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+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nUnsupervised dimensionality reduction is a key step in many applications, including visualization \\cite{maaten2008visualizing} \\cite{mcinnes2018umap}, clustering \\cite{cohen2015dimensionality} \\cite{niu2011dimensionality}, and preprocessing for downstream supervised learning \\cite{pechenizkiy2004pca}. Principal Component Analysis (PCA) is one well-known technique for dimensionality reduction, which notably makes no assumptions about the ordering of the samples in the data matrix $X \\in \\RR^{N \\times D}$. Multivariate Singular Spectrum Analysis (MSSA) \\cite{hassani2013multivariate} is an extension of PCA for time series data, which been successfully applied in applications like signal decomposition and forecasting \\cite{hassani2009forecasting} \\cite{mahmoudvand2015forecasting} \\cite{patterson2011multivariate}. In MSSA, each row is read at a certain time step, and thus is influenced by the ordering of the samples. MSSA works primarily by identifying key oscillatory modes in a signal, which also makes it useful as a general-purpose signal denoiser. However, MSSA (like PCA, upon which it is based) is limited to finding the principal components that capture the maximal variance in the data. In situations where the information of interest explains little overall variance, these methods fail to reveal it. Recently, extensions like contrastive PCA (cPCA) \\cite{abid2018exploring, zou2013contrastive,ge2016rich} have shown that utilizing a background dataset $Y \\in \\RR^{M \\times D}$ can help better discover structure in the foreground (target) $X$ that is of interest to the analyst.\n\n\\begin{figure}[htb]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{diagram}\n    \\caption{Schematic illustrating the relations among PCA, cPCA, MSSA, and cMSSA.}\n    \\label{fig:diagram}\n\\end{figure}\n\nContrastive Multivariate Singular Spectrum Analysis (cMSSA) generalizes cPCA and applies it to time series data. Figure~\\ref{fig:diagram} visualizes the relationships between the four methods. As a contrastive method, cMSSA emphasizes salient and unique sub-signals in time series data rather than just the sub-signals comprise the majority of the structure. So while standard MSSA is useful for denoising a signal, cMSSA additionally ``denoises'' signals of structured but irrelevant information.\n\n\\section{Contrastive Multivariate Singular Spectrum Analysis}\\label{cmssa}\n\n\\textbf{Standard MSSA}\nConsider a centered one-channel times series $\\mathbf{x} \\in \\RR^T$. We construct a Hankel matrix $H_\\mathbf{x} \\in \\RR^{T' \\times W}$ with window size $W$ as follows:\n\\[\nH_\\mathbf{x} = \n\\begin{pmatrix}\nx_1 & x_2 & \\ldots & x_W \\\\\nx_2 & x_3 & \\ldots & x_{W+1} \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\nx_{T'} & x_{T'+1} & \\ldots & x_T \\\\\n\\end{pmatrix}\n\\]\nwhere $T' = T-W+1$.\nTo extend to the multivariate case, let $X \\in \\RR^{T \\times D}$ be a $D$-channel time series that runs for $T$ steps. We construct the Hankelized matrix $H_X$ with window $W$ by horizontally concatenating the per-channel Hankel matrices into a $T'$-by-$DW$ matrix:\n$H_X = [H_{\\mathbf{x}^{(1)}} ; H_{\\mathbf{x}^{(2)}} ; \\ldots ; H_{\\mathbf{x}^{(D)}}]$. Next we compute the covariance matrix $C_X \\in \\RR^{DW\\times DW}$ for $H_X$. The next step is to perform the eigendecomposition on $C_X$, yielding $DW$ eigenvectors. Of these we take the top $K$ vectors with the largest corresponding eigenvalues. We denote $\\mathbf{e}^{(k)}$ as the eigenvector with the $k$th largest eigenvalue. We collect the vectors into a matrix $E \\in \\RR^{DW \\times K}$.\n\nTo transform our original time series $X$, we have two options: (a) Project $X$ into the principal component (PC) space defined by $E$:\n$A = H_X E$ or (b) use $A$ to compute the $k$th reconstructed component (RC) $R^{(k)}$ as done in the SSA literature:\n\n\\[\nR^{(k)}_{tj} = \\frac{1}{W_t} \\sum^{U_t}_{t' = L_t} A_{t-t'+1, k} \\cdot \\mathbf{e}^{(k)}_{(j-1)W + t'}\n\\]\n\nwhere $L_t = \\max(1, t-T+W)$, $U_t = \\min(t, W)$, and $W_t = U_t - L_t + 1$. The rows of $R^{(k)}$ are indexed by time $t \\in \\{1,\\ldots,T\\}$ and the columns by channel $j \\in \\{1,\\ldots,D\\}$. Summing up the reconstructed components reproduces a denoised version of the original signal. For our purposes, we opt instead to take the horizontal concatenation of the reconstructed components as the second transform:\n$R = [R^{(1)} ; R^{(2)} ; \\ldots ; R^{(K)}].$\nTo handle multiple time series, one simply vertically stacks each Hankelized matrix. The algorithm proceeds identically from there.\n\n\\bigskip\n\\noindent \\textbf{Contrastive MSSA}\nThe modification to MSSA we introduce is via a new variable $\\alpha \\geq 0$ we call the \\emph{contrastive} hyperparameter. We construct $H_Y$ for another $D$-channel times series $Y$ (the background data) via the same process. It is not required that $X$ and $Y$ run for the same number of time steps, only that their channels are aligned. We compute a contrastive covariance matrix $C = C_X - \\alpha C_Y$ and perform the eigendecomposition on $C$ instead of $C_X$. The intuition for this is that by subtracting out a portion of the variance in $Y$, the remaining variance in $X$ is likely to be highly specific to $X$ but not $Y$.\nThis is the key additional mechanism behind cMSSA --- if $\\alpha = 0$, then no contrast is performed, and cMSSA reduces down to just MSSA.\n\n\\begin{algorithm}[htb]\n\\caption{Spectral $\\alpha$-Search\n}\n\\label{algo:gen_alpha}\n\\begin{algorithmic}[1]\n\n\\Require Minimum $\\alpha$ to consider $\\alpha_{\\min}$, maximum $\\alpha$ to consider $\\alpha_{\\max}$, number of $\\alpha$s to consider $n$, number of $\\alpha$s to return $m$, foreground signal $X$, background signal $Y$, window $W$, and number of components $K$.\n\n\\Procedure{}{}\n    \\Let{$Q$}{\\textsc{LogSpace}($\\alpha_{\\min}$, $\\alpha_{\\max}$, $n$) $\\cup \\{0\\}$}\n    \\For{$\\alpha^{(i)} \\in Q$}\n        \\Let{$H_X$, $H_Y$}{\\textsc{Hankel}($X$, $W$), \\textsc{Hankel}($Y$,$W$)}\n        \\Let{$C_X$, $C_Y$}{\\textsc{Cov}($H_X$), \\textsc{Cov}($H_Y$)}\n        \\Let{$E^{(i)}$}{\\textsc{EigenDecomp}($C_X - \\alpha^{(i)}C_Y$, $K$)}\n    \\EndFor\n    \\Let{$S$}{\\textsc{Empty}($\\RR^{n+1 \\times n+1}$)}\n    \\For{$i \\in \\{1, \\ldots, n+1\\}$, $j \\in \\{i, \\ldots, n+1\\}$}\n        \\Let{$S_{i,j}, S_{j,i}$}{$\\left\\lVert {E^{(i)}}^T E^{(j)} \\right\\rVert_*$}\n    \\EndFor\n    \\Let{$Z$}{\\textsc{SpectralCluster}($S$, $Q$, $m$)}\n    \\Let{$Q^*$}{\\{0\\}}\n    \\For{$z \\in Z$}\n        \\If{$0 \\notin z$}\n            \\Let{$\\alpha^*$}{\\textsc{ClusterMediod}($z$, $S$)}\n            \\Let{$Q^*$}{$Q^* \\cup \\{\\alpha^*\\}$}\n        \\EndIf\n    \\EndFor\n    \\\\\n    \\Return{$Q^*$, set of $m$ best $\\alpha$s, including zero.}\n\\EndProcedure\n\\end{algorithmic}\n\\end{algorithm}\n\nThe choice of $\\alpha$ is non-trivial.  Algorithm~\\ref{algo:gen_alpha} outlines a routine for auto-selecting a small number of promising values for $\\alpha$. Because cMSSA is designed to assist data exploration, Algorithm~\\ref{algo:gen_alpha} uses spectral clustering to identify a diverse set of $\\alpha$ values corresponding to diverse eigenspaces and representations of  $X$. The procedure works by first generating a large number of $\\alpha$s spread evenly in log-space. For each candidate $\\alpha$, we use cMSSA to compute its corresponding eigenvector matrix $E$. The procedure then performs spectral clustering, which requires a pairwise distance matrix as input. The distance metric used takes the nuclear norm of the matrix computed by multiplying the eigenvector matrices $E$ for any pair of $\\alpha$s. After specifying the number of clusters desired, we take the mediod $\\alpha$ of each cluster and return them as output. We always include 0 in this set, as the analyst may want to perform their analysis without contrast as control.\n\n\\section{Experiments}\\label{experiements}\n\n\\textbf{Synthetic example}\nTo illustrate cMSSA, we present a simple synthetic example. We generate an artificial one-channel signal $Y$ by sampling 500 sinusoids with different frequencies, amplitudes, phases, and vertical shifts. White Gaussian noise sample from $\\mathcal{N}(0,1)$ is added in as well. We generate $X$ in the same manner, but add in a very specific sub-signal (Figure~\\ref{fig:syn_sub}) that has comparatively low variance compared to the whole time series. The signals $X$ and $Y$ are generated independently as to rule out simple signal differencing as an explanation. We take $X$ as foreground and $Y$ as background.\n\nWe set $W=100$, $\\alpha=2$, and use only the top $K=2$ RCs. Fig.~\\ref{fig:syn_exp} displays the reconstructions computed by MSSA versus cMSSA, alongside the sub-signal that was injected into $X$. Specifically, we see that the cMSSA reconstruction shown in Fig.~\\ref{fig:syn_x_rcs_contrast} yields a noisy approximation of the sub-signal of interest,  Fig.~\\ref{fig:syn_sub}. The variance of the noise here is comparable to the variance of the sub-signal---more noise would eventually overpower cMSSA's ability to extract the sub-signal.\n\n\\begin{figure}[htb]\n\\centering\n\\begin{subfigure}{\\columnwidth}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{syn_sub}\n    \\caption{Sub-signal specific to the foreground data $X$, which is of much lower amplitude than the other sinusoidal sub-signals in $X$.}\n    \\label{fig:syn_sub}\n\\end{subfigure}\n\\begin{subfigure}{\\columnwidth}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{syn_x_rcs}\n    \\caption{Without contrast, the reconstructed time series consists of the high-amplitude sinusoidal sub-signals in $X$.}\n    \\label{fig:syn_x_rcs}\n\\end{subfigure}\n\\begin{subfigure}{\\columnwidth}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{syn_x_rcs_contrast}\n    \\caption{With contrast, the reconstructed time series is able to identify the unique sub-signal in $X$.}\n    \\label{fig:syn_x_rcs_contrast}\n\\end{subfigure}\n\n\\caption{Results of a synthetic experiment that demonstrates that cMSSA is able to identify unique sub-signals in a time series, even when they are of much lower amplitude than background components.}\n\n\\label{fig:syn_exp}\n\\end{figure}\n\n\\bigskip\n\\noindent \\textbf{Clustering of electrocardiograms}\nThe data used in this experiment is taken from the public MHEALTH dataset \\cite{banos2014mhealthdroid}. In the dataset, 10 individuals were asked to perform 12 physical activities as several sensors recorded varied motion data. The researchers also collected two-lead electrocardiogram (ECG) readings, which we take as dual-channel time series data. In addition to the 12 activities, there is a 13th \\textsc{NULL} class that represents ECG signals collected between each activity but which don't have labels themselves. To increase the number of individual time series, we partition each one in half. \n\n    \n    \n\nFor our experiments, the foreground data are all time series labelled as either \\textsc{Jogging}, \\textsc{Running}, \\textsc{Jumping}, or \\textsc{Cycling}, 20 time series each for a total of 80. These four, being the more cardio-intensive of the 12, had much more signal activity that would be needed to be sifted through, exactly the type of environment cMSSA is intended to handle. For background data, we take all 272 time series belonging to the \\textsc{NULL} class.\n\nTo evaluate the effectiveness of cMSSA over its non-contrastive counterpart, we run both cMSSA and MSSA with a variety of hyperparameter settings. For each fitted model, we transform the foreground data to both the PC and RC spaces. Once the transformations are had, we perform  spectral clustering into 4 clusters and compare the resulting clusters to the activity labels on the time series data, which were hitherto withheld from the algorithms. There are 3 hyperparameters: the window size $W \\in \\{8, 16, 32, 84, 18\\}$, the number of desired components $K \\in \\{1,2,4,6,8,10,12,14,16,18,20\\}$, and the contrastive parameter $\\alpha$. We set $K$ only if the value is less than or equal to $DW$ (where $D=2$ in this case). For $\\alpha$, we used our automatic routine to compute five key values to try for each setting of $W$ and $K$. For each run of the routine, a total of 300 candidate $\\alpha$s we considered, with the minimum and maximum $\\alpha$s being $10^{-3}$ and $10^{3}$, respectively. Of the five ultimately returned, one was zero, representing standard MSSA. Altogether, we run 530 experiments, 106 of which are standard MSSA, and the remaining cMSSA.\n\nThe spectral clustering requires an affinity matrix $S \\in \\RR^{N \\times N}$ which contains the similarities between any pair of time series, where $N$ is the number of times series we wish to cluster. Let $X^{(i)}$ and $X^{(j)}$ be two time series. Using the FastDTW metric \\cite{salvador2007toward} with a euclidean norm\\footnote{FastDTW is not a symmetric metric, so we take the minimum between the two orderings of the operands.}, we define the similarity $S_{ij}$ to be\n$\n\\frac{1}{\\textsc{FastDTW}(X^{(i)}, X^{(j)}) + 1}.\n$\nThe cluster evaluation uses the well-rounded BCubed metric \\cite{amigo2009comparison} to compute the precision, recall, and F1 harmonic mean for a particular cluster prediction. We also perform the evaluation in the model-free sense where we simply cluster the time series with no transformation as a basic baseline.\n\n\\begin{table}[htb]\n  \\caption{Best cMSSA and MSSA results in terms of maximum F1 score. Model-free clustering baseline also included. For the best MSSA and cMSSA models (with $\\alpha$ automatically selected via Algorithm 1), PC transform outperformed RC transform. Best result per metric (precision, recall, or F1) is bolded.}\n  \\label{tab:mhealth_best}\n  \\centering\n    \\begin{tabular}{l | c c c}\n    \\toprule\n    Model &\n    $W$ &\n    $K$ &\n    P \/ R \/ F1 \\\\\n    \\midrule\n    \n    Model-free & - & - & 50.49 \/ 48.82 \/ 49.54 \\\\\n    MSSA & 16 & 16 & 57.67 \/ 64.63 \/ 60.95 \\\\\n    cMSSA ($\\alpha = 12.41$) & 128 & 1 & \\textbf{65.44} \/ \\textbf{75.88} \/ \\textbf{70.27} \\\\\n    \n    \\bottomrule\n    \\end{tabular}\n\\end{table}\n\nTable~\\ref{tab:mhealth_best} reports the best representative contrastive and non-contrastive models, comparing both to the model-free baseline. We observe a number of things. First, both MSSA and cMSSA outperform the model-free baseline. Second, cMSSA has 9-10 point gains over cMSSA in each of precision, recall, and F1. Third, both find that using $A$ over $R$ as the transform yielded better results.\nFinally, of the $DW$ number of PCs available, MSSA gets its best performance using half (16 out of 32), while cMSSA only uses one PC out of the maximum of 256 available. This highlights an interesting efficiency of cMSSA. By filtering out unnecessary components, the remaining not only account for less signal variance, but provide diminishing returns with each subsequent component used.\n\n\\begin{figure}[htb]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{f1_vs_F1_A}\n    \\caption{Plot of paired F1 scores. Each point is for a particular setting of $W$ and $K$. The contrastive F1 score used is the maximum of the four runs (one per automatically selected $\\alpha$) for that setting of the hyperparams. $x=y$ line drawn as guidance. The points look at only those where the transform used is $A$.}\n    \\label{fig:f1_vs_F1_A}\n\\end{figure}\n\nFigure~\\ref{fig:f1_vs_F1_A} shows a more granular view of the general gains to be had from using cMSSA. For a particular setting of $W$ and $K$, we plot the F1 score for the non-contrastive case vs the contrastive case. Due to the four values of $\\alpha$s used in the contrastive case, we take the model that had the greatest F1.  Points below the diagonal line mean that the contrast was useful for a particular setting of the hyperparameters.\n\n\\begin{figure}[htb]\n\\begin{subfigure}{0.46\\columnwidth}\n\\begin{subfigure}{\\columnwidth}\n    \\includegraphics[width=\\columnwidth]{random_cycling_k16_no_contrast_best_compare}\n    \\label{fig:random_cycling_k16_no_contrast_best_compare}\n\\end{subfigure}\n\\begin{subfigure}{\\columnwidth}\n    \\includegraphics[width=\\columnwidth]{random_jumping_k16_no_contrast_best_compare}\n    \\label{fig:random_jumping_k16_no_contrast_best_compare}\n\\end{subfigure}\n\\begin{subfigure}{\\columnwidth}\n    \\includegraphics[width=\\columnwidth]{random_jogging_k16_no_contrast_best_compare}\n    \\label{fig:random_jogging_k16_no_contrast_best_compare}\n\\end{subfigure}\n\\begin{subfigure}{\\columnwidth}\n    \\includegraphics[width=\\columnwidth]{random_running_k16_no_contrast_best_compare}\n    \\label{fig:random_running_k16_no_contrast_best_compare}\n\\end{subfigure}\n\\end{subfigure}\n\\qquad\n\\begin{subfigure}{0.46\\columnwidth}\n\\begin{subfigure}{\\columnwidth}\n    \\includegraphics[width=\\columnwidth]{random_cycling_k1_contrast_best_compare}\n    \\label{fig:random_cycling_k1_contrast_best_compare}\n\\end{subfigure}\n\\begin{subfigure}{\\columnwidth}\n    \\includegraphics[width=\\columnwidth]{random_jumping_k1_contrast_best_compare}\n    \\label{fig:random_jumping_k1_contrast_best_compare}\n\\end{subfigure}\n\\begin{subfigure}{\\columnwidth}\n    \\includegraphics[width=\\columnwidth]{random_jogging_k1_contrast_best_compare}\n    \\label{fig:random_jogging_k1_contrast_best_compare}\n\\end{subfigure}\n\\begin{subfigure}{\\columnwidth}\n    \\includegraphics[width=\\columnwidth]{random_running_k1_contrast_best_compare}\n    \\label{fig:random_running_k1_contrast_best_compare}\n\\end{subfigure}\n\\end{subfigure}\n\n\\caption{Reconstructed time series after performing MSSA ($W=16$, $K=16$) and cMSSA ($W=128$, $K=1$, $\\alpha = 12.41$). Each row is the same random signal reconstructed using MSSA (left) and cMSSA (right), one row per activity. The two colors correspond to the dual recording channels. \n}\n\\label{fig:random_compare}\n\\end{figure}\n\nFinally, Figure~\\ref{fig:random_compare} shows a visual comparison of MSSA versus cMSSA, using their respective hyperparameters settings as shown in Table~\\ref{tab:mhealth_best}. Each row depicts how a random signal is processed with contrast on or off. We immediately see that cMSSA finds simpler signals than those found by MSSA. In the case of MSSA, the processed signals do not look substantially different from the originals. This is due to the fact that the high variance signals are shared across activities, so MSSA favors them during reconstruction. This is not the case with cMSSA, which identifies the differentiating signals that can disambiguate the activities.\n\n\\section{Conclusion}\n\nWe have developed cMSSA, a general tool for dimensionality reduction and signal decomposition of temporal data. By introducing a background dataset, we can efficiently identify sub-signals that are enhanced in one time series data relative to another. In an empirical experiment, we find that for virtually any setting of the hyperparameters, cMSSA is more effective at unsupervised clustering than MSSA, contingent on appropriate choices for the foreground and background data. It is worth emphasizing that cMSSA is an unsupervised learning technique. It does \\emph{not} aim to discriminate between time series signals, but rather discover structure and sub-signals within a given time series more effectively by using a second time series as background. This distinguishes it from various discriminant analysis techniques for time series that are based on spectral analysis \\cite{maharaj2014discriminant}, \\cite{krafty2016discriminant}.\n\nSome basic heuristics should be kept in mind when choosing to use cMSSA. First, the data ideally should exhibit periodic behavior, as MSSA (and by extension, cMSSA) is particularly well suited to finding oscillatory signals. Second, the data of interest $X$ and background $Y$ should not be identical, but should share common structured signal such that the contrast retains some information in the foreground. As an example, the ECG foreground data consisted of subjects performing very specific activities, whereas the background consisted of unlabelled ECG signals in which the participants performed no specific activity. We would expect a good amount of overlap in signal variance, but signals specific to the four activities would be under-represented in the background. Thus contrast is a plausible way to extract this signal.\n\nFinally, we note that the only hyper parameter of cMSSA is the contrast\nstrength, $\\alpha$. In our default algorithm, we developed an automatic\nsubroutine that selects\nthe most informative values of $\\alpha$. The experiments\nperformed used the automatically\ngenerated values. We believe that this default will be sufficient\nin many use cases of cMSSA, but the user may also\nset specific values for $\\alpha$ if more granular exploration\nis desired.\n\n\n\n\n\\vfill\\pagebreak\n\n\\nocite{*}\n\\bibliographystyle{IEEEbib}\n\\input{Template.bbl}\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \n\\IEEEPARstart{C}{ell}-free massive multiple-input multiple-output (MIMO) systems have been proposed to effectively alleviate intercell interference by coordinating a large number of distributed access points (APs), which are connected through fronthaul links to the central processing unit (CPU) \\cite{NAY+17, NAM+17, ZBM+20}. Intelligent reflecting surface (IRS), also known as reconfigurable intelligent surface (RIS), has been considered as one of the prospective multiple antenna technologies for beyond the fifth-generation (5G) networks \\cite{ZBM+20}. By adjusting phase shifts of the IRS elements, the propagation environment can be favorably manipulated.