n\\begin{equation}\\label{eq:radialequation} \n \\left(- \\frac{1}{2} \\frac{d^2}{d\\rho_1^2} - \\frac{1}{2} \\frac{d^2}{d\\rho_2^2} + V_1(\\rho_1) + V_2(\\rho_2) + V_{12}(\\rho_1,\\rho_2) - E \\right) u(\\rho_1,\\rho_2) = f(\\rho_1,\\rho_2), \\quad \\forall \\rho_1,\\rho_2 \\in [0,\\infty[,\n\\end{equation}\nwhere $V_1(\\rho_1)$ and $V_2(\\rho_2)$ represents the one-particle\npotentials and $V_{12}(\\rho_1,\\rho_2)$ the two-particle repulsion.\nThis model is known as a $s$-wave or Temkin-Poet model\n\\cite{temkin1962nonadiabatic, poet1978exact}.\n\nBefore the photo-ionization, the atom is in a two-particle ground\nstate. In this $s$-wave model, it is the eigenstate of\n\\begin{equation}\n \\left(- \\frac{1}{2} \\frac{d^2}{d\\rho_1^2} - \\frac{1}{2} \\frac{d^2}{d\\rho_2^2} +\n V_1(\\rho_1) + V_2(\\rho_2) + V_{12}(\\rho_1,\\rho_2)\n \\right)\\phi_0(\\rho_1,\\rho_2) = \\lambda_0 \\phi_0(\\rho_1,\\rho_2).\n\\end{equation}\n with the lowest energy. Simultaniously,\nthere are one-particle states that are eigenstates of\n\\begin{equation}\\label{eq:eigenstate_H1}\n \\left(-\\frac{1}{2} \\frac{d^2}{d\\rho_1^2} + V_1(\\rho_1) \\right) \\phi_i(\\rho_1) = \\mu_i \\phi_i(\\rho_1)\n\\end{equation}\nand\n\\begin{equation}\n \\label{eq:eigenstate_H2}\n \\left(- \\frac{1}{2} \\frac{d^2}{d\\rho_2^2} + V_2(\\rho_2) \\right) \\varphi_i(\\rho_2) = \\nu_i \\varphi_i(\\rho_2).\n\\end{equation}\n\nWhen \\eqref{eq:radialequation} is solved with the energy $E = \\hbar\n\\nu + \\lambda<0$, there is only single ionization. Only one of the two\ncoordinates $\\rho_1$ or $\\rho_2$ can become large and the solution,\nas can be seen in Figure~\\ref{fig:singleionization}, is\nlocalized along both axis. The solution is a product of an outgoing\nwave in one coordinate and a bound state in the other\ncoordinate. For example, along the $\\rho_2$-axis, the solution is\ndescribed by $A_i(\\rho_2)\\phi_i(\\rho_1)$, where $A_i(\\rho_2)$ is a\none-dimensional outgoing wave, with an energy $E-\\mu_i$ and\n$\\phi_i(\\rho_1)$ is a bound state of \\eqref{eq:eigenstate_H1} in the\nfirst coordinate with energy $\\mu_i$. Similary, there is a wave,\nalong the $\\rho_1$ axis, that is an outgoing wave of the form\n$B_l(\\rho_1) \\varphi_l(\\rho_2)$, with a scattering wave in the first coordinate, $\\rho_1$,\nand a bound state in the coordinate $\\rho_2$, solution of\n\\eqref{eq:eigenstate_H2}.\n\nWhen \\eqref{eq:radialequation} is solved with energy $E = \\hbar \\nu +\n\\lambda \\ge 0$ there is also double ionization and both coordinates\n$\\rho_1$ and $\\rho_2$ can become large. We see, in\nFigure~\\ref{fig:doubleionization}, a (spherical) wave in the middle of\nthe domain, where both coordinates can be become large. To describe\nthis solution the full coordinate space is necessary. Note that\nthese solutions still show single ionization along the axes. Even for\n$E> 0$, one particle can take away all the energy and leave the other\nparticle as a bound state.\n\n\\begin{figure}\n\t\\centering\n\t\\begin{subfigure}{.45\\textwidth}\n\t\t\\centering\n\t\t\\input{singleionization2d-contourf-fullrank.tikz}\n\t\t\\caption{Single ionization}\n\t\t\\label{fig:singleionization}\n\t\\end{subfigure}\n\t\\begin{subfigure}{.45\\textwidth}\n\t\t\\centering\n\t\t\\input{doubleionization2d-contourf-fullrank.tikz}\n\t\t\\caption{Double ionization}\n\t\t\\label{fig:doubleionization}\n\t\\end{subfigure} \\caption{Left: When the energy $E = \\hbar \\nu + \\lambda_0 <0$, there is\n only single ionization. The solution is then localized along the\n edges, where the solution is a combination of an outgoing wave in\n the $\\rho_1$ and a bound state in $\\rho_2$, or vice-versa. Right: For\n energy $E>0$, there is, in addition to single ionization with\n solution localized along the edges, a double ionization wave where\n both coordinates can become large.\n \\label{fig:single_vs_double}}\n\\end{figure}\n\n\n\\begin{figure}\n\t\\centering\n\t\\begin{subfigure}{.3\\textwidth}\n\t\t\\centering\n\t\t\\input{wave2d-contourf-rank=1.tikz}\n\t\t\\caption{Rank-1 approximation}\n\t\\end{subfigure}\n\t\\begin{subfigure}{.3\\textwidth}\n\t\t\\centering\n\t\t\\input{wave2d-contourf-rank=2.tikz}\n\t\t\\caption{Rank-2 approximation}\n\t\\end{subfigure}\n\t\\begin{subfigure}{.3\\textwidth}\n\t\t\\centering\n\t\t\\input{wave2d-contourf-rank=3.tikz}\n\t\t\\caption{Rank-3 approximation}\n\t\\end{subfigure}\n\t\\\\~\\\\\n\t\\begin{subfigure}{.3\\textwidth}\n\t\t\\centering\n\t\t\\input{wave2d-contourf-rank=4.tikz}\n\t\t\\caption{Rank-4 approximation}\n\t\\end{subfigure}\n\t\\begin{subfigure}{.3\\textwidth}\n\t\t\\centering\n\t\t\\input{wave2d-contourf-rank=5.tikz}\n\t\t\\caption{Rank-5 approximation}\n\t\\end{subfigure}\n\t\\begin{subfigure}{.3\\textwidth}\n\t\t\\centering\n\t\t\\input{wave2d-contourf-fullrank.tikz}\n\t\t\\caption{Full rank wave}\n\t\\end{subfigure}\n\t\\caption{Contour plots of the double ionization wave function (bottom, right) and low-rank approximations for increasing rank.}\n\\end{figure}\n\n\n\\subsection{Coupled channel model for single ionization waves}\n\\label{sec:singleionization}\nIn this section, we write the single ionization solution as a low-rank\ndecomposition and derive the equations for the low-rank\ncomponents. When there is only single ionization, the total wave can\nbe written as\n\\begin{equation}\\label{expansion}\n u(\\rho_1,\\rho_2) = \\sum_{m=1}^{M} \\phi_m(\\rho_1) A_m(\\rho_2) + \\sum_{l=1}^L B_l(\\rho_1)\\varphi_l(\\rho_2),\n\\end{equation}\nwhere $\\phi_m(\\rho_1)$ and $\\varphi_l(\\rho_2)$ are the bound state eigenstates,\ndefined in \\eqref{eq:eigenstate_H1} and \\eqref{eq:eigenstate_H2}. The\nfirst term is localized along the $\\rho_2$-axis, the second term is\nlocalized along the $\\rho_1$-axis with $\\mu_i < 0$ and $\\nu_i<0$.\n\nAs discussed in \\cite{cools2016fast}, this expansion is not unique. We\ncan add multiples of $\\gamma_i \\varphi_i(\\rho_2)$ to $A_i(\\rho_2)$ and\nsimultaneously subtract $\\gamma_l \\phi_l(\\rho_1)$ from $B_l(\\rho_1)$\nwithout contaminating the result. Indeed, for any choice of $\\gamma_i\n\\in \\mathbb{C}$ and $L=M$ holds that\n\\begin{equation}\nu(\\rho_1,\\rho_2) = \\sum_{m=1}^{M} \\phi_m(\\rho_1) \\left(A_m(\\rho_2) + \\gamma_m \\varphi_m(\\rho_2) \\right) + \\sum_{l=1}^L \\left(B_l(\\rho_1)-\\gamma_i \\phi_i(\\rho_1))\\varphi_l(\\rho_2) \\right) = u(\\rho_1,\\rho_2).\n\\end{equation}\nTo make the expansion unique, \\cite{cools2016fast} chooses to select $A_i\n\\perp \\varphi_j$ when $j \\ge i$ and $B_i\\perp \\phi_m$ when $i \\ge j$.\n\nIn this paper, we choose to make the functions in the set $\\{\\phi_{i\n \\in\\{1,\\ldots,m\\}}, B_{l \\in \\{1,\\ldots, L \\}}\\}$ orthogonal. We\nalso assume that $V_{12}(\\rho_1,\\rho_2) \\approx \\sum_{i=1}^L\\sum_{l}^M\n\\phi_i(\\rho_1) \\varphi(\\rho_2) \\int\n\\phi_i^*(\\rho_1)\\varphi^*_l(\\rho_2) V_{12}(\\rho_1,\\rho_2) d\\rho_1\nd\\rho_2.$\n\nGiven a function $f(\\rho_1,\\rho_2)$, a right hand side, we can now\nderive the equations for $A_m$ and $B_l$. When we insert the low-rank\ndecomposition of the expansion \\eqref{expansion}, in the 2D Hamiltonian\n\\eqref{eq:radialequation}, multiply with $\\phi_i$ and integrate over\n$\\rho_1$ we find\n\\begin{equation}\n (H_2 + \\mu_j -E) A_j(\\rho_2) + \\sum_{l=1}^M V_{ij}(\\rho_2) A_l(\\rho_2) = \\int_0^\\infty \\phi^*_j(\\rho_1) f(\\rho_1,\\rho_2) d\\rho_1, \\quad \\text{for} \\quad j=1, \\ldots, M \\quad \\text{and}\\quad \\forall \\rho_2 \\in [0,L],\n\\end{equation}\nwith\n\\begin{equation}\n V_{ij}(\\rho_2)= \\int_0^\\infty \\phi^*_j(\\rho_1) V_{12}(\\rho_1,\\rho_2) \\phi_l(\\rho_1) d\\rho_1 .\n\\end{equation}\nWe have used that $\\phi_i \\perp B_l$ to eliminate the second term in\nthe expansion \\eqref{expansion}.\n\nSimilarly, for $B_l$, we find\n\\begin{equation}\n \\begin{aligned}\n (H_1 + \\nu_l -E) B_l(\\rho_1) + \\sum_{m=1}^L W_{l}(\\rho_1) B_k(\\rho_1) = \\int_0^\\infty\\!\\!\\!\\! \\varphi_l(\\rho_2) \\left(f(\\rho_1,\\rho_2)-\\sum_i^M \\phi_i \\phi^* f(\\rho_1,\\rho_2) \\right) d\\rho_2 \\quad\\\\\n \\text{for} \\quad i=1,\\ldots, M, \\quad \\forall \\rho_1 \\in [0,\\infty[\n \\end{aligned}\n\\end{equation}\nwhere\n\\begin{equation}\n W_{lk}(\\rho_1):=\\left(\\int_0^\\infty \\!\\!\\!\\! \\varphi^*_l(\\rho_2) V_{12}(\\rho_1,\\rho_2) \\varphi_k(\\rho_2) d\\rho_2 \\right).\n\\end{equation}\n\n\\section{Low-rank matrix representation of a 2D wave function that includes both single and double ionization}\n\\label{sec:lowrank}\n\\subsection{Low rank of the double ionization solution}\nWe now discuss the main result of the paper. We will derive a\ncoupled channel equation that gives a low-rank approximation for the\ndouble ionization wave function, as shown in Figure~\\ref{fig:doubleionization}.\n \nIn section \\ref{sec:singleionization} we have shown\nthat the single ionization wave can be represented by a low-rank\ndecomposition. In this section, we show that also the double\nionization wave can be written as a similar low-rank decomposition.\n\nWe first illustrate that the solution of a 2D driven Schr\\\"odinger equation\nthat contains both single and double ionization, it is a solution of\n\\eqref{eq:radialequation} with $E>0$, can be represented by a similar low-rank\ndecomposition.\n\nIn Figure~\\ref{fig:wavefunction} we solve the Helmholtz equation with a\nspace-dependent wave number, $k(x,y)$, in the first quadrant where $x\n\\ge 0$ and $y \\ge 0$. The equation is\n\\begin{equation}\\label{driven}\n\\left( - \\Delta_2 - k^2(x,y) \\right) u_{sc}(x,y) = f(x,y),\n\\end{equation}\nwhere $\\Delta_2$ is the 2D Laplacian and the solution $u_{sc}$\nsatisfies homogeneous boundary conditions $u_{sc}(x,0)=0$ for all\n$x\\ge 0$ and $u_{sc}(0,y)=0$ for all $y \\ge 0$. On the other\nboundaries we have outgoing boundary conditions.\n\nThe right hand side $f(x,y)$ has a support that is limited to $[0,b]^2\n\\subset [0,L]^2 \\subset \\mathbb{R}_+^2$, i.e. $f(x,y)=0$, for all\n$x\\ge b$ or $y\\ge b$.\n\nThe wave number $k(x,y)$ can be split in a constant part, $k_0^2$ and\nvariable part $\\chi(x,y)$. The variable part is also only non-zero on\n $[0,b[^2$\n\\begin{equation}\n k^2(x,y) = \\begin{cases}\n k_0^2 \\left(1 + \\chi(x,y) \\right) \\quad &\\text{if} \\quad x r_1$.\nIn a similar way, equations for $\\mv{U}_2$ and $\\mv{U}_3$ are\nderived by multiplying \\eqref{eq:start3d} with the other factor\nmatrices in the appropriate directions:\n\\begin{equation}\\label{eq:eqnU2}\n\t\\left\\{\n\t\\mv{I} \\otimes \\conj{\\left(-\\mv{D}_{yy} - k^2 \\mv{I} \\right)} +\n\t\\left[ \n\t\\conj{-\\left(\\mv{I} \\otimes \\mv{U}_1^\\H \\mv{D}_{xx} \\mv{U}_1\\right)} - \\conj{\\left(\\mv{U}_3^\\H \\mv{D}_{zz} \\mv{U}_3 \\otimes \\mv{I} \\right)}\n\t\\right] \\otimes \\mv{I}\n\t\\right\\} \\mvec{\\underbrace{\\conj{\\mv{U}_2}\\mv{G}_{(2)}}_{\\mv{X}_2}} \n\t= \\mvec{\\mv{F}_{(2)} \\left(\\mv{U}_3 \\otimes \\mv{U}_1\\right)},\n\\end{equation}\nand\n\\begin{equation}\\label{eq:eqnU3}\n\t\\left\\{\n\t\\mv{I} \\otimes \\conj{\\left(-\\mv{D}_{zz} - k^2 \\mv{I} \\right)} +\n\t\\left[- \n\t\\conj{\\left(\\mv{I} \\otimes \\mv{U}_1^\\H \\mv{D}_{xx} \\mv{U}_1\\right)} - \\conj{\\left(\\mv{U}_2^\\H \\mv{D}_{yy} \\mv{U}_2 \\otimes \\mv{I} \\right)}\n\t\\right] \\otimes \\mv{I}\n\t\\right\\} \\mvec{\\underbrace{\\conj{\\mv{U}_3}\\mv{G}_{(3)}}_{\\mv{X}_3}} \n\t= \\mvec{\\mv{F}_{(3)} \\left(\\mv{U}_2 \\otimes \\mv{U}_1\\right)}.\n\\end{equation}\nAlternating between solving for $\\mv{U}_1$, $\\mv{U}_2$ and $\\mv{U}_3$\nusing \\eqref{eq:eqnU1}, \\eqref{eq:eqnU2} or \\eqref{eq:eqnU3} results\nin algorithm that approximates the low-rank solutions for three\ndimensional problems as given in \\eqref{eq:driven3d}. This algorithm\nis summarized in Algorithm~\\ref{alg:const3dv1}.\nAlso in the three dimensional case the orthogonality of the columns \nof $\\mv{U}_1$, $\\mv{U}_2$ and $\\mv{U}_3$ are maintained by \nadditional QR factorizations.\n\\begin{algorithm}[t]\n \\SetAlgoLined\n\t[$\\mt{G}, \\mv{U}_1, \\mv{U}_2, \\mv{U}_3$] = hosvd(initial guess)\\;\n\t\\While{not converged}{\n\t\t\\For{i = 1, 2, 3}{\n\t\t\tSolve for $\\mv{X}_i = \\conj{\\mv{U}_i}\\mv{G}_{(i)} \\in \\mathbb{C}^{n \\times r^{d-1}}$ using \\eqref{eq:eqnU1}, \\eqref{eq:eqnU2} or \\eqref{eq:eqnU3}\\;\n\t\t\t$\\conj{\\mv{U}_{i}} \\mv{G}_{(i)} = \\qr{\\mv{X}_i(:,~ 1:r_i), 0}$;\n\t\t}\n\t}\n\t$\\mt{G} = \\texttt{reconstruct} \\left[\\mv{G}_{(i)}, ~i\\right]$\\;\n\t$\\mt{M} = \\mt{G} \\times_1 \\mv{U}_1 \\times_2 \\mv{U}_2 \\times_3 \\mv{U}_3$\\;\n \\caption{Solve for the low-rank tensor decomposition of the solution $\\mt{M}$ of a 3D Helmholtz with constant wave number (version 1).}\n \\label{alg:const3dv1}\n\\end{algorithm}\nObserve that we solve for a large matrix $\\mv{X}_i \\in \\mathbb{C}^{n_i \\times\n r_1r_2r_3\/r_i}$. So, in general the rank of this matrix could be\n$\\min\\left(n_i,~ r_1r_2r_3\/r_i\\right)$. But it is also known that\n$\\mv{X}_i = \\conj{\\mv{U}_i}\\mv{G}_{(i)}$ which leads to the fact that\nthe rank of $\\mv{X}_i$ should be at most $r_i$. Selecting the\nfirst $r_i$ columns of $\\mv{X}_i$ and computing its QR decomposition\nis sufficient to derive a new orthonormal basis as factor matrix\n$\\conj{\\mv{U}}_i$.