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+{"text":"\\section{Introduction}\nMany recent studies have been done on the construction, behavior, and performance of reservoir computers.  Two popular reservoir structures are the echo state network (ESN) and the single nonlinear node with delay line~\\cite{appeltantetal}.  Jaeger provided a mathematical description and analysis of echo states, in which case the reservoir is assumed to have a finite number of randomly interconnected internal nodes~\\cite{jaeger}.  While much work has been done to construct, test, and analyze single-node reservoir schemes~\\cites{appeltant,appeltantetal,paquot,duport,brunner}, there remains a lack of understanding and an absence of a rigorous mathematical analysis of the underlying dynamics of such a system.\n\nWe seek to provide some mathematical insight into the dynamics of the single-node reservoir computer.  This analysis is done in two parts.  Section \\ref{results1} describes how the variation in outputs (in the sense of the $\\ell^2$ norm) is bounded above by a constant multiple of the variation of time-series input vectors.  To this end, we draw on basic mathematical tools to prove something of a Lipschitz continuity condition on the entire system:\n\\newtheorem*{main_result}{Theorem \\ref{main_result}}\n\\begin{main_result}\nSuppose $f:\\mathbb{R}\\rightarrow \\mathbb{R}$ is $L$-Lipschitz with $\\alpha L < \\nicefrac{1}{\\sqrt{2}},$ and define $y_u:\\mathbb{Z}_{\\geq 0} \\rightarrow \\mathbb{R}$ by $y_u(t)=\\sum_{k=0}^{N} w_k x_k^{(u)}(t)$ for an input spike train $u\\in\\mathbb{R}^M.$  Then\n\\begin{center}\n$\\| y_u-y_v \\|_{\\ell^2(\\mathbb{R}^M)}^2 \\leq |w|^2 M(L\\beta)^2 (N+1) \\left(1+\\frac{2}{1-2\\alpha^2 L^2} \\right) \\| u-v \\|_{\\ell^2(\\mathbb{R}^M)}^2$\n\\end{center}\nfor all inputs $u,v.$\n\\end{main_result}\n\nSection \\ref{classification_section} provides a translation of some well known classification metrics~\\cites{goodman,gibbons} to the context of the single-node with delay line reservoir structure.  That section also contains a brief review and analysis of the importance of selecting an injective nonlinear sigmoid function for the single  nonlinear input node.  In particular, we show that classification mishaps are possible if the injectivity condition is disregarded, or if the input data are not properly pre-conditioned.\n\n\n\n\n\n\n\n\n\n\\section{Preliminaries}\\label{preliminaries}\n\n\\subsection{Notation and definitions}\\label{notation_subsection}\n\nLet $t\\in \\mathbb{Z}_{\\geq 0}$ denote discrete time and let $\\alpha, \\beta \\in (0,1)$.  An \\textit{input} is a vector $u \\in \\mathbb{R}^m$ where the $0$th entry is defined as $u(0)=0.$  Since we only consider reservoirs with a single nonlinear node and delay line, the term \\textit{reservoir} will be reserved for that structure.\n\\begin{definition}\\label{resdef}\nLet $f:\\mathbb{R}\\rightarrow\\mathbb{R}$ be non-linear.  