\n\nCell-free MIMO systems powered by IRSs have been recently introduced to further enhance the performance of cell-free MIMO systems at low and affordable cost and energy consumption by integrating multiple IRSs with the cell-free MIMO systems. The existing works, which studied the active and passive beamforming design, focused on the instantaneous performance metrics, i.e., instantaneous sum-rate \\cite{ZD21, ZDZ+21, HYX+21} or energy efficiency \\cite{ZDZ+21a, LND20X}. Theses works adopted alternating optimization algorithms, in which the active and passive beamformers are derived based on instantaneous channel state information (I-CSI). Thus, these algorithms would incur huge channel acquisition complexity and related pilot overhead since I-CSI is required for all AP-UE, AP-IRS, and IRS-UE links separately. Moreover, these algorithms would incur immense computational complexity and enormous fronthaul signaling overhead, since the active and passive beamformers need to be computed many times and transferred over fronthaul links for each coherence time. It is also challenging to control the IRSs in real-time, which requires stringent time synchronization \\cite{ZBM+20}. These disadvantages can be alleviated by designing a passive beamformer based on long-term channel statistics \\cite{HTJ+19, ZWZ+21}.\n\nTo the best of our knowledge, this paper is the first attempt for the cell-free MIMO systems powered by IRSs to consider the max-min achievable rate, where the achievable rate is a lower bound of the average rate. This metric aims to provide uniform performance and thus is widely used in the cell-free MIMO systems \\cite{NAY+17, NAM+17}. Moreover, we propose a novel non-iterative two-timescale algorithm to obtain 1) short-term active precoders for the APs that depend on I-CSI, 2) long-term power allocation for the APs using statistical CSI (S-CSI), and 3) long-term passive beamformers for the IRSs using S-CSI.\n\nThe rest of this paper is organized as follows. In Section~\\ref{sec:model}, we present the system model and formulate the max-min achievable rate optimization problem. In Section \\ref{sec:two-step}, we propose a two-step algorithm to solve the problem. Simulation results are provided to evaluate the proposed algorithm in Section~\\ref{sec:result}. Finally, we conclude the paper in Section \\ref{sec:conclusion}.\n\n\\textit{Notation:} Vectors and matrices are denoted by lower-case and upper-case boldface letters. The notations $\\otimes$ and $\\odot$ denote the Kronecker product and Hadamard product. $|\\cdot|$ and $\\angle(\\cdot)$ return the magnitude and angle of a complex argument. $\\mathbb{E}\\{\\cdot\\}$ represents the expectation operator. For a square matrix $\\bS$, $\\trace(\\bS)$ denotes the trace operation, and $\\bS \\succeq 0$ means that $\\bS$ is positive semidefinite. A random variable $x \\sim \\mathcal{CN}(m, \\sigma^2)$ is circularly symmetric complex Gaussian (CSCG) distributed with mean~$m$ and variance~$\\sigma^2$. \n\t\n\\section{System Model} \\label{sec:model}\n\nWe consider a downlink cell-free MIMO system powered by IRSs as illustrated in Fig. \\ref{fig:model}, where $L$ APs and $R$ IRSs are distributed to cooperatively serve $K$ single-antenna user equipments (UEs). All APs and IRSs are connected by wired or wireless fronthaul links to the CPU, which coordinates them. Each AP is equipped with $M$ antennas, and each IRS is comprised of $N$ passive reflecting elements. \n\n\\begin{figure}[t!] \n\t\\centering\n\t\\includegraphics[width=0.7 \\columnwidth]{fig1_model.eps}\n\t\\caption{A downlink cell-free MIMO system powered by IRSs.}  \\label{fig:model}\n\\end{figure}\n\nWe assume that the data symbols for $K$ UEs $\\bs \\in \\mathbb{C}^{K \\times 1}$ are transmitted from all APs \\cite{NAY+17, NAM+17}. The transmit signal from AP~$l$ is given by \n\\begin{align}\n\\bx_l = \\textstyle\\sum_{k=1}^K \\bw_{l,k} s_k, \n\\end{align}\nwhere $\\bw_{l,k} \\in \\mathbb{C}^{M \\times 1}$ and $s_k$ are the active beamforming vector and data symbol for UE $k$. Assuming $\\mathbb{E}\\{|s_k|^2\\} = 1$ for all~$k$, the transmit power constraint of  AP $l$ can be written as\n\\begin{align}\n\\textstyle \\sum_{k=1}^K \\mathbb{E}\\{ \\norm{\\bw_{l,k}}^2 \\} \\le \\bar{P}_l,\n\\end{align}\nwhere $\\bar{P}_l$ denotes the maximum transmit power of AP~$l$.\n\nThe channel between an AP and a UE consists of the direct (AP-UE) channel and the $R$ reflection (AP-IRS-UE) channels.\\footnote{Note that the signals reflected by the IRSs twice or more are weak enough to be neglected due to the harsh propagation loss of multiple hops \\cite{ZD21, ZDZ+21, HYX+21, ZDZ+21a, LND20X}.} Then, the overall channel from AP~$l$ to UE~$k$ can be expressed as\n\\begin{align}\n\\bh_{l,k}^H = \\bd_{l,k}^H + \\textstyle \\sum_{r=1}^R \\bv_{r,k}^H \\boldsymbol{\\Theta}_r \\bG_{l,r}, \\label{eq:overallCh}\n\\end{align}\nwhere $\\bd_{l,k}^H \\in \\mathbb{C}^{1 \\times M}$, $\\bG_{l,r} \\in \\mathbb{C}^{N \\times M}$, and $\\bv_{r,k}^H \\in \\mathbb{C}^{1 \\times N}$ denote the channel from AP~$l$ to UE~$k$, from AP~$l$ to IRS~$r$, and from IRS~$r$ to UE~$k$, respectively. The reflection coefficient matrix of IRS~$r$ is denoted by $\\boldsymbol{\\Theta}_r = \\diag(\\theta_{r,1}, \\cdots, \\theta_{r,N}) \\in \\mathbb{C}^{N \\times N}$, where  $|\\theta_{r,n}| = 1, \\forall r,n$ represents the unit-modulus constraint on the IRSs elements.\n\nWe assume the Rician fading channel model for all channels \\cite{ZD21, ZDZ+21, ZDZ+21a, WZ19}. Specifically, the channel between AP~$l$ and UE~$k$ is given by\n\\begin{align}\n\\bd_{l,k} &= \\textstyle \\sqrt{\\xi_{l,k}^\\mathrm{d}} \\sqrt{\\frac{\\beta_\\mathrm{d}}{1+\\beta_\\mathrm{d}}}\\bar{\\bd}_{l,k}' + \\sqrt{\\xi_{l,k}^\\mathrm{d}} \\sqrt{\\frac{1}{1+\\beta_\\mathrm{d}}}\\tilde{\\bd}_{l,k}' \\notag \\\\\n&= \\bar{\\bd}_{l,k} + \\tilde{\\bd}_{l,k}, \\label{eq:ricianCh}\n\\end{align}\nwhere $\\bar{\\bd}_{l,k}'$, $\\tilde{\\bd}_{l,k}'$, and $\\beta_\\mathrm{d}$ denote the line-of-sight (LoS) component, non-line-of-sight (NLoS) component, and Rician K-factor of the channel~$\\bd_{l,k}$, respectively. The channel~$\\bd_{l,k}$ includes the distance-dependent path loss $\\xi_{l,k}^\\mathrm{d}$. The AP-IRS and IRS-UE channels follow the same model as in \\eqref{eq:ricianCh} with proper notation changes. It is assumed that the NLoS components of all AP-UE, AP-IRS, and IRS-UE channels are independent each other, and each NLoS component has independent and identically distributed (i.i.d.) $\\mathcal{CN}(0, 1)$ entries.\n\nThe received signal at UE $k$ can be written as\n\\begin{align}\ny_k &= \\textstyle \\sum_{l=1}^L \\bh_{l,k}^H \\bx_l + z_k \\notag \\\\\n\t&= \\bh_{k}^H \\bw_{k} s_k + \\textstyle \\sum_{k' \\neq k}^K \\bh_{k}^H \\bw_{k'} s_{k'} + z_k,\n\\end{align}\nwhere $\\bh_k = [\\bh_{1,k}^T, \\cdots, \\bh_{L,k}^T]^T \\in \\mathbb{C}^{LM \\times 1}$, $\\bw_k = [\\bw_{1,k}^T, \\cdots, \\bw_{L,k}^T]^T \\in \\mathbb{C}^{LM \\times 1}$, and $z_k \\sim \\mathcal{CN}(0, \\sigma^2)$ is the i.i.d. complex additive white Gaussian noise (AWGN). To analyze the theoretic performance gain with the IRSs, we assume that the perfect I-CSI of direct and reflection channels is available at the CPU. We also assume that the UE $k$ has the knowledge of the average of effective channel $\\mathbb{E}\\{ \\bh_{k}^H \\bw_{k}\\}$ and adopt the \\emph{hardening bound}, which is widely used in the massive MIMO literature \\cite{BS20a}. Then, the achievable rate of UE $k$ is $\\log_2(1+\\mathsf{SINR}_k)$, where the effective signal-to-interference-plus-noise ratio (SINR) of UE~$k$ is given by\n\\begin{align}\n\\mathsf{SINR}_k = \\frac{ | \\mathbb{E}\\{ \\bh_{k}^H \\bw_{k}\\} |^2 }\n{ \\textstyle \\sum_{k'=1}^K \\mathbb{E}\\{ |\\bh_{k}^H \\bw_{k'}|^2 \\} - | \\mathbb{E}\\{ \\bh_{k}^H \\bw_{k}\\} |^2 + \\sigma^2}. \\label{eq:SINR}\n\\end{align}\n\nIn this paper, we aim to maximize the minimum achievable rate by jointly designing active and passive beamformers subject to the per AP transmit power constraint and unit-modulus constraint on the IRSs elements. This optimization problem can be formulated as\n\\begin{alignat}{3}\n& \\max_{ \\{\\bw_{k}\\}, \\{\\boldsymbol{\\Theta}_r\\} } ~ && \\min_{k} \\log_2(1+\\mathsf{SINR}_k) \\label{eq:prbA} \\\\\n&~ \\quad~~ \\mathrm{s.t.} &&~ C_1: \\textstyle \\sum_{k=1}^K \\mathbb{E}\\{ \\norm{\\bw_{l,k}}^2 \\} \\le \\bar{P}_l, && ~~\\forall l, \\notag \\\\\n&&&~ C_2: |\\theta_{r,n}| = 1, && ~~\\forall r, n, \\notag\n\\end{alignat}\nwhere $\\{\\bw_{k}\\}$ and $\\{\\boldsymbol{\\Theta}_r\\}$ represent the active and passive beamformers, respectively. The joint optimization of the problem \\eqref{eq:prbA} is very challenging since the active and passive beamformers are tightly coupled.\n\n\n\n\\section{Proposed Two-Step Algorithm} \\label{sec:two-step}\nIn this section, we propose a suboptimal two-step algorithm to solve the problem \\eqref{eq:prbA}. We first design an active beamforming technique, which consists of active precoding and power allocation. Then, we design a passive beamforming technique based on S-CSI. Finally, we summarize the proposed algorithm.\n\n\n\n\\subsection{Active Beamforming Design} \nWe decompose an active beamformer into a short-term active precoder and long-term power allocation to reduce computational complexity and fronthaul signaling overhead. We considered a zero-forcing (ZF) precoder since it shows better max-min rate performance than conjugate beamforming precoder in cell-free MIMO systems \\cite{NAM+17}. Moreover, the ZF precoder eliminates inter-user interference, and thus it makes the optimal power allocation simple.\n\nWe can express the received signal $\\by$ for $K$~UEs as\n\\begin{align}\n\\by = \\bH^H\\bW\\bs + \\bz,\n\\end{align}\nwhere $\\by = [y_1, \\cdots, y_K]^T \\in \\mathbb{C}^{K \\times 1}$, $\\bH = [\\bh_1, \\cdots, \\bh_K] \\in \\mathbb{C}^{LM \\times K}$, $\\bW = [\\bw_1, \\cdots, \\bw_K] \\in \\mathbb{C}^{LM \\times K}$, and $\\bz = [z_1, \\cdots, z_K]^T \\in \\mathbb{C}^{K \\times 1}$.\nThe active beamformer can be set as $\\bW = \\widetilde{\\bW} \\bP^{\\frac{1}{2}}$ with the ZF precoder $\\widetilde{\\bW} = \\bH \\left( \\bH^H\\bH \\right)^{-1}$, where the related condition $LM \\ge K$ can be easily fulfilled in the cell-free MIMO systems \\cite{NAM+17}. The long-term power allocation $\\bP = \\diag(p_1, \\cdots, p_K) \\in \\mathbb{C}^{K \\times K}$ is applied to all APs.\n\n\nWith the ZF precoder and long-term power allocation, the effective SINR \\eqref{eq:SINR} is simply reduced to $\\frac{p_k}{\\sigma^2}$. With a fixed passive beamformer, the problem \\eqref{eq:prbA} boils down to the long-term power allocation problem as\n\\begin{alignat}{3}\n& \\cP_1: && \\max_{ \\bP } ~ && \\min_{k} \\frac{p_k}{\\sigma^2} \\label{eq:prb1} \\\\\n&&&~~ \\mathrm{s.t.} &&~ C_1: \\textstyle \\sum_{k=1}^K p_k \\mathbb{E}\\{ \\norm{\\tilde{\\bw}_{l,k}}^2 \\} \\le \\bar{P}_l, ~\\forall l, \\notag\n\\end{alignat} \nwhere $\\widetilde{\\bW} = [\\tilde{\\bw}_1, \\cdots, \\tilde{\\bw}_K] \\in \\mathbb{C}^{LM \\times K}$ and $\\tilde{\\bw}_k = [\\tilde{\\bw}_{l,k}^T, \\cdots, \\tilde{\\bw}_{L,k}^T]^T \\in \\mathbb{C}^{LM \\times 1}$.\n\n\nThe objective function in the problem \\eqref{eq:prb1} forces the power allocation for all UEs to be the same\\footnote{Note that the instantaneous transmit power for UE~$k$ at AP~$l$ is equal to $p_k\\norm{\\tilde{\\bw}_{l,k}}^2$, and thus the actual transmit power is different per UE.}, i.e., $p_1 = \\cdots = p_K = p^{\\mathrm{opt}}$. Under the typical condition that $\\bar{P}_1 = \\cdots = \\bar{P}_L = \\bar{P}$ with a fixed $\\bar{P}$, the optimal power allocation $p^{\\mathrm{opt}}$ is determined by the AP that consumes the largest power for the active precoder, i.e., $\\max_l \\textstyle \\sum_{k=1}^K \\mathbb{E}\\{ \\norm{\\tilde{\\bw}_{l,k}}^2 \\}$. As the largest power for the active precoder reduces, the optimal power allocation increases, and thus the minimum achievable rate improves accordingly.\n\n\n\\subsection{Passive Beamforming Design} \nBased on the proposed active beamforming design, we find that the passive beamformers are irrelevant to the objective function and only related to the transmit power constraint and unit-modulus constraint in the problem (7). Therefore, we can design a long-term passive beamformer to minimize the largest power for the active precoder. The corresponding optimization problem can be formulated as\n\\begin{alignat}{2}\n& \\min_{ \\boldsymbol{\\theta} } ~ && \\max_{l} \\textstyle \\sum_{k=1}^K\\mathbb{E}\\{ \\norm{\\tilde{\\bw}_{l,k}}^2 \\} \\\\\n&~~ \\mathrm{s.t.} &&~ C_2: |\\theta_{r,n}| = 1, ~\\forall r, n, \\notag\n\\end{alignat}\nwhere $\\boldsymbol{\\theta} = \\boldsymbol{\\Theta}^H\\bone_{RN} \\in \\mathbb{C}^{RN \\times 1}$ and $\\boldsymbol{\\Theta} = \\diag(\\boldsymbol{\\Theta}_1, \\cdots, \\boldsymbol{\\Theta}_R) \\in \\mathbb{C}^{RN \\times RN}$.\n\nTo the best of our knowledge, there is no closed-form expression of $\\mathbb{E}\\{ \\norm{\\tilde{\\bw}_{l,k}}^2 \\}$ in terms of the long-term passive beamformer~$\\boldsymbol{\\theta}$. It is worth noting, however, that the transmit power reduces as the channel gain increases \\cite{WZ19}. Considering this fact, we propose a suboptimal optimization problem that maximizes the minimum average channel gain by passive beamforming at the IRSs as \n\\begin{alignat}{2}\n& \\max_{ \\boldsymbol{\\theta} } ~ && \\min_{k} \\textstyle \\sum_{l=1}^L\\mathbb{E}\\{ \\norm{\\bh_{l,k}}^2 \\} \\label{eq:prbB} \\\\\n&~~ \\mathrm{s.t.} &&~ C_2: |\\theta_{r,n}| = 1, ~\\forall r, n. \\notag\n\\end{alignat}\t\t\n\nBy exploiting S-CSI, the average channel gain of UE~$k$ can be expressed as an explicit function of~$\\boldsymbol{\\theta}$ that is given as\n\\begin{align}\n\\textstyle \\sum_{l=1}^L\\mathbb{E}\\{ \\norm{\\bh_{l,k}}^2 \\}\n= \\boldsymbol{\\theta}^H \\bA_k \\boldsymbol{\\theta} + \\boldsymbol{\\theta}^H \\bb_k + \\bb_k^H \\boldsymbol{\\theta} + c_k, \\label{eq:avChGain}\n\\end{align}\nwhere $\\bA_k$, $\\bb_k$, and $c_k$ are defined in Appendix A. Since $\\bA_k~\\succeq~0$, the average channel gain is a convex function of~$\\boldsymbol{\\theta}$. However, the problem \\eqref{eq:prbB} is a non-convex optimization problem since the objective function is not a concave function of~$\\boldsymbol{\\theta}$, and the unit-modulus constraint is not a convex set.\n\nWe apply semidefinite relaxation (SDR) to convert the non-convex problem \\eqref{eq:prbB} to a convex problem \\cite{NAK+20}. At first, by introducing an auxiliary variable~$q$, the average channel gain of UE~$k$ can be rewritten as \n\\begin{align}\n\\bar{\\boldsymbol{\\theta}}^H \\boldsymbol{\\Psi}_k \\bar{\\boldsymbol{\\theta}} + c_k,\n\\end{align}\t\nwhere $\\bar{\\boldsymbol{\\theta}} = \\begin{bmatrix} \\boldsymbol{\\theta} \\\\ q \\end{bmatrix}$ and $\\boldsymbol{\\Psi}_k = \\begin{bmatrix} \\bA_k & \\bb_k \\\\ \\bb_k^H & 0 \\end{bmatrix} \\succeq 0$. Note that $\\bar{\\boldsymbol{\\theta}}^H \\boldsymbol{\\Psi}_k \\bar{\\boldsymbol{\\theta}} = \\trace(\\boldsymbol{\\Psi}_k \\bar{\\boldsymbol{\\theta}} \\bar{\\boldsymbol{\\theta}}^H)$. We define $\\bar{\\boldsymbol{\\Theta}} = \\bar{\\boldsymbol{\\theta}} \\bar{\\boldsymbol{\\theta}}^H$, where $\\bar{\\boldsymbol{\\Theta}} \\succeq 0$ and $\\mathrm{rank}(\\bar{\\boldsymbol{\\Theta}})=1$. By relaxing the rank-one constraint on $\\bar{\\boldsymbol{\\Theta}}$, which is non-convex, the problem \\eqref{eq:prbB} can be reformulated as \n\\begin{alignat}{3}\n& \\cP_2: && \\max_{ \\bar{\\boldsymbol{\\Theta}} } ~ && \\min_{k} \\trace(\\boldsymbol{\\Psi}_k \\bar{\\boldsymbol{\\Theta}}) + c_k  \\label{eq:prb2} \\\\\n&&&~~ \\mathrm{s.t.} &&~ [\\bar{\\boldsymbol{\\Theta}}]_{i,i} = 1 , i = 1, \\ldots, RN+1, \\notag \\\\\n&&&&&~ \\bar{\\boldsymbol{\\Theta}} \\succeq 0. \\notag\n\\end{alignat}\n\nSince the problem \\eqref{eq:prb2} is a convex semidefinite program (SDP), it can be efficiently solved by existing convex optimization solvers. If the optimal $\\bar{\\boldsymbol{\\Theta}}^{\\mathrm{opt}}$ is a rank-one matrix, then the optimal $\\bar{\\boldsymbol{\\theta}}^{\\mathrm{opt}}$ is derived by taking the eigenvector corresponding to the maximum eigenvalue of $\\bar{\\boldsymbol{\\Theta}}^{\\mathrm{opt}}$. Otherwise, Gaussian randomization is applied to find $\\bar{\\boldsymbol{\\theta}}^{\\mathrm{opt}}$ \\cite{NAK+20}. Finally, the optimal solution of the problem \\eqref{eq:prbB} is recovered by taking ${\\boldsymbol{\\theta}}^{\\mathrm{opt}} = \\exp\\left( j\\angle \\left( \\left[ \\frac{\\bar{\\boldsymbol{\\theta}}^{\\mathrm{opt}}}{\\bar{\\theta}_{RN+1}^{\\mathrm{opt}}} \\right]_{(1:RN)} \\right) \\right)$, where $[\\bx]_{(1:RN)}$ denotes the vector that contains the first $RN$ entries in $\\bx$, and $\\bar{\\theta}_{RN+1}^{\\mathrm{opt}}$ is the last element of~$\\bar{\\boldsymbol{\\theta}}^{\\mathrm{opt}}$.\n\n\\subsection{Overall Algorithm Description} \n\\begin{algorithm}[t]\n\t\\caption{Proposed Two-Step Algorithm}\n\t\\hspace{1mm} \\textbf{Input}: S-CSI $\\{ \\bar{\\bd}_{l,k}, \\bar{\\bG}_l, \\bar{\\bv}_{k} \\}$ \\\\\n\t\\hspace*{1mm} \\textbf{Step 1}: Passive beamforming design\n\t\\begin{itemize} \t\t\t\n\t\t\\item Solve the problem $\\cP_2$ to obtain the optimal long-term passive beamformer ${\\boldsymbol{\\theta}}^{\\mathrm{opt}}$.\n\t\\end{itemize}\n\t\\hspace{1mm} \\textbf{Step 2}: Active beamforming design\n\t\\begin{itemize} \n\t\t\\item Apply the ZF precoder to instantaneous channels with the given ${\\boldsymbol{\\theta}}^{\\mathrm{opt}}$.\n\t\t\\item Solve the problem $\\cP_1$ to obtain the optimal long-term power allocation $\\bP^{\\mathrm{opt}}$.\n\t\\end{itemize}\n\\end{algorithm} \nIn the previous subsections, we first explained the active beamforming design and then described the passive beamforming design. The proposed algorithm, however, actually operates as summarized in Algorithm 1. First, the optimal long-term passive beamformer ${\\boldsymbol{\\theta}}^{\\mathrm{opt}}$ is obtained by solving $\\cP_2$ based on the S-CSI $\\{ \\bar{\\bd}_{l,k}, \\bar{\\bG}_l, \\bar{\\bv}_{k} \\}$. Then, the ZF precoder is applied to instantaneous channels with the given~${\\boldsymbol{\\theta}}^{\\mathrm{opt}}$, and the optimal long-term power allocation $\\bP^{\\mathrm{opt}}$ is derived by solving $\\cP_1$.\n\n\n\n\\section{Simulation Results} \\label{sec:result}\nIn this section, we provide simulation results to validate the minimum achievable rate performance of the proposed two-step algorithm. We consider a hotspot deployment scenario, where the UEs are placed in a hotspot, the APs are deployed a little far from the hotspot, and the IRSs are installed on a circle surrounding the hotspot in order to improve the rate performance \\cite{ZD21, ZDZ+21a}. This scenario is illustrated in Fig. \\ref{fig:hotspot}, where $L=4$ APs are located at $(0,0)$, $(D,0)$, $(D,D)$, and $(0,D)$, respectively. Up to $R=8$ IRSs are placed on a circle centered at $(d,d)$ with radius $r$, and $K=4$ UEs are uniformly distributed within the circle. We simulate three cases of $d = \\{40, 60, 120 \\}$~m with $r=30$~m and $D=300$~m.\n\n\t\n\n\n\\begin{figure}[h] \n\t\\centering\n\t\\includegraphics[width=0.5 \\columnwidth]{fig2_hotspot.eps}\n\t\\caption{Hotspot deployment scenario.} \\label{fig:hotspot}\n\\end{figure}\n\nAll APs are equipped with uniform linear arrays (ULAs) at a height of $10$~m with $M = \\{4, 8, 16\\}$ transmit antennas. Uniform planar array (UPA) is installed at a height of $5$~m for each IRS with $N = \\{8, 16, 32, 64, 128\\}$ elements. All UEs have single antenna, which is placed at a height of $1.5$~m \\cite{M.2412}. We evaluate three cases of $R = \\{2, 4, 8\\}$, where when $R=2$, only the first and fifth IRSs are present, and when $R=4$, the odd-numbered IRSs are present. It is assumed that all IRSs are deployed on building facades \\cite{ZBM+20} and look towards UEs. We consider an AP-IRS blockage model that when a signal from an AP arrives to the back of an IRS, then this signal is not reflected to the UEs, e.g., the signals from the first AP to the first, second, and eighth IRSs are blocked.\n\nThe distance-dependent path loss of all channels is modeled as $\\xi(d_{\\mathrm{link}}) = \\xi_0d_{\\mathrm{link}}^{-\\alpha_\\mathrm{X}}$, where $\\xi_0$ is the path loss at the reference distance $1$~m, $\\alpha_\\mathrm{X}$ denotes the path loss exponent of the channel~$\\bX$, and $d_{\\mathrm{link}}$ represents three-dimensional distance of a channel link considering vertical difference among the APs, IRSs, and UEs. We set $\\xi_0 = -30$~dB, $\\alpha_\\mathrm{d} = 3.4$, $\\alpha_\\mathrm{v} = \\alpha_\\mathrm{G} = 2.2$, $\\beta_\\mathrm{d} = -5$~dB, and $\\beta_\\mathrm{v} = \\beta_\\mathrm{G} = 5$~dB considering that the AP-UE channels would suffer from severer attenuation than the AP-IRS and IRS-UE channels \\cite{ZWZ+21}. Other system parameters are set as follows: $\\bar{P}_l = \\{20, 30, 40\\}$~dBm for all~$l$, $\\sigma^2 = -97$~dBm assuming $10$~MHz of system bandwidth, and $7$~dB of noise figure \\cite{M.2412}. We simulate $1,000$ uniform UE drops and generate $1,000$ independent instantaneous channels for each UE drop. \n\nWe consider three benchmark schemes. One is \\emph{No-IRS} and another is \\emph{Random Passive Beamforming}: the long-term passive beamformers at the IRSs are randomly selected. The active beamforming for both benchmarks is the same as that of the proposed algorithm. To the best of our knowledge, there are no other schemes that can be directly applied to the max-min achievable rate problem in the cell-free MIMO systems powered by IRSs. Instead, we compare with the third benchmark scheme of \\emph{Sum-Rate-Max} that maximizes instantaneous sum-rate by utilizing an alternating optimization algorithm to derive the active and passive beamformers [4].