\n\n\nFinally, observe that solving for $\\mv{X}_i$ using \\eqref{eq:eqnU1},\n\\eqref{eq:eqnU2} or \\eqref{eq:eqnU3} is computationally not\nefficient. In all iterations, we solve for a total of $d nr^{d-1}$\nunknowns, while there are only $r^d + dnr$ unknowns in the Tucker\ntensor factorization. Furthermore, solving equations \\eqref{eq:eqnU1},\n\\eqref{eq:eqnU2} and \\eqref{eq:eqnU3} is also expensive. Indeed,\ncomputing a symmetric reverse Cuthill-McKee permutation of the system\nmatrix one observes a matrix with a bandwidth $\\Oh{r^{d-1}}$. For\nexample when $d = 3, n_i = n = 168, r_i = r = 18$ one obtains the\nsparsity pattern on the diagonal of the matrix as shown in Figure\n\\ref{fig:spytop_symrcm_UGsystem_alg1}. So solving a system as given in\n\\eqref{eq:eqnU1}, \\eqref{eq:eqnU2} or \\eqref{eq:eqnU3} has a\ncomputational cost of $\\Oh{nr^{2(d-1)}}$.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\begin{subfigure}{0.4\\textwidth}\n\t\t\\includegraphics[width=\\textwidth]{spytop_symrcm_UGsystem_alg1.png}\n\t\t\\caption{Sparsity pattern of the top of the symmetric reverse Cuthill-McKee permutation of the system matrix to solve for $\\mv{X}_i$ using \\eqref{eq:eqnU1}. Note: only the first 4.5 of the 168 blocks are shown.}\n\t\t\\label{fig:spytop_symrcm_UGsystem_alg1}\n \t\\end{subfigure}\n \\quad\n\t\\begin{subfigure}{0.4\\textwidth}\n\t\t\\includegraphics[width=\\textwidth]{spytop_symrcm_Gsystem_alg2.png}\n\t\t\\caption{Sparsity pattern of the symmetric reverse Cuthill-McKee permutation of the system matrix to solve for $\\mv{G}_{(1)}$ using \\eqref{eq:eqnGv2}.}\n\t\t\\label{fig:spytop_symrcm_Gsystem_alg2}\n\t\\end{subfigure}\n\t\\caption{Sparsity patterns of the symmetric reverse Cuthill-McKee permutation of certain system matrices ($d = 3, n = 168, r = 18$).}\n\\end{figure}\n\n\n\\subsubsection{Solving for the basis functions and the core tensor separately (version 2)}\\label{sec:version2}\nTo circumvent solving the large systems in \\eqref{eq:eqnU1},\n\\eqref{eq:eqnU2} and \\eqref{eq:eqnU3}, we can pre-compute the\nQR factorization of the unfolding of the core tensor, $\\mv{G}_{(i)}$,\nand project the equations onto the obtained $\\mv{Q}_i$. Indeed, this\nwill further reduce the number of unknowns in these linear systems to exactly\nthe number of unknowns that are needed for the factor matrices\n$\\conj{\\mv{U}}_i$, for $i = 1,2,3$.\n\nLet us discuss the details. We start again from equation \\eqref{eq:firstUnfolding} and use the QR\nfactorization of $\\mv{G}_{(1)}^\\H$, $\\mv{Q}_1 \\mv{R}_1^\\H = \\qr{\\mv{G}^\\H_{(1)}}$. This yields\n\\begin{equation*}\n\t\\conj{\\left(-\\mv{D}_{xx} - k^2 \\mv{I} \\right) \\mv{U}_1} \\mv{R}_1\\mv{Q}_1^\\H\n\t- \\conj{\\mv{U}_1} \\mv{R}_1\\mv{Q}_1^\\H \\left(I \\otimes \\mv{U}_2^\\H \\mv{D}_{yy} \\mv{U}_2\\right)^\\H\n\t- \\conj{\\mv{U}_1} \\mv{R}_1\\mv{Q}_1^\\H \\left(\\mv{U}_3^\\H \\mv{D}_{zz} \\mv{U}_3 \\otimes I\\right)^\\H\n\t= \\mv{F}_{(1)} \\left(\\mv{U}_3 \\otimes \\mv{U}_2\\right).\n\\end{equation*}\nPost multiplication of this equation by $\\mv{Q}_1$ yields\n\\begin{equation*}\n\t\\conj{\\left(-\\mv{D}_{xx} - k^2 \\mv{I} \\right) \\mv{U}_1} \\mv{R}_1\n\t- \\conj{\\mv{U}_1} \\mv{R}_1\\mv{Q}_1^\\H \\left(I \\otimes \\mv{U}_2^\\H \\mv{D}_{yy} \\mv{U}_2\\right)^\\H \\mv{Q}_1\n\t- \\conj{\\mv{U}_1} \\mv{R}_1\\mv{Q}_1^\\H \\left(\\mv{U}_3^\\H \\mv{D}_{zz} \\mv{U}_3 \\otimes I\\right)^\\H \\mv{Q}_1\n\t= \\mv{F}_{(1)} \\left(\\mv{U}_3 \\otimes \\mv{U}_2\\right)\\mv{Q}_1.\n\\end{equation*}\nTo solve this equation for $\\mv{U}_1$, it is written is vectorized form\nas\n\\begin{equation}\\label{eq:eqnU1v2}\n\\left\\{\n\\mv{I} \\otimes \\conj{\\left(-\\mv{D}_{xx} - k^2 \\mv{I} \\right)} +\n\\mv{Q}_1^{\\mkern-1.5mu\\mathsf{T}}\\left[ -\\conj{\\left(\\mv{I} \\otimes \\mv{U}_2^\\H \\mv{D}_{yy} \\mv{U}_2\\right)} - \\conj{\\left(\\mv{U}_3^\\H \\mv{D}_{zz} \\mv{U}_3 \\otimes \\mv{I} \\right)}\n\\right]\\conj{\\mv{Q}_1} \\otimes \\mv{I}\n\\right\\} \\mvec{\\underbrace{\\conj{\\mv{U}_1}\\mv{R}_1}_{\\mv{X}_1}} = \\mvec{\\mv{F}_{(1)} \\left(\\mv{U}_3 \\otimes \\mv{U}_2\\right) \\mv{Q}_1}.\n\\end{equation}\nIn a similar way, the update equations for $\\mv{U}_2$ and $\\mv{U}_3$ are\nderived by multiplying \\eqref{eq:start3d} with the other factor\nmatrices in the appropriate dimensions and using the QR factorizations\nof $\\mv{G}_{(i)}^\\H$:\n\\begin{equation}\\label{eq:eqnU2v2}\n\\left\\{\n\\mv{I} \\otimes \\conj{\\left(-\\mv{D}_{yy} - k^2 \\mv{I} \\right)} +\n\\mv{Q}_2^{\\mkern-1.5mu\\mathsf{T}}\\left[- \n\\conj{\\left(\\mv{I} \\otimes \\mv{U}_1^\\H \\mv{D}_{xx} \\mv{U}_1\\right)} - \\conj{\\left(\\mv{U}_3^\\H \\mv{D}_{zz} \\mv{U}_3 \\otimes \\mv{I} \\right)}\n\\right]\\conj{\\mv{Q}_2} \\otimes \\mv{I}\n\\right\\} \\mvec{\\underbrace{{\\conj{\\mv{U}_2}\\mv{R}_2}}_{\\mv{X}_2}} \n= \\mvec{\\mv{F}_{(2)} \\left(\\mv{U}_3 \\otimes \\mv{U}_1\\right) \\mv{Q}_2}.\n\\end{equation}\nand\n\\begin{equation}\\label{eq:eqnU3v2}\n\\left\\{\n\\mv{I} \\otimes \\conj{\\left(-\\mv{D}_{zz} - k^2 \\mv{I} \\right)} +\n\\mv{Q}_3^{\\mkern-1.5mu\\mathsf{T}}\\left[ -\n\\conj{\\left(\\mv{I} \\otimes \\mv{U}_1^\\H \\mv{D}_{xx} \\mv{U}_1\\right)} - \\conj{\\left(\\mv{U}_2^\\H \\mv{D}_{yy} \\mv{U}_2 \\otimes \\mv{I} \\right)}\n\\right]\\conj{\\mv{Q}_3} \\otimes \\mv{I}\n\\right\\} \\mvec{\\underbrace{\\conj{\\mv{U}_3}\\mv{R}_3}_{\\mv{X}_3}} \n= \\mvec{\\mv{F}_{(3)} \\left(\\mv{U}_2 \\otimes \\mv{U}_1\\right) \\mv{Q}_3}.\n\\end{equation}\nAll these equations are cheap to solve. Indeed,\n$\\mvec{\\conj{\\mv{U}_i}\\mv{R}_i}$ has length $n_i r_i$. Computing a\nsymmetric reverse Cuthill-McKee permutation of these system matrices one\nobserves a matrix with a bandwidth $\\Oh{r}$, so solving these\nequations has a computational cost $\\Oh{nr^2}$.\n\nOf course, this only updates the factor matrices as basis vectors in each\ndirection. As a single final step, we still have to compute the core\ntensor $\\mt{G}$. This will be the computationally most expensive\npart.\n\nCore tensor $\\mt{G}$ can be obtained by multiplying \\eqref{eq:start3d}\nwith all the $d$ factor matrices in the matching directions.\nUnfolding this equation in a certain direction (eg. the first folding)\nleads again to a matrix equation. In vectorized form, it is given by\n\\begin{equation}\\label{eq:eqnGv2}\n\t\\left\\{\n\t\\mv{I} \\otimes \\conj{\\mv{U}_1^\\H\\left(-\\mv{D}_{xx} - k^2 \\mv{I} \\right)\\mv{U}_1} +\n\t\\left[ \n\t-\\conj{\\left(\\mv{I} \\otimes \\mv{U}_2^\\H \\mv{D}_{yy} \\mv{U}_2\\right)} - \\conj{\\left(\\mv{U}_3^\\H \\mv{D}_{zz} \\mv{U}_3 \\otimes \\mv{I} \\right)}\n\t\\right] \\otimes \\mv{I}\n\t\\right\\} \\mvec{\\mv{G}_{(1)}} = \\mvec{\\mv{U}_1^{\\mkern-1.5mu\\mathsf{T}} \\mv{F}_{(1)} \\left(\\mv{U}_3 \\otimes \\mv{U}_2\\right)}.\n\\end{equation}\nIndeed, considering again an example where $d = 3, n_i = n = 168, r_i\n= r = 18$ one obtains a matrix with a sparsity pattern that is shown\nin Figure~\\ref{fig:spytop_symrcm_Gsystem_alg2}. Hence, this matrix has\nnot a limited bandwidth anymore. It coupled all functions to all other\nfunctions. Although this equation has to be solved only once in the \nalgorithm, when the rank increases, it will rapidly dominate the computational \ncost of this algorithm.\n\n\n\\begin{algorithm}[t]\n\t[$\\mt{G}, \\mv{U}_1, \\mv{U}_2, \\mv{U}_3$] = hosvd(initial guess)\\;\n\t\\While{not converged}{\n\t\t\\For{i = 1, 2, 3}{\n\t\t\t$\\mv{Q}_i \\widetilde{\\mv{R}} = \\qr{\\mv{G}_{(i)}^\\H, 0}$\\;\n\t\t\tSolve for $\\mv{X}_i = \\conj{\\mv{U}_i}\\mv{R}_i \\in \\mathbb{C}^{n_i \\times r_i}$ using \\eqref{eq:eqnU1v2}, \\eqref{eq:eqnU2v2} or \\eqref{eq:eqnU3v2}\\;\n\t\t\t$\\conj{\\mv{U}_{i}} \\mv{R}_i = \\qr{\\mv{X}_i, 0}$\\;\n\t\t\t$\\mt{G} = \\texttt{reconstruct} \\left[\\mv{R}_i\\mv{Q}_i^\\H, ~i\\right]$;\n\t\t}\n\t}\n\tSolve for $\\mv{G}_{(1)} \\in \\mathbb{C}^{r_1 \\times r_2r_3}$ using \\eqref{eq:eqnGv2}\\;\n\t$\\mt{G} = \\texttt{reconstruct} \\left[\\mv{G}_{(1)}, ~1\\right]$\\;\n\t$\\mt{M} = \\mt{G} \\times_1 \\mv{U}_1 \\times_2 \\mv{U}_2 \\times_3 \\mv{U}_3$\\;\n \\caption{Solve for the low-rank tensor decomposition of the solution $\\mt{M}$ of a 3D Helmholtz problem with constant wave number (version 2).}\n \\label{alg:const3dv2}\n\\end{algorithm}\n\n\n\n\\subsubsection{Efficient combination of version 1 and version 2 into new algorithm (version 3)}\\label{sec:version3}\nIn the first version of the algorithm, (see section~\\ref{sec:version1}),\nan update for $\\mv{G}_{(i)}$ is computed for each direction in each\niteration. This leads to a too expensive algorithm.\nThen we changed the algorithm such that the costs for the updates in each\ndirection is reduced, (see section~\\ref{sec:version2}). But, in that version\nalmost all information for a full update of core tensor $\\mt{G}$ is\nlost. Therefore a final, but potential too expensive, equation needs to be solved.\n\nObserve that the expensive computation for the full core tensor, in \nversion 2, can now be replaced by a single solve per iteration as done in\nversion 1. This leads to a third version of the algorithm.\nIt avoids repeatedly solving the large systems (like version 1) and it\ndoes not solve too expensive systems (like version 2). The computational\ncomplexity of this algorithm is equal to the complexity of version 1,\nso $\\Oh{nr^{2(d-1)}}$.\nFurthermore, the systems that need to be solved, each iteration, have \nexactly the same number of unknowns as the representation of the \ntensor in low-rank Tucker tensor format.\nIn summary, this final version of the algorithm is given by Algorithm~\\ref{alg:const3dv3}.\n\n\\begin{algorithm}[t]\n\t[$\\mt{G}, \\mv{U}_1, \\mv{U}_2, \\mv{U}_3$] = hosvd(initial guess)\\;\n\t\\While{not converged}{\n\t\t\\For{i = 1, 2}{\n\t\t\t$\\mv{Q}_i \\widetilde{\\mv{R}} = \\qr{\\mv{G}_{(i)}^\\H, 0}$\\;\n\t\t\tSolve for $\\mv{X}_i = \\conj{\\mv{U}_i}\\mv{R}_i \\in \\mathbb{C}^{n_i \\times r_i}$ using \\eqref{eq:eqnU1v2} or \\eqref{eq:eqnU2v2}\\;\n\t\t\t$\\conj{\\mv{U}_{i}} \\mv{R}_i = \\qr{\\mv{X}_i, 0}$\\;\n\t\t\t$\\mt{G} = \\texttt{reconstruct} \\left[\\mv{R}_i\\mv{Q}_i^\\H, ~i\\right]$;\n\t\t}\n\t\tSolve for $\\mv{X}_3 = \\conj{\\mv{U}_3}\\mv{G}_{(3)} \\in \\mathbb{C}^{n_3 \\times r^{d-1}}$ using \\eqref{eq:eqnU3}\\;\n\t\t$\\conj{\\mv{U}_{3}} \\mv{G}_{(3)} = \\qr{\\mv{X}_3, 0}$\\;\n\t\t$\\mt{G} = \\texttt{reconstruct} \\left[\\mv{G}_{(3)},~3\\right]$\\;\n\t}\n\t$\\mt{M} = \\mt{G} \\times_1 \\mv{U}_1 \\times_2 \\mv{U}_2 \\times_3 \\mv{U}_3$\\;\n \\caption{Solve for the low-rank tensor decomposition of the solution $\\mt{M}$ of a 3D Helmholtz problem with constant wave number (version 3).}\n \\label{alg:const3dv3}\n\\end{algorithm}\n\n\\subsubsection{Numerical comparison of three versions for 3D Helmholtz equation}\nConsider a three dimensional domain $\\Omega = [-10, ~10]^3$ that is\ndiscretized with $M=100$ equidistant mesh points per direction in the\ninterior of the domain. The domain is extended with exterior complex\nscaling to implement the absorbing boundary conditions. Hence, in total\nthere are $n = n_x = n_y = n_z = 168$ unknowns per direction. As\nconstant wave number we use $\\omega = 2$ and a right hand side $f(x,y,z)\n= -e^{-x^2-y^2-z^2}$.\nBy symmetry, we expect a low rank factorization with a equal low-rank \nin each direction, so we fix $r = r_x = r_y = r_x$.\n\nThe convergence of the residuals of the three versions are given in\nthe left column of Figure~\\ref{fig:residual+runtime-const3d}.\nIt is clear that all three versions converge to a good low-rank \napproximation of the full solution. By increasing the maximal\nattainable rank $r$, a better low-rank solution is obtained, as \nexpected. Remarkably, for $r=30$, in version 2, the final residual\nis larger then the residuals obtained by both other algorithms while\nthe compute-time for version 2 is larger than the other algorithms.\n\n\\begin{figure}\n\t\\centering\n\t\\input{residual-v1-M=100.tikz} ~ \\input{runtime-v1-3d-M=100.tikz} \\par\n\t\\input{residual-v2-M=100.tikz} ~ \\input{runtime-v2-3d-M=100.tikz} \\par\n\t\\input{residual-v3-M=100.tikz} ~ \\input{runtime-v3-3d-M=100.tikz}\n \\caption{Left: Plot of residual per iteration for constant wave number in 3D Helmholtz problem.\n \tRight: Plot of runtime of most time consuming parts for constant wave number in 3D Helmholtz problem. Both problems have $M=100$.\n \tTop: Algorithm~\\ref{alg:const3dv1} (version 1),\n \tmiddle: Algorithm~\\ref{alg:const3dv2} (version 2),\n \tbottom: Algorithm~\\ref{alg:const3dv3} (version 3).}\n\t\\label{fig:residual+runtime-const3d}\n\\end{figure}\n\nThe compute-time for the most time-consuming parts in the different\nversions of the algorithm can be measured as a function of the maximal\nattainable rank $r$. For the three versions of the algorithm the\nruntimes are shown in the right column of\nFigure~\\ref{fig:residual+runtime-const3d}. For all parts the expected\nand measured dependence on the rank $r$ are given. For all versions of\nthe algorithm 10 iterations are applied.