A \\textit{reservoir of length} $N$, \\textit{nonlinearity} $f$, \\textit{input gain} $\\beta,$ \\textit{and feedback gain} $\\alpha,$ denoted $X(N,f,\\beta,\\alpha),$ is a set of $N$ functions $X=\\{x_1,x_2,\\dots,x_N\\},$ where $ x_k:\\mathbb{Z}_{\\geq 0} \\rightarrow \\mathbb{R},$ defined by the relations \\begin{equation}\\label{resdefeqn}\n\\begin{split}\nx_k(0) &= 0~\\text{for all}~k\\geq 0 \\\\\nx_1(t) &= f(\\alpha x_N(t-1) + \\beta u(t))~\\text{for}~t\\geq 1\\\\\nx_{k+1}(t) &= x_k(t-1)~\\text{for all}~k~\\text{and for}~t\\geq 1\n\\end{split}\n\\end{equation}\nWe will have occasion to use a modified version of Definition \\ref{resdef} in which $x_1(t)= f(\\alpha x_N(t) + \\beta u(t))$ instead of $x_1(t)= f(\\alpha x_N(t-1) + \\beta u(t))$ in \\eqref{resdefeqn}.\n\\end{definition}\nCall $x(t)=\\{x_1(t),x_2(t),\\dots,x_N(t)\\}$ the \\textit{state of the reservoir at time} $t.$  A vector\/set of \\textit{weights} is $(w_1,w_2,\\dots,w_n)\\in \\mathbb{R}^N,$ where $w_k$ is the weight of the ``node\" $x_k.$  Given a set of weights, the \\textit{output of the reservoir at time $t$} is defined as the sum of weighted node states $y(t) = \\sum_{k=1}^N w_k x_k(t).$  Suppose we are working with a given data set of (masked) inputs, call it $\\mathcal{\\tilde{U}}$.  Put $M=\\max\\{m:u\\in\\mathbb{R}^m\\}$. The respective reservoir outputs will be considered over the interval of discrete time $[1,M].$  In order to compare two inputs $u$ and $v$ it is convenient to simply extend all input vectors of dimension less than $M$.  To this end, if $u\\in\\mathbb{R}^m$ and if $m<M,$ identify $u$ with $(u(1),u(2),\\dots,u(m),0,\\dots,0)\\in\\mathbb{R}^M.$  From this point forward we will write $\\mathcal{U}$ as the set of all inputs after this elongation process is performed.  Writing $y_u(t)$ for the reservoir output at time $t$ with respect to an input $u,$ a discrete time interval $[1,M]$ yields an output vector $y_u \\in \\mathbb{R}^{M}.$  So it is reasonable to consider the $\\ell^2$ norm of $u$ or $y_u,$ denoted $\\|u\\|_{\\ell^2(\\mathbb{R}^M)}$, or just $\\|u\\|_2$ where the dimension of $u$ is clear from the context.\n\n\\begin{center}\n\t\\begin{figure}[h]\n\t\t\\includegraphics[width=0.99\\textwidth]{reservoir_picture.png}\\caption{(Brunner et al., 2013)}\\label{caption}\n\t\\end{figure}\n\\end{center}\n\n\\begin{center}\n    \\begin{tabular}{ | l | l | l | p{5cm} |}\n    \\hline\n    Notation & Meaning \\\\ \\hline \\hline\n    $u(t)$ & coordinate $t$ of input $u$ \\\\ \\hline\n    $x_n^{(u)}(t)$ & state of $n$th node at time $t$ with input $u$ \\\\ \\hline\n    $\\alpha$ & feedback gain \\\\ \\hline\n\t\t$\\beta$ & input gain \\\\ \\hline\n\t\t$w_i$ & weight of $i$th node state \\\\ \\hline\n    \\end{tabular}\n\\end{center}\n\n\n\n\\subsection{Weight training}\nWhen preparing a reservoir to perform a task, it is necessary to choose a set of weights.  We assume the existence of a data set consisting of a collection of inputs.  One way or another, these data are used to determine (train) appropriate weights.  