\n\nFig. \\ref{fig:cdf} depicts the empirical cumulative distribution functions (CDFs) of the minimum achievable rate by varying the maximum transmit power $\\bar{P}_l$. The proposed scheme provides a significant gain over No-IRS and the random passive beamforming for all values of $\\bar{P}_l$. Specifically for $\\bar{P}_l=20$~dBm, the median rate gains of the proposed scheme over No-IRS are equal to $3.4\\%$, $7.1\\%$, and $12.7\\%$ with $N=32$, $64$, and $128$, respectively. By doubling the number of IRS elements, the performance gain is almost doubled. Furthermore, the proposed scheme achieves comparable performance to the Sum-Rate-Max, which requires much higher computational complexity and signaling overhead than the proposed scheme.\n\n\\begin{figure}[t] \n\t\\centering\n\t\\includegraphics[width=0.95 \\columnwidth]{fig3_cdf.eps}\n\t\\caption{CDFs of the minimum achievable rate by varying the maximum transmit power $\\bar{P}_l$ with $d=40$~m, $M=8$, and $R=4$.} \\label{fig:cdf}\n\\end{figure}\n\nIn Fig.~\\ref{fig:rate}, we plot the minimum achievable rate versus the number of IRS elements~$N$ by varying the center of hotspot~$d$, the number of AP transmit antennas~$M$, and the number of IRSs $R$ with $\\bar{P}_l=20$~dBm. For all cases, the minimum achievable rate of the proposed scheme significantly improves with~$N$ and outperforms that of both benchmark schemes.\nFig.~\\ref{fig:rate}(a) shows that the performance of all schemes decreases as $d$ increases, i.e., the UEs move towards the center of service area. This is attributed to the fact that the received signal power from the first AP, which is the dominant AP to the UEs, decreases. However, the performance gain of the proposed scheme over No-IRS increases with $d$, i.e., when $N = 128$, the gains are equal to $12.5\\%$, $12.9\\%$, and $16.1\\%$ for $d=40$~m, $60$~m, and $120$~m, respectively.\nIn Fig.~\\ref{fig:rate}(b), it is seen that the smaller $M$, the lower the performance of all schemes.\nThe proposed scheme, however, provides higher performance gain over No-IRS as $M$ decreases, i.e., when $N = 128$, the gains are equal to $10.4\\%$, $12.5\\%$, and $12.8\\%$ for $M=16$, $8$, and $4$, respectively. \nFig.~\\ref{fig:rate}(c) shows that the performance of the proposed scheme increases with $R$ as expected. \nWhen the total number of IRS elements $RN$ is the same, similar performance is observed, showing that the proposed scheme is robust to IRS deployment scenarios. \n\n\\begin{figure*}[t] \n\t\\centering\n\t\\includegraphics[width=0.9 \\textwidth]{fig4_rate.eps}\n\t\\caption{The minimum achievable rate vs. the number of IRS elements $N$: (a) Varying $d$ with $R=4$, $M=8$; (b) Varying $M$ with $d=40$~m, $R=4$; (c) Varying $R$ with $d=40$~m, $M=8$.} \n\t\\label{fig:rate}\n\\end{figure*}\t\n\n\n\\section{Conclusion} \\label{sec:conclusion}\nIn this paper, we considered a joint beamforming framework in a cell-free MIMO system powered by IRSs. We formulated a maximization of minimum achievable rate problem and proposed a novel non-iterative two-timescale algorithm that derives the long-term passive beamformers and power allocation and short-term active precoders by exploiting S-CSI. Simulation results revealed that the proposed scheme can significantly improve the minimum achievable rate of the cell-free MIMO systems powered by IRSs compared to the benchmark schemes.\n\n\n\n\\begin{appendices}\n\\section{Derivation of \\eqref{eq:avChGain}}\nThe overall channel from AP~$l$ to UE~$k$ in \\eqref{eq:overallCh} can be rewritten as \n\\begin{align}\n\\bh_{l,k}^H = \\boldsymbol{\\theta}^H \\bV_k^H \\bG_l + \\bd_{l,k}^H, \n\\end{align}\nwhere, $\\bG_l = [\\bG_{l,1}^T, \\cdots, \\bG_{l,R}^T]^T \\in \\mathbb{C}^{RN \\times M}$, $\\bV_k^H = \\diag(\\bv_k^H) \\in \\mathbb{C}^{RN \\times RN}$, and $\\bv_k^H = [\\bv_{1,k}^H, \\cdots, \\bv_{R,k}^H] \\in \\mathbb{C}^{1 \\times RN}$. By decomposing the Rician fading channels into the LoS and NLoS components in \\eqref{eq:ricianCh}, the average channel gain from AP~$l$ to UE~$k$ can be expressed as\n\\begin{align}\n&\\mathbb{E} \\{ \\norm{\\bh_{l,k}}^2 \\} \\notag \\\\\n&~~= \\mathbb{E} \\left\\{ \\left\\| (\\bar{\\bG}_l^H + \\tilde{\\bG}_l^H)(\\bar{\\bV}_k + \\tilde{\\bV}_k) \\boldsymbol{\\theta} \n\t+ (\\bar{\\bd}_{l,k} + \\tilde{\\bd}_{l,k}) \\right\\|^2 \\right\\} \\notag \\\\\n&~~= \\boldsymbol{\\theta}^H \\bA_{l,k} \\boldsymbol{\\theta} + \\boldsymbol{\\theta}^H \\bb_{l,k} + \\bb^H_{l,k}\\boldsymbol{\\theta} + c_{l,k},\n\\end{align}\nwhere $\\bb_{l,k} = \\bar{\\bV}_k^H \\bar{\\bG}_l \\bar{\\bd}_{l,k}$, $c_{l,k} = \\norm{\\bar\\bd_{l,k}}^2 + \\frac{M\\xi_{l,k}^\\mathrm{d}}{1+\\beta_\\mathrm{d}}$, and $\\bA_{l,k}$ is defined below in \\eqref{eq:A_lk}, where $\\boldsymbol{\\Xi}_k^\\mathrm{v} = \\diag{(\\xi_{1,k}^\\mathrm{v}, \\cdots, \\xi_{R,k}^\\mathrm{v}) \\in \\mathbb{C}^{R \\times R}}$, $\\boldsymbol{\\Xi}_l^\\mathrm{G} = \\diag{(\\xi_{l,1}^\\mathrm{G}, \\cdots, \\xi_{l,R}^\\mathrm{G}) \\in \\mathbb{C}^{R \\times R}}$, and $\\bA_{l,k} \\succeq 0$. All the variables $\\bA_{l,k}$, $\\bb_{l,k}$, and $c_{l,k}$ are expressed in terms of the S-CSI $\\{ \\bar{\\bd}_{l,k}, \\bar{\\bG}_l, \\bar{\\bv}_{k} \\}$ and path loss $\\{ \\xi_{l,k}^\\mathrm{d}, \\boldsymbol{\\Xi}_l^\\mathrm{G}, \\boldsymbol{\\Xi}_k^\\mathrm{v} \\}$ of all channel links \\cite{HTJ+19, ZWZ+21}. The details are omitted here due to the space limitation. Finally, the average channel gain of UE~$k$ can be written as\t\n\\begin{align}\n\\textstyle \\sum_{l=1}^L\\mathbb{E}\\{ \\norm{\\bh_{l,k}}^2 \\}\n= \\boldsymbol{\\theta}^H \\bA_k \\boldsymbol{\\theta} + \\boldsymbol{\\theta}^H \\bb_k + \\bb^H_k \\boldsymbol{\\theta} + c_k,\n\\end{align}\t\nwhere $\\bA_k = \\textstyle \\sum_{l=1}^L \\bA_{l,k}$, $\\bb_k = \\textstyle \\sum_{l=1}^L \\bb_{l,k}$, and $c_k = \\textstyle \\sum_{l=1}^L c_{l,k}$.\n\n\\begin{figure*} [b]\t\n\t\\hrule \\vspace{5mm}\t\n\t\\begin{align}\n\t\t\\bA_{l,k} = \\bar{\\bV}_k^H \\left( \\bar{\\bG}_l \\bar{\\bG}_l^H + \\frac{M}{1+\\beta_\\mathrm{G}} \\boldsymbol{\\Xi}_l^\\mathrm{G} \\otimes \\bI_N \\right) \\bar{\\bV}_k\n\t\t+ \\left( \\frac{1}{1+\\beta_\\mathrm{v}} \\boldsymbol{\\Xi}_k^\\mathrm{v} \\otimes \\bI_N \\right) \\odot \\left( \\bar{\\bG}_l \\bar{\\bG}_l^H \\right) \n\t\t+ \\frac{1}{1+\\beta_\\mathrm{v}}\\frac{M}{1+\\beta_\\mathrm{G}} \\boldsymbol{\\Xi}_k^\\mathrm{v} \\boldsymbol{\\Xi}_l^\\mathrm{G} \\otimes \\bI_N \\label{eq:A_lk}\n\t\\end{align}\n\\end{figure*}\n\n\\end{appendices}\n\t\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nCondensed matter provides a platform to realize many physical objects in\nother subjects such as Majorana and Dirac Weyl fermions which are proposed\nin particle physics but not be discovered in nature \\cite{XWAN, LLU, SMH,\nHWENG, SYXU, BQLV, LLU2, VMO, SNA}. Another example is the Dirac monopole,\nwhich is a point source of a magnetic field proposed by Dirac \\cite{PAM}. It\nhas a quantum analogy in quantum physics, where the Berry curvature of\nenergy band acts as the magnetic field generated by degeneracy points as\nDirac monopoles \\cite{DXIAO}.\\ As the extension of degeneracy points, nodal\nloops as closed $1$-dimensional ($1$D) manifolds in $3$D momentum space can\nbe classified as nodal rings \\cite{AAB}, nodal chains \\cite{TB}, nodal links\n\\cite{ZYAN}, and nodal knots \\cite{RBI}. It has been extensively studied\nboth theoretically\\cite{XQSUN, SNIE, JAHN, CFANG1, TK, JYL, CFANG2, PYC,\nWCHEN, MEZAWA, YZHOU, ZYANG, MXIAO} and experimentally \\cite{QXU, RYU, QYAN,\nGCHANG, XFENG}. In the recent work, it has turned out that the relation\nbetween degeneracy lines and the corresponding polarization field in the\nparameter space is topologically isomorphic to Biot-Savart law in\nelectromagnetism \\cite{RWANG}.\n\n\\begin{figure}[tbp]\n\\includegraphics[ bb=112 370 500 623, width=0.45\\textwidth, clip]{1.eps}\n\\caption{Schematic illustration of the aim of present work. (a) We consider\na $2$D system with the Bloch Hamiltonian related to two periodic\nvector functions $\\mathbf{r}_{1}(k_{x})$\nand $\\mathbf{r}_{2}(k_{y})$ in auxiliary\nspace, which correspond to two knots. The topological index of the energy\nband is determined by the relations of two knots: The Chern number of the\nband equals to the linking number of two knots.\\ Several representative\nconfigurations of $[\\mathbf{r}_{1}(k_{x}),$ $\\mathbf{r}_{2}(k_{y})]$ is\npresented. Here $\\mathbf{r}_{2}$ is a trefoil knot, while three types of \n\\mathbf{r}_{1}$\\ are taken as simple loops, but at different positions,\nresulting linking numbers $\\mathcal{N}=0$, $1$, and $2$, respectively. The\ncorresponding Bloch Hamiltonians describe the systems with Chern numbers \nc=0 $, $1$, and $2$, respectively. (b) The main purpose of this work. For a \n1$D model with a fixed function $\\mathbf{r}_{2}(k_{y})$,  $\\mathbf{r}_{2}(k_{y})$ is\nreferred as to the degeneracy circuit. For an\narbitrary point $\\mathbf{r}_{1}(k_{x})$, the polarization filed $\\mathbf{P}\n\\mathbf{r}_{1})$ obeys the Biot-Savart law for magnetic field arising from\nthe degeneracy loop as a current loop. Polarization field $\\mathrm{d}\\mathbf\nP}$ at point $\\mathbf{r}_{1}$\\ generated from an infinitesmall length of\ndegeneracy line $\\mathrm{d}\\mathbf{r}_{2}$ at $\\mathbf{r}_{2}$, has the\nidentical form with the Biot-Savart Law related magnetic field generated\nfrom the current loop. Finding the polarization field $\\mathbf{P}$ at\narbitrary point $\\mathbf{r}_{1} $ resulting from a degeneracy line can be\nsimply obtained by the Biot-Savart law in electromagnetism.}\n\\label{fig1}\n\\end{figure}\n\nIn this work, we provide another quantum analogy of classical\nelectromagnetism. We consider a class of Bloch Hamiltonians, which contains\ntwo periodic vector functions with respect to two independent variables,\nsuch as momentum $k_{x}$\\textbf{\\ }and\\textbf{\\ }$k_{y}$ for a $2$D lattice\nsystem, respectively. These two periodic vector functions correspond to two\nknots in $3$D auxiliary space (see Fig. \\ref{fig1}(a)). The Bloch vector is\nthe difference of two vectors. When we only consider one of two knots, the\nsystem reduces to a $1$D lattice system. The Zak phase and polarization\nfield at a fixed point in $3$D auxiliary space can be obtained.\\ We show\nexactly that the knot as a degeneracy line has a simple relation of its\ncorresponding polarization field, obeying the Biot-Savart law: The\ndegeneracy line acts as a current-carrying wire, while the polarization\nfield corresponds to the generated magnetic field. The relationship between\ntwo knots can be characterized by applying the Amp\\`{e}re's circuital law on\nthe field integral arising from one knot along another knot. For a\nnontrivial topological system, the integral is nonzero, due to the fact that\ntwo Bloch knots entangle with each other, forming a link with the linking\nnumber being the value of Chern number of the energy band. In Fig. \\ref{fig1\n, we schematically illustrate the main conclusion of this work.\n\nWe propose two lattice models to exemplify the application of our approach.\nThe first one is an extended QWZ model. We show that the Bloch Hamiltonian\nis an example of our concerned system. Two knots of the original QWZ model\nsimply reduce to two circles. The second one is a time-dependent quasi-$1$D\nmodel with magnetic flux. In this case, the Amp\\`{e}re circulation integral\nis equivalent to the topological invariant. In the aid of the Biot-Savart\nlaw, the pumping charge acts as a dynamic measure of the Chern number. We\nperform numerical simulation for several representative quasi-adiabatic\nprocesses to demonstrate this application.\n\nThe remainder of this paper is organized as follows. In Sec. \\ref{Model with\ndegeneracy loop}, we present a class of models, whose Bloch Hamiltonian\nrelates to two knots. In Sec. \\ref{Kitaev model on square lattice} We\npropose the extended QWZ model to exemplify the application of our approach.\nSec. \\ref{Ladder system} gives another example, which is a time-dependent\nquasi-$1$D model with magnetic flux. Sec. \\ref{Pumping charge} devotes to a\ndynamic measure of Chern number, the pumping charge, which can be computed\nnumerically for several representative quasi-adiabatic processes to\ndemonstrate our work. Finally, we present a summary and discussion in Sec.\n\\ref{Summary}.\n\n\\section{Double-knot model}\n\n\\label{Model with degeneracy loop}\n\nConsider a Bloch Hamiltonian $h_{\\mathbf{k}}$ in the for\n\\begin{eqnarray}\nh_{\\mathbf{k}} &=&\\left(\n\\begin{array}{cc}\n\\left( z_{1}-z_{2}\\right) & x_{1}-x_{2}-i\\left( y_{1}-y_{2}\\right) \\\\\nx_{1}-x_{2}+i\\left( y_{1}-y_{2}\\right) & -\\left( z_{1}-z_{2}\\right\n\\end{array\n\\right)  \\notag \\\\\n&=&\\left[ \\mathbf{r}_{1}(k_{x})\\mathbf{-r}_{2}(k_{y})\\right] \\mathbf{\\cdot\n\\sigma },  \\label{hk}\n\\end{eqnarray\nwhich is the starting point of our study. It is consisted of two periodic\nvector functions $\\mathbf{r}_{1}(k_{x})=\\mathbf{r}_{1}(2\\pi +k_{x})=x_{1\n\\mathbf{i}+y_{1}\\mathbf{j}+z_{1}\\mathbf{k}$\\textbf{\\ }and\\textbf{\\ }$\\mathbf\nr}_{2}(k_{y})=\\mathbf{r}_{2}(2\\pi +k_{y})=x_{2}\\mathbf{i}+y_{2}\\mathbf{j\n+z_{2}\\mathbf{k}$, representing two knots (loops) in $3$D auxiliary space.\nHere $\\mathbf{\\sigma =(}\\sigma _{x},\\sigma _{y},\\sigma _{z}\\mathbf{)}$ are\nPauli matrices and $h_{\\mathbf{k}}$\\ represents a class of models, which is\nreferred\\ as to double-knot (double-loop) model. Matrix $h_{\\mathbf{k}}$\\\ncan take the role of a core matrix of crystalline system for non-interacting\nHamiltonian, or Kitaev Hamiltonian. We note that the spectrum of $h_{\\mathbf\nk}}$\\ is two-band and the gap closes when two knots have crossing points.\nThe aim of this work is to reveal the feature of the system which is\noriginated from the character of two knots.\n\nTo this end, we first consider the case with a fixed $k_{x}$. Then the model\nonly contains a point $\\mathbf{r}_{1}$ and a knot $\\mathbf{r}_{2}\\left(\nk_{y}\\right) $. The Hamiltonian reduces t\n\\begin{equation}\nh_{k_{y}}=\\left[ \\mathbf{r}_{1}\\mathbf{-r}_{2}(k_{y})\\right] \\mathbf{\\cdot\n\\sigma },\n\\end{equation\nwhich is a $1$D system in real space. Here $r_{2}(k_{y})$\\ is a degeneracy\nline, at which the gap closes.\\ The solution of equation \nh_{k_{y}}\\left\\vert u_{\\pm }^{k_{y}}\\right\\rangle =\\varepsilon _{k_{y}}^{\\pm\n}\\left\\vert u_{\\pm }^{k_{y}}\\right\\rangle $ has the form\n\\begin{equation}\n\\left\\vert u_{+}^{k_{y}}\\right\\rangle =\\left(\n\\begin{array}{c}\n\\cos \\frac{\\theta _{k_{y}}}{2}e^{-i\\varphi _{k_{y}}} \\\\\n\\sin \\frac{\\theta _{k_{y}}}{2\n\\end{array\n\\right) ,\\left\\vert u_{-}^{k_{y}}\\right\\rangle =i\\left(\n\\begin{array}{c}\n-\\sin \\frac{\\theta _{k_{y}}}{2} \\\\\n\\cos \\frac{\\theta _{k_{y}}}{2}e^{i\\varphi _{k_{y}}\n\\end{array\n\\right)\n\\end{equation\nwith $\\varepsilon _{k_{y}}^{\\pm }=\\pm \\left\\vert \\mathbf{r}_{1}\\mathbf{-r\n_{2}\\left( k_{y}\\right) \\right\\vert $, where the azimuthal and polar angles\nare defined a\n\\begin{equation}\n\\cos \\theta _{k_{y}}=\\frac{z_{1}-z_{2}}{\\left\\vert \\mathbf{r}_{1}\\mathbf{-r\n_{2}\\right\\vert },\\tan \\varphi _{k_{y}}=\\frac{y_{1}-y_{2}}{x_{1}-x_{2}}.\n\\end{equation\nFor this $1$D system, the corresponding Zak phases for upper and lower bands\nare defined a\n\\begin{equation}\n\\mathcal{Z}_{\\pm }=\\frac{i}{2\\pi }\\int_{-\\pi }^{\\pi }\\left\\langle u_{\\pm\n}^{k_{y}}\\right\\vert \\frac{\\partial }{\\partial k_{y}}\\left\\vert u_{\\pm\n}^{k_{y}}\\right\\rangle \\mathrm{d}k_{y}.\n\\end{equation\nIt is well known that the Zak phase is gauge-dependent and the present\nexpression of $\\left\\vert u_{\\pm }^{k_{y}}\\right\\rangle $\\ results i\n\\begin{equation}\n\\mathcal{Z}=\\mathcal{Z}_{+}=-\\mathcal{Z}_{-}=\\frac{1}{2\\pi }\\oint_{\\mathrm{L\n}\\cos ^{2}\\frac{\\theta _{k_{y}}}{2}\\mathrm{d}\\varphi _{k_{y}},\n\\end{equation\nwhere \\textrm{L} denotes the integral loop about the solid angle.\nAccordingly, the polarization vector field is defined a\n\\begin{equation}\n\\mathbf{P}=-\\mathbf{\\nabla }\\mathcal{Z},\n\\end{equation\nwhere $\\mathbf{\\nabla }$ is the nabla operato\n\\begin{equation}\n\\mathbf{\\nabla }=(\\frac{\\partial }{\\partial x_{1}}\\mathbf{i}+\\frac{\\partial\n}{\\partial y_{1}}\\mathbf{j}+\\frac{\\partial }{\\partial z_{1}}\\mathbf{k}),\n\\end{equation\nwith unitary vectors $\\mathbf{i}$, $\\mathbf{j}$, and $\\mathbf{k}$ in $3$D\nauxiliary space. Straightforward derivation (see Appendix) shows that\n\n\\begin{equation}\n\\mathbf{P}=\\frac{1}{4\\pi }\\oint_{\\mathrm{L}}\\frac{\\mathrm{d}\\mathbf{r\n_{2}\\times \\left( \\mathbf{r}_{1}-\\mathbf{r}_{2}\\right) }{\\left\\vert \\mathbf{\n}_{1}-\\mathbf{r}_{2}\\right\\vert ^{3}},  \\label{Polarization}\n\\end{equation\nwhere $\\mathrm{L}$ denotes the integral loop about the degeneracy loop. It\nis clear that if we consider a degeneracy loop as current-carrying wire with\nsteady current strength $I=1\/\\mu _{0}$, flowing in the direction of\nincreasing $k_{y}$ from $0$\\ to $2\\pi $, the field $\\mathbf{P}$ is identical\nto the magnetic field generated by the current loop,\\ where $\\mu _{0}$\\ is\nthe vacuum permittivity of free space. Since the Eq. (\\ref{Polarization})\nholds for an arbitrary loop $\\mathrm{L}$, one can have its differential for\n\\begin{equation}\n\\mathrm{d}\\mathbf{P}=\\frac{1}{4\\pi }\\frac{\\mathrm{d}\\mathbf{r}_{2}\\times\n\\left( \\mathbf{r}_{1}-\\mathbf{r}_{2}\\right) }{\\left\\vert \\mathbf{r}_{1}\n\\mathbf{r}_{2}\\right\\vert ^{3}},  \\label{Biot-Savart law}\n\\end{equation\nwhich is illustrated in Fig. \\ref{fig1}. It indicates that the relationship\nbetween $\\mathbf{P}$ and the degeneracy loop obeys the Biot-Savart law. It\nreveals the topological characteristics of the degeneracy lines in a clear\nphysical picture. We will regard degeneracy loops as a \\textit{band\ndegeneracy circuit}. This result helps us to determine the polarization of\nany loops in the auxiliary space. In addition, the Amp\\`{e}re circulation\nintegral $\\oint_{\\ell }\\mathbf{P(\\mathbf{r})\\cdot }\\mathrm{d}\\mathbf{r}$\nalong a loop $\\ell $\\ has clear physical means: (i) It equals to the sum of\nthe current through the surface spanned by the loop $\\ell $; (ii) It is the\npumping charge for the adiabatic passage $\\ell $.\n\n\\begin{figure*}[tbp]\n\\includegraphics[ bb=3 414 620 765, width=0.92\\textwidth, clip]{links.eps}\n\\caption{Schematic several representative configurations of double-knot \n\\left\\{ \\mathbf{r}_{1}(k_{x}),\\mathbf{r}_{2}(k_{y})\\right\\} $ for the\nextended QWZ model. The plots are obtained from parameter equations in Eq. \n\\protect\\ref{XYZ}) with parameters indicated in the panels. The arrows on\nthe loops indicate the directions of the knots with various topologies. The\ncorresponding Chern numbers are labeled, that match the linking numbers\nexactly.}\n\\label{fig2}\n\\end{figure*}\n\nNow we go back to $h_{\\mathbf{k}}$, taking the loop $\\ell $\\ as the knot \n\\mathbf{r}_{1}(k_{x})$, which has no crossing point on the knot $\\mathbf{r\n_{2}(k_{y})$. We find that the corresponding Amp\\`{e}re circulation integral\nis connected to the topology of two knots and the band structure of the\nsystem\n\\begin{equation}\n-\\oint_{\\ell }\\mathbf{P(\\mathbf{r})\\cdot }\\mathrm{d}\\mathbf{r=}c=\\mathcal{N\n\\mathbf{.}  \\label{CN}\n\\end{equation\nHere the Chern number for lower band is defined as \\cite{XLQI, GYCHO\n\\begin{equation}\nc=\\frac{1}{4\\pi }\\int_{0}^{2\\pi }\\int_{0}^{2\\pi }\\frac{\\mathbf{r}^{\\prime }}\n\\left\\vert \\mathbf{r}^{\\prime }\\right\\vert ^{3}}\\mathbf{\\cdot }\\left( \\frac\n\\partial \\mathbf{r}^{\\prime }}{\\partial k_{x}}\\times \\frac{\\partial \\mathbf{\n}^{\\prime }}{\\partial k_{y}}\\right) \\mathrm{d}k_{x}\\mathrm{d}k_{y},\n\\end{equation\nwith $\\mathbf{r}^{\\prime }=\\mathbf{r}_{1}-\\mathbf{r}_{2}$, which also equals\nto the linking number \\cite{RICCA} of two knots $\\mathbf{r}_{1}(k_{y})$ and \n\\mathbf{r}_{2}(k_{x})$\n\\begin{equation}\n\\mathcal{N}=\\frac{1}{4\\pi }\\int_{0}^{2\\pi }\\int_{0}^{2\\pi }\\frac{\\mathbf{r\n^{\\prime }}{\\left\\vert \\mathbf{r}^{\\prime }\\right\\vert ^{3}}\\mathbf{\\cdot \n\\left( \\frac{\\partial \\mathbf{r}_{1}}{\\partial k_{x}}\\times \\frac{\\partial\n\\mathbf{r}_{2}}{\\partial k_{y}}\\right) \\mathrm{d}k_{x}\\mathrm{d}k_{y}.\n\\end{equation\nThese relations are evident demonstrations of the system's topological\nfeature and clearly reveal the physical significance of the Amp\\`{e}re\ncirculation integral $\\oint_{\\ell }\\mathbf{P(\\mathbf{r})\\cdot }\\mathrm{d\n\\mathbf{r}$. Furthermore, it corresponds to the jump of Zak phase for an\nadiabatic passage along a knot, which\\ can be measured by the Thouless\npumping charge in a quasi $1$D system. In the following, we present two\nexamples to illustrate our results.