\n\n\\begin{figure}\n\t\\centering\n \t\\input{runtimes-v123-M=100-iter=4.tikz}\n \\caption{Plot of runtime for 4 iteration with constant wavenumber in 3D using the three different algorithms ($M=100$)\n \t\\label{fig:runtime-const3d}}\n\\end{figure}\n\n\nComparing the total runtime for the three different versions one obtains\nresults as shown in Figure~\\ref{fig:runtime-const3d}. Indeed, as\nexpected version 3 is approximately 3 times faster than version 1 and\nthe runtime scales similar in rank $r$. Further, for small rank $r$\nversion 2 is faster than both other versions. But when the rank\nincreases the expensive solve for the core tensor $\\mt{G}$ starts to\ndominate the runtime. The total runtime will increase dramatically.\n\n\n\\subsection{Projection operator for constant wave number}\nAlso in three dimensions we can write the linear systems\n\\eqref{eq:eqnU1} for $\\mv{U}_1$, \\eqref{eq:eqnU2} for \n$\\mv{U}_2$ and \\eqref{eq:eqnU3} for $\\mv{U}_3$ as\nprojection operators applied to the residual of the tensor equation,\n\\eqref{eq:start3d}.\n\nConsider a tensor $\\mt{M}$ in Tucker format and factorized as $\\mt{M}\n= \\mt{G} \\times_1 \\mv{U}_1 \\times_2 \\mv{U}_2 \\times_3 \\mv{U}_3$, with\nunknowns $\\mt{G}, \\mv{U}_1, \\mv{U}_2$ and $\\mv{U}_3$. Discretization\nof \\eqref{eq:driven3d} leads to a linear operator $\\mathcal{L}$\napplied on tensors. Its matrix representation $\\mv{L}$ has a sum\nof Kronecker products structure, as given in\n\\eqref{eq:operatorLasSumKroneckerProducts}.\n\nSolving for an unknown factors $\\mv{U}_1$, $\\mv{U}_2$ or $\\mv{U}_3$\n(and the core-tensor $\\mt{G}$) using \\eqref{eq:eqnU1},\n\\eqref{eq:eqnU2} or \\eqref{eq:eqnU3} can be interpreted as a\nprojection operator applied on the residual. For example,\n\\eqref{eq:eqnU1} can be interpreted as\n\\begin{equation}\n\t\\conj{\\left(\\mv{U}_3^\\H \\otimes \\mv{U}_2^\\H \\otimes \\mv{I} \\right) \\mv{L} \\left(\\mv{U}_3 \\otimes \\mv{U}_2 \\otimes \\mv{I}\\right)} \\mvec{\\conj{\\mv{U}_1}\\mv{G}_{(1)}} = \\left( \\mv{U}_3^{\\mkern-1.5mu\\mathsf{T}} \\otimes \\mv{U}_2^{\\mkern-1.5mu\\mathsf{T}} \\otimes \\mv{I} \\right) \\mvec{\\mv{F}_{(1)}}.\n\\end{equation}\nThe residual, in tensor format, is given by\n\\begin{equation}\n\t\\begin{split}\n\t\t\\mt{R} &= \\mt{F} - \\mathcal{L}\\mt{M}, \\\\\n\t\t&= \\mt{F} - \\mt{G} \\times_1 \\left(-\\mv{D}_{xx} - k^2 \\mv{I}\\right) \\mv{U}_1 \\times_2 \\mv{U}_2 \\times_3 \\mv{U}_3 + \\mt{G} \\times_1 \\mv{U}_1 \\times_2 \\mv{D}_{yy}\\mv{U}_2 \\times_3 \\mv{U}_3 + \\mt{G} \\times_1 \\mv{U}_1 \\times_2 \\mv{U}_2 \\times_3 \\mv{D}_{zz}\\mv{U}_3.\n\t\\end{split}\n\\end{equation}\nWriting this tensor equation in the first unfolding leads to the following matrix equation\n\\begin{equation}\n\t\t\\mv{R}_{(1)} = \\mv{F}_{(1)} - \\conj{\\left(-\\mv{D}_{xx} - k^2 \\mv{I}\\right)\\mv{U}_1}\\mv{G}_{(1)} \\left(\\mv{U}_3 \\otimes \\mv{U}_2 \\right)^\\H + \\conj{\\mv{U}_1}\\mv{G}_{(1)} \\left(\\mv{U}_3 \\otimes \\mv{D}_{yy}\\mv{U}_2 \\right)^\\H + \\conj{\\mv{U}_1}\\mv{G}_{(1)} \\left(\\mv{D}_{zz}\\mv{U}_3 \\otimes \\mv{U}_2 \\right)^\\H,\n\\end{equation}\nwhich can be vectorized as\n\\begin{equation}\n\t\\begin{split}\n\t\t\\mvec{\\mv{R}_{(1)}} &= \\mvec{\\mv{F}_{(1)}} - \\left(\\conj{\\left(\\mv{U}_3 \\otimes \\mv{U}_2 \\otimes \\left(-\\mv{D}_{xx}-k^2\\mv{I}\\right)\\right)} - \\conj{\\left(\\mv{U}_3 \\otimes \\mv{D}_{yy}\\mv{U}_2 \\otimes \\mv{I}\\right)} - \\conj{\\left(\\mv{D}_{zz}\\mv{U}_3 \\otimes \\mv{U}_2 \\otimes \\mv{I}\\right)}\\right)\\mvec{\\conj{\\mv{U}_1}\\mv{G}_{(1)}} \\\\\n\t\t&= \\mvec{\\mv{F}_{(1)}} - \\conj{\\mv{L} \\left(\\mv{U}_3 \\otimes \\mv{U}_2 \\otimes \\mv{I}\\right)}\\mvec{\\conj{\\mv{U}_1}\\mv{G}_{(1)}} \\\\\n\t\t&= \\mvec{\\mv{F}_{(1)}} - \\conj{\\mv{L} \\left(\\mv{U}_3 \\otimes \\mv{U}_2 \\otimes \\mv{I}\\right)} \\left[\\conj{\\left(\\mv{U}_3^\\H \\otimes \\mv{U}_2^\\H \\otimes \\mv{I} \\right) \\mv{L} \\left(\\mv{U}_3 \\otimes \\mv{U}_2 \\otimes \\mv{I}\\right)}\\right]^{-1}\\left( \\mv{U}_3^{\\mkern-1.5mu\\mathsf{T}} \\otimes \\mv{U}_2^{\\mkern-1.5mu\\mathsf{T}} \\otimes \\mv{I} \\right) \\mvec{\\mv{F}_{(1)}} \\\\\n\t\t&= P_{23}\\mvec{\\mv{F}_{(1)}},\n\t\\end{split}\n\\end{equation}\nwhere operator $P_{23}$ is given by\n\\begin{equation}\\label{eq:projector3dconst}\n\t\\begin{split}\n\t\tP_{23} &= \\mv{I} - \\conj{\\mv{L} \\left(\\mv{U}_3 \\otimes \\mv{U}_2 \\otimes \\mv{I}\\right)} \\left[\\conj{\\left(\\mv{U}_3^\\H \\otimes \\mv{U}_2^\\H \\otimes \\mv{I} \\right) \\mv{L} \\left(\\mv{U}_3 \\otimes \\mv{U}_2 \\otimes \\mv{I}\\right)}\\right]^{-1}\\left( \\mv{U}_3^{\\mkern-1.5mu\\mathsf{T}} \\otimes \\mv{U}_2^{\\mkern-1.5mu\\mathsf{T}} \\otimes \\mv{I} \\right) \\\\\n\t\t&= \\mv{I} - \\mv{X}.\n\t\\end{split}\n\\end{equation}\nThis operator $P_{23}$ is indeed a projection operator. Observe that the terms between the two inverses cancel against one of the inverse factors:\n\\begin{equation*}\n\t\\begin{split}\n\t\t\\mv{X}^2 &= \\conj{\\mv{L} \\left(\\mv{U}_3 \\otimes \\mv{U}_2 \\otimes \\mv{I}\\right)} \\left[\\conj{\\left(\\mv{U}_3^\\H \\otimes \\mv{U}_2^\\H \\otimes \\mv{I} \\right) \\mv{L} \\left(\\mv{U}_3 \\otimes \\mv{U}_2 \\otimes \\mv{I}\\right)}\\right]^{-1}\\left( \\mv{U}_3^{\\mkern-1.5mu\\mathsf{T}} \\otimes \\mv{U}_2^{\\mkern-1.5mu\\mathsf{T}} \\otimes \\mv{I} \\right)\\conj{\\mv{L} \\left(\\mv{U}_3 \\otimes \\mv{U}_2 \\otimes \\mv{I}\\right)} \\left[\\conj{\\left(\\mv{U}_3^\\H \\otimes \\mv{U}_2^\\H \\otimes \\mv{I} \\right) \\mv{L} \\left(\\mv{U}_3 \\otimes \\mv{U}_2 \\otimes \\mv{I}\\right)}\\right]^{-1}\\left( \\mv{U}_3^{\\mkern-1.5mu\\mathsf{T}} \\otimes \\mv{U}_2^{\\mkern-1.5mu\\mathsf{T}} \\otimes \\mv{I} \\right) \\\\\n\t\t&= \\conj{\\mv{L} \\left(\\mv{U}_3 \\otimes \\mv{U}_2 \\otimes \\mv{I}\\right)} \\left[\\conj{\\left(\\mv{U}_3^\\H \\otimes \\mv{U}_2^\\H \\otimes \\mv{I} \\right) \\mv{L} \\left(\\mv{U}_3 \\otimes \\mv{U}_2 \\otimes \\mv{I}\\right)}\\right]^{-1}\\left( \\mv{U}_3^{\\mkern-1.5mu\\mathsf{T}} \\otimes \\mv{U}_2^{\\mkern-1.5mu\\mathsf{T}} \\otimes \\mv{I} \\right) \\\\\n\t\t&= \\mv{X}.\n\t\\end{split}\n\\end{equation*}\nThis operator is a natural extension to higher dimensions of the two\ndimensional operators as derived in section \\ref{sec:projection2d}.\n\nA similar derivation results in projection operators $P_{13}$ and $P_{12}$ for the updates in $\\mv{U}_2$ and $\\mv{U}_3$, respectively.\n\\begin{equation}\\label{eq:projectors3dconst}\n\t\\begin{split}\n\t\tP_{23} &= \\mv{I} - \\conj{\\mv{L} \\left(\\mv{U}_3 \\otimes \\mv{U}_2 \\otimes \\mv{I}\\right)} \\left[\\conj{\\left(\\mv{U}_3^\\H \\otimes \\mv{U}_2^\\H \\otimes \\mv{I} \\right) \\mv{L} \\left(\\mv{U}_3 \\otimes \\mv{U}_2 \\otimes \\mv{I}\\right)}\\right]^{-1}\\left( \\mv{U}_3^{\\mkern-1.5mu\\mathsf{T}} \\otimes \\mv{U}_2^{\\mkern-1.5mu\\mathsf{T}} \\otimes \\mv{I} \\right), \\\\\n\t\tP_{13} &= \\mv{I} - \\conj{\\mv{L} \\left(\\mv{U}_3 \\otimes \\mv{I} \\otimes \\mv{U}_1\\right)} \\left[\\conj{\\left(\\mv{U}_3^\\H \\otimes \\mv{I} \\otimes \\mv{U}_1^\\H \\right) \\mv{L} \\left(\\mv{U}_3 \\otimes \\mv{I} \\otimes \\mv{U}_1\\right)}\\right]^{-1}\\left( \\mv{U}_3^{\\mkern-1.5mu\\mathsf{T}} \\otimes \\mv{I} \\otimes \\mv{U}_1^{\\mkern-1.5mu\\mathsf{T}} \\right), \\\\\n\t\tP_{12} &= \\mv{I} - \\conj{\\mv{L} \\left(\\mv{I} \\otimes \\mv{U}_2 \\otimes \\mv{U}_1\\right)} \\left[\\conj{\\left(\\mv{I} \\otimes \\mv{U}_2^\\H \\otimes \\mv{U}_1^\\H \\right) \\mv{L} \\left(\\mv{I} \\otimes \\mv{U}_2 \\otimes \\mv{U}_1\\right)}\\right]^{-1}\\left( \\mv{I} \\otimes \\mv{U}_2^{\\mkern-1.5mu\\mathsf{T}} \\otimes \\mv{U}_1^{\\mkern-1.5mu\\mathsf{T}} \\right).\n\t\\end{split}\n\\end{equation}\nThe successive application of these projection operators on the\nresidual results in an updated residual that lies in the intersection\nof all subspaces.\n\n\n\\subsection{Helmholtz equation with space-dependent wave number}\nThe presented algorithms with constant wave number can be\nextended to space-dependent wave numbers. So, lets\nconsider a 3D Helmholtz problem where $\\mt{K} = k^2(x,y,z)$ represents the space-dependent\nwave number on the discretized mesh.\n\nFurther, we assume that a Canonical Polyadic decomposition of the\nspace-dependent wave number tensor \n$\\mt{K}$ is known, i.e.\n\\begin{equation}\\label{eq:CP4K}\n\t\\mt{K} = \\sum_{i=1}^s \\sigma_i \\left(\\mv{v}_i^{(1)} \\circ \\mv{v}_i^{(2)} \\circ \\cdots \\circ\t\\mv{v}_i^{(d)}\\right),\n\\end{equation}\nwhere $s \\in \\mathbb{N}_+$ is the CP-rank of $\\mt{K}$ and $\\mv{v}_i^{(j)} \\in \\mathbb{C}^{n_j}$ for $i = 1,2,\\ldots, s; j = 1,2,\\ldots, d$ are vectors. Further, $\\sigma_i$ is a tensor generalization of a singular value and $\\circ$ denotes the vector outer product.\n\nThe application of the space-dependent Helmholtz operator $\\mathcal{L}$ on tensor $\\mt{M}$ is given by\n\\begin{equation}\\label{eq:start3dvar}\n\t\\begin{split}\n\t\t\\mathcal{L} \\mt{M} &= \\mt{F} \\\\\n\t\t\\mathcal{L} \\mt{M} &= -\\mt{G} \\times_1 \\mv{D}_{xx} \\mv{U}_1 \\times_2 \\mv{U}_2 \\times_3 \\mv{U}_3\\\\\n\t\t&- \\mt{G} \\times_1 \\mv{U}_1 \\times_2 \\mv{D}_{yy} \\mv{U}_2 \\times_3 \\mv{U}_3\\\\\n\t\t&- \\mt{G} \\times_1 \\mv{U}_1 \\times_2 \\mv{U}_2 \\times_3 \\mv{D}_{zz} \\mv{U}_3\\\\\n\t\t&- \\mt{K} \\circ \\left(\\mt{G} \\times_1 \\mv{U}_1 \\times_2 \\mv{U}_2 \\times_3 \\mv{U}_3 \\right)\\\\\n\t\t&= \\mt{F},\n\t\\end{split}\n\\end{equation}\nwhere $\\mv{U}_i^\\H \\mv{U}_i = \\mv{I}$ for $i = 1,2,3$ and $\\mt{F}$\nis a tensor representation of the right hand side function $f$ \ndiscretized on the used grid.\nHere $\\circ$ denotes the Hadamard product for tensors.\n\nIn a similar way as in the three dimensional constant wave number\ncase, we can derive equations to iteratively solve for the factors\n$\\mv{U}_1$, $\\mv{U}_2$ and $\\mv{U}_3$. We start from \\eqref{eq:start3dvar} and multiply with\n$\\mv{U}_2$ and $\\mv{U}_3$ in the second and third direction,\nrespectively. Using that the columns of $\\mv{U}_i$ are orthonormal,\nthe following expression is derived:\n\\begin{equation*}\n\t\\mathcal{L} \\mt{M} \\times_2 \\mv{U}_2^\\H \\times_3 \\mv{U}_3^\\H = -\\mt{G} \\times_1 \\mv{D}_{xx}\n\t- \\mt{G} \\times_1 \\mv{U}_1 \\times_2 \\mv{U}_2^\\H\\mv{D}_{yy} \\mv{U}_2\n\t- \\mt{G} \\times_1 \\mv{U}_1 \\times_3 \\mv{U}_3^\\H\\mv{D}_{zz} \\mv{U}_3\n\t- \\left[\\mt{K} \\circ \\left(\\mt{G} \\times_1 \\mv{U}_1 \\times_2 \\mv{U}_2 \\times_3 \\mv{U}_3 \\right) \\right] \\times_2 \\mv{U}_2^\\H \\times_3 \\mv{U}_3^\\H\n\\end{equation*}\nWritten in the first unfolding, the multiplication with $\\mv{U}_2^\\H$ and $\\mv{U}_3^\\H$ in, respectively, the second and third direction is equivalent to post-multiplication with the matrix\n$\\left(\\mv{I} \\otimes \\mv{U}_2^\\H \\right)^\\H \\left(\\mv{U}_3^\\H \\otimes \\mv{I} \\right)^\\H = \\left(\\mv{U}_3^\\H \\otimes \\mv{U}_2^\\H \\right)^\\H = \\left(\\mv{U}_3 \\otimes \\mv{U}_2 \\right)$.\n\nMost of the terms are equal to the case where we had a constant wave\nnumber, see also \\eqref{eq:start3d}. Let us focus on the last term \nthat contains the Hadamard product with the space-dependent wave number, i.e.:\n\\begin{equation}\\label{eq:hadamardProduct}\n\t\\begin{split}\n\t\t\\mt{K} &\\circ \\left(\\mt{G} \\times_1 \\mv{U}_1 \\times_2 \\mv{U}_2 \\times_3 \\mv{U}_3 \\right).\n\t\\end{split}\n\\end{equation}\nFor Hadamard products of tensors, $\\mt{Z} = \\mt{X} \\circ \\mt{Y}$, the following property for the $k$-th unfolding holds $\\mv{Z}_{(k)} = \\mv{X}_{(k)} \\circ \\mv{Y}_{(k)}$.\nThus, written in the first unfolding \\eqref{eq:hadamardProduct} is given by\n\\begin{equation}\\label{eq:hadamardProduct1stUnfolding}\n\t\\begin{split}\n\t\t\\mv{K}_{(1)} &\\circ \\mv{M}_{(1)}\\\\\n\t\t\\mv{K}_{(1)} &\\circ \\left( \\conj{\\mv{U}_1} \\mv{G}_{(1)} \\left( \\mv{U}_3 \\otimes \\mv{U}_2 \\right)^\\H \\right).\n\t\\end{split}\n\\end{equation}\n\nAs the Hadamard product-term \\eqref{eq:hadamardProduct1stUnfolding} is written in the first unfolding and multiplication with $\\mv{U}_2^\\H$ and $\\mv{U}_3^\\H$ in respectively the second and third dimension results in\n\\begin{equation}\\label{eq:lastTerm}\n\t\\left[\\underbrace{\\mv{K}_{(1)}}_{\\mv{K}} \\circ \\underbrace{\\conj{\\mv{U}_1}\\mv{G}_{(1)}}_{\\mv{U}} \\underbrace{\\left( \\mv{U}_3 \\otimes \\mv{U}_2 \\right)^\\H}_{\\mv{V}^\\H} \\right] \\underbrace{\\left(\\mv{U}_3 \\otimes \\mv{U}_2 \\right)}_{\\mv{V}}.\n\\end{equation}\n\nThe derivation of the other terms of \\eqref{eq:start3dvar} are equal to the constant wave number case.\nThe equation in the first unfolding leads to a matrix equation:\n\\begin{equation}\\label{eq:firstUnfoldingVar}\n\t-\\conj{\\mv{D}_{xx}\\mv{U}_1} \\mv{G}_{(1)}\n\t- \\conj{\\mv{U}_1} \\mv{G}_{(1)} \\left(I \\otimes \\mv{U}_2^\\H \\mv{D}_{yy} \\mv{U}_2\\right)^\\H\n\t- \\conj{\\mv{U}_1} \\mv{G}_{(1)} \\left(\\mv{U}_3^\\H \\mv{D}_{zz} \\mv{U}_3 \\otimes I\\right)^\\H\n\t- \\left[\\mv{K}_{(1)} \\circ \\conj{\\mv{U}_1}\\mv{G}_{(1)} \\left( \\mv{U}_3 \\otimes \\mv{U}_2 \\right)^\\H \\right] \\left(\\mv{U}_3 \\otimes \\mv{U}_2 \\right)\n\t= \\mv{F}_{(1)} \\left(\\mv{U}_3 \\otimes \\mv{U}_2\\right).\n\\end{equation}\n\nVectorization of the last term, i.e. \\eqref{eq:lastTerm}, results again in an expression for the space-dependent wave number of the form $\\left(\\mv{K} \\circ \\mv{U}\\mv{V}^\\H\\right)\\mv{V}$, similar to the two dimensional case which was given in \\eqref{eq:2dvarwavenr}.