To train a set of weights, one needs a set of inputs and a corresponding set of target outputs $\\{\\hat{y}(t)\\}$ to be compared to the actual machine outputs.  That is, one finds $\\{w_k\\}$ such that the variation between $y(t)=\\sum_{k=1}^N w_k x_k(t)$ and $\\hat{y}(t)$ is minimized.\n\nThere is a variety of methods to train weights that is used in the literature including least squares, ridge regression, minimization of mean square error, and minimization of normalized mean square error~\\cites{appeltant,paquot,duport}.  Ridge regression is commonly used to accomodate systems with several parameters to avoid overfitting weights to training data.  The goal is to ensure that the system does not categorize two distinct members of the same input class as ``different.\"    One of the most popular benchmark tasks for testing a reservoir computer is NARMA, wherein the Normalized Root Mean Square Error (NRMSE) is minimized~\\cite{appeltant}.\n\nThe authors of \\cite{prater} developed an algorithm for computing what are called Dantzig selectors, which were defined by Candes and Tao in \\cite{candes}.  Let $\\delta >0,~ X\\in M_{n\\times p}(\\mathbb{R}),$ and $y\\in \\mathbb{R}^n.$  Let $D$ be a diagonal matrix where $D(k,k)$ is the $\\ell^2$ norm of the $k$th column of $X.$ The Dantzig selector, denoted $\\hat{\\beta}\\in\\mathbb{R}^p$, solves the following optimization problem:\n\\begin{equation*}\n\\hat{\\beta}\\in \\text{argmin}\\{\\|\\beta\\|_1 : D^{-1} X^T(X\\beta - y)\\|_{\\infty} \\leq \\delta\\}.\n\\end{equation*}\nThis concept can be used to train reservoir node weights by storing the states of the reservoir over time as the columns of $X$; that is, $X(:,t)=X(t)$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{An upper bound on input $\\mapsto$ output error}\\label{results1}\n\nTo ensure that an input $\\mapsto$ output system is reasonably accurate, it is desirable to have a growth condition that places an upper bound on the distortion of distances between two outputs with respect to the distance between their corresponding inputs.  The following is a classical definition that describes such a condition.\n\n\\begin{definition}\\label{Lipschitz_definition}\nSuppose $X$ and $Y$ are metric spaces and $L\\geq 0.$  A mapping $f:X\\rightarrow Y$ is \\textit{$L$-Lipschitz} if\n\\begin{equation*}\nd_Y(f(z),f(w)) \\leq L d_X(z,w)\n\\end{equation*}\nfor all $z,w\\in X.$  The function $f$ is \\textit{Lipschitz} if it is $L$-Lipschitz for some $L\\geq 0.$\n\\end{definition}\n\\noindent If $f:\\mathbb{R}\\rightarrow\\mathbb{R}$ is differentiable, then $f'$ is bounded if and only if $f$ is Lipschitz.  In fact, if $f:\\mathbb{R}\\rightarrow \\mathbb{R}$ is any Lipschitz function, then $f$ is absolutely continuous, and is differentiable except on a set $E\\subset \\mathbb{R}$ of Lebesgue measure $0.$  Definition \\ref{Lipschitz_definition} says that $f$ can only increase distances by a bounded amount.\n\nIn this section we assume that there is a specific task at hand, and that target outputs have been defined and used to train a set of weights.  