\n\n\\section{Extended QWZ model}\n\n\\label{Kitaev model on square lattice}\n\nIn this section, we consider a model, which is an extension of QWZ model\nintroduced by Qi, Wu and Zhang \\cite{QWZ}, to illustrate our result. The\nBloch Hamiltonian is\n\n\\begin{equation}\nh_{\\mathbf{k}}=B_{x}\\sigma _{x}+B_{y}\\sigma _{y}+B_{z}\\sigma _{z},\n\\end{equation\nwhere the field components ar\n\\begin{equation}\n\\left\\{\n\\begin{array}{l}\nB_{x}=\\sin k_{x}+\\lambda \\sin \\left( 2k_{x}\\right) \\\\\nB_{y}=\\sin k_{y}+\\lambda \\sin \\left( 2k_{y}\\right) \\\\\nB_{z}=u+\\cos k_{x}+\\cos k_{y} \\\\\n+\\lambda \\left[ \\cos \\left( 2k_{x}\\right) +\\cos \\left( 2k_{y}\\right) \\right\n\\end{array\n\\right. .  \\label{XYZ}\n\\end{equation\nIt reduces to original QWZ model when taking $\\lambda =0$.\n\nNow we rewrite it in the for\n\\begin{equation}\nh_{\\mathbf{k}}=\\left[ \\mathbf{r}_{1}(k_{x})\\mathbf{-r}_{2}(k_{y})\\right]\n\\mathbf{\\cdot \\sigma },\n\\end{equation\nwhere two vector functions ar\n\\begin{equation}\n\\left\\{\n\\begin{array}{l}\n\\mathbf{r}_{1}=(\\sin k_{x}+\\lambda \\sin \\left( 2k_{x}\\right) ,0,u+\\cos\nk_{x}+\\lambda \\cos \\left( 2k_{x}\\right) ) \\\\\n\\mathbf{r}_{2}=-(0,\\sin k_{y}+\\lambda \\sin \\left( 2k_{y}\\right) ,\\cos\nk_{y}+\\lambda \\cos \\left( 2k_{y}\\right) \n\\end{array\n\\right. .\n\\end{equation\nIt is clear that $\\mathbf{r}_{1}(k_{x})$\\ and $\\mathbf{r}_{2}(k_{y})$\\\nrepresent two limacons within $xz$ and $yz$ plane, respectively. When taking\n$\\left\\vert \\lambda \\right\\vert <0.5$, the crossing point of the limacon\ndisappears. Particularly, when taking $\\lambda =0$, limacons reduce to\ncircles. The radiuses of two circles are both $1$, but the centers are \n(0,0,u)$\\ and $(0,0,0)$, respectively. Chern numbers can be easily obtained\nfrom the linking numbers of these two circles: $c=0$, for $\\left\\vert\nu\\right\\vert >2$, and $c=\\pm 1$, for $0<\\pm u<2$. When taking $\\left\\vert\n\\lambda \\right\\vert >0.5$, the crossing point of the limacon appears. Since\nlimacons with crossing point cannot be classified as knots, we add\nperturbation terms $\\kappa \\sin 2k_{x}$\\ to $r_{1y}$ and $\\kappa \\sin 2k_{y}$\nto\\ $r_{2x}$\\ to untie the crossing point ($\\left\\vert \\kappa \\right\\vert\n\\ll 1$), then limacons become knots again. The possible linking numbers of\nsuch two knots are still equal to the Chern numbers $c=0$, $\\pm 1$, $\\pm 3$,\nand $\\pm 4$. The absence of $c=\\pm 2$ is due to the fact that we take the\nidentical $\\lambda $ in the expressions of $\\mathbf{r}_{1}(k_{x})$ and \n\\mathbf{r}_{2}(k_{y})$. In Fig. \\ref{fig2}, we plot some representative\nconfigurations to demonstrate this point. Comparing to the direct\ncalculation of Chern number from the Berry connection, the example shows\nthat the Chern number can be easily obtained by the geometrical\nconfigurations hidden in the Bloch Hamiltonian.\n\n\\section{Ladder system}\n\n\\label{Ladder system}\n\nAs a simple application of our result, we consider a quasi $1$D system with\nperiodically time-dependent parameters. The Bloch Hamiltonian has the for\n\\begin{equation}\nh_{k}(t)=\\left[ \\mathbf{r}(t)\\mathbf{-r}_{c}(k)\\right] \\mathbf{\\cdot \\sigma \n,\n\\end{equation\nwhere $\\mathbf{r}(t)=\\mathbf{r}(t+T)$ represents a loop $\\ell $ without\ncrossing point on the degeneracy loop $\\mathbf{r}_{c}(k)$. The result\nobtained above still apply to the case of replacing $(k_{x},k_{y})$\\ with \n(t,k)$, and replacing $\\left\\vert u_{\\pm }^{\\mathbf{k}}\\right\\rangle $ with \n|u_{\\pm }^{k}(t)\\rangle $ accordingly. In this section we will demonstrate\nour result and its physical implications through an alternative\ntight-binding model, which is two coupled SSH chains, or a ladder system\nwith staggered magnetic flux, on-site potential and long range hopping\nterms. These ingredients allow the system to support multiple types of\ndegeneracy loops with different geometric topologies.\n\nWe consider a ladder system which is illustrated in Fig. \\ref{fig3},\nrepresented by the Hamiltonian\n\n\\begin{eqnarray}\n&&H_{\\text{L}}=\\sum_{j=1}^{N}\\{r_{\\bot }e^{i\\phi }c_{2j}^{\\dag\n}c_{2j-1}+\\alpha c_{2j-1}^{\\dag }c_{2\\left( j+1\\right) }+\\beta c_{2\\left(\nj+1\\right) -1}^{\\dag }c_{2j}  \\notag \\\\\n&&+\\mu c_{2\\left( j+2\\right) -1}^{\\dag }c_{2j}+\\nu c_{2j-1}^{\\dag\n}c_{2\\left( j+2\\right) }+i\\kappa \\lbrack c_{2\\left( j+3\\right) -1}^{\\dag\n}c_{2j-1}  \\notag \\\\\n&&-c_{2\\left( j+3\\right) }^{\\dag }c_{2j}]+\\text{\\textrm{H.c.}\n\\}+z\\sum_{j=1}^{2N}\\left( -1\\right) ^{j+1}c_{j}^{\\dag }c_{j},\\label{ladd}\n\\end{eqnarray\non a $2N$ lattice. Here $c_{j}^{\\dag }$ is the creation operator of a\nfermion at the $j$th site with the periodic boundary condition \nc_{2N+1}=c_{1}$. The inter-sublattice hopping amplitudes are $\\left( \\alpha\n,\\beta ,\\mu ,\\nu \\right) $ and the intra-sublattice hopping amplitude is \n\\kappa $.\\ Besides, two time-dependent parameters, $2\\phi (t)$ is the\nstaggered magnetic flux threading each plaquette and $z(t)$\\ is the strength\nof staggered potentials. The ladder system is essentially two coupled SSH\nchains. As a building block of the system, the SSH model \\cite{SSH} has\nserved as a paradigmatic example of the $1$D system supporting topological\ncharacter \\cite{Zak}. It has an extremely simple form but well manifests the\ntypical feature of topological insulating phase, and the transition between\nnon-trivial and trivial topological phases, associated with the number of\nzero energy and edge states as the topological invariant \\cite{Asboth}. It\nhas been demonstrated that all the parameters of this model can be easily\naccessed within the existing technology of cold-atomic experiments \\cit\n{Clay,Ueda,Jo}. We schematically illustrate this model in Fig.~\\ref{fig3}.\nWe introduce the fermionic operators in $k$ space\n\\begin{equation}\n\\left\\{\n\\begin{array}{l}\na_{k}=\\frac{1}{\\sqrt{N}}\\sum_{j=1}^{N}e^{-ikj}c_{2j-1} \\\\\nb_{k}=\\frac{1}{\\sqrt{N}}\\sum_{j=1}^{N}e^{-ikj}c_{2j\n\\end{array\n\\right. ,  \\label{Fourier 1}\n\\end{equation\nand the wave vector $k=\\pi (2n-N)\/N$, $(n=0,1,...,N-1)$. Then we hav\n\\begin{equation}\nH_{\\text{L}}=\\sum_{k}(a_{k}^{\\dagger },b_{k}^{\\dagger })h_{k}\\left(\n\\begin{array}{c}\na_{k} \\\\\nb_{k\n\\end{array\n\\right) ,\n\\end{equation\n\\begin{figure}[tbp]\n\\includegraphics[ bb=130 440 476 540, width=0.5\\textwidth, clip]{3.eps}\n\\caption{Schematics of the two coupled SSH chains with staggered flux and\npotential. The system consists of two sublattices A and B with on-site\npotentials $z$\\ and $-z$, indicated by filled and empty circles,\nrespectively. Hopping amplitudes along each chain are staggered by $\\protec\n\\alpha $ (blue solid line) and $\\protect\\beta $ (blue dotted line). The\ninterchain hopping amplitude is $r_{\\perp }$ (thick black line) associated\nwith a phase factor and interchain diagonal hopping amplitude $\\protect\\mu $\n(gray solid line), $\\protect\\nu $ (gray dotted line) and $i\\protect\\kappa $\n(light green solid line). The red arrows indicate the hopping directions for\ncomplex amplitudes, which are induced by the staggered flux threading each\nplaquettes (arrow circles).}\n\\label{fig3}\n\\end{figure}\nwhere the core matrix has the for\n\\begin{equation}\nh_{k}=\\left(\n\\begin{array}{cc}\nz+2\\kappa \\sin \\left( 3k\\right) & R(\\phi ,k) \\\\\nR^{\\ast }(\\phi ,k) & -z-2\\kappa \\sin \\left( 3k\\right\n\\end{array\n\\right) ,\n\\end{equation\nand the off-diagonal matrix element i\n\\begin{equation}\nR(\\phi ,k)=r_{\\bot }e^{-i\\phi }+\\alpha e^{ik}+\\beta e^{-ik}+\\mu e^{-2ik}+\\nu\ne^{2ik}.\n\\end{equation\nTaking\n\n\\begin{equation}\nx+iy=r_{\\bot }e^{i\\phi },\n\\end{equation\nthe parameter equations for degeneracy loop i\n\\begin{equation}\n\\left\\{\n\\begin{array}{l}\nx_{c}=-\\left( \\alpha +\\beta \\right) \\cos k-\\left( \\mu +\\nu \\right) \\cos\n\\left( 2k\\right) \\\\\ny_{c}=-\\left( \\beta -\\alpha \\right) \\sin k-\\left( \\mu -\\nu \\right) \\sin\n\\left( 2k\\right) \\\\\nz_{c}=-2\\kappa \\sin \\left( 3k\\right\n\\end{array\n\\right. ,  \\label{trefoil knot}\n\\end{equation\nwhich is plotted in Fig. \\ref{fig4} for the case with parameters $\\alpha\n=\\mu =0.5$, $\\beta =1$, $\\nu =1.5$, and $\\kappa =0.1$. One can see that the\ndegeneracy curve is a trefoil knot. Intuitively, it should result in\ntopological features with indices $2$, $1$, and $0$. We will demonstrate\nthis point in the next section.\n\n\\begin{figure*}[tbp]\n\\includegraphics[ bb=19 8 1241 380, width=0.92\\textwidth, clip]{fig4a.eps}\n\\includegraphics[ bb=19 8 1241 380, width=0.92\\textwidth, clip]{fig4b.eps}\n\\includegraphics[ bb=19 8 1241 380, width=0.92\\textwidth, clip]{fig4c.eps}\n\\caption{ (a1-a3) Schematics of three adiabatic passages in $3$D auxiliary\nspace for pumping charge. The degeneracy curve (red) is a trefoil knot with\nparameters $\\protect\\alpha =\\protect\\mu =0.5$, $\\protect\\beta =1$, $\\protec\n\\nu =1.5$, and $\\protect\\kappa =0.1$ of the system in Eq. (\\protect\\ref{ladd\n) (Fig. \\protect\\ref{fig3}). The adiabatic passages are straight lines\n(blue)\\ at positions $(x,y):$ (a1) $(3.80,0)$, (a2) $(1.15,0.84)$, and (a3) \n(0.40,0.01)$, respectively. (b1-b3) and (c1-c3)\\ are plots of current and\nthe corresponding total charge transfer for quasi-adiabatic process. The\nresults are obtained by numerically exact diagonalization method for the\nsystem in Eq. (\\protect\\ref{ladd}) with $N=100 $. The speed of time\nevolution is $\\protect\\omega =1\\times 10^{-3}$. It indicates that the\ntopological invariant can be obtained by dynamical process.}\n\\label{fig4}\n\\end{figure*}\n\n\\section{Pumping charge}\n\n\\label{Pumping charge}\n\nFor a $2$D system with Bloch Hamiltonian in the form of Eq. (\\ref{hk}), the\nphysical and geometric meanings of\\ Chern number is well established. For a\nquasi $1$D system with Bloch Hamiltonian in the form of Eq. (\\ref{hk}) by\nreplacing $(k_{x},k_{y})$\\ with $(t,k)$, the Chern number is connected to an\nadiabatic passage driven by the parameters from $t$\\ to $t+T$, or a periodic\nloop $\\mathbf{r=r}(t)$ in auxiliary space. In a $1$D model, it has been\nshown that the adiabatic particle transport over a time period takes the\nform of the Chern number\\ and it is quantized \\cite{DXIAO}. The pumped\ncharge counts the net number of degeneracy point enclosed by the loop. This\ncan be extended to the loop $\\mathbf{r=r}(t)$\\ in the present model.\n\nActually, one can rewrite Eq. (\\ref{CN}) in the for\n\\begin{equation}\nc=\\mathcal{N}=-\\oint_{\\ell }\\mathbf{P(\\mathbf{r})\\cdot }\\frac{\\partial\n\\mathbf{\\mathbf{r}}}{\\partial t}\\mathrm{d}t\\mathbf{.}\n\\end{equation\nwhere $\\mathbf{\\mathbf{r}}$\\ (or $r_{\\bot },\\phi ,$ and $z$) is periodic\nfunction of time $t$. Furthermore, we can find out the physical meaning of\nthe Chern number by the relation\n\\begin{equation}\nc=\\int_{0}^{T}\\mathcal{J}(t)\\mathrm{d}t,\n\\end{equation\nwher\n\\begin{equation}\n\\mathcal{J}=\\frac{i}{2\\pi }\\int_{0}^{2\\pi }[(\\partial _{t}\\langle\nu_{-}^{k}|)\\partial _{k}|u_{-}^{k}\\rangle -(\\partial _{k}\\langle\nu_{-}^{k}|)\\partial _{t}|u_{-}^{k}\\rangle ]\\mathrm{d}k\n\\end{equation\nis the adiabatic current. Then $c$ is pumped charge of all channel $k$\ndriven by the time-dependent Hamiltonian varying in a period, which can be\nmeasured through a quasi adiabatic process.\n\nInspired by these analysis,\\ we expect that the Chern number can be unveiled\nby the pumping charge of all the energy levels. This can be done in\nsingle-particle sub-space. The accumulated charge passing the unit cell $l$\nduring the period $T$ i\n\\begin{equation}\nQ_{l}=\\sum_{k}\\int_{0}^{T}j_{l}\\mathrm{d}t,\n\\end{equation\nwhere current across two neighboring unit cells i\n\\begin{eqnarray}\n&&j_{l}=\\frac{1}{i}\\left\\langle u_{-}^{k}\\left( t\\right) \\right\\vert [\\alpha\na_{j}^{\\dag }b_{j+1}+\\beta b_{j}^{\\dag }a_{j+1}+\\mu b_{j}^{\\dag }a_{j+2}+\n\\notag \\\\\n&&\\nu a_{j}^{\\dag }b_{j+2}-i\\kappa a_{j}^{\\dag }a_{j+3}+i\\kappa b_{j}^{\\dag\n}b_{j+3}-\\text{\\textrm{H.c.}}]\\left\\vert u_{-}^{k}\\left( t\\right)\n\\right\\rangle .\n\\end{eqnarray}\n\n\nAs we mentioned above, there are three types of adiabatic loop $\\mathbf{r=r\n(t)$ in auxiliary space, with pumping charges $Q_{l}=0$, $1$, and $2$,\nrespectively. In general, three periodic functions $r_{\\bot }(t)$, $\\phi (t)\n, and $z(t)$\\ should be taken to measure the pumping charge. However, a\nquasi adiabatic loop is tough to be realized in practice. Thanks to the\nBiot-Savart law for the field $\\mathbf{P}(r)$, we can take the adiabatic\npassage along a straight line with fixed $r_{\\bot }$ and $\\phi $, since the\nfield $\\mathbf{P}$\\ far from the trefoil knot $\\mathbf{r}_{c}(k)$ has no\ncontribution to the Amp\\`{e}re circulation integral, or the pumping charge.\n\nWe consider the case by taking $z=\\omega t$ with $\\omega \\ll 1$. According\nto the analysis above, if $t$ varies from $-\\infty $ to $\\infty $, $Q_{l}$\\\nshould be $0$, $1$, and $2$, respectively. To examine how the scheme works\nin practice, we simulate the quasi-adiabatic process by computing the time\nevolution numerically for finite system. In principle, for a given initial\neigenstate $\\left\\vert u_{-}^{k}\\left( 0\\right) \\right\\rangle $, the time\nevolved state under a Hamiltonian $H_{\\text{L}}\\left( t\\right) $ i\n\\begin{equation}\n\\left\\vert \\Phi \\left( t\\right) \\right\\rangle =\\mathcal{T}\\{\\exp\n(-i\\int_{0}^{t}H_{\\text{L}}\\left( t\\right) \\mathrm{d}t)\\left\\vert\nu_{-}^{k}\\left( 0\\right) \\right\\rangle \\},\n\\end{equation\nwhere $\\mathcal{T}$ is the time-ordered operator. In low speed limit $\\omega\n\\rightarrow 0$, we hav\n\\begin{equation}\nf\\left( t\\right) =\\left\\vert \\langle u_{-}^{k}\\left( t\\right) \\left\\vert\n\\Phi \\left( t\\right) \\right\\rangle \\right\\vert \\rightarrow 1,\n\\end{equation\nwhere $\\left\\vert u_{-}^{k}\\left( t\\right) \\right\\rangle $\\ is the\ncorresponding instantaneous eigenstate of $H_{\\text{L}}\\left( t\\right) $.\nThe computation is performed by using a uniform mesh in the time\ndiscretization for the time-dependent Hamiltonian $H_{\\text{L}}$. In\norder to demonstrate a quasi-adiabatic process, we keep $f\\left( t\\right)\n>0.9$\\ during the whole process by taking sufficient small $\\omega $. Fig.\n\\ref{fig4} plots the simulations of particle current and the corresponding\ntotal probability, which shows that the obtained dynamical quantities are in\nclose agreement with the expected Chern number.\n\n\\section{Summary and discussion}\n\n\\label{Summary}\n\nWe have analyzed a family of $2$D tight-binding model with various Chern\nnumbers, which are directly connected to the topology of two knots. When\nreduced to $1$D single-knot degeneracy model, a polarization vector field\ncan be established for a gapped band. We have exactly shown an interesting\nanalogy between the topological feature of the band and classical\nelectromagnetism: polarization vector field acts as the static magnetic\nfield generated by the degeneracy knot as a current circuit. It indicates\nthat there is a quantum analogy of\\ Biot-Savart law in quantum matter.\nBefore ending this paper, we would like to point out that our findings also\nreveal the topological feature hidden in the case with zero Chern number. In\nFig. \\ref{fig2}(a) and (b), we find out that though the linking numbers of\nthese two sets of loops are zero, the configurations are different. It\nshould imply certain topological feature in a single direction, which will\nbe investigated in future work. This finding extends the understanding of\ntopological feature in matter and provides methodology and tool for dealing\nwith the calculation and detection of Chern numbers.\n\n\\section{Appendix: Proof of the Biot-Savart law}\n\nIn this appendix, we provide the proof of Eq. (\\ref{Polarization}) in the\nmain text. To this end, we first revisit the Biot-Savart law\\ for a current\ncarrying loop, and then compare it with the polarization field in the\npresent work.\n\n\\subsection{The magnetic field}\n\nConsider a current carrying loop $\\mathrm{L}$\\ with current strength $I=1\n\\mu\n_{0}$, which is described by a periodic function $\\mathbf{r}_{2}\\left(\nk_{y}\\right) =x_{2}\\mathbf{i+}y_{2}\\mathbf{j+}z_{2}\\mathbf{k}$ in a $3$D\nspace. Here \n\\mu\n_{0}$ is the vacuum permittivity of free space and the current\\ flows in the\ndirection of increasing $k_{y}$ from $0$ to $2\\pi $. According to the\nBiot-Savart law, the magnetic field $\\mathbf{B}$ at position $\\mathbf{r\n_{1}=x_{1}\\mathbf{i+}y_{1}\\mathbf{j+}z_{1}\\mathbf{k}$ generated by the loop \n\\mathrm{L}$ is\n\n\\begin{equation}\n\\mathbf{B}=\\frac{1}{4\\pi }\\oint_{\\mathrm{L}}\\frac{\\mathbf{r}_{2}-\\mathbf{r\n_{1}}{\\left\\vert \\mathbf{r}_{1}-\\mathbf{r}_{2}\\right\\vert ^{3}}\\times\n\\mathrm{d}\\mathbf{r}_{2}.\n\\end{equation\nFor the sake of simplicity we only give the proof for $\\mathbf{B}$\\ and \n\\mathbf{P}$\\ in the $x$ component as an example. The explicit form of the\ncomponent i\n\\begin{equation}\nB_{x}=\\frac{1}{4\\pi }\\oint_{\\mathrm{L}}\\frac{\\left( y_{2}-y_{1}\\right)\n\\mathrm{d}z_{2}\\mathbf{-}\\left( z_{2}-z_{1}\\right) \\mathrm{d}y_{2}}\n\\left\\vert \\mathbf{r}_{1}-\\mathbf{r}_{2}\\right\\vert ^{3}}.\n\\end{equation\nAccording to the Stokes' theorem, the line integral of $B_{x}$ can be\nexpressed as a double integra\n\\begin{eqnarray}\nB_{x} &=&\\frac{1}{4\\pi }\\iint\\nolimits_{\\mathrm{S}}[\\frac{3\\left(\nx_{2}-x_{1}\\right) ^{2}-\\left\\vert \\mathbf{r}_{1}-\\mathbf{r}_{2}\\right\\vert\n^{2}}{\\left\\vert \\mathbf{r}_{1}-\\mathbf{r}_{2}\\right\\vert ^{5}}\\mathrm{d\ny_{2}\\mathrm{d}z_{2}  \\notag \\\\\n&&-\\frac{3\\left( x_{2}-x_{1}\\right) \\left( y_{1}-y_{2}\\right) }{\\left\\vert\n\\mathbf{r}_{1}-\\mathbf{r}_{2}\\right\\vert ^{5}}\\mathrm{d}z_{2}\\mathrm{d}x_{2}\n\\notag \\\\\n&&-\\frac{3\\left( x_{2}-x_{1}\\right) \\left( z_{1}-z_{2}\\right) }{\\left\\vert\n\\mathbf{r}_{1}-\\mathbf{r}_{2}\\right\\vert ^{5}}\\mathrm{d}x_{2}\\mathrm{d\ny_{2}],\n\\end{eqnarray}\nwhere $S$ represents a smooth surface spanned by the loop $L$.\n\n\\subsection{The polarization vector field}\n\nNow we turn to the quantum analogy of Biot-Savart law. For a fixed $k_{x}$, \nh_{\\mathbf{k}}$\\ reduces to a $1$D system $h_{k_{y}}$, and the corresponding\nZak phases for upper and lower bands are defined a\n\\begin{equation}\n\\mathcal{Z}_{\\pm }=\\frac{i}{2\\pi }\\int_{-\\pi }^{\\pi }\\left\\langle u_{\\pm\n}^{k_{y}}\\right\\vert \\frac{\\partial }{\\partial k_{y}}\\left\\vert u_{\\pm\n}^{k_{y}}\\right\\rangle \\mathrm{d}k_{y},\n\\end{equation\nwhich is gauge-dependent. For the present expression of $\\left\\vert u_{\\pm\n}^{k_{y}}\\right\\rangle $, we have\n\\begin{equation}\n\\mathcal{Z}=\\mathcal{Z}_{+}=-\\mathcal{Z}_{-}=\\frac{1}{2\\pi }\\oint_{\\mathrm{L\n}\\cos ^{2}\\frac{\\theta }{2}\\mathrm{d}\\varphi ,\n\\end{equation\nwhere $\\mathrm{L}$ denotes loop $\\mathbf{r}_{2}(k_{y})$\\ an\n\\begin{equation}\n\\cos \\theta =\\frac{z_{1}-z_{2}}{\\left\\vert \\mathbf{r}_{1}-\\mathbf{r\n_{2}\\right\\vert },\\tan \\varphi =\\frac{y_{1}-y_{2}}{x_{1}-x_{2}}.\n\\end{equation\nThe polarization vector field is defined a\n\\begin{equation}\n\\mathbf{P}=-\\mathbf{\\nabla }\\mathcal{Z},  \\label{P1}\n\\end{equation\nwhere $\\mathbf{\\nabla }$\\ is the nabla operato\n\\begin{equation}\n\\mathbf{\\nabla }=(\\frac{\\partial }{\\partial x_{1}}\\mathbf{i}+\\frac{\\partial\n}{\\partial y_{1}}\\mathbf{j}+\\frac{\\partial }{\\partial z_{1}}\\mathbf{k}),\n\\end{equation\nwith unitary vectors $\\mathbf{i}$, $\\mathbf{j}$, and $\\mathbf{k}$ in $3$D\nauxiliary space. We note the fact that\n\n\\begin{equation}\n\\mathbf{k\\cdot \\lbrack }\\oint_{\\mathrm{L}}\\frac{\\mathbf{r}_{2\\bot }-\\mathbf{\n}_{1\\bot }}{\\left\\vert \\mathbf{r}_{2\\bot }-\\mathbf{r}_{1\\bot }\\right\\vert\n^{2}}\\times \\mathrm{d}\\left( \\mathbf{r}_{2\\bot }-\\mathbf{r}_{1\\bot }\\right)\n]=\\oint_{\\mathrm{L}}\\mathrm{d}\\varphi =2\\pi w,\n\\end{equation\nwhere $w$ is winding number of the integral loop $\\mathbf{r}_{2\\bot\n}(k_{y})=x_{2}\\mathbf{i}+y_{2}\\mathbf{j}$\\ around the point $\\mathbf{r\n_{1\\bot }=x_{1}\\mathbf{i}+y_{1}\\mathbf{j}$. Then the Zak phase can be\nrewritten a\n\\begin{equation}\n\\mathcal{Z}=\\frac{\\mathbf{k}}{4\\pi }\\mathbf{\\cdot }\\oint_{\\mathrm{L}}\\left[\n1+\\frac{z_{1}-z_{2}}{\\left\\vert \\mathbf{r}_{1}-\\mathbf{r}_{2}\\right\\vert \n\\right] \\frac{\\mathbf{r}_{2\\bot }-\\mathbf{r}_{1\\bot }}{\\left\\vert \\mathbf{r\n_{2\\bot }-\\mathbf{r}_{1\\bot }\\right\\vert ^{2}}\\times \\mathrm{d}\\left(\n\\mathbf{r}_{2\\bot }-\\mathbf{r}_{1\\bot }\\right) .\n\\end{equation\nThe projection of the polarization vector field $\\mathbf{P}$ in the $x$\ndirection is represented a\n\\begin{equation}\nP_{x}=-\\frac{\\partial }{\\partial x_{1}}\\mathcal{Z}=\\frac{1}{4\\pi }\\oint_\n\\mathrm{L}}\\left( G\\mathrm{d}x_{2}+Q\\mathrm{d}y_{2}+R\\mathrm{d}z_{2}\\right) ,\n\\end{equation\nwher\n\\begin{eqnarray}\nG &=&-\\frac{\\left( z_{1}-z_{2}\\right) \\left( x_{2}-x_{1}\\right) \\left(\ny_{2}-y_{1}\\right) }{\\left\\vert \\mathbf{r}_{1}-\\mathbf{r}_{2}\\right\\vert\n^{3}\\left\\vert \\mathbf{r}_{2\\bot }-\\mathbf{r}_{1\\bot }\\right\\vert ^{2}} \\\\\n&&-\\left( 1+\\frac{z_{1}-z_{2}}{\\left\\vert \\mathbf{r}_{1}-\\mathbf{r\n_{2}\\right\\vert }\\right) \\frac{2\\left( x_{2}-x_{1}\\right) (y_{2}-y_{1})}\n\\left\\vert \\mathbf{r}_{2\\bot }-\\mathbf{r}_{1\\bot }\\right\\vert ^{4}},  \\notag\n\\end{eqnarray\nan\n\\begin{eqnarray}\nQ &=&\\left( 1+\\frac{z_{1}-z_{2}}{\\left\\vert \\mathbf{r}_{1}-\\mathbf{r\n_{2}\\right\\vert }\\right) \\frac{\\left[ \\left( x_{2}-x_{1}\\right) ^{2}-\\left(\ny_{2}-y_{1}\\right) ^{2}\\right] }{\\left\\vert \\mathbf{r}_{2\\bot }-\\mathbf{r\n_{1\\bot }\\right\\vert ^{4}}  \\notag \\\\\n&&+\\frac{\\left( z_{1}-z_{2}\\right) \\left( x_{2}-x_{1}\\right) ^{2}}\n\\left\\vert \\mathbf{r}_{1}-\\mathbf{r}_{2}\\right\\vert ^{3}\\left\\vert \\mathbf{r\n_{2\\bot }-\\mathbf{r}_{1\\bot }\\right\\vert ^{2}}.