\nUsing again \\eqref{eq:vecThirdTerm}, the vectorization of this expression is given by\n\\begin{equation}\\label{eq:tmp1}\n\t\\left(\\mv{U}_3^{\\mkern-1.5mu\\mathsf{T}} \\otimes \\mv{U}_2^{\\mkern-1.5mu\\mathsf{T}} \\otimes \\mv{I} \\right) \\diag{\\mvec{\\mv{K}_{(1)}}} \\conj{\\left(\\mv{U}_3 \\otimes \t\\mv{U}_2 \\otimes \\mv{I} \\right)} \\mvec{\\conj{\\mv{U}_1}\\mv{G}_{(1)}}.\n\\end{equation}\n\nBecause $\\mt{K}$ is known in a Canonical Polyadic tensor (CP tensor) decomposition\\footnote{Otherwise a Canonical Polyadic tensor decomposition can be computed using for example an CP-ALS algorithm \\cite{kolda2009tensor}.}, as given in \\eqref{eq:CP4K}, we have\n\\begin{equation}\n\t\\diag{\\mvec{\\mv{K}_{(1)}}} = \\sum_{i=1}^{s} \\sigma_i \\diag{\\mv{v}_{i}^{(3)}} \\otimes \\diag{\\mv{v}_{i}^{(2)}} \\otimes \\diag{\\mv{v}_{i}^{(1)}}.\n\\end{equation}\n\nSo, using the CP tensor representation of the space-dependent wave number the vectorization in \\eqref{eq:tmp1} simplifies even further:\n\\begin{equation*}\n\t\\sum_{i=1}^{s} \\sigma_i \\left(\\mv{U}_3^{\\mkern-1.5mu\\mathsf{T}} \\otimes \\mv{U}_2^{\\mkern-1.5mu\\mathsf{T}} \\otimes \\mv{I} \\right) \\left[ \\diag{\\mv{v}_{i}^{(3)}} \\otimes \\diag{\\mv{v}_{i}^{(2)}} \\otimes \\diag{\\mv{v}_{i}^{(1)}} \\right] \\conj{\\left(\\mv{U}_3 \\otimes \\mv{U}_2 \\otimes \\mv{I}\\right)} \\mvec{\\conj{\\mv{U}_1}\\mv{G}_{(1)}}\n\\end{equation*}\nwhich reduced to\n\\begin{equation*}\n\t\\underbrace{\\sum_{i=1}^{s} \\sigma_i\n\t\t\\left(\\mv{U}_3^{\\mkern-1.5mu\\mathsf{T}} \\diag{\\mv{v}_{i}^{(3)}} \\conj{\\mv{U}_3} \\right) \\otimes\n\t\t\\left(\\mv{U}_2^{\\mkern-1.5mu\\mathsf{T}} \\diag{\\mv{v}_{i}^{(2)}} \\conj{\\mv{U}_2} \\right) \\otimes\n\t\t\\left(\\diag{\\mv{v}_{i_1}^{(1)}}\\right)}_{K_1} \\mvec{\\conj{\\mv{U}_1}\\mv{G}_{(1)}}.\n\\end{equation*}\nIn this way the $K_1$ operator is defined and can be applied to $\\mvec{\\conj{\\mv{U}_1}\\mv{G}_{(1)}}$. Observe that this expansion is only advantageous if the space-dependent wave number has low rank, which is typical the case for our applications.\n\nIn a similar way, the $K_2$ and $K_3$ operators can be derived:\n\\begin{equation}\\label{eq:Koperators}\n\t\\begin{split}\n\t\tK_1 &= \\sum_{i=1}^{s}\\sigma_i \\left(\\mv{U}_3^{\\mkern-1.5mu\\mathsf{T}} \\diag{\\mv{v}_{i}^{(3)}} \\conj{\\mv{U}_3} \\right) \\otimes \\left(\\mv{U}_2^{\\mkern-1.5mu\\mathsf{T}} \\diag{\\mv{v}_{i}^{(2)}} \\conj{\\mv{U}_2} \\right) \\otimes \\left(\\diag{\\mv{v}_{i}^{(1)}}\\right) \\\\ \n\t\tK_2 &= \\sum_{i=1}^{s}\\sigma_i \\left(\\mv{U}_3^{\\mkern-1.5mu\\mathsf{T}} \\diag{\\mv{v}_{i}^{(3)}} \\conj{\\mv{U}_3} \\right) \\otimes \\left(\\mv{U}_1^{\\mkern-1.5mu\\mathsf{T}} \\diag{\\mv{v}_{i}^{(1)}} \\conj{\\mv{U}_1} \\right) \\otimes \\left(\\diag{\\mv{v}_{i}^{(2)}}\\right) \\\\\n\t\tK_3 &= \\sum_{i=1}^{s}\\sigma_i \\left(\\mv{U}_2^{\\mkern-1.5mu\\mathsf{T}} \\diag{\\mv{v}_{i}^{(2)}} \\conj{\\mv{U}_2} \\right) \\otimes \\left(\\mv{U}_1^{\\mkern-1.5mu\\mathsf{T}} \\diag{\\mv{v}_{i}^{(1)}} \\conj{\\mv{U}_1} \\right) \\otimes \\left(\\diag{\\mv{v}_{i}^{(3)}}\\right).\n\t\\end{split}\n\\end{equation}\n\nSo, we find the following linear system to solve for $\\mvec{\\conj{\\mv{U}_1}\\mv{G}_{(1)}}$:\n\\begin{equation}\\label{eq:eqnU1var}\n\t\\left\\{\n\t-\\mv{I} \\otimes \\conj{\\mv{D}_{xx}} -\n\t\\left[ \n\t\\conj{\\left(\\mv{I} \\otimes \\mv{U}_2^\\H \\mv{D}_{yy} \\mv{U}_2\\right)} - \\conj{\\left(\\mv{U}_3^\\H \\mv{D}_{zz} \\mv{U}_3 \\otimes \\mv{I}\\right)}\n\t\\right] \\otimes \\mv{I}\n\t- K_1\n\t\\right\\}\n\t\\mvec{\\underbrace{\\conj{\\mv{U}_1}\\mv{G}_{(1)}}_{\\mv{X}_1}} = \\mvec{\\mv{F}_{(1)} \\left(\\mv{U}_3 \\otimes \\mv{U}_2\\right)}.\n\\end{equation}\nObserve this is a square system with $n_1 \\times r_2 r_3$ unknowns\n(where the solution in matrix form $\\mv{X}_1$ is typical for rank $r >\nr_1$). In a similar way, update equations for $\\mv{U}_2$ and\n$\\mv{U}_3$ are derived by multiplying \\eqref{eq:start3dvar} with the\nother factor matrices in the appropriate directions:\n\\begin{equation}\\label{eq:eqnU2var}\n\t\\left\\{\n\t-\\mv{I} \\otimes \\conj{\\mv{D}_{yy}} +\n\t\\left[ -\n\t\\conj{\\left(I \\otimes \\mv{U}_1^\\H \\mv{D}_{xx} \\mv{U}_1\\right)} - \\conj{\\left(\\mv{U}_3^\\H \\mv{D}_{zz} \\mv{U}_3 \\otimes \\mv{I}\\right)}\n\t\\right] \\otimes \\mv{I}\n\t- K_2\n\t\\right\\}\n\t\\mvec{\\underbrace{\\conj{\\mv{U}_2}\\mv{G}_{(2)}}_{\\mv{X}_2}} = \\mvec{\\mv{F}_{(2)} \\left(\\mv{U}_3 \\otimes \\mv{U}_1\\right)}.\n\\end{equation}\nand\n\\begin{equation}\\label{eq:eqnU3var}\n\t\\left\\{-\\mv{I} \\otimes \\conj{\\mv{D}_{zz}} +\n\t\\left[-\n\t\\conj{\\left(\\mv{I} \\otimes \\mv{U}_1^\\H \\mv{D}_{xx} \\mv{U}_1\\right)} - \\conj{\\left(\\mv{U}_2^\\H \\mv{D}_{yy} \\mv{U}_2 \\otimes I\\right)}\n\t\\right] \\otimes I\n\t- K_3\n\t\\right\\} \\mvec{\\underbrace{\\conj{\\mv{U}_3}\\mv{G}_{(3)}}_{\\mv{X}_3}} = \\mvec{\\mv{F}_{(3)} \\left(\\mv{U}_2 \\otimes \\mv{U}_1\\right)}.\n\\end{equation}\nAlternating between solving for $\\mv{U}_1$, $\\mv{U}_2$ and $\\mv{U}_3$ using \\eqref{eq:eqnU1var}, \\eqref{eq:eqnU2var} or \\eqref{eq:eqnU3var} results in an algorithm to approximate low-rank tensor solutions for three dimensional problems as given in \\eqref{eq:driven3d}. Also in this case the orthogonality of the columns of $\\mv{U}_1$, $\\mv{U}_2$ and $\\mv{U}_3$ are maintained by additional QR factorizations.\nSo, we derive the algorithm as formulated in Algorithm \\ref{alg:var3dv1}. The generalization for dimensions $d > 3$ is straight forward.\n\n\\begin{algorithm}[t]\n\t\\SetAlgoLined\n\t[$\\mt{G}, \\mv{U}_1, \\mv{U}_2, \\mv{U}_3$] = hosvd(initial guess)\\;\n\t[$\\mv{\\Sigma}, \\mv{V}_1, \\mv{V}_2, \\mv{V}_3$] = cp\\_als($\\mt{K}$)\\;\n\t\\While{not converged}{\n\t\t\\For{i = 1, 2, 3}{\n\t\t\tCompute $K_i$ using \\eqref{eq:Koperators}\\;\n\t\t\tSolve for $\\mv{X}_i = \\conj{\\mv{U}_i}\\mv{G}_{(i)} \\in \\mathbb{C}^{n_i \\times r^{d-1}}$ using \\eqref{eq:eqnU1var}, \\eqref{eq:eqnU2var} or \\eqref{eq:eqnU3var}\\;\n\t\t\t$\\conj{\\mv{U}_{i}} \\mv{G}_{(i)} = \\qr{\\mv{X}_i(:,~ 1:r_i), 0}$;\n\t\t}\n\t}\n\t$\\mt{G} = \\texttt{reconstruct} \\left[\\mv{G}_{(i)}, ~i\\right]$\\;\n\t$\\mt{M} = \\mt{G} \\times_1 \\mv{U}_1 \\times_2 \\mv{U}_2 \\times_3 \\mv{U}_3$\\;\n\t\\caption{Solve for the low-rank tensor decomposition of the solution $\\mt{M}$ of a 3D Helmholtz problem with space-dependent wave number (version 1).}\n\t\\label{alg:var3dv1}\n\\end{algorithm}\n\nSimilar to the discussion for the constant wave number algorithms,\nobserve that we solve again for a large matrix $\\mv{X}_i \\in \\mathbb{C}^{n_i \\times\tr_1r_2r_3\/r_i}$.\nSo, in general the rank of this matrix could be\n$\\min\\left(n_i,~ r_1r_2r_3\/r_i\\right)$. But it is also known that\n$\\mv{X}_i = \\conj{\\mv{U}_i}\\mv{G}_{(i)}$ leads to the fact that\nthe rank of $\\mv{X}_i$ should be at most $r_i$. So selecting the\nfirst $r_i$ columns of $\\mv{X}_i$ and computing its QR decomposition\nis sufficient to derive a new orthonormal basis as factor matrix\n$\\conj{\\mv{U}}_i$.\n\nAlgorithm \\ref{alg:var3dv1} is exactly the space-dependent wave number\nequivalent of Algorithm \\ref{alg:const3dv1}. The same ideas can be applied\nto derive space-dependent wave number alternatives of the algorithms\ncorresponding to version 2 and 3.\nAgain, to circumvent solving large systems, we can pre-compute the \nQR factorization of $\\mv{G}_{(i)}$ and project these equations onto \nthe obtained $\\mv{Q}_i$. Indeed, this will reduce the number of \nunknowns in these linear systems to exactly the number of unknowns \nas needed for the factor matrices $\\conj{\\mv{U}}_1$ and\n$\\conj{\\mv{U}}_2$.\n\nLet us discuss the details. We start again from equation \\eqref{eq:firstUnfoldingVar} and use the QR\nfactorization of $\\mv{G}_{(1)}^\\H$, $\\mv{Q}_1 \\mv{R}_1^\\H = \\qr{\\mv{G}^\\H_{(1)}}$. This yields\n\\begin{equation*}\n\t-\\conj{\\mv{D}_{xx}\\mv{U}_1} \\mv{R}_1\\mv{Q}_1^\\H\n\t- \\conj{\\mv{U}_1} \\mv{R}_1\\mv{Q}_1^\\H \\left(I \\otimes \\mv{U}_2^\\H \\mv{D}_{yy} \\mv{U}_2\\right)^\\H\n\t- \\conj{\\mv{U}_1} \\mv{R}_1\\mv{Q}_1^\\H \\left(\\mv{U}_3^\\H \\mv{D}_{zz} \\mv{U}_3 \\otimes I\\right)^\\H\n\t- \\left[\\mv{K}_{(1)} \\circ \\conj{\\mv{U}_1}\\mv{R}_1\\mv{Q}_1^\\H \\left( \\mv{U}_3 \\otimes \\mv{U}_2 \\right)^\\H \\right] \\left(\\mv{U}_3 \\otimes \\mv{U}_2 \\right)\n\t= \\mv{F}_{(1)} \\left(\\mv{U}_3 \\otimes \\mv{U}_2\\right).\n\\end{equation*}\nPost multiplication of the left hand side of this equation by $\\mv{Q}_1$ yields\n\\begin{equation*}\n\t-\\conj{\\mv{D}_{xx}\\mv{U}_1} \\mv{R}_1\n\t- \\conj{\\mv{U}_1} \\mv{R}_1\\mv{Q}_1^\\H \\left(I \\otimes \\mv{U}_2^\\H \\mv{D}_{yy} \\mv{U}_2\\right)^\\H\\mv{Q}_1\n\t- \\conj{\\mv{U}_1} \\mv{R}_1\\mv{Q}_1^\\H \\left(\\mv{U}_3^\\H \\mv{D}_{zz} \\mv{U}_3 \\otimes I\\right)^\\H\\mv{Q}_1\n\t- \\left[\\mv{K}_{(1)} \\circ \\conj{\\mv{U}_1}\\mv{R}_1\\mv{Q}_1^\\H \\left( \\mv{U}_3 \\otimes \\mv{U}_2 \\right)^\\H \\right] \\left(\\mv{U}_3 \\otimes \\mv{U}_2 \\right)\\mv{Q}_1.\n\n\\end{equation*}\nTo solve this equation for $\\mv{U}_1$, it is written in vectorized form as\n\\begin{equation}\\label{eq:eqnU1v2forvar}\n\t\\left\\{\n\t-\\mv{I} \\otimes \\conj{\\mv{D}_{xx}} +\n\t\\mv{Q}_1^{\\mkern-1.5mu\\mathsf{T}}\\left[- \n\t\\conj{\\left(\\mv{I} \\otimes \\mv{U}_2^\\H \\mv{D}_{yy} \\mv{U}_2\\right)} - \\conj{\\left(\\mv{U}_3^\\H \\mv{D}_{zz} \\mv{U}_3 \\otimes \\mv{I} \\right)}\n\t\\right]\\conj{\\mv{Q}_1} \\otimes \\mv{I}\n\t-\\mv{Q}_1^{\\mkern-1.5mu\\mathsf{T}} K_1 \\conj{\\mv{Q}_1}\n\t\\right\\} \\mvec{\\underbrace{\\conj{\\mv{U}_1}\\mv{R}_1}_{\\mv{X}_1}} = \\mvec{\\mv{F}_{(1)} \\left(\\mv{U}_3 \\otimes \\mv{U}_2\\right) \\mv{Q}_1}.\n\\end{equation}\nIn a similar way, the update equations for $\\mv{U}_2$ and $\\mv{U}_3$ are\nderived by multiplying \\eqref{eq:start3dvar} with the other factor\nmatrices in the appropriate dimensions and using the QR factorizations\nof $\\mv{G}_{(i)}^\\H$:\n\\begin{equation}\\label{eq:eqnU2v2forvar}\n\t\\left\\{\n\t-\\mv{I} \\otimes \\conj{\\mv{D}_{yy}} +\n\t\\mv{Q}_2^{\\mkern-1.5mu\\mathsf{T}}\\left[ \n\t-\\conj{\\left(\\mv{I} \\otimes \\mv{U}_1^\\H \\mv{D}_{xx} \\mv{U}_1\\right)} - \\conj{\\left(\\mv{U}_3^\\H \\mv{D}_{zz} \\mv{U}_3 \\otimes \\mv{I} \\right)}\n\t\\right]\\conj{\\mv{Q}_2} \\otimes \\mv{I}\n\t-\\mv{Q}_2^{\\mkern-1.5mu\\mathsf{T}} K_2 \\conj{\\mv{Q}_2}\n\t\\right\\} \\mvec{\\underbrace{{\\conj{\\mv{U}_2}\\mv{R}_2}}_{\\mv{X}_2}} \n\t= \\mvec{\\mv{F}_{(2)} \\left(\\mv{U}_3 \\otimes \\mv{U}_1\\right) \\mv{Q}_2}.\n\\end{equation}\nand\n\\begin{equation}\\label{eq:eqnU3v2forvar}\n\t\\left\\{\n\t-\\mv{I} \\otimes \\conj{\\mv{D}_{zz}} +\n\t\\mv{Q}_3^{\\mkern-1.5mu\\mathsf{T}}\\left[ \n\t-\\conj{\\left(\\mv{I} \\otimes \\mv{U}_1^\\H \\mv{D}_{xx} \\mv{U}_1\\right)} - \\conj{\\left(\\mv{U}_2^\\H \\mv{D}_{yy} \\mv{U}_2 \\otimes \\mv{I} \\right)}\n\t\\right]\\conj{\\mv{Q}_3} \\otimes \\mv{I}\n\t\\right\\} \\mvec{\\underbrace{\\conj{\\mv{U}_3}\\mv{R}_3}_{\\mv{X}_3}} \n\t-\\mv{Q}_3^{\\mkern-1.5mu\\mathsf{T}} K_3 \\conj{\\mv{Q}_3}\n\t= \\mvec{\\mv{F}_{(3)} \\left(\\mv{U}_2 \\otimes \\mv{U}_1\\right) \\mv{Q}_3}.\n\\end{equation}\n\nAll these equations are cheap to solve. Indeed,\n$\\mvec{\\conj{\\mv{U}_i}\\mv{R}_i}$ has length $n_i r_i$. Computing a\nsymmetric reverse Cuthill-McKee permutation of the system matrix one\nobserves a matrix with a bandwidth $\\Oh{r}$, so solving these\nequations has a computational cost $\\Oh{nr^2}$.\n\nAlternating between solving for $\\mv{U}_1$, $\\mv{U}_2$ and $\\mv{U}_3$ using \\eqref{eq:eqnU1v2forvar}, \\eqref{eq:eqnU2v2forvar} or \\eqref{eq:eqnU3var} results again in an algorithm to approximate low-rank solutions for three dimensional space-dependent Helmholtz problems. Also in this case the orthogonality of the columns of $\\mv{U}_1$, $\\mv{U}_2$ and $\\mv{U}_3$ are maintained by additional QR factorizations.\nSo, we derive the algorithm as formulated in Algorithm \\ref{alg:var3dv3}.\nAlgorithm \\ref{alg:var3dv3} is exactly the space-dependent wave number equivalent of Algorithm \\ref{alg:const3dv3}.\n\n\\begin{algorithm}[t]\n\t\\SetAlgoLined\n\t[$\\mt{G}, \\mv{U}_1, \\mv{U}_2, \\mv{U}_3$] = hosvd(initial guess)\\;\n\t[$\\mv{\\Sigma}, \\mv{V}_1, \\mv{V}_2, \\mv{V}_3$] = cp\\_als($\\mt{K}$)\\;\n\t\\While{not converged}{\n\t\t\\For{i = 1, 2}{\n\t\t\tCompute $K_i$ using \\eqref{eq:Koperators}\\;\n\t\t\t$\\mv{Q}_i \\widetilde{\\mv{R}} = \\qr{\\mv{G}_{(i)}^\\H, 0}$\\;\n\t\t\tSolve for $\\mv{X}_i = \\conj{\\mv{U}_i}\\mv{R}_i \\in \\mathbb{C}^{n_i \\times r_i}$ using \\eqref{eq:eqnU1v2forvar} or \\eqref{eq:eqnU2v2forvar}\\;\n\t\t\t$\\conj{\\mv{U}_{i}} \\mv{R}_i = \\qr{\\mv{X}_i, 0}$\\;\n\t\t\t$\\mt{G} = \\texttt{reconstruct} \\left[\\mv{R}_i\\mv{Q}_i^\\H, ~i\\right]$;\n\t\t}\n\t\tCompute $K_3$ using \\eqref{eq:Koperators}\\;\n\t\tSolve for $\\mv{X}_3 = \\conj{\\mv{U}_3}\\mv{G}_{(3)} \\in \\mathbb{C}^{n_3 \\times r^{d-1}}$ using \\eqref{eq:eqnU3var}\\;\n\t\t$\\conj{\\mv{U}_{3}} \\mv{G}_{(3)} = \\qr{\\mv{X}_3, 0}$\\;\n\t\t$\\mt{G} = \\texttt{reconstruct} \\left[\\mv{G}_{(3)},~3\\right]$\\;\n\t}\n\t$\\mt{M} = \\mt{G} \\times_1 \\mv{U}_1 \\times_2 \\mv{U}_2 \\times_3 \\mv{U}_3$\\;\n\t\\caption{Solve for the low-rank tensor decomposition of the solution $\\mt{M}$ of a 3D Helmholtz problem with space-dependent wave number (version 3).