It is also assumed that for the set of inputs $\\mathcal{U}$, each $u\\in\\mathcal{U}$ has been standardized to have dimension $M$ as described in Subsection \\ref{notation_subsection}.  In other words, we only consider inputs over the discrete time interval $[1,M]\\cap \\mathbb{N}$.\n\nFor the purpose of classification in a reservoir system, it is desirable that the distance between two outputs is controlled by the distance between their respective inputs.  The Lipschitz condition clearly describes this property.  The main result in this section is that the function $u\\mapsto y_u$ is Lipschitz, where the Lipschitz constant depends on $N,\\alpha, \\beta,$ and $M.$  This result does not appear to be optimal, however, since the Lipschitz constant is directly proportional to the dimension of the input vectors and the number of nodes in the reservoir.  Recall Definition \\ref{resdef}:\n\n\\begin{definition}\\label{1} For a system with $N+1$ nodes and input $u,$\n\\begin{equation}\n\\begin{split}\nx_0(t) &= f( \\alpha x_N(t-1) + \\beta u(t)) \\\\\nx_{n+1}(t) &= x_n(t-1),~~n=0,\\dots,N-1 \\\\\nx_n(0) &= 0\\\\\nx_0(1) &= f(0)\\\\\n\\end{split}\n\\end{equation}\n\\end{definition}\n\n The following couple of modest observations will be useful for proving Theorem \\ref{main_result}.\n\n\\begin{fact}\\label{2}\nIf $0\\leq t \\leq N$ then $x_N(t)=0$ and $x_N(N+t)=f(\\beta u(t))$.\n\\end{fact}\n\n\\begin{corollary}\\label{3}\nIf $t\\in [1,N]\\cap \\mathbb{Z}$ then $x_0(t) = f(\\beta u(t))$.\n\\end{corollary}\n\\begin{proof}\nSince $t-1\\leq N,$ Definition \\ref{1} and Fact \\ref{2} yield\n\\begin{equation*}\n\\begin{split}\nx_0(t) &= f(\\alpha x_N(t-1) + \\beta u(t))\\\\\n&= f(\\alpha x_{N-t+1}(0) + \\beta u(t))\\\\\n&= f(\\beta u(t)). \\qedhere\n\\end{split}\n\\end{equation*}\n\\end{proof}\n\nIt would be convenient to have a Lipschitz continuity condition on the function $u(t)\\mapsto y_u(t)$ where $t$ is fixed.  This idea seems naive, however, since we do not expect the difference of reservoir outputs $|y_u(t)-y_v(t)|$ to depend entirely on the difference in one entry of the inputs $|u(t)-v(t)|.$  That is, we do not know how to capture the behavior of the reservoir simply by looking at the input difference at a particular time step.  An approach to predicting classification accuracy involves the concept of ``separation\" of inputs and reservoir states, which will be discussed in Section \\ref{classification_section}.\n\n\\begin{theorem}\\label{main_result}\nSuppose $X(N,f,\\beta,\\alpha)$ is a reservoir and $f:\\mathbb{R}\\rightarrow \\mathbb{R}$ is $L$-Lipschitz with $\\alpha L < \\nicefrac{1}{\\sqrt{2}}.$  Then for two inputs $u$ and $v,$\n\\begin{center}\n$\\| y_u-y_v \\|_{\\ell^2(\\mathbb{R}^M)}^2 \\leq |w|^2 M(L\\beta)^2 (N+1) \\left(1+\\frac{2}{1-2\\alpha^2 L^2} \\right) \\| u-v \\|_{\\ell^2(\\mathbb{R}^M)}^2.$\n\\end{center}\n\\end{theorem}\n\\begin{remark}\nThe bars $\\|\\cdot\\|$ indicate the length of the vector as a time series, whereas the bars $|\\cdot|$ are used for all other lengths.