\n\\end{eqnarray\nBy the Stokes' theorem, the line integral of $P_{x}$ can be expressed as a\ndouble integral\n\n\\begin{eqnarray}\nP_{x} &=&\\frac{1}{4\\pi }\\iint\\nolimits_{\\mathrm{S}}[\\frac{3\\left(\nx_{2}-x_{1}\\right) ^{2}-\\left\\vert \\mathbf{r}_{1}-\\mathbf{r}_{2}\\right\\vert\n^{2}}{\\left\\vert \\mathbf{r}_{1}-\\mathbf{r}_{2}\\right\\vert ^{5}}\\mathrm{d\ny_{2}\\mathrm{d}z_{2}  \\notag \\\\\n&&-\\frac{3\\left( x_{2}-x_{1}\\right) \\left( y_{1}-y_{2}\\right) \\allowbreak }\n\\left\\vert r_{2}-r_{1}\\right\\vert ^{5}}\\mathrm{d}z_{2}\\mathrm{d}x_{2}  \\notag\n\\\\\n&&-\\frac{3\\left( x_{1}-x_{2}\\right) \\left( z_{2}-z_{1}\\right) }{\\left\\vert\n\\mathbf{r}_{1}-\\mathbf{r}_{2}\\right\\vert ^{5}}\\mathrm{d}x_{2}\\mathrm{d\ny_{2}],\n\\end{eqnarray\nwhich results i\n\\begin{equation}\nP_{x}=B_{x}.\n\\end{equation\nSimilarly, the projection of polarization vector field $\\mathbf{P}$ and\nmagnetic field $\\mathbf{B}$ in the $y$ and $z$ direction can be calculated\nin the same way. Eventually, we can come to a conclusio\n\\begin{equation}\n\\mathbf{P}=\\mathbf{B}=\\frac{1}{4\\pi }\\oint_{\\mathrm{L}}\\frac{\\mathbf{r}_{2}\n\\mathbf{r}_{1}}{\\left\\vert \\mathbf{r}_{1}-\\mathbf{r}_{2}\\right\\vert ^{3}\n\\times \\mathrm{d}\\mathbf{r}_{2}.\n\\end{equation}\n\n\\acknowledgments This work was supported by the National Natural Science\nFoundation of China (under Grant No. 11874225)..\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Ice model} \\label{sec:mathmodel}\n\n\\subsection{The full Stokes (FS) equations}\n\\label{sec:Stokes}\nWe use the FS equations in 2D with coordinates $\\mathbf x=(x, z)^T$ for modeling of the flow of an ice sheet \\cite{Hutter83}.\nThese nonlinear partial differential equations (PDEs) in the interior of the ice $\\Omega$ are given by \n\\begin{equation}\n\\begin{cases}\n    \\nabla\\cdot\\mathbf{u}=0,\\\\\n    - \\nabla\\cdot{\\mathbf{\\mathbb{\\sigma}}} =\\rho \\mathbf{g},\n \\end{cases}\n  \\label{eq:FS}\n\\end{equation}\n  where the stress tensor is ${\\mathbf{\\mathbb{\\sigma}}} = 2\\eta(\\mathbf{u})\\mathbf{\\mathbb{\\tau}}(\\mathbf{u})-p\\mathbb{I}$. The symmetric strain rate tensor is defined by\n\\begin{equation}\\label{eq:taudef}\n  \\mathbf{\\mathbb{\\tau}}(\\mathbf{u})=\\frac{1}{2}(\\nabla\\mathbf{u}+\\nabla\\mathbf{u}^T)=\\left(\\begin{array}{cc}\\tau_{11}&\\tau_{12}\\\\\\tau_{12}&\\tau_{22}\\end{array}\\right),\n\\end{equation}\n$\\mathbb{I}$ is the identity matrix, and the viscosity is defined by Glen's flow law\n\\begin{equation}\\label{eq:visc}\n  \\eta(\\mathbf{u})=\\frac{1}{2}\\left(\\mathcal{A}(T^\\prime)\\right)^{-\\frac{1}{n}}\\mathbf{\\mathbb{\\tau}}_e^{\\frac{1-n}{n}},\\qquad \\mathbf{\\mathbb{\\tau}}_e = \\sqrt{\\frac{1}{2}\\text{tr}(\\mathbf{\\mathbb{\\tau}}(\\mathbf{u})\\mathbf{\\mathbb{\\tau}}(\\mathbf{u}))}.\n\\end{equation}\n\nHere ${\\mathbf{u}}=(u, w)^T$ is the vector of velocities, $\\rho$ is the density of the ice, $p$ denotes the pressure, and the gravitational acceleration in the $z$-direction is denoted by ${\\bf g}$. The rate factor $\\mathcal{A}(T^\\prime)$ describes how the viscosity depends on the pressure melting point corrected temperature $T^\\prime$. For isothermal flow assumed here, the rate factor $\\mathcal{A}$ is constant. Finally, $n$ is usually taken to be 3.\n\n\n\\subsection{Boundary conditions}\n\\begin{figure}[htbp]\n\\center\n  \\includegraphics[width=0.45\\textwidth]{.\/Figures\/GL} \n  \\caption{A two dimensional schematic view of a marine ice sheet.}\n\\label{fig:ice}\n\\end{figure}\n\n\nAt the boundary $\\Gamma$ of the ice we define the normal outgoing vector $\\mathbf{n}$ and tangential vector $\\mathbf{t}$, see Figure~\\ref{fig:ice}.\nIn a 2D case considered here, $y$ is constant in the figure. The upper boundary is denoted by $\\Gamma_s$ and the lower boundary is $\\Gamma_b$.\nAt $\\Gamma_s$ and $\\Gamma_{bf}$, the floating part of $\\Gamma_b$, we have that \n\\begin{equation}\n{\\mathbf{\\mathbb{\\sigma}}}{\\mathbf{n}}=\\mathbf{f}_s.\n\\label{eq:bc_s}\n\\end{equation}\nThe ice is stress-free at $\\Gamma_s$, $\\mathbf{f}_s=0$, and $\\mathbf{f}_s=-p_w\\mathbf{n}$ at the ice\/ocean interface $\\Gamma_{bf}$ where $p_w$ is the\nwater pressure. Let\n\\[\n   \\mathbf{\\mathbb{\\sigma}}_{\\mathbf{n}\\mathbf{t}}=\\mathbf{t}\\cdot\\mathbf{\\mathbb{\\sigma}}\\mathbf{n},\\; \\mathbf{\\mathbb{\\sigma}}_{\\mathbf{n}\\bn}=\\mathbf{n}\\cdot\\mathbf{\\mathbb{\\sigma}}\\mathbf{n},\\; u_\\mathbf{t}=\\mathbf{t}\\cdot\\mathbf{u}.\n\\]\nThen for the slip boundary $\\Gamma_{bg}$, the part of $\\Gamma_b$ where the ice is grounded, we have a friction law for the sliding ice\n  \\begin{equation}\n      {\\mathbf{\\mathbb{\\sigma}}}_{\\mathbf{n}\\mathbf{t}} + \\beta(\\mathbf{u},\\mathbf x) u_\\mathbf{t}=0,\\quad u_\\mathbf{n}=\\mathbf{n}\\cdot\\mathbf{u}=0, \\quad -{\\mathbf{\\mathbb{\\sigma}}}_{\\mathbf{n}\\bn}\\geq p_w.       \\label{eq:BCGI}\n    \\end{equation} \nThe type of friction law is determined by the friction coefficient $\\beta$.\nThere is a balance between ${\\mathbf{\\mathbb{\\sigma}}}_{\\mathbf{n}\\bn}$ and $p_w$ at $\\Gamma_{bf}$ and the contact is friction-free, $\\beta=0$,\n            \\begin{equation}\n           {\\mathbf{\\mathbb{\\sigma}}}_{\\mathbf{n}\\mathbf{t}} = 0, \\qquad -{\\mathbf{\\mathbb{\\sigma}}}_{\\mathbf{n}\\bn}= p_w.\n              \\label{eq:BCFI}\n          \\end{equation}\nThe GL is located where the boundary condition switches from $\\beta>0$ and $u_\\mathbf{n}=0$ on $\\Gamma_{bg}$ to $\\beta=0$ and a free $u_\\mathbf{n}$ on $\\Gamma_{bf}$. In 2D,\nthe GL is the point $(x_{GL}, z_{GL})$ between $\\Gamma_{bg}$ and $\\Gamma_{bf}$. \n\nWith the ocean surface at $z=0$, $p_w=-\\rho_w g z_b$ where $\\rho_w$ is the density of sea water, $z_b$ is the $z$-coordinate of $\\Gamma_b$, and $g$ is the gravitation constant.           \n\n\n\\subsection{The free surface equations}\n\\label{sec:height}\n\nThe boundaries $\\Gamma_s$ and $\\Gamma_b$ are time-dependent and move according to two free surface equations. The boundary $\\Gamma_{bg}$ follows the\nfixed bedrock with coordinates $(x, b(x))$.\n\nThe $z$-coordinate of the free surface position $z_s(x,t)$ at $\\Gamma_s$ (see Fig. \\ref{fig:ice}) is the solution of an advection equation\n\\begin{equation}\n  \\frac{\\partial z_s}{\\partial t}+u_s \\frac{\\partial z_s}{\\partial x}-w_s=a_s,\n\\label{eq:freeSurface}\n\\end{equation}\nwhere $a_s$ denotes the net surface accumulation\/ablation of ice and ${\\mathbf{u}}_s=(u_s, w_s)^T$ the velocity at the free surface in contact with the atmosphere. Similarly, the $z$-coordinate for the lower surface $z_b$ of the floating ice at $\\Gamma_{bf}$ satisfies\n\\begin{equation}\n  \\frac{\\partial z_b}{\\partial t}+u_b \\frac{\\partial z_b}{\\partial x}-w_b=a_b,\n\\label{eq:lowerSurface}\n\\end{equation}\nwhere $a_b$ is the net accumulation\/ablation at the lower surface and ${\\mathbf{u}}_b=(u_b, w_b)^T$ the velocity of the ice at $\\Gamma_{bf}$. On $\\Gamma_{bg}$, $z_b=b(x)$.\n\nThe thickness of the ice is denoted by $H=z_s-z_b$ and depends on $(x, t)$. \n\n\\subsection{The solution close to the grounding line}\n\\label{sec:GLsol}\n\nThe 2D solution of the FS equations in Eq. \\eqref{eq:FS} with a constant viscosity, $n=1$ in Eq. \\eqref{eq:visc}, is expanded in small parameters in \\cite{Schoof11}. The solutions in different\nregions around the GL are connected by matched asymptotics. Upstream of the GL at the bedrock, $x<x_{GL}$, the leading terms in the expansion satisfy\na simple equation in scaled variables close to the GL. Across the GL, $u$, the flux of ice $uH$, and the depth integrated normal or longitudinal stress $\\tau_{11}$ in Eq. \\eqref{eq:taudef} are continuous.\nBy adding higher order terms, it is shown that the upper surface slope is continuous and Archimedes' flotation condition\n\\begin{equation}\n  H\\rho=-z_b\\rho_w\n\\label{eq:flot}\n\\end{equation}\nis not satisfied immediately downstream of the GL.\nA rapid variation in the vertical velocity $w$ in a short interval at the GL causes oscillations in the upper surface as observed also in FS simulations in \\cite{Durand09b}.\n\nIn \\cite[Ch. 4.3]{Schoof11}, the solution to the FS in 2D is expanded in two parameters $\\nu$ and $\\epsilon$. The aspect ratio of the ice $\\nu$ is the quotient\nbetween a typical scale of the height of the ice $\\mathcal{H}$ and a length scale $\\mathcal{L}$, $\\nu=\\mathcal{H}\/\\mathcal{L}$, and $\\epsilon$\nis $\\nu$ times the quotient between the longitudinal and the shear stresses $\\tau_{11}$ and $\\tau_{12}$ in Eq. \\eqref{eq:taudef}. \nIf $\\nu^{5\/2}\\ll \\epsilon\\ll 1$ then in a boundary layer close to the GL and $x<x_{GL}$ the \nleading terms in the solution in scaled variables satisfy\n\\begin{equation}\n  \\tau_{22}-p={\\sigma}_{22}=\\rho g(z-z_s).\n\\label{eq:perturbsol}\n\\end{equation}\nOn floating ice $\\tau_{22}-p+p_w=0$ and the flotation criterion Eq. \\eqref{eq:flot} is fulfilled, and on the bedrock $\\tau_{22}-p+p_w<0$, see Eq. \\eqref{eq:bc_s} and \\eqref{eq:BCFI}.\n\nIntroduce the notation\n\\begin{equation}\n  \\chi(x, z)=\\tau_{22}-p+p_w=\\rho g(z-z_s(x))-\\rho_wgz_b(x),\n\\label{eq:chidef}\n\\end{equation}\nand approximate $z_s$ and $z_b$ linearly in $x$ in the vicinity of $x_{GL}$ and let $H_{bw}$ be the thickness of the ice below the water surface. \nThen \n\\begin{equation}\n  \\chi(x,z_b)=-g(\\rho H-\\rho_w H_{bw})\n\\label{eq:chidef2}\n\\end{equation}\nis linear in $x$. If $x<x_{GL}$ then $\\chi<0$ in the neighborhood of $x_{GL}$ on $\\Gamma_{bg}$ and if $x>x_{GL}$ then $\\chi=0$ and Eq. \\eqref{eq:flot} holds true\non $\\Gamma_{bf}$.\nIn numerical experiments with the linear FS $(n=1)$ in \\cite{NoWi08}, $\\chi(x, z_b)$ in the original variables varies linearly in $x$ for $x<x_{GL}$. \nIn Sect. \\ref{sec:subgrid}, $\\chi(x, z_b)$ is used to estimate the GL position.\n\n\n\\section{Discretization by FEM}\n\\label{sec:discr}\n\nIn this section we state the weak form of Eq. \\eqref{eq:FS} and introduce the spatial FEM discretization used for Eq. \\eqref{eq:FS} and give the time-discretization of Eq. (\\ref{eq:freeSurface}) and (\\ref{eq:lowerSurface}). \n\n\\subsection{The weak form of the FS equations}\n\\label{sec:weakFS}\n\nWe start by defining the mixed weak form of the FS equations.\nIntroduce $k=1+1\/n$, $k^*=1+n$ and the spaces\n    \\begin{equation}\\label{eq:spaces}\n        {\\boldsymbol{V}}_k= \\{\\mathbf{v}:\\mathbf{v}\\in (W^{1,k}(\\Omega))^2 \\}, \\quad\n        Q_{k^*}= \\{q: q\\in L^{k^*}(\\Omega)\\},\n    \\end{equation}\nsee, e.g. \\cite{chen2013well,martin2014four}.\nThe weak solution $(\\mathbf{u},p)$ of Eq. \\eqref{eq:FS} is obtained as follows.\nFind $(\\mathbf{u},p)\\in{\\boldsymbol{V}}_k\\times Q_{k^*}$ such that for all $(\\mathbf{v},q)\\in \\boldsymbol{V}_k\\times Q_{k^*}$ the equation\n  \\begin{equation}\\label{eq:FSweakform}\n    A((\\mathbf{u}, p), (\\mathbf{v},q)) +B_\\Gamma(\\mathbf{u}, p, \\mathbf{v}) + B_\\mathcal{N}(\\mathbf{u}, \\mathbf{v}, q) =F(\\mathbf{v}),\n  \\end{equation} \n is satisfied, where \n  \\begin{equation}\n    \\begin{split}\n      &A((\\mathbf{u}, p), (\\mathbf{v},q))=\\int_{\\Omega}2\\eta(\\mathbf{u})\\mathbf{\\mathbb{\\tau}}(\\mathbf{u}) :\\mathbf{\\mathbb{\\tau}}(\\mathbf{v})\\ \\ \\text{d}\\fatx  - b(\\mathbf{u},q) -b(\\mathbf{v}, p),\\\\\n      &b(\\mathbf{u},q) = \\int_{\\Omega}q\\nabla\\cdot \\mathbf{u}\\ \\ \\text{d}\\fatx,\\\\\n      &B_\\Gamma(\\mathbf{u}, p, \\mathbf{v})= \\int_{\\Gamma_{bg}} \\left(-\\mathbf{\\mathbb{\\sigma}}_{\\mathbf{n}\\bn}(\\mathbf{u}, p) \\mathbf{n}\\cdot\\mathbf{v} +\\beta  \\mathbf{u} \\cdot\\mathbf{v}\\right)\\,\\ \\text{d}s,\\\\\n      &   B_\\mathcal{N}(\\mathbf{u}, \\mathbf{v}, q) = -\\int_{\\Gamma_{bg}} \\mathbf{\\mathbb{\\sigma}}_{\\mathbf{n}\\bn}(\\mathbf{v}, q) \\mathbf{n}\\cdot\\mathbf{u} \\ \\text{d}s +\\gamma_0\\int_{\\Gamma_{bg}}\\frac{1}{h}(\\mathbf{n}\\cdot\\mathbf{u})(\\mathbf{n}\\cdot\\mathbf{v}) \\ \\text{d}s,\\\\\n      &F(\\mathbf{v})=\\int_{\\Omega}\\rho\\mathbf g\\cdot\\mathbf{v}\\ \\ \\text{d}\\fatx -\\int_{\\Gamma_{bf}} p_w\\mathbf{n}\\cdot\\mathbf{v}\\,\\ \\text{d}s. \\nonumber  \n    \\end{split}\n  \\end{equation}\n\nThe last term in $B_\\mathcal{N}$ is added in the weak form in Nitsche's method \\cite{Nitsche} to impose the Dirichlet condition $u_\\mathbf{n}=0$ weakly on $\\Gamma_{bg}$. \nIt can be considered as a penalty term. \nThe size of the positive parameter $\\gamma_0$  depends on the application and $h$ is a measure of the mesh size on $\\Gamma_b$. \nThe first term in $B_{\\mathcal{N}}$ symmetrizes the boundary term $B_\\Gamma+B_\\mathcal{N}$ on $\\Gamma_{bg}$ and vanishes when $u_\\mathbf{n}=0$. \n\n\n\n\\subsection{The discretized FS equations}\n\\label{sec:discrFS}\n\nWe employ linear Lagrange elements with Galerkin Least Square (GLS) stabilization \\cite{franca1992stabilized,helanow2018stabilized} to avoid spurious oscillations in the pressure using the standard setting in Elmer\/ICE \\cite{ElmerDescrip} approximating solutions in the spaces ${\\boldsymbol{V}}_k$ and $ M_{k^*}$ in Eq. \\eqref{eq:spaces}.\n\nThe mesh is constructed from a footprint mesh on the bottom surface and then extruded with the same number of layers in the vertical direction according to the thickness of the ice.\nTo simplify the implementation in 2D, the footprint mesh on the bottom surface consists of $N+1$ nodes $x_i,\\; i=0,\\ldots,N,$ with a constant mesh size $\\Delta x$.\n\nIn general, the GL is somewhere in the interior of an element $\\mathcal{E}_i=[x_i,\\, x_{i+1}]$ and it crosses the element boundaries as it moves forward in the advance phase and backward in the retreat phase of the ice. The advantage with Nitsche's way of formulating the boundary conditions is that if $x_{GL}\\in \\mathcal{E}_i$ then the\nboundary integral over $\\mathcal{E}_i$ can be split into two parts in Eq. \\eqref{eq:FSweakform} such that $[x_i,\\, x_{GL}]\\in \\Gamma_{bg}$ and $[x_{GL},\\, x_{i+1}]\\in \\Gamma_{bf}$ as follows\n \\begin{equation}\\label{eq:Nitscheint}\n\\begin{split}\n   &{\\int_{\\mathcal{E}_i} B_\\Gamma +B_\\mathcal{N}\\, \\ \\text{d}s} \\\\\n   =& {\\int_{[x_i,\\, x_{GL}]}-({\\mathbf{\\mathbb{\\sigma}}}_{\\mathbf{n}\\bn}(\\mathbf{u}, p) \\mathbf{n}\\cdot\\mathbf{v}+{\\mathbf{\\mathbb{\\sigma}}}_{\\mathbf{n}\\bn}(\\mathbf{v}, q) \\mathbf{n}\\cdot\\mathbf{u})} \\\\\n   &{+ \\beta\\mathbf{u}\\cdot\\mathbf{v}+\\frac{\\gamma_0}{h}(\\mathbf{n}\\cdot\\mathbf{u})(\\mathbf{n}\\cdot\\mathbf{v})\\, \\ \\text{d}s+\\int_{[x_{GL},\\, x_{i+1}]} p_w\\mathbf{n}\\cdot\\mathbf{v}\\, \\ \\text{d}s.}\n\\end{split}\n  \\end{equation}\nThere is a change of boundary conditions in the middle of the element $\\mathcal{E}_i$ where the GL is located. With a strong formulation of $u_\\mathbf{n}=0$ the basis functions in ${\\boldsymbol{V}}_s$ share\nthis property and the condition changes from the grounded node $x_i$ where the basis function satisfies $u_\\mathbf{n}=0$ and the floating node at $x_{i+1}$ with a free $u_\\mathbf{n}$ without \ntaking the position of the GL inside $\\mathcal{E}_i$ into account.\n\nThe resulting system of non-linear equations form a nonlinear complementarity problem \\cite{CKPS}. The distance $d$ between the base of the ice and the bedrock at time $t$ and at\n$x$ is $d=z_b(x, t)-b(x)\\ge 0$. If $d>0$ on $\\Gamma_{bf}$ then the ice is not in contact with the bedrock and $\\mathbf{\\mathbb{\\sigma}}_{\\mathbf{n}\\bn}+p_w=0$ and if $\\mathbf{\\mathbb{\\sigma}}_{\\mathbf{n}\\bn}+p_w<0$ on $\\Gamma_{bg}$ then the ice and the bedrock are in contact and $d=0$. Hence, the\ncomplementarity relation in the vertical direction is\n \\begin{equation}\\label{eq:complv}\n\\begin{array}{ll}\n   z_b(x, t)-b(x)\\ge 0,\\; \\mathbf{\\mathbb{\\sigma}}_{\\mathbf{n}\\bn}+p_w\\le 0,\\\\ \n  (z_b(x, t)-b(x))(\\mathbf{\\mathbb{\\sigma}}_{\\mathbf{n}\\bn}+p_w)=0\\;\\textrm{on}\\;\\Gamma_b.\n\\end{array}\n  \\end{equation}\nThe contact friction law is such that $\\beta>0$ when $x<x_{GL}$ and  $\\beta=0$ when $x>x_{GL}$. The complementarity relation along the slope at $x$ is then the non-negativity of $d$ and \n \\begin{equation}\\label{eq:compls}\n   \\beta\\ge 0,\\; \\beta(x, t)(z_b(x, t)-b(x))=0\\;\\textrm{on}\\;\\Gamma_b.\n  \\end{equation}\nIn particular, these relations are valid at the nodes $x=x_j$, $j=0,1,\\dots,N$.\n \n\nThe complementarity condition also holds for $u_\\mathbf{n}$ and $\\sigma_{\\mathbf{n}\\bn}$ such that\n \\begin{equation}\\label{eq:complu}\n\\begin{array}{ll}\n   \\mathbf{\\mathbb{\\sigma}}_{\\mathbf{n}\\bn}+p_w\\le 0,\\\\ \n  u_\\mathbf{n}(\\mathbf{\\mathbb{\\sigma}}_{\\mathbf{n}\\bn}+p_w)=0\\;\\textrm{on}\\;\\Gamma_b,\n\\end{array}\n  \\end{equation}\nwithout any sign constraint on $u_\\mathbf{n}$ except for the retreat phase when the ice leaves the ground and $u_\\mathbf{n}<0$. \n\n\nSimilar implementations for contact problems using Nitsche's method are found in \\cite{chouly2017overview,chouly2017nitsche}, where the unknowns in the PDEs are the displacement fields\ninstead of the velocity in Eq. \\eqref{eq:FS}.\nAnalysis in \\cite{chouly2017overview} suggests that Nitsche's method for the contact problem can provide a stable numerical solution with an optimal convergence rate.\n\nThe nonlinear equations for the nodal values of $\\mathbf{u}$ and $p$ are solved by Newton iterations. The system of linear equations in every Newton iteration is solved iteratively by using the\nGeneralised Conjugate Residual (GCR) method in Elmer\/ICE. The condition on $d_j$ in a node $x_j$ is used for a so called grounded mask, which is computed at each timestep and not changed during the nonlinear iterations.\n\n\n\n\\subsection{Discretization  of  the advection equations}\\label{sec:updlower}\n\nThe advection equations for the moving ice boundary in Eq. \\eqref{eq:freeSurface} and \\eqref{eq:lowerSurface} are discretized in time by a finite difference method and in\nspace by FEM with linear Lagrange elements for $z_s$ and $z_b$. A stabilization term is added,  making the spatial discretization behave \nlike an upwind scheme in the direction of the velocity as implemented in Elmer\/ICE.\n\n\nThe advection equations Eq. \\eqref{eq:freeSurface} and Eq. \\eqref{eq:lowerSurface} are integrated in time by a semi-implicit method of first order accuracy. \nLet $c=s$ or $b$. Then the solution is advanced\nfrom time $t^n$ to $t^{n+1}=t^n+\\Delta t$ with the timestep $\\Delta t$ by \n\\begin{equation}\\label{eq:zint}\n   z_c^{n+1}=z_c^n+\\Delta t(a_c^n-u_c^n \\frac{\\partial{z_{c}^{n+1}}}{\\partial x}+w_c^n).\n\\end{equation}\nThe spatial derivative of $z_c$ is approximated by FEM. A system of linear equations is solved at $t^{n+1}$ for $z_c^{n+1}$. This time discretization and its properties are \ndiscussed in \\cite{cheng2017accurate}.\n\nA stability problem in $z_b$ is encountered in the boundary condition at $\\Gamma_{bf}$ in \\cite{Durand09b}. \nIt is solved by expressing $z_b$ in $p_w$ at $\\Gamma_{bf}$ with a damping term in \\cite{Durand09b}.\nAn alternative interpretation of the idea in \\cite{Durand09b} and an explanation follow below.\n\n\nThe relation between $u_\\mathbf{n}$ and $u_\\mathbf{t}$ at $\\Gamma_{bf}$ and $\\mathbf{u}_b=\\mathbf{u}(x, z_b(x))$ is\n\\begin{equation}\n  \\mathbf{u}_b=\\left(\\begin{array}{c} u_b \\\\ w_b\\end{array}\\right)=\\left(\\begin{array}{c} z_{bx} \\\\ -1\\end{array}\\right)\\frac{u_\\mathbf{n}}{\\sqrt{1+z_{bx}^2}}\n        +\\left(\\begin{array}{c} 1 \\\\ z_{bx}\\end{array}\\right)\\frac{u_\\mathbf{t}}{\\sqrt{1+z_{bx}^2}},\n\\label{eq:udef}\n\\end{equation}\nwhere $z_{bx}$ denotes $\\partial z_b\/\\partial x$. Insert $u_b$ and $w_b$ from Eq. \\eqref{eq:udef} into Eq. \\eqref{eq:lowerSurface} to obtain\n\\begin{equation}\n  \\frac{\\partial z_b}{\\partial t}=a_b-u_\\mathbf{n}\\sqrt{1+z_{bx}^2},\n\\label{eq:zbeq2}\n\\end{equation}\nInstead of discretizing Eq. \\eqref{eq:zbeq2} explicitly at $t^n$ with $u_\\mathbf{n}^{n-1}$ to determine $p_w^n$, the base coordinate is updated implicitly\n\\begin{equation}\n  z_{b}^n=z_{b}^{n-1}+\\Delta t\\left(a_b^n-u_\\mathbf{n}^n\\sqrt{1+z_{bx}^2}\\right)\n\\label{eq:zbimpl}\n\\end{equation}\nin the solution of Eq. \\eqref{eq:FSweakform}.\n\nAssume that $z_{bx}$ is small.\nThe timestep restriction in Eq. \\eqref{eq:zbimpl} is estimated by considering a 2D slab of the floating ice of width $\\Delta x$ and thickness $H$. Newton's law of motion yields\n\\[\n    M \\dot{u}_\\mathbf{n}= M g-\\Delta x p_w,\n\\]\nwhere $M=\\Delta x(z_s-z_b)\\rho$ is the mass of the slab. Divide by $M$, integrate in time for $u_\\mathbf{n}(t^m)$, let $m=n$ or $n-1$, and approximate the integral by the trapezoidal rule for the quadrature to obtain\n\\[\n\\begin{split}\n      u_\\mathbf{n}(t^m)&=\\displaystyle{\\int_0^{t^m} g+\\frac{g\\rho_w}{\\rho}\\frac{z_b}{z_s-z_b}\\,\\textrm{d}s} \\\\\n      &\\approx \\displaystyle{gt^m+\\frac{g\\rho_w}{\\rho}\\sum_{i=0}^m\\alpha_i\\frac{z_b^i}{z_s^i-z_b^i}\\Delta t,}\\\\\n\\end{split}\n\\]  \n\\[\n      \\alpha_i=0.5, i=0, m,\\quad  \\alpha_i=1, i=1,\\ldots,m-1. \n\\]\nThen insert $u_\\mathbf{n}^m$ into Eq. \\eqref{eq:zbimpl}. All terms in $u_\\mathbf{n}^m$ from timesteps $i<m$ are collected in the sum $\\Delta t F^{m-1}$. \nThen Eq. \\eqref{eq:zbimpl} can be written\n\\begin{equation}\n  z_{b}^n=z_{b}^{n-1}-\\Delta t^2\\frac{g\\rho_w}{2\\rho}\\frac{z_b^m}{z_s^m-z_b^m}+\\Delta t\\left(a_b^n-gt^{m}-\\Delta t F^{m-1}\\right).