}\n\t\\label{alg:var3dv3}\n\\end{algorithm}\n\n\n\\subsection{Projection operator for space-dependent wave number}\nConsider a tensor $\\mt{M}$ in Tucker tensor format and factorized as\n$\\mt{M} = \\mt{G} \\times_1 \\mv{U}_1 \\times_2 \\mv{U}_2 \\times_3 \\mv{U}_3$,\nwith unknowns $\\mt{G}, \\mv{U}_1, \\mv{U}_2$ and $\\mv{U}_3$.\nDiscretization of \\eqref{eq:driven3d} with a space-dependent wave number \nleads to a linear operator $\\mathcal{L}$ applied on tensors. Its matrix\nrepresentation $\\mv{L}$ has again a structure as given in\n\\eqref{eq:operatorLasSumKroneckerProducts}.\n\nSolving for the unknown factors $\\mv{U}_1$, $\\mv{U}_2$ or $\\mv{U}_3$\n(and the core-tensor $\\mt{G}$) using \\eqref{eq:eqnU1var},\n\\eqref{eq:eqnU2var} or \\eqref{eq:eqnU3var} can, again, be interpreted as a\nprojection operator applied on the residual. For example,\n\\eqref{eq:eqnU1var} can be interpreted as\n\\begin{equation}\n\t\\conj{\\left(\\mv{U}_3^\\H \\otimes \\mv{U}_2^\\H \\otimes \\mv{I} \\right) \\mv{L} \\left(\\mv{U}_3 \\otimes \\mv{U}_2 \\otimes \\mv{I}\\right)} \\mvec{\\conj{\\mv{U}_1}\\mv{G}_{(1)}} = \\left( \\mv{U}_3^{\\mkern-1.5mu\\mathsf{T}} \\otimes \\mv{U}_2^{\\mkern-1.5mu\\mathsf{T}} \\otimes \\mv{I} \\right) \\mvec{\\mv{F}_{(1)}},\n\\end{equation}\nThe residual in tensor format is given by\n\\begin{equation}\n\t\\begin{split}\n\t\t\\mt{R} = \\mt{F} &- \\mathcal{L}\\mt{M}, \\\\\n\t\t= \\mt{F} \n\t\t&+ \\mt{G} \\times_1 \\mv{D}_{xx} \\mv{U}_1 \\times_2 \\mv{U}_2 \\times_3 \\mv{U}_3 \\\\\n\t\t&+ \\mt{G} \\times_1 \\mv{U}_1 \\times_2 \\mv{D}_{yy}\\mv{U}_2 \\times_3 \\mv{U}_3 \\\\\n\t\t&+ \\mt{G} \\times_1 \\mv{U}_1 \\times_2 \\mv{U}_2 \\times_3 \\mv{D}_{zz}\\mv{U}_3 \\\\\n\t\t&+ \\mt{K} \\circ \\left(\\mt{G} \\times_1 \\mv{U}_1 \\times_2 \\mv{U}_2 \\times_3 \\mv{U}_3 \\right).\n\t\\end{split}\n\\end{equation}\n\nWriting this tensor equation in the first unfolding leads to the following matrix equation\n\\begin{equation}\n\t\\begin{split}\n\t\\mv{R}_{(1)} = \\mv{F}_{(1)} \n\t&+ \\conj{\\mv{D}_{xx}\\mv{U}_1}\\mv{G}_{(1)} \\left(\\mv{U}_3 \\otimes \\mv{U}_2 \\right)^\\H \\\\\n\t&+ \\conj{\\mv{U}_1}\\mv{G}_{(1)} \\left(\\mv{U}_3 \\otimes \\mv{D}_{yy}\\mv{U}_2 \\right)^\\H \\\\\n\t&+ \\conj{\\mv{U}_1}\\mv{G}_{(1)} \\left(\\mv{D}_{zz}\\mv{U}_3 \\otimes \\mv{U}_2 \\right)^\\H \\\\\n\t&+ \\mv{K}_{(1)} \\circ \\left( \\conj{\\mv{U}_1} \\mv{G}_{(1)} \\left( \\mv{U}_3 \\otimes \\mv{U}_2 \\right)^\\H \\right)\n\t\\end{split}\n\\end{equation}\nwhich can be matricized as\n\\begin{equation*}\n\t\\begin{split}\n\t\t\\mvec{\\mv{R}_{(1)}} &= \\mvec{\\mv{F}_{(1)}} \\\\\n\t\t&- \\left(-\\conj{\\left(\\mv{U}_3 \\otimes \\mv{U}_2 \\otimes \\mv{D}_{xx}\\right)} \n\t\t- \\conj{\\left(\\mv{U}_3 \\otimes \\mv{D}_{yy}\\mv{U}_2 \\otimes \\mv{I}\\right)} \n\t\t- \\conj{\\left(\\mv{D}_{zz}\\mv{U}_3 \\otimes \\mv{U}_2 \\otimes \\mv{I}\\right)} \n\t\t- \\diag{\\mvec{\\mv{K}_{(1)}}} \\conj{\\left(\\mv{U}_3 \\otimes \t\\mv{U}_2 \\otimes \\mv{I} \\right)}\\right)\\mvec{\\conj{\\mv{U}_1}\\mv{G}_{(1)}}.\n\t\\end{split}\n\\end{equation*}\nRewriting this results in exactly the same structure and projection operator as in the constant wave number case:\n\\begin{equation}\n\t\\begin{split}\n\t\t\\mvec{\\mv{R}_{(1)}} &= \\ldots \\\\\n\t\t&= \\mvec{\\mv{F}_{(1)}} - \\conj{\\mv{L} \\left(\\mv{U}_3 \\otimes \\mv{U}_2 \\otimes \\mv{I}\\right)}\\mvec{\\conj{\\mv{U}_1}\\mv{G}_{(1)}} \\\\\n\t\t&= \\mvec{\\mv{F}_{(1)}} - \\conj{\\mv{L} \\left(\\mv{U}_3 \\otimes \\mv{U}_2 \\otimes \\mv{I}\\right)} \\left[\\conj{\\left(\\mv{U}_3^\\H \\otimes \\mv{U}_2^\\H \\otimes \\mv{I} \\right) \\mv{L} \\left(\\mv{U}_3 \\otimes \\mv{U}_2 \\otimes \\mv{I}\\right)}\\right]^{-1}\\left( \\mv{U}_3^{\\mkern-1.5mu\\mathsf{T}} \\otimes \\mv{U}_2^{\\mkern-1.5mu\\mathsf{T}} \\otimes \\mv{I} \\right) \\mvec{\\mv{F}_{(1)}} \\\\\n\t\t&= P_{23}\\mvec{\\mv{F}_{(1)}}\n\t\\end{split}\n\\end{equation}\nwhere projection operator $P_{23}$ is similar to the projector in the constant wave number case, see \\eqref{eq:projector3dconst}, and now given by\n\\begin{equation}\n\t\\begin{split}\n\t\tP_{23} &= \\mv{I} - \\conj{\\mv{L} \\left(\\mv{U}_3 \\otimes \\mv{U}_2 \\otimes \\mv{I}\\right)} \\left[\\conj{\\left(\\mv{U}_3^\\H \\otimes \\mv{U}_2^\\H \\otimes \\mv{I} \\right) \\mv{L} \\left(\\mv{U}_3 \\otimes \\mv{U}_2 \\otimes \\mv{I}\\right)}\\right]^{-1}\\left( \\mv{U}_3^{\\mkern-1.5mu\\mathsf{T}} \\otimes \\mv{U}_2^{\\mkern-1.5mu\\mathsf{T}} \\otimes \\mv{I} \\right) \\\\\n\t\t&= \\mv{I} - \\mv{X}.\n\t\\end{split}\n\\end{equation}\n\nA similar derivation results in projection operators $P_{13}$ and $P_{12}$ for the updates in $\\mv{U}_2$ and $\\mv{U}_3$, respectively. Both are also the same as in the constant wave number case, as given in \\eqref{eq:projectors3dconst}.\n\n\n\\section{Numerical results} \\label{sec:NumericalResults}\nIn this section, we demonstrate the promising results of the derived\nalgorithms with some numerical experiments in two and three\ndimensions. Furthermore, we consider discretizations of the Helmholtz\nequation with constant and space-dependent wave numbers.\n\n\\subsection{2D Helmholtz problem with space-dependent wave number}\nFirst, we consider a 2D Helmholtz problem with a space-dependent wave \nnumber given by $k^2(x,y) = 2 + e^{-x^2 -y^2}$.\n\nFor this example the two dimensional domain $\\Omega = [-10, ~10]^2$ is discretized with $M=1000$ equidistant mesh points per direction in the interior of the domain. Further it is extended with exterior complex scaling to implement the absorbing boundary conditions. In total, the number of discretization points per directions equals $n = n_1 = n_2 = 1668$. As external force $f(x, y) = -e^{-x^2-y^2}$ is applied.\n\nIn this space-dependent wave number example it is known that the matrix representation of the semi-exact solution of the Helmholtz equation on the full grid has a low rank. Indeed, approximating the semi-exact solution with a low-rank matrix with rank $r=17$ is in this case sufficient to obtain an error below the threshold $\\tau = 10^{-6}$.\n\nStarting with an random (orthonormalized) initial guess for $\\mv{V}^{(0)} \\in \\mathbb{C}^{n \\times r}$ only a small number of iterations of Algorithm \\ref{alg:2d} is needed to obtain an error similar to the specified threshold $\\tau$. As shown in Figure~\\ref{fig:error+singularvalues-var2d-r=17} both the residual and the error with respect to the semi-exact solution decay in only a few iterations (i.e. in this example 4-8 iterations) to a level almost similar to the expected tolerance.\n\nThe singular values of the approximation $\\mv{A}^{(k)} =\n\\mv{U}^{(k)}{\\mv{R}^{(k)}}^\\H {\\mv{V}^{(k)}}^\\H$ in iteration $i$ can\nbe computed and are shown for increasing iterations in\nFigure~\\ref{fig:error+singularvalues-var2d-r=17}. As expected the low-\nrank approximations recover the singular values of the full grid\nsemi-exact solution. In fact $\\mv{R}^{(k)}$ converges towards\n$\\text{diag}(\\sigma_i)$.\n \n\n\\begin{figure}\n \\centering\n \\begin{tabular}{cc}\n\t\\input{error-variable-2d-r=17.tikz} & \n\t\\input{singularvalues-variable-2d.tikz}\n \\end{tabular}\n \\caption{Plot of error and residual (left) and singular values (right) per iteration for space-dependent wave number in 2D Helmholtz problem ($M=1000, r=17$).}\n \\label{fig:error+singularvalues-var2d-r=17}\n\\end{figure}\n\nThe numerical rank of the matrix representation of the solution of a\nHelmholtz problem with a space-dependent wave number is unknown in\nadvance. But, the presented algorithm is stable with respect to over-\nand underestimation of the numerical rank of the solution. Figure\n\\ref{fig:error+residuals-var2d-M=1000} shows both the error and\nresidual per iteration and illustrates this statement by approximating\nthe same semi-exact solution with increasing ranks $r \\in \\{ 12, 18, 24, 36 \\}$.\n\n\\begin{figure}\n \\centering\n \\begin{tabular}{cc}\n\t\\input{error-variable-2d-M=1000.tikz} &\n\t\\input{residuals-variable-2d-M=1000.tikz}\n \\end{tabular}\n \\caption{Plot of errors (left) and residuals (right) per iteration for space-dependent wave number in 2D Helmholtz problem with increasing ranks ($M=1000$)}\n \\label{fig:error+residuals-var2d-M=1000}\n\\end{figure}\n\n\\begin{figure}\n\t\\centering\n \\begin{tabular}{cc}\n\t\\input{runtime-variable-2d-M=1000.tikz} &\n\t\\input{runtime-variable-2d-M=1000+loglog.tikz}\n \\end{tabular}\n \\caption{Plot of runtime for 10 iterations for space-dependent wave number in 2D Helmholtz problem with increasing ranks ($M=1000$). Left: linlin-scale, right: loglog-scale.}\n \\label{fig:runtime-variable-M=1000}\n\\end{figure}\n\nIn contrast to the constant coefficient wave number case the\nconvergence with space-dependent wave number depends also\non the maximal attainable rank. For increasing maximal attainable\nranks the number of needed iterations decreases. This is especially\nobserved when the error is considered, but it can also be seen in the\nfigure where the residuals are shown, Fig.~\\ref{fig:error+residuals-var2d-M=1000}.\n\n\n\n\\subsection{3D Helmholtz problem with space-dependent wave number}\nIn this example we solve a 3D Helmholtz problem with a space-dependent\nwave number discretized on a DVR-grid \\cite{rescigno2000numerical}. All\nthree versions of the 3D algorithm for space-dependent wave numbers\ncan successfully be applied.\n\nFirst, to reduce computational cost of construction of the operators $K_1$, $K_2$ and $K_3$, see \\eqref{eq:Koperators}, a CP-decomposition of the space-dependent wave number is constructed.\nAs shown in Figure \\ref{fig:cprank_wavenumber} the space-dependent wave number can be well-approximated by a small number of rank-1 tensors. For the examples discussed in this section we used a CP-rank $s = 32$ to approximate this space-dependent wave number. Hence, the error in approximating the wave number is approximately $\\Oh{10^{-4}}$.\n\nFor all three versions of the algorithm we use 10 iterations of the algorithm to converge to the low-rank solution. For example if we compute the low-rank solution (with $r = r_x = r_y = r_z = 16$) the residual after each iteration for all algorithms is shown in Figure \\ref{fig:residual-all-orderDvr=7-solrank=16-waverank=32}.\n\n\\begin{figure}\n\t\\centering\n\t\\begin{subfigure}{.45\\textwidth}\n\t\t\\centering\n\t\t\\input{cprank-orderDvr=7-wavenumber.tikz} \\\\\n\t\t\\caption{CP-rank of space-dependent wave number for 3D Helmholtz problem.}\n\t\t\\label{fig:cprank_wavenumber}\n\t\\end{subfigure}\n\t\\quad\n\t\\begin{subfigure}{.45\\textwidth}\n\t\t\\centering\n\t\t\\input{residual-all-orderDvr=7-solrank=16-waverank=32.tikz} \\\\\n\t\t\\caption{Residual per iteration ($r = 16, s = 32$).}\n\t\t\\label{fig:residual-all-orderDvr=7-solrank=16-waverank=32}\n\t\\end{subfigure}\n\t\\caption{Low rank approximation to space-dependent wave number and residuals for version 1, version 2 and version 3 of 3D Helmholtz problem with space-dependent wave number (orderDvr = 7).}\n\t\\label{fig:residual-all-orderDvr=7}\n\\end{figure}\n\n\\begin{figure}\n\t\\centering\n\t\\begin{subfigure}{.45\\textwidth}\n\t\t\\centering\n\t\t\\input{residual-all-orderDvr=7-waverank=32.tikz} \\\\\n\t\t\\caption{orderDvr = 7}\n\t\t\\label{fig:residual-all-orderDvr=7-waverank=32}\n\t\\end{subfigure}\n\t\\quad\n\t\\begin{subfigure}{.45\\textwidth}\n\t\t\\centering\n\t\t\\input{residual-all-orderDvr=14-waverank=32.tikz} \\\\\n\t\t\\caption{orderDvr = 14}\n\t\t\\label{fig:residual-all-orderDvr=14-waverank=32}\n\t\\end{subfigure}\n\t\\caption{Residual after iteration 10 iterations for all three versions of algorithm with $s = 32$.}\n\t\\label{fig:residual-all-waverank=32}\n\\end{figure}\n\n\nIf we increase the maximal attainable rank $r$ of the low-rank approximation, indeed the residual decreases as shown in Figure \\ref{fig:residual-all-orderDvr=7-waverank=32}. The residual for version 1 and version 3 are good, while version 2 cannot compete with both other versions by reducing the residual as far as the other versions. Therefore version 1 or version 3, as given in Algorithm~\\ref{alg:var3dv1} or Algorithm~\\ref{alg:var3dv3} are preferred.\n\nConsidering the runtimes of the three versions, similar results as before are observed. In this experiment with \\mbox{orderDvr=7} the number of gridpoints equals to $n = 41$. For version 1 and 3, again a runtime of $\\Oh{nr^4}$ is observed. The runtime for version 2 splits into two parts: $\\Oh{nr^2 + r^9}$. Due to the small rank $r$ and the large number of iterations in these examples algorithm 2 is the fastest version. The runtimes for version 1 and version 3 differ indeed approximately a factor $d$, which makes version 3 better then version 1. The runtimes with orderDvr = 7 (i.e. $n=41$) are shown in Figure \\ref{fig:runtimes-all-orderDvr=7-waverank=32+loglog} and with orderDvr = 14 (i.e. $n=90$) are shown in Figure \\ref{fig:runtimes-all-orderDvr=14-waverank=32+loglog}.\n\nComparing the runtimes for orderDvr=7 and orderDvr=14 we see for version 2 (when the rank gets larger) indeed approximately the same runtime independent of orderDvr. Also versions 1 and 3 consume approximately twice as much time which is as expected by the linear dependence on $n$ for both algorithms.