\n\\end{remark}\n\\begin{proof}\nFor all $t,$\n\\begin{equation}\n\\begin{split}\\label{4}\n|y_u(t)-y_v(t)| &= \\left|\\sum_{i=0}^N w_i (x_i^{(u)}(t) - x_i^{(v)}(t))\\right| \\\\\n&\\leq \\sum_{i=0}^N |w_i| \\cdot |x_i^{(u)}(t) - x_i^{(v)}(t)|.\n\\end{split}\n\\end{equation}\n\nIf $t=0$ there is nothing to prove.  We will deal with special time intervals and achieve the above estimation on each interval.\n\n\\begin{enumerate}[(I)]\n\\item ${1\\leq t \\leq N+1}$:\\\\\n\\begin{center}\n$x_0(t) = f(\\alpha x_N(t-1) + \\beta u(t)) = f(\\beta u(t))$ \\\\\n\\begin{displaymath}\n  x_1(t)=\\left\\{\n  \\begin{array}{lr}\nf(\\beta u(t-1) & ,~ t\\geq 2 \\\\\n0 & ,~ t=1\n\\end{array}\n\\right.\n\\end{displaymath}\n\\\\\n\\begin{displaymath}\nx_2(t)=\\left\\{\n\\begin{array}{lr}\nf(\\beta u(t-2) & ,~ t\\geq 3 \\\\\n0 & ,~ t=1,2\n\\end{array}\n\\right.\n\\end{displaymath}\n\\vdots \\\\\n\\begin{displaymath}\nx_i(t) = \\left\\{\n\\begin{array}{lr}\nf(\\beta u(t-i)) & ,~ t\\geq i\\\\\n0 & ,~ t<i\n\\end{array}\n\\right.\n\\end{displaymath}\n\\end{center}\nIn this case,\n\\begin{equation}\n\\begin{split}\\label{5}\n\\sum_{i=0}^N |w_i|\\cdot|x_i^{(u)}(t)-x_i^{(v)}(t)|\n&\\leq \\sum_{i=0}^N |w_i|\\cdot |f(\\beta u(t-i))-f(\\beta v(t-i))| \\\\\n&\\leq \\sum_{i=0}^N |w_i|\\cdot L\\beta|u(t-i)-v(t-i)|\\\\\n&\\leq L\\beta \\sum_{i=0}^N |w_i| \\cdot |u(t-i)-v(t-i)|.\n\\end{split}\n\\end{equation}\nIt follows from \\eqref{4}, \\eqref{5}, and the Schwarz inequality that\n\\begin{equation}\n\\begin{split}\\label{6}\n|y_u(t)-y_v(t)|^2 &\\leq L^2 \\beta^2 \\left(\\sum_{i=0}^N |w_i| \\cdot |u(t-i)-v(t-i)| \\right)^2 \\\\\n&\\leq L^2 \\beta^2 \\sum_{i=0}^N |u(t-i)-v(t-i)|^2 \\sum_{i=0}^N |w_i|^2 \\\\\n&= |w|^2 L^2 \\beta^2 \\sum_{i=0}^N |u(t-i)-v(t-i)|^2.\n\\end{split}\n\\end{equation}\n\nWrite $C=|w|^2 L^2 \\beta^2 $.  Note that the function $t\\mapsto t-i$ is linear and hence injective.  Therefore equation \\eqref{6} gives\n\\begin{equation*}\n\\begin{split}\n\\sum_t |y_u(t)-y_v(t)|^2 &\\leq \\sum_t C\\sum_{i=0}^N |u(t-i)-v(t-i)|^2 \\\\\n&= C\\sum_{i=0}^N \\sum_t |u(t-i)-v(t-i)|^2 \\\\\n&\\leq C\\sum_{i=0}^N \\|u-v\\|_{\\ell^2(\\mathbb{R}^{N+2})}^2 \\\\\n&= C(N+1) \\|u-v\\|_{\\ell^2(\\mathbb{R}^{N+2})}^2,\n\\end{split}\n\\end{equation*}\nso that\n\\begin{equation}\\label{7}\n\\|y_u-y_v\\|_{\\ell^2(\\mathbb{R}^{N+2})}^2 \\leq |w|^2 (N+1) (L \\beta)^2 \\|u-v\\|_{\\ell^2(\\mathbb{R}^{N+2})}^2.\n\\end{equation}\n\n\\item $N+1\\leq t\\leq 2N+2$\n\nNote that for $0\\leq k \\leq N$ we have\n\\begin{equation}\n\\begin{split}\nx_0(N+1+k)&=f(\\alpha x_N(N+k) + \\beta u(N+k+1))\\\\\n&= f(\\alpha x_0(k) + \\beta u(N+k+1))\\\\\n&= f(\\alpha f(\\beta u(k)) + \\beta u(N+1+k)).\n\\end{split}\n\\end{equation}\nSo the node states can be written as follows:\n\\begin{center}\n$x_0(t)=f(\\alpha f(\\beta u(t-N-1)) + \\beta u(t))$ \\\\\n\n\\begin{displaymath}\nx_1(t)= \\left\\{\n\\begin{array}{lr}\nf(\\alpha f(\\beta u(t-N-2)) + \\beta u(t-1)) & ,~ t\\geq N+3\\\\\nf(\\beta u(t-1)) & ,~ t=N+1,N+2\n\\end{array}\n\\right.