\n\\label{eq:zbimpl2}\n\\end{equation}\nFor small changes in $z_b$ in Eq. \\eqref{eq:zbimpl2}, the explicit method with $m=n-1$ is stable when $\\Delta t$ is so small that\n\\begin{equation}\n |1-\\Delta t^2\\frac{g\\rho_w}{2H\\rho}|\\le 1.\n\\label{eq:stabconde}\n\\end{equation}\nWhen $H=100$ m on the ice shelf, $\\Delta t< 6.1$ s which is far smaller than the stable steps for Eq. \\eqref{eq:zint}.\nChoosing the implicit scheme with $m=n$, the bound on $\\Delta t$ is \n\\begin{equation}\n1\/|1+\\Delta t^2\\frac{g\\rho_w}{2H\\rho}|\\le 1,\n\\label{eq:stabcondi}\n\\end{equation}\ni.e. there is no bound on positive $\\Delta t$ for stability but accuracy will restrict $\\Delta t$. \n\n\nMuch longer stable timesteps are possible at the surface and the base of the ice with a semi-implicit method Eq. \\eqref{eq:zint}\nand a fully implicit method Eq. \\eqref{eq:zbimpl} compared to an explicit method. \nFor example, the timestep for the problem in Eq. \\eqref{eq:zint} with 1~km mesh size can be up to a couple of months.\nTherefore, we use the scheme in Eq. \\eqref{eq:zint} for Eq. \\eqref{eq:freeSurface} and \\eqref{eq:lowerSurface} and \nthe scheme in Eq. \\eqref{eq:zbimpl} for Eq. \\eqref{eq:zbeq2} and $p_w$ as in \\cite{Durand09b}. The difference between the approximations of $z_b$ in Eq. \\eqref{eq:zint} and \\eqref{eq:zbimpl} is of $\\mathcal{O}(\\Delta t^2)$.\n\n\n\\section{Subgrid modeling around grounding line}\n\\label{sec:subgrid}\n\nThe basic idea of the subgrid method for the FS equations in this paper follows the GL parameterization for SSA in \\cite{Seroussi14} and the analysis for FS in \\cite{Schoof11}.\nThe GL is located at the position where the ice is on the ground and the flotation criterion is perfectly satisfied such that $\\sigma_{\\mathbf{n}\\bn}=-p_w$.\nIn the Stokes equations, the hydrostatic assumption may not be valid, so the exact GL position can not be determined by simply checking the total thickness of the ice $H$ against the depth below sea level $H_{bw}=-z_b$.\nInstead, the flotation criterion is computed by comparing the water pressure with the normal stress component orthogonal to the boundary as indicated by the first order analysis in Sect. \\ref{sec:GLsol}. The indicator is here defined by $\\chi(x)=\\sigma_{\\mathbf{n}\\bn}+p_w$ which vanishes on the floating ice and is approximately $\\tau_{22}-p+p_w$ and negative on the ground since the slope of the bedrock is small.\n\n\nTypically, at the lower surface of the floating ice where $z_b(x,t)> b(x)$, as the blue line in Fig. \\ref{fig:GL}, the boundary conditions are given by Eq. \\eqref{eq:BCFI}, and where the ice is in contact with the bedrock, as the red line in Fig. \\ref{fig:GL}, the boundary conditions are given by Eq. \\eqref{eq:BCGI}. \nHowever, there is another case as shown in Fig. \\ref{fig:GL2} when the net force at $x_i$ is pointing inward, namely $\\sigma_{\\mathbf{n}\\bn}(x_i)+p_w(x_i)>0$.\nThen, the floating boundary condition Eq. \\eqref{eq:BCFI} should be imposed up until the node $x_{i-1}$.\nThis can happen at some point due to the low spatial and temporal resolutions, but the node $x_i$ will move upward as long as $\\mathbf{u}\\cdot\\mathbf{n}<0$, or the net force switches signs and the condition transforms into the case in Fig. \\ref{fig:GL} when $\\sigma_{\\mathbf{n}\\bn}(x_i)+p_w(x_i)<0$.\nDenote the situation in Fig. \\ref{fig:GL} by case {\\romannumeral 1}, and the one in Fig. \\ref{fig:GL2} by case {\\romannumeral 2}.\nWe call the node `grounded' when it is in contact with the bedrock with net force from the ice pointing outward ($\\sigma_{\\mathbf{n}\\bn}+p_w<0$), and `floating' when the net force is pointing inward ($\\sigma_{\\mathbf{n}\\bn}+p_w\\geq0$).\nThe element which contains both grounded and floating nodes is called the GL element and the grounded node in it is called the last grounded node and the floating one is called the first floating node.\n\nIn coarse meshes, the true position of the GL is generally not in one of the nodes, but usually between the last grounded and the first floating nodes. \nInstead of refining the mesh around GL, which would lead to very small time steps for stability reasons, we will here introduce a subgrid model for the GL element.\n\n\nWe let $\\chi(x)=\\sigma_{\\mathbf{n}\\bn}(x)+p_w(x)$ and assume that it is linear as in Eq. \\eqref{eq:chidef} to determine the position of the GL, $x_{GL}$, in the GL element. \nIn case {\\romannumeral 2}, the GL is located between $x_{i-1}$ and $x_i$ even though the whole element $[x_{i-1},x_i]$ is geometrically grounded.\nThe equation $\\chi(x_{GL})=0$ is solved by linear interpolation between $\\chi(x_{i-1})<0$ and $\\chi(x_i)>0$ yielding a unique solution satisfying $x_{i-1}<x_{GL}<x_i$, depicted as the red dot in the lower panel of Fig. \\ref{fig:GL2}.\n\n\nThere is a more complicated situation in case {\\romannumeral 1}, where $\\chi(x_i)<0$ but $\\chi(x_{i+1})=0$ due to the floating boundary condition.\nA correction of $\\chi$ is made by using $\\tilde{\\chi}(x)=\\sigma_{\\mathbf{n}\\bn}(x)+p_b(x)$ where $p_b(x)=-\\rho_w gb(x)$ is the water pressure on the bedrock.\nFor $x>x_i$, we have $b(x)<z_b(x)$ and $p_b(x)>p_w(x)$.\nTherefore, $\\tilde{\\chi}(x_{i+1})>\\chi(x_{i+1})=0$ and  $\\tilde{\\chi}(x_i)=\\chi(x_i)<0$.\nThen, a linear interpolation between $\\tilde{\\chi}(x_i)$ and $\\tilde{\\chi}(x_{i+1})$ guarantees a unique solution of $\\tilde{\\chi}(x_{GL})=0$ in the GL element $[x_i,x_{i+1}]$, see Fig. \\ref{fig:GL}.\nIn case {\\romannumeral 2}, $p_b$ can also be used since $p_b(x)=p_w(x)$ as long as the element is on the bedrock.\n\nConceptually, the linear interpolation of the function $\\tilde{\\chi}(x)$ can be considered separately by looking at the two linear functions $\\sigma_{\\mathbf{n}\\bn}(x)$ and $p_b(x)$. \nAs the GL always rests on the bedrock, $p_b(x_{GL})=p_w(x_{GL})$ is actually an exact representation of the water pressure imposed on the ice at GL, although geometrically $z_b(x_{GL})$ may not coincide with $b(x_{GL})$, especially on coarse meshes.\nThis also leads to the fact that the interpolated normal stress $\\sigma_{\\mathbf{n}\\bn}(x_{GL},z_b(x_{GL}))$ is a first order approximation of the normal stress at the exact GL position $(x_{GL},b(x_{GL}))$.\n\nThis correction is not necessary when the GL is advancing since the implicit treatment of the bottom surface is equivalent to additional water pressure at the stress boundary as discussed in Sect. \\ref{sec:updlower}.\n\n\n\nAfter the GL position is determined, the domains $\\Gamma_{bg}$ and $\\Gamma_{bf}$ are separated at $x_{GL}$ as in Eq. \\eqref{eq:Nitscheint} and the integrals are calculated with a high-order integration scheme as in \\cite{Seroussi14} to achieve a better resolution within the element shown in Figures \\ref{fig:GL} and \\ref{fig:GL2}.\nFor a smoother transition of $\\beta$ at $GL$, the slip coefficient is multiplied by 1\/2 at the whole GL element before integrating using the high order scheme.\n\n\n\nThe penalty term from Nitsche's method restricts the motion of the element in the normal direction. It should only be imposed on the element which is fully on the ground.\nOn the contrary, in case {\\romannumeral 1}, the GL element $[x_i,x_{i+1}]$ is not in contact with the bedrock as in Fig. \\ref{fig:GL}, so only the floating boundary condition should be used on the element $[x_i,x_{i+1}]$.\nAdditionally, the implicit representation of the bottom surface in Eq. \\eqref{eq:zbimpl} also implies that the case {\\romannumeral 2}\\, with retreating GL should be merged to case {\\romannumeral 1}\\, since the surface is leaving the bedrock and the normal velocity should not be forced to zero.\nTo summarize, Nitsche's penalty term should be imposed on all the fully grounded elements and partially on the GL element in the advance phase.\n\n\n\\begin{figure}[htbp]\n\\center\n  \\includegraphics[width=0.45\\textwidth,page=1]{.\/Figures\/subgrid} \n  \\caption{Schematic figure of Grounding Line in case {\\romannumeral 1}.\n  Upper panel: the last grounded and first floating nodes as defined in Elmer\/ICE. \n  Lower panel: linear interpolation to compute a more accurate position of the Grounding Line.}\n  \\label{fig:GL}\n\\end{figure}\n\n\\begin{figure}[htbp]\n\\center\n  \\includegraphics[width=0.45\\textwidth,page=2]{.\/Figures\/subgrid} \n  \\caption{Schematic figure of Grounding Line in case {\\romannumeral 2}.\n  Upper panel: the last grounded and first floating nodes as defined in Elmer\/ICE. \n  Lower panel: linear interpolation to compute a more accurate position of the Grounding Line.}\n  \\label{fig:GL2}\n\\end{figure}\n\n\nEquations (\\ref{eq:FS}), (\\ref{eq:freeSurface}), and (\\ref{eq:lowerSurface}) form a system of coupled nonlinear equations. They are solved in the same manner as in Elmer\/ICE v.8.3.\nThe $x_{GL}$ position is determined dynamically within every nonlinear iteration when solving the FS equations and the high order integrations are based on the current $x_{GL}$.\nThe nonlinear FS is solved with fixed-point iterations to $10^{-5}$ relative error with a limit of maximal 25 nonlinear iterations and the grounded condition is set if the distance between of the bottom surface and the bedrock is smaller than $10^{-3}$~m.\n\n\n\n\\section{Results} \\label{sec:results}\n\nThe numerical experiments follow the MISMIP benchmark \\cite{MISMIP} and comparison is made with the results in \\cite{gagliardini2016impact}.\nUsing the experiment MISMIP 3a, the setups are exactly the same as in the advancing and retreating simulations in \\cite{gagliardini2016impact}.\nThe experiments are run with spatial resolutions of $\\Delta x=4$~km, 2~km and 1~km with 20 vertical extruded layers.\nThe timestep is $\\Delta t=0.125$~year for all the three resolutions to eliminate time discretization errors when comparing different spatial resolutions.\n\n\nThe dependence on $\\gamma_0$ for the retreating ice is shown in Fig. \\ref{fig:gammas} with $\\gamma_0$ between $10^4$ and $10^9$.\nThe estimated GL positions do not vary with different choices of $\\gamma_0$ from $10^5$ to $10^8$ which suggests a suitable range of $\\gamma_0$.\nIf $\\gamma_0$ is too small ($\\gamma_0\\ll10^4$), oscillations appear in the estimated GL positions. \nIf $\\gamma_0$ is too large ($\\gamma_0\\gg10^8$), then more nonlinear iterations are needed for each time step.\nThe same dependency of $\\gamma_0$ is observed for the advance experiments and for different mesh resolutions as well.\nFor the remaining experiments, we fix $\\gamma_0=10^6$.\n\n\\begin{figure}[htbp]\n\\center\n  \\includegraphics[width=0.4\\textwidth]{.\/Figures\/MISMIP_3_gammas.pdf} \n  \\caption{The MISMIP 3a retreat experiment with $\\Delta x=1000$~m for different choices of $\\gamma_0$ in\n    the time interval $[0,10000]$ years.}\n\\label{fig:gammas}\n\\end{figure}\n\n\nThe GL position during 10000 years in the advance and retreat phases are displayed in Fig. \\ref{fig:MISMIP3} for different mesh sizes.\nThe range of the results from \\cite{gagliardini2016impact} with mesh resolutions $\\Delta x=25$ and $50$~m are shown as background shaded regions with colors purple and pink.\nWe achieve similar GL migration results both for the advance and retreat experiments with at least 20 times larger mesh sizes.\n\n\\begin{figure}[htbp]\n\\center\n  \\includegraphics[width=0.45\\textwidth]{.\/Figures\/MISMIP_3.pdf} \n  \\caption{The MISMIP 3a experiments for the GL position when $t\\in[0, 10000]$ with $\\Delta x=4000, 2000$ and $1000$~m for the advance (solid) and retreat (dashed) phases. \n  The shaded regions indicate the range of the results in \\cite{gagliardini2016impact} with $\\Delta x=50$~m in red and $\\Delta x=25$~m in blue.   }\n\\label{fig:MISMIP3}\n\\end{figure}\n\n\nWe observed oscillations at the top surface near the GL in all the experiments as expected from \\cite{Durand09b, Schoof11}.\nA zoom-in plot of the surface elevation with $\\Delta x=1$~km at $t=10000$ years is shown to the left in Fig. \\ref{fig:surfOsc}, where the red dashed line indicates the estimated GL position. \nObviously, the estimated GL position does not coincide with any nodes even at the steady state.\n\\begin{figure}[htbp]\n\\center\n  \\includegraphics[width=0.45\\textwidth]{.\/Figures\/retreatDetails} \n  \\caption{Details of the solutions for the retreat experiment with $\\Delta x=1$~km after 10000 years. \n  The solid dots represent the nodes of the elements and the vertical, red, dashed lines indicate the GL position.\n  \\emph{Left panel}: The oscillations at top surface near GL.\n  \\emph{Right panel}: The flotation criterion is evaluated by $H_{bw}\/H$. The ratio between $\\rho\/\\rho_w$ is drawn in a horizontal, purple, dash-dotted line. \n  }\n\\label{fig:surfOsc}\n\\end{figure}\n\n\nThe ratio between the thickness below sea level $H_{bw}$ and the ice thickness $H$ is shown in Fig. \\ref{fig:surfOsc}.\nThe horizontal, purple, dash-dotted line indicates the ratio of $\\rho\/\\rho_w$ and the estimated GL is located at the red, dashed line.\nThis result confirms that the hydrostatic assumption $H\\rho=H_{bw}\\rho_w$ is not valid in the FS equations for $x>x_{GL}$ close to the GL and at the GL position, cf. \\cite{Durand09b, Schoof11}. For $x<x_{GL}$ we have that $H_{bw}\/H<\\rho\/\\rho_w$ since $H_{bw}$ decreases and $H$ increases. The conclusion from numerical experiments in \\cite{van2018dynamically} is that the hydrostatic assumption and the SSA equations approximate the FS equations well for the floating ice beginning at a short distance away from the GL. \n\n\n\nThe top and bottom surface velocity solutions from the retreat experiment are shown in Fig. \\ref{fig:velocity} with $\\Delta x=1$~km after 10000 years.\nThe horizontal velocities on the two surfaces are similar with negligibly small differences on the floating ice.\nThe vertical velocities $w$ on the top (orange line) and bottom surface (blue line) at the GL are almost discontinuous as analyzed in \\cite{Schoof11}.\nWith the subgrid method, the rapid variation is resolved by the 1 km mesh size.\n\n\n\\begin{figure}[htbp]\n\\center\n  \\includegraphics[width=0.45\\textwidth]{.\/Figures\/retreatVelocity.eps} \n  \\caption{The velocities $u$ (upper panel) and $w$ (lower panel) on the top (orange) and bottom (blue) surface of the ice in the retreat experiment with $\\Delta x=1$~km after 10000 years. The red, dashed line indicates the GL position. The vertical velocity $w$ is zoomed-in close to the GL.}\n\\label{fig:velocity}\n\\end{figure}\n\n\n\\section{Discussion}\\label{sec:disc}\n\nSeroussi et al \\cite{Seroussi14} describe four different subgrid models for the friction in SSA and evaluate them in a FEM discretization on a triangulated, planar domain. \nThe flotation criterion is applied\nat the nodes of the triangles. Depending on how many of the nodes that are floating, the amount of friction in the triangle is determined. Also, a higher order polynomial integration \nover the triangles in FEM allows an inner structure in the triangular element.\n\nOur method can be extended to a triangular mesh covering $\\Gamma_b$ in the following way. \nThe condition on $\\chi$ in Eq. \\eqref{eq:chidef} is applied on the edges of each triangle $\\mathcal{T}$ in the mesh. If $\\chi<0$ in all three nodes \nthen $\\mathcal{T}$ is grounded. If $\\chi\\ge 0$ in all nodes then $\\mathcal{T}$ is floating. The GL passes inside \n$\\mathcal{T}$ if $\\chi$ has a different sign in one of the nodes. Then the GL crosses the two edges where\n$\\chi<0$ in one node and $\\chi\\ge 0$ in the other node. In this way, a continuous reconstruction of a piecewise linear GL is possible on $\\Gamma_b$. \nThe FEM approximation is modified in the same manner as in Sect.~\\ref{sec:subgrid} with Nitsche's method. \n\nAn alternative to subgrid modeling is to introduce dynamic adaptation of the mesh on $\\Gamma_b$ with a refinement at the GL as in e.g. \\cite{Cornford13, Drouet13, Gladstone10a}.\nIn general, a fine mesh is needed along the GL and in an area surrounding it. Since the GL moves long distances at least in simulations of palaeo-ice sheets, \nthe adaptation should be dynamic, permit refinement and coarsening \nof the mesh, and be based on some estimate of the model inaccuracy. Furthermore, shorter timesteps are necessary for stability when the mesh size is smaller in a mesh adaptive  method. Introducing a time dependent mesh adaptivity into an existing code requires a substantial coding effort and will\nincrease the computational work considerably. Subgrid modeling is easier to implement and the increase in computing time is small. \n\n\n\\conclusions\\label{sec:concl}\n\nSubgrid models at the GL have been developed and tested in the SSA model for 2D flow in \\cite{Gladstone10b} and for 3D flow in \\cite{Seroussi14}, for the friction in the \nvertically integrated model BISICLES \\cite{Cornford13} for 3D flow in \\cite{Cornford16},\nand for the PISM model mixing SIA with SSA in 3D in \\cite{Feldmann14}. Here we propose a subgrid model in 2D for the FS equations implemented in Elmer\/ICE that can be extended to 3D.\nThe mesh is static and the moving GL position within one element is determined by linear interpolation with an auxiliary function $\\tilde{\\chi}$ based on the theory in \\cite{Schoof11}. \nOnly in that element, the FEM discretization is modified.\n \nThe method is applied to the simulation of an ice sheet in 2D with an advancing GL and one with a retreating GL. The data for the tests are the same as in one of the MISMIP examples \\cite{MISMIP} and in \\cite{gagliardini2016impact}. Comparable results to \\cite{gagliardini2016impact} are obtained with subgrid modeling with more than 20 times larger mesh sizes. A larger mesh size also allows a longer timestep for the time integration. Without further knowledge of the basal conditions and detailed models at the GL, solving $\\tilde{\\chi}(x)=0$ provides a good approximation of the GL position.\n\n\n\n\n\\codeavailability{  \nThe FS sub-grid model is implemented based on Elmer\/ICE Version: 8. 3(Rev: f6bfdc9) with the scripts at \\url{https:\/\/github.com\/enigne\/elmerfem\/tree\/NitscheMethodAdvance} and \\url{https:\/\/github.com\/enigne\/elmerfem\/tree\/NitscheMethodRetreat}.\n}\n\n\\authorcontribution{GC developed the model code and performed the simulations. GC and PL contributed to the theory of the paper. GC, PL and LvS contributed to the development of the method and the writing of the paper} \n\n\\competinginterests{The authors declare that they have no conflict of interest.}\n\n\\begin{acknowledgements}\nThis work has been supported by Nina Kirchner's Formas grant 2017-00665 and the Swedish e-Science initiative eSSENCE.\nWe are grateful to Thomas Zwinger for advise and help in the implementation of the subgrid model in Elmer\/ICE.\nThe computations were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC) at the PDC Center for High Performance Computing, KTH Royal Institute of Technology.\n\\end{acknowledgements}\n\n\n\n\n\\bibliographystyle{copernicus}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec1}\nA challenge is an online competition that uses data, truth and metrics to evaluate the performance of automatic algorithms, submitted by participants, with respect to a research problem. In the medical image analysis field, close to 150 challenges have been organized thus far \\citep{Ginneken18}. \n\nWell designed challenges have the potential to unite and focus the field on creating solutions to important problems as well as generating new insights and identifying common themes for future research. Several breakthrough technologies can attribute the recognition of their capabilities to challenges. The ImageNet challenge revealed the utility of neural networks \\citep{Russakovsky2015}, and in the medical image analysis field the Camelyon Challenge, for example, showed that deep learning algorithms can, under certain conditions, outperform a panel of 11 pathologists in detecting lymph node metastases on whole-slide pathology images \\citep{bejnordi2017diagnostic}. Furthermore, the indirect impact of challenges should not be underestimated. For example, the Retrospective Image Registration Experiment challenge revealed the utility of mutual information \\citep{West97comparison}, and the NetFlix challenge boosted the field of recommender systems \\citep{gomez16netflix,Netflix17}. From the field of robotics it is also evident that competitions such as the RoboCup \\citep{kitano97robocup} can be a catalyst for innovation \\citep{hart18,pavez18} as well as interdisciplinary teamwork.  \n\nPoorly designed and poorly documented challenges, however, can equally impede a field, e.g. lead to false performance expectations and promote unfruitful research directions. The potential detriment of challenges has inspired a critical assessment of the proposal and review processes for challenges at the international Medical Image Computer and Computer-Assisted Intervention (MICCAI) conference and has led to a definition of best practices for challenge design \\citep{Reinke18, lena18}. The main criticism raised in that assessment is based on the observation that small changes to the underlying challenge data have the potential to drastically change the rank order in the leaderboard. Our framework emphasizes that this criticism is exacerbated by the practice of leaderboard climbing \\citep{Hardt15,whitehill17}, which is characterized by participants focusing on incrementally improving metric results rather than gaining new insights or solving the challenge's driving problem. \n\nChallenge leaderboards are valuable, but it is critical to understand the scope of a challenge when interpreting its leaderboard. Leaderboards do provide a quick overview of algorithm performance and can reveal trends and boost collaboration as well as competition; however, they are also constrained by the design of the underlying data, truth and metrics. These design constraints limit the conclusions that \\emph{can} be drawn from a challenge, i.e., a challenge's design limits its scope. The theoretical framework for challenge design that we are proposing can be used to clearly describe a challenge's scope, i.e., what conclusions can be and are resolved by a challenge. Thus, the aim of the framework is twofold: first, to guide challenge organizers in challenge design and description, and second, to clarify the scope of a challenge for challenge participants and the community at large. The latter is critical in order to moving beyond a challenge's leaderboard, i.e., to understanding the relevance of a challenge and identifying commonalities and differences across multiple challenges.