\n\n\\begin{figure}\n\t\\centering\n\t\\begin{subfigure}{.45\\textwidth}\n\t\t\\centering\n\t\t\\input{runtimes-all-orderDvr=7-waverank=32+loglog.tikz} \\\\\n\t\t\\caption{orderDvr = 7}\n\t\t\\label{fig:runtimes-all-orderDvr=7-waverank=32+loglog}\n\t\\end{subfigure}\n\t\\quad\n\t\\begin{subfigure}{.45\\textwidth}\n\t\t\\centering\n\t\t\\input{runtimes-all-orderDvr=14-waverank=32+loglog.tikz} \\\\\n\t\t\\caption{orderDvr = 14}\n\t\t\\label{fig:runtimes-all-orderDvr=14-waverank=32+loglog}\n\t\\end{subfigure}\n\t\\caption{Runtime of 10 iterations for all three versions of algorithm for 3D Helmholtz with space-dependent wave number of low-rank $s = 32$.}\n\t\\label{fig:runtimes-all-waverank=32}\n\\end{figure}\n\nAn impression of the low-rank approximation to the wave function is shown in Figures \\ref{fig:impression3d} and \\ref{fig:impression3dv2}. In this impression the single, double and triple ionization are visible and can be represented by a low-rank wave function.\n\n\\begin{figure}\n\t\\centering\n\t\\begin{subfigure}{\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=0.85\\textwidth]{wave2d-LRimpression.png}\n\t\t\\caption{Impression of a low-rank matrix approximation in 2D.}\n\t\t\\label{fig:impression2d}\n\t\\end{subfigure}\n\t\\par\n\t\\begin{subfigure}{\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=0.9\\textwidth]{wave3d-LRimpression.png}\n\t\t\\caption{Impression of a low-rank tensor approximation in 3D.}\n\t\t\\label{fig:impression3d}\n\t\\end{subfigure}\n\t\\caption{Impressions of a low-rank approximation of a matrix and a Tucker tensor representing the wave function as solution to a 2D and 3D Helmholtz problem with a space-dependent wave number.}\n\\end{figure}\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.9\\textwidth]{wave3d+nocube+c1+c2-v2.png}\n\t\\caption{Visualization of a 3D wave as low-rank approximation\n to a 3D Helmholtz problem with space-dependent wave number\n with single, double and triple ionization.}\n\t\\label{fig:impression3dv2}\n\\end{figure}\n\n\n\\section{Discussion and conclusions}\\label{sec:discussion}\nIn this paper we have analyzed the scattering solutions of a driven\nSchr\\\"odinger equations. These describe a break-up reaction where a\nquantum system is fragmented into multiple fragments. These problems\nare equivalent to solving a Helmholtz equations with space-dependent wave\nnumbers.\n\nWe have shown, first in 2D and then in 3D, that the wave function of\nmultiple ionization can be well approximated by a low-rank solution.\nIn 2D, the waves can be represented as a product of two low-rank \nmatrices and 3D as a low-rank Tucker tensor decomposition.\n\nWe propose a method that determines these low-rank components\nof the solution directly. We write the solution as a product of low-rank\ncomponents and assume that a guess for all but one component is given.\nWe then write a linear system for the remaining unknown component.\nThis is the repeated until each of the components is updated.\n\nThis procedure can be interpreted as a series of projections of the\nresidual on a subspaces and a correction within that subspace.\n\nIn theory, the generalization for dimensions $d > 3$ is straightforward. But\nfor dimensions $d>3$ it starts to be beneficial to change to a Tensor\nTrain factorization \\cite{oseledets2011tensor}. It is expected that similar \nstrategy can also be applied to tensors in Tensor Train format.\n\nAs demonstrated by the numerical experiments, the presented algorithms\nare able to exploit the low-rank structure of the solutions. This\ngives the advantage to reduce the number of unknowns and shorten the\ncomputational time to solve the Helmholtz equation.\n\nIn two dimensions, the low-rank representation of the solution can be\nrepresented by only $2nr$ unknowns instead of the full grid of\n$n^2$ unknowns. Also the linear systems to solve per iteration have\nonly $nr$ unknowns.\n\nIn high-dimensional Helmholtz equations, the low-rank Tucker tensor\ndecomposition represents the solution with $\\Oh{r^d + dnr}$\nunknowns. So, the total number of unknowns is reduced, but it is still\nexponential in the dimension $d$. For increasing dimensions this leads,\nagain, to systems with a number of unknowns exponential in $d$. Maybe\nother Tucker-like tensor decompositions with a number of unknowns only\npolynomial in $d$ can resolve this problem and make the presented\nalgorithm also applicable for higher dimensions.\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{sec:intro}\n\tIn the past decade many surveys of the galactic plane have been carried out in the continuum, covering wavelengths from the millimetre regime to the infrared (IR).\n\tThey provide a complete picture of the dust emission, tracing both very cold material (at millimetre, sub-mm and far-IR wavelengths; for example ATLASGAL, \\citealt{Schuller+09_aap504_415}, and Hi-GAL, \\citealt{Molinari+10_pasp122_314}), and hot dust and PAHs (in the mid- and near-IR; for example MSX, \\citealt{Egan+03_AFRL}, MIPSGAL, \\citealt{Carey+09_pasp121_76}, WISE, \\citealt{Wright+10_aj140_1868}). The temperature, mass and column density of the dust can be estimated by constructing and modelling the spectral energy distribution of the thermal dust emission \\citep[SED; e.g.][]{Koenig+17_aap599_139}. The dust, however, constitutes only a minor fraction of the total mass of molecular clouds. One has to assume a gas-to-dust mass ratio ($\\ensuremath{\\gamma}$) to derive the mass and column density of molecular hydrogen. \n\tA direct, local determination shows that the hydrogen-to-dust mass ratio is $\\sim100$, corresponding to a gas-to-dust mass ratio $\\ensuremath{\\gamma}\\approx136$, when accounting for helium \\citep{Draine+07_apj663_866}. \n\tCurrent research uses a constant value of the gas-to-dust ratio irrespective of the galactocentric distance of the cloud \\citep[typically $100-150$, e.g.][]{Elia+13_apj772_45,Elia+17_mnras471_100,Koenig+17_aap599_139}, and while these values are reasonable within the solar circle they are not likely to be reliable for the outer parts of the disk, where the metallicity and average disk surface density might be substantially lower.\n\t\n\tHeavy elements are the main constituents of dust grains, and therefore when their abundance with respect to hydrogen changes, dust may be influenced too. Models combining chemical evolution of the Galaxy with dust evolution indeed suggest that $\\ensuremath{\\gamma}$ increases with decreasing metallicity $Z$ \\citep[][]{Dwek98_apj501_643,Mattsson+12_mnras423_38,HirashitaHarada17_mnras467_699}. This is also supported by observations in nearby galaxies \\citep[e.g.][]{Sandstrom+13_apj777_5}. \n\t\n\tExcept for a few cases, the data for external galaxies are averaged over the entire galaxy, and in all cases optically thick CO lines are used to obtain the mass of molecular gas. Moreover, in studies in which the gradient in $\\ensuremath{\\gamma}$ with $Z$ can be spatially resolved, the resolution is of the order of a kpc, introducing large uncertainties, for example, by assuming a uniform single temperature for dust or a specific calibration in deriving the metallicity \\citep[e.g.][]{Sandstrom+13_apj777_5}. As \\citet{Mattsson+12_mnras423_38} discuss, this could lead to a dust content which, in the central regions, often is larger than the amount of available metals in the interstellar medium (ISM).\n\t\n\tThe study of the metallicity-$\\ensuremath{\\gamma}$ relation in the Milky Way not only opens the possibility to have, for the first time, more accurate estimates of the amount of molecular gas in clouds,\n\tbut also provides the possibility to explore it on spatial scales and sensitivities that are extremely challenging to obtain, if not inaccessible, in galaxies other than our own. \\citet{Issa+90_aap236_237} studied how the gradient in gas-to-dust ratio depends on the galactocentric radius, but for a limited range of $D_{GC}$ ($9-11\\usk\\mathrm{kpc}$) and using optically thick CO lines to estimate the amount of molecular gas, via the integrated intensity of the CO (1--0) line-to-molecular mass conversion factor $X_\\mathrm{CO}$. \n\t\n\tIn this work we use a sample of 23 sources in the far outer Galaxy, complemented by 57 sources from the ATLASGAL TOP100 in the inner Galaxy (Fig.~\\ref{fig:distribution_clumps}) to expand this pioneering work, exploring the variation of $\\ensuremath{\\gamma}$ across the entire disk of the Milky Way. \n\tThis opens up the possibility of using the appropriate value of the gas-to-dust ratio to obtain more precise estimates of the very basic properties of molecular clouds throughout the Milky Way from publicly available surveys, such as the total mass and H$_{2}$ column density. From these quantities it is possible to derive molecular abundances and, in combination with complete surveys of the galactic disk, a reliable distribution of mass of molecular gas in the Milky Way.\n\t\n\n\\section{Observations and sample selection}\\label{sec:obs_and_sample}\n\n\tFrom the \\citet{WouterlootBrand89_aaps80_149} IRAS\/CO catalogue and that compiled by K\\\"onig et al. (in prep.)\\ using $^{12}$CO(2--1) and $^{13}$CO (2--1)\\footnote{Observed with the APEX-1 receiver at the Atacama Pathfinder Experiment (APEX) 12-m telescope.}, we selected a sample of 23 sources in the far outer Galaxy ($D_{GC} > 14\\usk\\mathrm{kpc}$; FOG) with the following criteria: i) the source must be associated with IR emission in WISE images ii) Herschel data must be available to estimate the dust content, and iii) the surface density of dust ($\\ensuremath{\\Sigma_\\mathrm{dust}}$) at the emission peak must exceed $3\\times10^{-5}\\usk\\gram\\usk\\centi \\metre^{-2}$, or $N_\\mathrm{H_{2}}=8.75\\times10^{20}\\usk\\centi \\metre^{-2}$ (i.e. $\\Sigma_{gas}\\sim19\\msun\\usk\\mathrm{pc}^{-2}$), assuming $\\ensuremath{\\gamma}=136$. \n\tAccording to the model of \\citet{HirashitaHarada17_mnras467_699}, the latter condition is sufficient to ensure that the vast majority of gas is in molecular form for $Z \\gtrsim 0.2 \\usk Z_{\\odot}$. \n\tIn the FOG, in fact, the metallicity ranges from $\\sim0.5\\,Z_{\\odot}$ at $D_{GC}\\sim14\\usk\\mathrm{kpc}$ to $\\sim0.2\\,Z_{\\odot}$ at $D_{GC}\\sim21\\usk\\mathrm{kpc}$ \\citep[using the results in][ see Eq.~\\ref{eq:Z_grad_MW}]{LuckLambert11_aj142_136}.\n\tObservations of the Magellanic Clouds also provide support for this statement. \n\tThe metallicities in the Large and Small Magellanic Clouds (LMC, SMC) are $Z = 0.5 \\, Z_{\\odot}$ and $Z = 0.2\\, Z_{\\odot}$, respectively \\citep{RussellDopita92_apj384_508}, encompassing the range of the far outer Galaxy. \n\tObservations of the atomic and molecular gas in these galaxies by \\citet{RomanDuval+14_apj797_86} demonstrate that the H\\textsc{i}--H$_{2}$ transition occurs at $\\approx30\\msun\\usk\\mathrm{pc}^{-2}$ in the LMC and $\\approx80\\msun\\usk\\mathrm{pc}^{-2}$ in the SMC. Our criterion on the surface density of dust, when using the gas-to-dust ratios estimated by \\citet{RomanDuval+14_apj797_86} in the Magellanic Clouds, exceeded these observed thresholds: $\\Sigma_{gas}\\approx70\\msun\\usk\\mathrm{pc}^{-2}$ for $Z=0.5\\, Z_{\\odot}$ and $\\sim230\\msun\\usk\\mathrm{pc}^{-2}$ for $Z=0.2\\, Z_{\\odot}$.\n\n\tThe selection of only IR-bright sources, \n\tstill associated with substantial molecular material, \n\timplies that we are dealing with clumps in a relatively advanced stage of the star formation process, when CO is not significantly affected by depletion \\citep{Giannetti+14_aap570_65}.\n\tThe sources have been followed-up with single-pointing observations centred on the dust emission peak, as identified in Herschel images, carried out using the APEX-1 receiver at APEX tuned to $218.09\\ensuremath{\\usk\\giga\\hertz}$, a setup which includes C$^{18}$O(2--1). Here, we have used this transition to estimate the total amount of H$_{2}$ at the position of the dust emission peak. The angular resolution of APEX at this frequency is $\\sim28\\arcsec$.\n\tObservations were performed between September 29 and October 15 2015, and between December 3 and 11 2015. The typical rms noise on the $T_\\mathrm{{A}}^{*}$-scale ranges between $10\\usk\\milli\\kelvin$ and $20\\usk\\milli\\kelvin$ at a spectral resolution of $0.4\\usk\\kilo\\metre\\usk\\second^{-1}$. \n\tWe converted the antenna temperature $T_\\mathrm{{A}}^{*}$ to main beam brightness temperature, $T_\\mathrm{{MB}}$, using $\\eta_{\\mathrm{MB}} = 0.75$.\n\t\n \\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=\\columnwidth]{figures\/source_distribution.pdf}\n\t\\caption{Distribution of the sources considered in this work. In white we show the sources in the FOG, and in red we show clumps from the TOP100 sample. The background image is an artist impression of the Milky Way as seen from the northern galactic pole (courtesy of NASA\/JPL-Caltech\/R. Hurt -- SSC\/Caltech). The Sun in at (0, 8.34) kpc.}\\label{fig:distribution_clumps}\n \n \\end{figure}\n\n\\section{Results}\\label{sec:results}\n\n\tAs a first step we constructed the SED for each of the sources in the FOG to obtain the peak mass surface density of the dust. \n\tFor the SED construction and fitting, we follow the procedure described in \\citet{Koenig+17_aap599_139} and adopted in \\citet{Giannetti+17_aap603_33} and \\citet{Urquhart+17_ArXiv}, with minor changes due to the absence of ATLASGAL images for the outer Galaxy. We considered the five Hi-GAL \\citep{Molinari+10_pasp122_314} bands (500, 350, 250, 160 and $70\\usk\\micro\\metre$) from the SPIRE \\citep{Griffin+10_aap518_3} and PACS \\citep{Poglitsch+10_aap518_2} instruments, to reconstruct the cold dust component of the SED. The contribution from a hot embedded component is estimated from mid-IR continuum measurements, using MSX \\citep{Egan+03_AFRL} and WISE \\citep{Wright+10_aj140_1868} images at 21, 14, 12 and $8\\usk\\micro\\metre$, and 24 and $12\\usk\\micro\\metre$, respectively.\n\t\n\tThe flux for each of the bands was calculated using an aperture-and-annulus scheme. The aperture is centred on the emission peak at $250\\usk\\micro\\metre$ and its size was set to three times the FWHM of a Gaussian fitted to the $250\\usk\\micro\\metre$ image. The background was calculated as the median flux over an annulus with inner and outer radii of 1.5 and 2.5 the aperture size, respectively. After being normalised to the area of the aperture, the background was subtracted from the flux within the aperture.\n\tThe uncertainties on the background-corrected fluxes were calculated summing in quadrature the pixel noise of the images and a flux calibration uncertainty. \n\tWe adopted a calibration uncertainty of $20\\%$ for the 350, 250, 160 and $70\\usk\\micro\\metre$ fluxes, and of $30\\%$ for the mid-IR bands. An uncertainty of $50\\%$ is assumed for the $500\\usk\\micro\\metre$ flux, due to the\n\tlarge pixel size, and for the $8\\usk\\micro\\metre$ flux, due to contamination from PAHs.\n\tThe grey-body plus black-body model was optimised via a $\\chi^{2}$ minimisation, and the uncertainties on the parameters were estimated propagating numerically the errors on the observables.\n\tDifferently from \\citet{Koenig+17_aap599_139} and \\citet{Urquhart+17_ArXiv}, we use the $350\\usk\\micro\\metre$ Herschel flux measurement to calculate the peak dust surface density of the clump, because the sources were not covered in ATLASGAL and because this image has a comparable resolution to our molecular-line observations. The method is discussed in more detail in \\citet{Koenig+17_aap599_139} and \\citet{Urquhart+17_ArXiv}, and we refer the interested reader to these publications.\n\n\tThe dust opacity and emissivity used are the same as in \\citet{Koenig+17_aap599_139}, that is, $\\kappa_{870\\usk\\micro\\metre}=1.85\\usk\\centi \\metre^{2}\\usk\\gram^{-1}$ and $\\beta=1.75$, respectively \\citep[see e.g.][]{Kauffmann+08_aap487_993}. The SEDs for the entire sample can be found in Fig.~\\ref{fig:seds}; an example is shown in Fig.~\\ref{fig:sed_example}. In addition to the mass surface density of dust at the far-IR peak, we derived the bolometric luminosity, the dust temperature and mass of the sources, as measured within the apertures listed in Table~\\ref{tab:sed_fit}, that contains the complete results of the SED fit.\n\t\n\tWe fitted the C$^{18}$O (2--1) line using MCWeeds \\citep{Giannetti+17_aap603_33} with the algorithm that makes use of the Normal approximation \\citep{gelman2003bayesian} to obtain the column density of carbon monoxide, under the assumption of LTE; the adopted partition function is reported in Table~\\ref{tab:part_funct}. Using the relation between the dust temperature and the excitation temperature of CO isotopologues found in \\citet[][see their Fig.~10]{Giannetti+17_aap603_33}, we estimated the excitation conditions for the sources in the FOG. We used this value of $T_\\mathrm{{ex}}$ as the most probable one in the prior, with a value of $\\sigma$ equal to the measured intrinsic scatter; all priors are fully described in Table~\\ref{tab:priors} and the results are listed in Table~\\ref{tab:mcweeds_fit_fog}. To exclude biases connected to the $T_\\mathrm{{d}}\\,vs.T_\\mathrm{{ex}}$ relation, we compared the column densities with those computed using the unmodified values of the dust temperature from the SED; this has only a minor impact on the derived quantities. In Appendix~\\ref{app:spectra} we show the fit results, superimposed on the observed spectra; an example is given in Fig.~\\ref{fig:fit_example}.\n\t\n\t\\begin{figure}\n\t\t\\centering\n\t\t\\includegraphics[width=0.5\\textwidth]{figures\/SEDs\/{WB89_789_SED}.pdf}\n\t\t\\caption{Top: example of the SED fit for WB$\\_$789, hosting one of the the furthest clusters from the galactic centre yet detected \\citep{BrandWouterloot07_aa464_909}. Extracted fluxes are indicated by the red crosses, and upper and lower limits are indicated by triangles pointing downwards and upwards, respectively. The best fit curve is indicated in blue, and the separate contributions of the grey and black bodies are shown by the green dashed lines. Bottom: Residuals calculated as $(S_{\\mathrm{obs}} - S_{\\mathrm{mod}}) \/ S_{\\mathrm{obs}}$. The SED and their residuals for all the other sources are included in Fig.~\\ref{fig:seds}.\\label{fig:sed_example}}\n\t\\end{figure}\n\n\t\\begin{figure}\n\t\t\\centering\n\t\t\\includegraphics[width=0.8\\columnwidth]{figures\/spectra\/post_fit_WB89_789_slice_0.pdf}\n\t\t\\caption{Example of the C$^{18}$O(2--1) observations for WB$\\_$789. We indicate in red the best fit from MCWeeds. The spectra and their fits for the entire sample are shown in Fig.~\\ref{fig:spectra}.\\label{fig:fit_example}}\n\t\\end{figure}\n\t\n\tIn order to study how the gas-to-dust ratio varies across the galactic disk, we complemented the FOG sample with sources selected from the TOP100 \\citep{Giannetti+14_aap570_65, Koenig+17_aap599_139}, a representative and statistically significant sample of high-mass star-forming clumps covering a wide range of evolutionary phases \\citep{Koenig+17_aap599_139, Giannetti+17_aap603_33}. These sources are among the brightest in their evolutionary class in the inner Galaxy.\n\tFor the 57 sources classified as H\\textsc{ii}\\ and IRb in the TOP100, we used the column density determinations from \\citet{Giannetti+14_aap570_65} to derive the H$_{2}$ column density. Among the isotopologues analysed in that work, we elected to use C$^{17}$O, because in these extreme sources C$^{18}$O can have a non-negligible optical depth, and because we have FLASH$^{+}$ \\citep{Klein+14_TransThzSciTech4_588} observations of C$^{17}$O (3--2) for the entire subsample. \n\t\n\tWe have $\\mathrm{C^{17}O(1-0)}$ for 17 of the selected sources in the TOP100, for which we were able to estimate the optical depth; $30\\%$ of the sample has an optical depth $\\approx0.1$, the remaining clumps have optical depths below this value.\n\tAssuming LTE, $T=30\\usk\\kelvin$, and $\\tau_\\mathrm{C^{17}O(1-0)}=0.1$, the optical depth of $\\mathrm{C^{17}O(3-2)}$ is about a factor of four to five higher than that of the (1--0) transition, leading to an underestimate of the carbon monoxide column density less than $\\sim30\\%$. Therefore the correction for opacity can be considered negligible (see Sect.~\\ref{sec:discussion}). The column density of C$^{17}$O is then converted to C$^{18}$O using $^{18}\\mathrm{O}\/^{17}\\mathrm{O} = 4$, according to \\citet{Giannetti+14_aap570_65}, as determined from the same sample. The peak surface densities (and total masses) of dust for the clumps in the TOP100 were taken from the results of \\citet{Koenig+17_aap599_139}. The properties of the sources extracted from the TOP100 are listed in Table~\\ref{tab:mcweeds_fit}.\n\t\n\tWe computed the mass surface density of the gas from the column density of C$^{18}$O, obtained via MCWeeds for the FOG sample, and of C$^{17}$O for the TOP100. When more than one velocity component was observed in the spectra of the CO isotopologues, the column densities were summed to obtain the total surface density of carbon monoxide along the line of sight, because all sources contribute to the observed continuum. This has the effect of introducing scatter in the value of $\\ensuremath{\\gamma}$ at a particular $D_{GC}$, as the clumps have different distances, but it only happens in a minor fraction of the sources (e.g. two in the FOG sample), and depends on the $D_{GC}$ of the sources. From the C$^{18}$O surface density, we derived the total mass surface density of the cloud ($\\Sigma$), accounting for helium, and assuming that the expected abundance of the C$^{18}$O is described by:\n\t\\begin{equation}\n\t\t\\chi_\\mathrm{C^{18}O}^\\mathrm{E} = \\frac{9.5 \\times \\pot{-5} \\times 10^{\\alpha(D_{GC} - R_\\mathrm{GC,\\odot})}}{^{16}\\mathrm{O}\/^{18}\\mathrm{O}} , \\label{eq:expected_ab}\n\t\\end{equation}\n\twhere $D_{GC}$ is expressed in $\\mathrm{kpc}$, $R_\\mathrm{GC,\\odot}=8.34 \\usk\\mathrm{kpc}$ \\citep{Reid+14_apj783_130}, and $\\alpha$ describes the C\/H gradient, taken to be $-0.08\\usk \\mathrm{dex}\\usk\\mathrm{kpc}^{-1}$ from \\citet{LuckLambert11_aj142_136}. We assumed that the CO abundance is controlled by the carbon abundance, because it is always less abundant than oxygen, and becomes progressively more so in the outer Galaxy.\n\tA smaller abundance of CO at lower $Z$ \n\tis consistent with observations of low-metallicity galaxies, where the detectable CO-emitting region is significantly smaller than the H$_{2}$ envelope \\citep{Elmegreen+13_nat495_487,Rubio+15_nat525_218}.\n\tThe oxygen isotopic ratio is commonly described by the relation $^{16}\\mathrm{O}\/^{18}\\mathrm{O} = 58.8 D_{GC} + 37.1$ \\citep{WilsonRood94_araa32_191}. On the one hand, independent measurements of $^{16}\\mathrm{OH}\/^{18}\\mathrm{OH}$ by \\citet{Polehampton+05_aap437_957}, despite finding consistent results with the previous works, do not strongly support such a gradient. On the other hand, \\citet{Wouterloot+08_aa487_237} find an even steeper gradient considering sources in the FOG, where C$^{18}$O is likely to be less abundant, if not for the oxygen isotopic ratio, then for selective photodissociation due to lower shielding of the dust, and self-shielding. \n\n\tFor the moment we ignore the effect of a gradient, and adopt the local CO\/C$^{18}$O ratio. We used $\\Sigma$, together with $\\ensuremath{\\Sigma_\\mathrm{dust}}$, as observed data for a JAGS\\footnote{\\url{http:\/\/mcmc-jags.sourceforge.net\/}} \\citep[Just Another Gibbs Sampler,][]{Plummer2003_IWDSC} model which derives the gas-to-dust ratios as $\\Sigma\/\\Sigma_{dust}$, and fits the points in a log-linear space, considering an intrinsic scatter. \n\tFigure~\\ref{fig:gtd_fit}a shows that the gas-to-dust ratio increases with galactocentric distance, with a gradient for $\\ensuremath{\\gamma} \\ensuremath{\\,vs.\\,}\\ D_{GC}$ described by:\n\t\\begin{equation}\n\t\tlog(\\ensuremath{\\gamma}) = \\left( 0.087\\left[ \\asymErr{+0.045}{-0.025} \\right]\\pm0.007\\right) \\, D_{GC} + \\left( 1.44 \\left[ \\asymErr{-0.45}{+0.21} \\right]\\pm0.03 \\right), \\label{eq:gtd_gradient}\n\t\\end{equation}\n\twhere $D_{GC}$ is expressed in $\\mathrm{kpc}$; we first indicate the systematic uncertainty between square brackets (discussed in the next section), and the statistical uncertainties afterwards. \n\tThis equation gives values for $\\ensuremath{\\gamma}$ at the Sun distance between $\\approx130$ and $\\approx 145$, well consistent with the local value of 136, considering the intrinsic scatter of the observed points (cf. Fig.~\\ref{fig:gtd_fit}) and the uncertainties in the derived relation.\n\tAs indicated in Fig.~\\ref{fig:gtd_fit}a, our results are, in general, valid only between $\\sim2\\usk\\mathrm{kpc}$ and $\\sim20\\usk\\mathrm{kpc}$ from the galactic centre, the range spanned by the sources in our sample. \n\t\n\tThe slope of the gradient is very close to that used in Eq.~\\ref{eq:expected_ab}, showing that C$^{18}$O behaves in a way comparable to the dust, with respect to metallicity. This implies that the results are closely linked to the assumed galactocentric carbon gradient. In the next section we discuss as limiting cases how the $\\ensuremath{\\gamma}$ gradient would change if the CO abundance follows the oxygen variation instead, and if C$^{18}$O is less abundant with respect to CO in the outer Galaxy, as a consequence of the $^{16}\\mathrm{O}\/^{18}\\mathrm{O}$ gradient, or of selective photodissociation (see also Fig.~\\ref{fig:gtd_fit}b, c). The effect of such systematic uncertainties causes the variations in the slope and intercept of Eq.~\\ref{eq:gtd_gradient} indicated in the square brackets.\n\n \\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{figures\/gtd_gradients.pdf}\n\t\\caption{Variation of the gas-to-dust ratio with galactocentric radius, for our fiducial case (Panel a), considering a CO\/C$^{18}$O galactocentric gradient (Panel b), and assuming that the abundance of CO follows the radial oxygen gradient, rather than the C\/H (Panel c). The thick blue lines indicate the best fit, reported in the bottom right corner; the $68\\%$ and $95\\%$ highest probability density intervals of the fit parameters are indicated by the light and dark yellow-shaded regions, respectively. The intrinsic scatter is indicated by the dashed lines. \n\tFor comparison with external galaxies $log$(O\/H) + 12 is shown on the top axis.