\n\nWe propose that to understand and define the scope of a challenge, it is critical to distinguish two fundamental types of challenges: 1) an \\textbf{insight challenge} that can be used to gain insight into a research problem by analyzing the performance of multiple algorithms, and 2) a \\textbf{deployment challenge} that can be used to test which algorithms successfully solve a research problem. A deployment challenge follows a quantitative research study design, whereas an insight challenge follows a qualitative design. Our work borrows heavily from research study design practices that are well described in other fields, such as educational research \\citep{Creswell02}.  We translate those practices into a framework for challenge design by viewing challenges as research studies on algorithm performance for a specific problem, as described next.\n\n\\section{Insight and Deployment Challenges}\n\\label{sec2}\nWhen challenges are conceptualized as studies of algorithm performance that address a specific problem, the complexity of defining and sampling the relevant problem space is revealed. Examples of research problems in medical image analysis include tissue segmentation, image registration, disease classification, pathology detection, and so on. Clearly, any given challenge will only cover a small portion of the space associated with any of those problems. That is, the scope of a challenge is constrained by its underlying data, truth and metrics, i.e., by its study design. This is illustrated in Figure \\ref{fig1}.\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=.3]{Challenge-Fig1.png}\n\\caption{\\label{fig1}Challenges can be conceptualized as statistical samples from a high dimensional space defined by the data, truth, and metrics of that challenge. \\textbf{Left}: Three challenges are illustrated as ellipses (e.g., iso-probability contours) in a data, truth, and metric space. \\textbf{Middle}: The individual ellipses represent Insight challenges that systematically explore regions within a problem space (i.e., use purposive sampling). \\textbf{Right}: The span of a problem space must be completely covered by a deployment challenge (i.e., use representative sample), for that challenge to determine whether a problem has been solved. The Middle and Right graphs emphasize how the data used in challenges must cover the problem space, but as illustrated in the graph on the Left, the metrics and truth used in challenges must also cover (i.e., be representative of) the relevant problem space.}\n\\end{figure}\n\nAs mentioned in the previous section, we propose to categorize challenges into two fundamental types that can be used to address a research problem: 1) an insight challenge that is based on a qualitative study design, and 2) a deployment challenge that is based on a quantitative study design. This categorization is driven by the two primary types of research study designs, that are distinguished by their use of qualitative or quantitative experiments \\citep{Onwuegbuzie07, Creswell02}. The main difference between a qualitative and a quantitative study design is whether or not the results of the challenge can be generalized to the statistical population of the research problem. In other words, whether the submitted algorithms can be expected to perform similarly on other data from the research problem, beyond the test data provided by the challenge.\n\nAlthough a challenge in itself is a quantitative system of comparison, most challenges that have been organized in the medical image analysis field thus far follow a qualitative experimental design in consideration of their underlying data, truth or metrics. For example, obtaining a randomized, representative set of data with gold standard truth for a medical image analysis problem for a quantitative experiment is often expensive, labor intensive, time consuming, and restricted by ethical concerns. Therefore most challenges incorporate indirect metrics, bronze standards for truth \\citep{Jannin02}, and\/or non-randomly selected datasets, which are the hallmarks of a qualitative experiment. As qualitative experiments, their leaderboards or subsequent tests cannot be used to draw quantitative conclusions about the research problem targeted by the challenge. Attempting to compute quantitative statistics from qualitative experiments is analogous to applying a t-test when the source data is not randomly sampled or does not have a normal distribution. This is also evident from the work by Reinke et al. \\citep{Reinke18}, that can be interpreted to show that a quantitative system should be used with caution for challenges.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[scale=.75]{Challenge-Fig2.png}\n\\caption{\\label{fig2}Example of potential statistical population variables in a medical image analysis problem that can influence algorithm performance, e.g., scanner protocols as well as a patient's age can influence automated brain tissue segmentation in MRI. The relevant variables define the domain of the statistical population that is relevant to a research problem. Note that not only the data, but also the truth and metrics are part of the statistical population of a challenge.}\n\\end{figure*}\n\nThe existence of constrains on the quantitative conclusions that can be drawn from qualitative challenges, however, does not mean that the leaderboards of qualitative challenges are not valuable. Whereas quantitative study design is focused on testing a hypothesis on a representative sample of the statistical population to ensure generalization, qualitative design is aimed at (1) discovering potential hypotheses and influential population variables from specific non-randomly sampled data and (2) uncovering the underlying reasoning \\citep{Foley2015, Hanson2011, Kaplan2005}. Qualitative and quantitative study design can strengthen each other \\citep{Shah06, brannen2017combining} and work together towards solving a research problem. Insight challenges, in particular, can be used to draw conclusions that lead to new insights with respect to algorithm performance for a certain research problem, when (1) the design choices for those insight challenges are openly described with respect to data sampling, ground truth protocol and performance metrics used; and when (2) those challenges clearly follow from a research problem, purpose statement and research question. Analysis of the challenge results, in combination with knowledge of the algorithms' methodologies, can lead to the discovery of new theories and research directions. The systematic analysis of qualitative data is commonly used in other fields, such as social sciences. An example of systematic qualitative analysis is the grounded theory approach \\citep{corbin1990grounded,Glaser68}.\n\nInsights from challenges can in turn inspire new challenges that address the same research problem from a different angle, e.g., that address a particularly interesting or difficult portion of the broader research problem. This is illustrated in Figure \\ref{fig1}, in which insight challenges occupy a limited part of the problem space that has three meta-axes: data, truth, and metrics. In reality, the problem space (statistical population) of a challenge (algorithm evaluation) is much more complex. A problem space is typically an extremely high-dimensional space that incorporates a multitude of statistical population variables.\n\nFigure \\ref{fig2} provides an example of potential statistical population variables associated with an algorithm evaluation problem space in medical image analysis, such as brain tissue segmentation in MRI. Algorithm performance can be influenced by variations in data, truth, and metrics. In terms of data, in this case the image, variations in image acquisition and the subject imaged can have a large influence on algorithm performance. Additionally, the truth that is provided for both training and testing the algorithm adds another layer of variation, in terms of labeling protocol (labeling software, manual \/ semi-automatic labeling) and observers (inter- and intra-observer variability). Insight challenges can provide insight into how algorithms are influenced by these variables in order to determine which variables should be taken into account when designing a deployment challenge. For example, some methods might be robust against certain types of pathology, while others are not. Furthermore, the metrics that are used to measure the distance between an algorithm's results and the provided truth must also be taken into account. Until there is consensus on one metric or a combination of metrics that work best for evaluating algorithms for a specific problem, there are a variety of metrics and ranking schemes that can be used or explored via insight challenges. \n\n\\section{Challenge Framework}\\label{sec3}\nFigure \\ref{fig3} provides a schematic overview of the proposed framework. The design of a challenge starts with a description of the \\textbf{problem} that it aims to address. As described in Section \\ref{sec2}, this problem has a certain problem space (Figure \\ref{fig1}), which encompasses the variations that are expected to impact algorithm performance (example Figure \\ref{fig2}). \n\nThe \\textbf{purpose statement} describes which part of the problem is going to be the focus of the challenge, i.e., which part of the problem space is covered by the challenge. As part of describing the problem space and purpose of a challenge, it is good practice to define which \\textbf{research questions} the challenge aims to answers \\cite{Creswell02}. Based on the purpose statement and the challenge type (insight or deployment), the \\textbf{data}, \\textbf{truth criteria}, and \\textbf{metrics} are then defined. \n\nThe truth criteria, determine the task to be addressed by the challenge, given the data. The combination of truth criteria and data provides the \\textbf{labeled sample} that can be used to train algorithms for a challenge's specific task and to test their performance. Division of the labeled samples into training and test data should follow the sampling scheme of either an insight or a deployment design (Sec \\ref{sec31},\\ref{sec32}). In both designs, the \\textbf{training data} should be representative of the \\textbf{test data}, and in case of a hold-out set, the \\textbf{hold-out data} for a challenge's private leaderboard should be representative of the test data used for the public leaderboard.\n\nA challenge's metrics determine the distance between the defined truth and the output of the \\textbf{algorithms} submitted by \\textbf{challenge participants} on the test data. A \\textbf{ranking heuristic} can combine various metrics into a single value that can be used for ranking algorithms for the \\textbf{leaderboard}. The ranking heuristic should be chosen so that it reflects the performance of algorithms with respect to the purpose statement. In this way, well-designed leaderboards can be of great value by providing quick insights into algorithm performance. \n\nAs mentioned in Section \\ref{sec2}, while leaderboards can be of great value, leaderboards should be interpreted with caution, e.g., with respect to their underlying data, truth criteria, and metrics. Sections \\ref{sec31} and \\ref{sec32} describe the interpretation of the leaderboard for a deployment or insight design. In general, leaderboards of insight challenges should be assessed as rough estimates of algorithm performance with respect to the challenge purpose, after which further \\textbf{analysis of algorithm results} is required to gain insight. The leaderboard of a deployment challenge should be more precise allowing for \\textbf{hypothesis testing} to check whether an algorithm solves the challenge's problem. \n\nOne important consideration in leaderboard design is the theory of static versus adaptive data analysis \\cite{Hardt15dec}. Static data analysis is the more traditional view that the test data can only be used once, in order to test generalization. This prevents leaking knowledge from the test data into algorithm design. In challenge design, static data analysis can be used for deployment challenges to ensure generalization and allow for testing whether a problem is solved. A hold-out test dataset can be selected that is sufficiently large and representative of the test data that is used for the public leaderboard. The disadvantage of this approach is that it requires significantly more data. Approaches like the re-usable hold-out \\cite{Dwork15, Hardt15Google} offer a potential solution to enable data re-use without exhausting the dataset, i.e., to preserve the opportunity to test for generalization. This approach offers adaptive data analysis while reducing the risk of overfitting on the test data. There are two forms of adaptivity \\cite{Hardt15dec}: 1) Algorithmic adaptivity and 2) Human adaptivity. \n\nThe practice of leaderboard climbing \\cite{Hardt15,whitehill17} is a form of algorithmic adaptivity. It is characterized by participants focusing on incrementally improving ranking results on the public leaderboard rather than advancing science or solving the challenge's driving problem. A potential way to combat this is to only update the leaderboard in case of a significant change \\cite{Blum15}, or use the method behind differential privacy \\cite{dwork2011}, similar to what is done with data for the re-usable hold-out, to add noise to the evaluation scores. At its core, differential privacy is a notion of stability requiring that any single sample should not influence the outcome of the analysis significantly \\cite{Hardt15Google}. Another option is to limit the number of allowed resubmissions, but this is harder to check and easier to circumvent by participants. \n\nHuman adaptivity is more complex to quantify. If humans work adaptively with data, they gain insight into the data \\cite{Hardt15dec}, and certain assumption can be made about the data that are taken into account in algorithm development. This could be beneficial, especially since humans are much better able to generalize than machine learning algorithms, but can affect algorithm validation. In terms of challenge design, hidden test data would aid in combating human adaptivity on the test data. This can be done by offering the opportunity to submit algorithms that are then run on the test data by the challenge organizers or by an automated challenge platform. Further research is required to determine an effective balance between avoiding human and algorithmic adaptivity and offering the opportunity for further analysis of algorithm results for insight challenges. \n\n\\subsection{Deployment Design}\\label{sec31}\nFor a deployment challenge, the goal is to make conclusions that generalize to the statistical population described by the problem and purpose statement, and, in particular, determine whether any of the submitted algorithms successfully solves the problem and purpose. Therefore the data sample should be representative of the statistical population, i.e., Figure \\ref{fig2}.\n\nFollowing a quantitative experimental design, a form of random data selection (\\textbf{probability sampling}) should be used, such as stratified random sampling \\cite{elfil2017sampling}; and the sample should be sufficiently large. As mentioned above, sampling of the training, test, and hold-out data should follow the same sampling method, and challenge organizers should make sure that the test and hold-out data are representative of the training data. For a deployment challenge, it is good practice to have a hold-out dataset with a \\textbf{private leaderboard}. \n\nIdeally, an \\textbf{hypothesis test} on the private leaderboard should determine whether an algorithm successfully solves the problem and purpose. To be able to assess whether a problem is solved, the truth criteria and metrics that are used should be representative of the problem as well. This is none-trivial and often requires various insight challenges and discussions with experts (both clinical and medical image analysis) in the field to reach a consensus. Multidisciplinary teamwork is essential in order to be able to design and set-up well designed deployment challenges. Metrics should be defined that fit clinical (practical) requirements \\cite{nikolov18} and that could be used to test which algorithms would be good enough to be used in clinical practice (see Section \\ref{sec4} for practical significance). Keeping these deployment challenges open (not time-bound) would allow for continuous monitoring (benchmarking) of state-of-the-art algorithms for specific problems, keeping track of how the field of research develops over time.\n\n\\subsection{Insight Design}\\label{sec32}\nFor an insight challenge, the goal is not to generalize to the statistical population, but to gain insight into the performance of algorithms, given specific data, truth, and metrics. Insight challenges can provide small non-randomly selected data. When little data is available, it is more beneficial from a research perspective, to select a targeted sample, following a qualitative experimental design. For qualitative research it is common to purposefully select subjects, groups, or settings to  maximize understanding of the phenomenon under investigation \\cite{Onwuegbuzie07}. \n\nIn challenge design, you could think of gaining insight into the influence of various population variables (data, metrics, and ground truth) on algorithm results. Next to purposeful or purposive sampling, many variations of \\textbf{non-probability sampling} exist. Like the definition of truth and metrics, the sampling method is closely linked to the purpose statement of a challenge. Some challenges have been set-up around a dataset that was already available (convenience sampling), in which case the purpose statement should clearly describe which insights are expected from this data, and truth criteria and metrics should be defined accordingly. \n\nMaking existing data available through a challenge can be extremely beneficial, since it could lead to new insights. Also, if one group starts with opening up their data for a certain research problem, others might follow (snowball sampling), covering a larger part of the problem space (Figure \\ref{fig1}). This might in turn inspire new \\textbf{research directions} that focus on specific sub-parts of the problem, using for example maximum variation sampling, critical case sampling, extreme case sampling, or homogeneous sampling. Ultimately leading to a collaborative effort to set-up a deployment challenge that takes all insights from previous challenges into account.\n\nSince insight challenges follow a qualitative experimental design, their leaderboard should be treated as a rough estimate of relative performance. Qualitative research is aimed at identifying underlying concepts or characteristics and aims to answer why a certain phenomenon may occur \\cite{lune2016qualitative}. Therefore, leaderboard climbing and overfitting on the test data of the public leaderboard is less of a concern for insight challenges, since incremental improvements on insight challenges do not aid in solving the underlying problem. Further analysis of the results beyond the leaderboard is essential to gain insight into algorithm performance with respect to the data, truth criteria and metrics used in a challenge. Therefore, tools should be offered to be able to gain this insight and identify overfitting on the test data or leaderboard climbing. Most importantly, the leaderboard should be analyzed with respect to the qualitative design of the challenge and can therefore not be used to draw conclusions about generalization (Section \\ref{sec4}).\n\n\\begin{figure*}\n\\centering\n\\includegraphics[scale=1.0]{Framework-Fig3.png}\n\\caption{\\label{fig3}Framework for challenge design.}\n\\end{figure*}\n\n\\section{Applying the Framework: Significance, Generalizability, and Saturation}\n\\label{sec4}\n\nIn addition to a challenge's data, truth, and metrics defining its type, they also impact its ``efficacy''.  Others have covered the difficulties associated with selecting data \\citep{Fan06}, truth \\citep{Rodriguez12}, and metrics \\citep{Fishbaugh17} to create ``effective'' challenges; however, much of that work has focused on maximizing a study's statistical significance.  