}\\label{fig:gtd_fit}\n \\end{figure*}\n\n\\section{Discussion}\\label{sec:discussion}\n\n\tIn this section, we discuss the uncertainties in the gas-to-dust ratio estimates and in its galactocentric gradient, why $\\ensuremath{\\gamma}$ has to be higher at large $D_{GC}$, and how it depends on metallicity. Estimates of the gas-to-dust ratio are difficult, resting on the derivation of surface densities of a tracer of H$_{2}$ (C$^{18}$O and C$^{17}$O in our case) and of the dust. Several assumptions introduce a systematic uncertainty in $\\ensuremath{\\gamma}$. For the surface density of molecular gas, the main sources of uncertainties are the canonical CO abundance, that can vary by a factor of two, the CO--C$^{18}$O conversion, discussed in Sect.~\\ref{sec:results}, and the assumption of LTE, which is likely less important, given the results of the comparison between temperatures derived from CH$_{3}$CCH and CO isotopologues in the TOP100 sample in \\citet{Giannetti+17_aap603_33}. Dust is more problematic, especially because its properties are poorly constrained. Opacity and emissivity are sensitive to the grain composition and size distribution:\n\tdistinct models can induce discrepancies in the estimated mass surface density of dust up to a factor of approximately three \\citep[see e.g.][]{OssenkopfHenning94_aap291_943, LiDraine01_apj554_778, Gordon+14_apj797_85, Gordon+17_apj837_98}.\n\tThe simple SED model adopted is a crude approximation as well: \n\ttemperature varies along the line of sight, and in the extreme case where the representative grey-body temperature changes from $\\approx20\\usk\\kelvin$ (the median in our FOG sample is $\\approx23\\usk\\kelvin$) to $\\approx 50\\usk\\kelvin$, the dust surface density changes by a factor of approximately five.\n\t\n\tPropagating these to the gas-to-dust ratio implies a global uncertainty of nearly a factor of six on $\\ensuremath{\\gamma}$ for each target. It is therefore relevant to test whether a simpler model with a constant $\\ensuremath{\\gamma}$ ratio is to be favoured over the proposed gradient. A Bayesian model comparison, which automatically takes into account the Ockham's Razor principle \\citep[e.g.][]{BolstadIBS}, shows that, in the unfavourable case that the CO abundance follows the oxygen gradient, the odds ratio is approximately eight in favour of the gradient model\\footnote{Considering a flat prior on the slope and intercept of the $log(\\ensuremath{\\gamma}) \\ensuremath{\\,vs.\\,} D_{GC}$ relation in the ranges $0-1$ and $0-4$, respectively.}, which is then to be preferred over a constant value of $\\ensuremath{\\gamma}$ across the entire disk.\n\t\n\tFactors that can change the slope of the $\\ensuremath{\\gamma} \\ensuremath{\\,vs.\\,} D_{GC}$ relation are the molecular gas- and the CO-dark gas fractions, the CO abundance gradient, and the dust model. Larger quantities of gas in atomic form, as well as more CO-dark gas at lower metallicities (due to a lower shielding of dust and self-shielding) would cause the relation to be steeper. However, because we target exclusively dense molecular clouds, the vast majority of gas should be in molecular form (see Sect.~\\ref{sec:obs_and_sample}).\n\tA larger fraction of CO-dark gas is evident for the low-metallicity galaxy WLM \\citep{Elmegreen+13_nat495_487,Rubio+15_nat525_218}; a less extreme, but analogous situation is possible for clouds at the edge of the Milky Way disk \\citep[in the FOG, $Z$ is larger than in WLM by a factor of between approximatley two and five, see][]{Leaman+12_apj750_33}. \n\n\tOn the other hand, the variation of dust composition and size distribution of grains tend to make the measured relation flatter. \n\tSilicates are likely to be more common in the outer Galaxy \\citep[e.g.][]{Carigi+05_apj623_213}; in this case the opacity would be lower, leading to an underestimate of the dust surface density. The models in \\citet{OssenkopfHenning94_aap291_943} show that a variation in the silicate-to-carbon fraction has a much smaller impact than the change in size distribution due to coagulation. \n\tIn the extreme case in which no coagulation takes place in the FOG, and it is, on the contrary, efficient in the inner Galaxy, the dust opacity changes by a factor of approximately two in the far-IR and submm regimes, reducing the mass surface densities by the same quantity. This effect has an impact similar in magnitude, but opposite, to that of the CO\/C$^{18}$O gradient. \n\n\tIn the outer Galaxy, where extinction is lower, dust grains can be more efficiently reprocessed. As a consequence, more carbon is present in the gas phase \\citep[e.g.][]{Parvathi+12_apj760_36}, effectively making the gradient in CO abundance shallower than the C\/H one, if it follows the gas-phase abundance of carbon. A limiting case is obtained by using the slope of the oxygen gradient in Eq.~\\ref{eq:expected_ab}, in which case the slope in Eq.~\\ref{eq:gtd_gradient} can be as shallow as $0.062 \\usk\\mathrm{dex}\\usk\\mathrm{kpc}^{-1}$. It is, however, unlikely that the increased abundance of C in the gas phase and the less efficient coagulation have such an important effect. Conversely, if we neglect these effects, but consider the CO\/C$^{18}$O gradient, we can obtain an upper limit for the gradient slope. Under these conditions, $\\ensuremath{\\gamma}$ varies by $0.132 \\usk\\mathrm{dex}\\usk\\mathrm{kpc}^{-1}$.\n\n\tThe CO\/C$^{18}$O abundance gradient, and larger fractions of CO-dark and atomic gas at lower metallicity \\citep[see e.g.][]{Elmegreen+13_nat495_487,Rubio+15_nat525_218} contrast the effects of the increased fraction of C in the gas phase, of dust size distribution and composition. \n\tFor simplicity, as a fiducial value, we have therefore considered the relation for which the variation in grain size distribution and the increased fraction of CO-dark gas cancel out the impact of the CO\/C$^{18}$O abundance gradient (Fig.~\\ref{fig:gtd_fit}a).\n\t\n\t\\medskip\n\t\n\tTheoretical considerations also indicate that the gas-to-dust ratio has to be higher in the FOG.\n\tIn the following, we show that at a distance of $\\sim15\\usk\\mathrm{kpc}$ the fraction of heavy elements locked into dust grains must be $80\\%$ to maintain $\\ensuremath{\\gamma}$ at the local value of $136$.\n\tFollowing \\citet{Mattsson+12_mnras423_38} we can conservatively use the O\/H gradient to obtain an approximation of the galactocentric metallicity behaviour. \n\tThe radial $Z$ gradient for our Galaxy can be reliably obtained via measurements of the abundance of heavy elements in Cepheids, which are young enough to represent the present-day composition. \n\tWe use the results from \\citet{LuckLambert11_aj142_136}, who consider a large number of Cepheids with $5\\usk\\mathrm{kpc}\\lesssimD_{GC}\\lesssim17\\usk\\mathrm{kpc}$, deriving for oxygen the gradient $d\\mathrm{[O\/H]}\/dD_{GC} = -0.056 \\,\\mathrm{dex}\\usk\\mathrm{kpc}^{-1}$. We obtain for $Z$:\n\t\\begin{equation}\n\t\tlog(Z) = -0.056 D_{GC} - 1.176\\label{eq:Z_grad_MW},\n\t\\end{equation} \n\twhich gives, at the location of the Sun, an H-to-metal mass ratio $\\sim44$. If approximately $40\\%$ of the heavy elements are locked into dust grains \\citep{Dwek98_apj501_643}, this implies $\\ensuremath{\\gamma}=110$, which is in very good agreement with the locally-estimated value of $136$.\n\t\t\n\tThe dust-to-gas mass ratio $Z_{d}$ is the inverse of $\\ensuremath{\\gamma}$, and the fraction of mass in heavy elements locked in dust grains, the dust-to-metal ratio, can be expressed as the ratio of $Z_{d}$ and the gas metallicity, that is, $Z_{d}\/Z$. A dust-to-metal ratio of one would imply that dust grains contain all elements heavier than helium.\n\tIf we were to assume that the gas-to-dust ratio remains constant to $\\ensuremath{\\gamma} \\equiv Z_{d}^{-1}=136$ (implying that progressively more heavy elements end up in dust grains), using Eq.~\\ref{eq:Z_grad_MW} we would see that the dust metallicity $Z_{d}\/Z$ reaches $80\\%$ at $D_{GC}\\approx15\\usk\\mathrm{kpc}$. In addition, the metallicity gradient is most likely steeper than the oxygen gradient \\citep[e.g.][]{Mattsson+12_mnras423_38}, moving this limit inwards, thus indicating that in the FOG the gas-to-dust ratio is bound to be higher.\n\t\n\tNow using our results for the increase of the gas-to-dust ratio with $D_{GC}$, the dust metallicity can be derived from Eqs.~\\ref{eq:gtd_gradient} and \\ref{eq:Z_grad_MW}:\n\t\\begin{equation}\n\t\tlog\\left( \\frac{Z_{d}}{Z} \\right) = \\left( -0.031 \\left[\\asymErr{+0.025}{-0.047} \\right] \\right) D_{GC} - \\left( 0.26 \\left[ \\asymErr{-0.21}{+0.45} \\right] \\right), \\label{eq:Zdust}\n\t\\end{equation}\n\twhich shows that the dust-to-metal ratio decreases with galactocentric radius. \n\tA decrease of the the dust-to-metal ratio is the most common situation in late-type galaxies and indicates that grain growth in the dense ISM dominates over dust destruction \\citep[e.g.][]{Mattsson+12_mnras423_38,Mattsson+14_mnras444_797}. This strongly reinforces the previous argument that a constant $\\ensuremath{\\gamma}=136$ cannot be sustained in the far outer Galaxy, because the metal-to-dust ratio virtually always decreases moving outwards in the disk, for Milky-Way type galaxies.\n\n\t\n\tA good test bench for Eq.~\\ref{eq:gtd_gradient} is represented by the Magellanic Clouds. Combining Eqs.~\\ref{eq:gtd_gradient} and \\ref{eq:Z_grad_MW}, and using the appropriate metallicity \\citep[$Z = 0.5 \\, Z_{\\odot}$ and $Z = 0.2\\, Z_{\\odot}$ for the Large and Small Magellanic Clouds, respectively;][]{RussellDopita92_apj384_508}, we obtain $\\ensuremath{\\gamma} \\sim 420\\asymErr{+250}{-110}$ and $\\ensuremath{\\gamma} = 1750\\asymErr{+4100}{-900}$, in excellent agreement with the results of \\citet{RomanDuval+14_apj797_86}.\n\t \n\\section{Summary and conclusions}\n\n\tWe combined our molecular-line surveys towards dense and massive molecular clouds in the inner- and far outer disk of the Milky Way to study how the gas-to-dust ratio $\\ensuremath{\\gamma}$ varies with galactocentric distance and metallicity.\n\tWe estimated conservative limits for the galactocentric gradient of gas-to-dust mass ratio, by considering multiple factors that influence its slope (see Sect.~\\ref{sec:discussion}), and defined, for simplicity, the fiducial value as the case where dust coagulation and the larger fraction of carbon in the gas phase in the FOG balance the CO\/C$^{18}$O abundance gradient.\n\tThe gas-to-dust mass ratio is shown to increase with $D_{GC}$ according to Eq.~\\ref{eq:gtd_gradient}, and this gradient is compared with that of metallicity, as obtained from Cepheids by \\citet{LuckLambert11_aj142_136}.\n\tThe variation in gas-to-dust ratio is steeper than that of $Z$ ($\\ensuremath{\\gamma} \\propto Z^{-1.4\\asymErr{+0.3}{-1.0}}$),\n\timplying that the the dust-to-metal ratio decreases with distance from the galactic centre. This indicates that dust condensation in the dense ISM dominates over dust destruction, which is typical of late-type galaxies like ours \\citep{Mattsson+12_mnras423_38,Mattsson+14_mnras444_797}. \n\tThe predictions obtained combining Eq.~\\ref{eq:gtd_gradient} and \\ref{eq:Z_grad_MW} for the metallicities of the Magellanic Clouds are in excellent agreement with the results on $\\ensuremath{\\gamma}$ in these galaxies by \\citet{RomanDuval+14_apj797_86}.\n\n\tThe use of Eq.~\\ref{eq:gtd_gradient} to calculate the appropriate value of $\\ensuremath{\\gamma}$ at each galactocentric radius is fundamental for the study of individual objects, allowing us to derive accurate H$_{2}$ column densities and total masses from dust continuum observations, as well as for any study that compares the properties of molecular clouds in the inner and outer Galaxy. This opens the way for a complete view of the galactic disk and of the influence of $Z$ on the physics and chemistry of molecular clouds. \n\n\\begin{acknowledgements}\nWe are thankful to Frank Israel for a discussion on the uncertainties involved in deriving the gas-to-dust ratio and to the anonymous referee that both helped to improve the quality and the clarity of this paper. This work was partly carried out within the Collaborative Research Council 956, sub-project A6, funded by the Deut\\-sche For\\-schungs\\-ge\\-mein\\-schaft (DFG). This paper is based on data acquired with the Atacama Pathfinder EXperiment (APEX). APEX is a collaboration between the Max Planck Institute for Radioastronomy, the European Southern Observatory, and the Onsala Space Observatory. \nThis research made use of Astropy, a community-developed core Python package for Astronomy \\citep[][\\url{http:\/\/www.astropy.org}]{astropy_2013}, of NASA's Astrophysics Data System, and of Matplotlib \\citep{Hunter_2007_matplotlib}. MCWeeds makes use of the PyMC package \\citep{Patil+10_jstatsoft35_1}.\n\\end{acknowledgements}\n\n \n\\bibliographystyle{bibtex\/aa}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}