There are three difficulties with focusing on statistical significance when optimizing a challenge: 1) statistical significance is associated with testing a hypothesis and therefore is only relevant to quantitative experiments; 2) the complex interrelationship between data, metrics, and truth is oversimplified by commonly used measures of statistical significance; and 3) statistical significance is not a measure of practical significance or generalizability. In this section we provide a brief introduction to statistical significance and then introduce practical significance, generalization theory, and data saturation as important yet often overlooked concepts that impact a challenges \"efficacy.\"\n\nTests of statistical significance typically assume (1) that the test data is from a random sampling of the target population and (2) that the observed errors have a known, e.g., normal, distribution with respect to each independent variate \\citep{arbuthnot1710argument}. The utility of these tests are degraded when these assumptions are incorrect, and in medical imaging, these assumptions are often incorrect: patients typically differ from one another with respect to a multitude of independent variables that are difficult to sample appropriately or that induce errors with non-normal distributions, e.g., with respect to age, gender, imaging device manufacturer.  Additionally, the definition of \"significant\" for any specific problem is highly sensitive to the (often arbitrarily) chosen values of $\\alpha$, the sample size, and the use of one-tailed or two-tailed tests.\n\nPractical significance, on the other hand, indicates if the findings have any decision-making utility. This implies a consideration of ``effect size'' \\citep{Hojat04} as well as statistical significance. In the medical field, practical significance is often referred to as ``clinical significance'', and that term more clearly reflects the fact that small yet statistically significant differences between methods may have no impact on treatment or outcome. For example, a small yet statistically significant improvement in tumor boundary estimation may not enable a reduction in the radiation therapy treatment margins around that tumor.  \n\nThere are numerous tests for practical significance, such as Scoring-Guide Scales and Meaningful Reference Group Comparisons \\citep{Greenland16}.  In scoring-guide scales, results from two methods are compared based on the percentage of individuals who are at each point on the scale, where that scale maps to the magnitude of clinical impact.  For example, scales can be defined, e.g., via retrospective clinical analyses or expert assessments, which map registration errors in radiation therapy treatment delivery to probability of tumor recurrence, and then those scales can be used to determine the clinical impact of one registration method versus another.  Somewhat similarly, in meaningful reference group comparisons, results from a method are compared to norms from chosen groups.  For examples, groups can be defined based on clinical expertise at tumor scoring, and the performance of a method can be equated to performing at the level of a novice, intermediate, or expert clinician. \n\nOne of the strengths of measures of practical significance is that computed differences between competing methods are easy to understand.  For example, when using scoring-guide scales, conclusions are of the form, ``for these data, tumor recurrence rates would increase by 10\\% if method Y is used''.  When using meaningful reference group comparisons, conclusions are of the form ``for these data, method Y performed at the level of an expert clinician''.  Additionally, many test of practical significance can be conducted even if the studies were qualitative, i.e., even if the test data was not representative of the complete clinical population.  Determining if the conclusions can be applied beyond the test data is accomplished via measures of generalizability. \n\nGeneralizability is concerned with indicating what are the potential sources of the observed differences between the two methods being compared and indicating the amount of difference that can be a attributed to a source \\citep{Li15}. Statistical significance tests may indicate that two diagnostic methods are not equivalent, but they will not inform the experimenter if their difference is driven by, for example, a patient's gender, by differences between the human experts who provided truth, or by the scanner on which some of the data was collected. \n\nIn this paper, we take a broad definition of generalizability to also include the concepts of validity (the appropriateness of the data, truth, and metrics for the research question \\citep{Leung2015}) and reproducibility (the ability to achieve consistent results under the described conditions \\citep{Leung2015}). Admittedly, the exact definitions and metrics of generalizability, validity, and reproducibility are the topics of extensive research and debate. For example, one accepted practice defines validity as described above as ``internal validity'' and combines the concepts of reproducibility and generalizability as ``external validity'' \\citep{Kukull12}.  We have chosen to combine these terms into a broad definition of generalizability, given that internal and external validity combine to impact how future researchers should interpret a challenge's published results.\n\nConceptually, testing for generalization is relatively straightforward.  Typically, analysis of variance (ANOVA) testing is used to partition observed variance into its primary components. The primary components can be separated into two categories: (1) those that arise from individual differences among the subjects and (2) those that arise from experimental ``facets'', e.g., differences between experimental factors such raters, imaging scanners, and such. By systematically accounting for variance, the ability of a study's results to be generalized to another case is revealed. \\citep{reynolds98} \\citep{Sinharay10}.   Also, by understanding a study's generalization limits (its biases and confounding facets), it is sometimes possible to adjust the study's results to reduce the impact of its facets and thereby improve the study's generalization capabilities \\citep{Jagger08}.  \n\nThe widespread misinterpretation of statistical significance as a measure of practical significance or generalization has prompted statistical journals to discourage the use of tests of statistical significance \\citep{wasserstein_asas_2016}. The Journal of Basic and Applied Social Psychology, for example, has gone so far as to ban the use of significance testing in its published papers. They instead require measures that focus on practical significance and generalizability \\citep{Trafimow15}.  The trend away from statistical significance is also being prompted by the increased acceptance of the concept of data saturation, as explained next.\n\nWe define data saturation, in the context of challenges, as encompassing insufficiencies in the data, metrics, and\/or truth that diminish the statistical and practical significance as well as the generalizability that can be achieved, even as sample size is increased.  Although the concept of data saturation has been hotly debated in the field of qualitative statistics \\citep{Saunders18}, the prevailing premise is that a problem is suffering from data saturation when the utility of additional data is limited by the additional data's redundancy with existing data.  These data redundancies may arise from limitations in sampling (e.g., purposeful versus random sampling), input detail (e.g., image resolution or contrast limitations make differentiation impossible), metric sensitivity and specificity (e.g., dice overlap is an incomplete measure of segmentation quality), and truth quality (e.g., expert assessments of breast density may be inconsistent).\n\nData saturation may be particularly relevant when trying to detect small yet statistically significant differences between algorithms using large databases, as is often the case with challenges and deep learning methods.  Consider, for example, the fact that a study has shown that it was possible to train a neural network to determine the source database of an image when using images from twelve popular, very large, ``real-world'' image databases \\citep{Shankar17}.  This study suggests that there are biases in these databases, despite their massive sizes, that may have arisen from their samplings of ``nature,'' their acquisition protocols, or the class labels they provide. These biases will carry to systems trained and tested using them.  As a result, statistically significant differences measured with any one or combination of these data may not equate to differences in real-world performance (i.e., practical significance), and perhaps even more importantly, the source of errors that may impact real-world performance (i.e., generalization) will not be revealed by the measures of statistical significance. \n\nConsider the toy example of a fictitious challenge whose problem is to distinguish between two different patient populations using a single feature.  In this one-dimensional feature space, each statistical population is a mixture of three Gaussians, see Fig. \\ref{fig4}.  Such mixture distributions may arise, for example, from genetic variations in each population or from the progression of a disease.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=.5]{Challenge-Fig4.png}\n\\caption{\\label{fig4} The two populations used in our study are mixtures of Gaussians.  Population A is shown in blue, and Population B is shown in green. Their component Gaussians are shown in dotted lines, and their combined distributions are shown in solid lines.  Population A approximates a normal distribution with p = 1e-5 using the Shapiro-Wilks test for normality.  Population B deviates from normal, with test for normality p = 3.1e-2.  These distributions were chosen to represent a problem that on the whole is well separated (offer a TPR of 0.93 at a FPR of 0.1)}\n\\end{figure}\n\nGiven 200 repeated random trials in which, for each trial, a random set of 200 training cases and a random set of 200 test cases were generated from altered versions of the statistical populations shown in \\ref{fig4}.  A linear classifier was then defined by determining the threshold value that produces a 0.1 false-positive rate (FPR) on the training data. The 0.1 FPR target was chosen to reflect the common condition that an automated clinical system must avoid unnecessarily treating healthy subjects (i.e., false positives), while not drastically under-diagnosing those who have the disease. Using the estimated training data threshold, we then computed the FPR and true positive rate (TPR) for the test data.  Additionally, we generated a set of 200 test cases from the actual (un-altered) statistical populations and computed the threshold's FPR and TPR values on that ``deployment'' data.  Using all 200 repeated runs, means and standard deviations of the altered data's and deployment data's testing FPRs and TPRs were recorded and were used to estimate the minimum statistically significant detectable difference at $\\alpha$ = 0.01 and power = 0.95 using two-sided t-tests.  \n\nIn Fig. \\ref{fig6} we illustrate the impact of Data Augmentation (adding 2\\% noise to the training cases) and Mechanical Turk (increasing sample size, yet having 2\\% of the training cases mislabeled).  While it is intuitive that these modifications resulted in non-optimal classifier design, in Fig. \\ref{fig6} we show that these modification also drastically reduce the statistically significant differences that can be detected for a given sample size.  By mislabeling only 2\\% of the data (e.g., 4 of 200 cases), instead of requiring 200 cases to achieve a 0.04 detectable difference between the performance of two classifiers, a challenge organizer would need to provide approximately 500 cases to achieve that same detectable difference, and if 2\\% noise is added to the data, then over 700 cases would be needed.  \n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=.5]{Challenge-Fig6.png}\n\\caption{\\label{fig6}The impact of Data Augmentation (adding 2\\% noise) and Mechanical Turk (mislabeling 2\\% of the cases) for increasing sample size is particularly relevant to the study of data saturation.  The green (bottom) curve is the detectable difference curve when the actual statistical population data is used for training and test.  The black (top) curve occurs if the training data has had 2\\% noise added to it.  The purple (middle) curve results when 2\\% of the training cases are mislabeled.  The solid horizontal line highlights how additional noise and mislabelings can drastically increase the number of samples needed to achieve a given detectable difference when comparing methods.}\n\\end{figure}\n\n\\section{Discussion \\& Future perspective}\nWith this paper, we aim to make researchers aware of the value and complexity of challenges and their design and provide a framework to put challenges in perspective and determine next steps in challenge design. Most challenges organized in the field of medical image analysis thus far, are insight challenges. The first challenges often used a form of convenience sampling, i.e., evaluation data that happened to be available in the organizers' group (e.g., from another study) was used for organizing a challenge. This already led to new insights into algorithm performance on this specific dataset. It was a huge step forward in comparison to researchers writing papers claiming that their algorithm performed best on their private dataset \\cite{price1986anything}, which did not give other researchers the opportunity to test their algorithm against the proposed algorithm. Later, a form of snowball sampling was used (i.e., groups, that knew each other or participated in a previous challenge, teamed up to share their data in a new challenge on the same topic), which led to insight into the performance of algorithms on data from different hospitals \\cite{menze2014multimodal,bejnordi17,bakas2018identifying}.\n\nWhat is currently lacking, is an overview of the problems that researchers in the field of medical image analysis are working on and how challenges relate to these research problems. The paper \"Grand Challenges in Biomedical Image Analysis\" \\cite{Ginneken18} is a excellent first step in providing an overview; however the quality of various challenges discussed in that paper is unclear, e.g., which part of the problem space (Figure \\ref{fig1}) each challenge covers and what insights can be gained from these challenges. Without these details, it is hard to determine new research directions that would drive the field forward. \n\nThe medical image analysis field needs tools that help guide researchers in proper challenge design and that generate an overview of the state-of-the-art for each problem. Ideally, multi-skilled teams should be organized that include web developers, research software engineers, researchers, statisticians and clinicians to join forces and set-up well-designed challenges that boost the field. Developing these tools and accomplishing these goals might be significantly accelerated by changes to the reward system used by funding agencies and journal article reviewers.\n\nWhereas insight challenges are very interesting from a research perspective, deployment challenges are perhaps particularly well suited to be set-up by companies, as part of their clinical trials or cohort studies. This could aid in bridging the gap between industry and academia. If data, truth criteria, and metrics are representative of the problem, direct assessment of algorithms from academia could result in more practical and accelerated use of academic algorithms. Some examples of first steps in this direction can be found on Kaggle (kaggle.com). Further research is required, in particular, to determine metrics that capture practical significance, e.g., aid in determining when an algorithm is good enough to be used in clinical practice?. For some problems, multiple insight challenges with targeted samples (purposive samples) should be set-up in order to better understand the statistical population variables that influence algorithm performance (e.g. generalization). Examples of this are extreme case sampling (e.g., determining what is the influence of extreme cases of pathology or artefacts?), critical case sampling (e.g., selecting forms of pathology that are clinically ambiguous but critical), or maximum variation sampling (e.g., maximizing acquisition variation or variation in pathology).  \n\nThe focus of challenges should be on solving or gaining insight into the challenge's target problem and purpose. Analysis of algorithm results beyond the leaderboard will aid in identifying leaderboard climbing, but future research should focus on finding better algorithms to detect or combat leaderboard climbing. Future research should also focus on the quality of the ground truth and how to assess whether algorithms outperform humans. Interesting examples are letting a panel of experts perform the same task as the algorithm with time limitations \\cite{bejnordi17} or presenting the false positives of the top performing algorithms to experts for re-evaluation \\cite{veta2015assessment}.  \n\nFurthermore, when designing a challenge, it is important to not only focus on statistical significance and sample size.  Practical significance and generalization must also be computed and reported, and therefore data saturation concerns must also be monitored. In particular for challenges, increasing sample size at the cost of adding noise or accidentally mislabeling even a small portion of the data can have drastic detrimental effects.   This is clearly demonstrated for the simple single-feature, two-class experiments presented in this paper, and the potential importance of these lessons only increases for problems with higher-dimensional feature spaces and complex decision boundaries, e.g., the types of problems that are now being investigated using deep learning and other data-drive machine learning methods.  The measured performance of an algorithm on an insight challenge may appear better or worse than the performance that would be achieved if that algorithm was deployed. Measures of statistical significant on challenges with a qualitative design will not indicate if an algorithm's performance on a challenge is close to or far from what its performance would be if deployed. Aspects of these shortcomings are also discussed in other publications in the machine learning field. For example, Masko et al. \\citep{Masko15} explored the detrimental effects on generalization when class priors are altered in a neural network's training data.\n\nFinally, more advanced infrastructures and additional funds are required to support open challenges (not time-bound) and submission of algorithms instead of algorithm results. In other words, enable continuous monitoring of state-of-the-art algorithms for various research problems (benchmarking), keeping track of how the field develops over time. Dynamic benchmarks, where data can be added after quality checks and algorithms can be re-run is an interesting future prospect to quickly track progress. An example of providing evaluation as a service for medical image registration is described in \\cite{marstal2019continuous}. Although many great initiatives exist, such as dream challenges (dreamchallenges.org), Sage bionetworks (sagebionetworks.org), codalab (codalab.org), COMIC (grand-challenge.org\/Create\\_your\\_own\\_challenge) and Kaggle, it remains a challenge to keep these platforms sustainable and to track progress in the field.\n\n\\section{Conclusion}\nThe challenge framework presented in this paper offers a unified approach to the design of qualitative and quantitative challenges, i.e., insight-generating and problem-solving challenges.  Both types of challenges are beneficial to research, but the design decisions that define a challenge must consistently hold to the challenge's objective of being qualitative or quantitative.  Furthermore, when data augmentation, mechanical turk, novice raters, and\/or simulated data is involved; care must be taken to ensure that their detrimental effects are closely assessed.  Additionally, assessing statistically significant differences on leaderboards should not be the end goal of a challenge.  Deployment challenges should have hold-out datasets on which generalization and practical significance are measured, and insight challenges should conduct qualitative analyses to determine fruitful research directions and to design effective future challenges.\n\n\\section{Acknowledgments}\nThe authors would like to acknowledge Bram van Ginneken and Tobias Heimann, who have made significant contributions to the concept of challenges in the medical image analysis field, and to all challenge organizers who invested their time in open science to advance this concept. For dr. A.M. Mendrik, this work was funded, in part, by IMDI Grant 104002002 (Brainbox) from ZonMw, the Netherlands Organisation for Health Research and Development, and in part, by the Netherlands eScience Center, Research and Development budget. For dr. S.R. Aylward, this work was funded, in part, by National Institute Of General Medical Sciences (NIGMS) and the National Institute Of Biomedical Imaging And Bioengineering (NIBIB) via NIH grant R01EB021396 and by the National Institute of Neurological Disorders and Stroke via NIH grant R42NS086295.\n\n\\bibliographystyle{unsrt}  \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}