\n\\end{displaymath}\n\\\\\n\n\\begin{displaymath}\nx_2(t)= \\left\\{\n\\begin{array}{lr}\nf(\\alpha f(\\beta u(t-N-3)) + \\beta u(t-2)) & ,~ t\\geq N+4\\\\\nf(\\beta u(t-2)) & ,~ t=N+1,N+2,N+3\n\\end{array}\n\\right.\n\\end{displaymath}\n\\vdots\n\\begin{displaymath}\nx_i(t)= \\left\\{\n\\begin{array}{lr}\nf(\\alpha f(\\beta u(t-N-(i+1)) + \\beta u(t-i)) & ,~ t\\geq N+i+2\\\\\nf(\\beta u(t-i)) & ,~ t=N+1,N+2,\\dots,N+i+1\n\\end{array}\n\\right.\n\\end{displaymath}\n\\end{center}\n\nFix $t$ and write $t=N+2+i_0$ for some $i_0 \\in \\{0,1,\\dots,N\\}.$  Then\n\n\\begin{displaymath}\nx_i(t) = \\left\\{\n\\begin{array}{lr}\nf(\\alpha f(\\beta u(t-N-i-1)) + \\beta u(t-i)) & ,~ 0\\leq i\\leq i_0 \\\\\nf(\\beta (t-i)) & ,~ i_0 < i \\leq N \\\\\n\\end{array}\n\\right.\n\\end{displaymath}\nSo $|y_u(t) - y_v(t)|$ is bounded above by\n\\begin{equation}\n\\begin{split}\\label{10}\n& \\sum_{0\\leq i\\leq i_0} |w_i| \\cdot |f(\\alpha f(\\beta u(t-N-i-1)) + \\beta u(t-i)) - [f(\\alpha f(\\beta v(t-N-i-1)) + \\beta v(t-i))]| \\\\\n&\\hspace{1cm} + \\sum_{i_0 < i \\leq N} |w_i| \\cdot |f(\\beta u(t-i)) - f(\\beta v(t-i))|\n\\end{split}\n\\end{equation}\nAlso\n\\begin{equation}\n\\begin{split}\\label{8}\n& |f(\\alpha f(\\beta u(t-N-i-1)) + \\beta u(t-i)) - f(\\alpha f(\\beta v(t-N-i-1)) + \\beta v(t-i))| \\\\\n&\\leq L|\\alpha f(\\beta u(t-N-i-1)) + \\beta u(t-i) - [\\alpha f(\\beta v(t-N-i-1)) + \\beta v(t-i)]| \\\\\n&= L|\\alpha[f(\\beta u(t-N-i-1))-f(\\beta v(t-N-i-1))] + \\beta[u(t-i)-v(t-i)]| \\\\\n&\\leq L\\left[ \\alpha|f(\\beta u(t-N-i-1))-f(\\beta v(t-N-i-1))| + \\beta|u(t-i)-v(t-i)| \\right] \\\\\n&\\leq L[ \\alpha L \\beta |u(t-N-i-1)-v(t-N-i-1)| + \\beta|u(t-i)-v(t-i)| ] \\\\\n&\\leq \\alpha L^2 \\beta |u(t-N-i-1)-v(t-N-i-1)| + L\\beta|u(t-i)-v(t-i)|.\n\\end{split}\n\\end{equation}\nThus the square of the quantity \\eqref{10} is bounded above by\n\\begin{equation}\n\\begin{split}\\label{9a}\n& [~~\\alpha L^2 \\beta \\sum_{0\\leq i\\leq i_0} |w_i| \\cdot |u(t-N-i-1)-v(t-N-i-1)| + L\\beta \\sum_{i_0 < i \\leq N} |w_i| \\cdot |u(t-i)-v(t-i)| ~~]^2 \\\\\n&\\leq 2(L\\beta)^2 [~~ (\\alpha L)^2 \\left(\\sum_{0\\leq i\\leq i_0} |w_i| \\cdot |u(t-N-i-1)-v(t-N-i-1)| \\right)^2 \\\\\n&\\hspace{1cm} + \\left(\\sum_{i_0 < i \\leq N} |w_i| \\cdot |u(t-i)-v(t-i)|\\right)^2 ] \\\\\n&\\leq 2(L\\beta)^2 [~~ (\\alpha L)^2 \\sum_{0 \\leq i \\leq i_0} |w_i|^2 |u(t-N-i-1)-v(t-N-i-1)|^2 \\\\\n&\\hspace{1cm} + \\sum_{i_0 < i \\leq N} |w_i|^2 \\cdot |u(t-i)-v(t-i)|^2 ~~ ]\n\\end{split}\n\\end{equation}\nIt now follows that\n\\begin{equation}\n\\begin{split}\n\\|y_u - y_v\\|_{\\ell^2(\\mathbb{R}^{N+2})}^2 &\\leq (L\\beta)^2 (N+1) (2(\\alpha L)^2 +2)|w|^2 \\cdot \\|u-v\\|_{\\ell^2(\\mathbb{R}^M)}^2\n\\end{split}\n\\end{equation}\nNote that we have used the Schwarz inequality and the convexity of the function $x\\mapsto x^2$ to obtain \\eqref{9a}.\n\n\\item $2N+2 \\leq t \\leq 3N+3$ \\\\\n\n\\begin{center}\n$x_0(t) = f(\\alpha f(\\alpha f(\\beta u(t-2N-2)) + \\beta u(t-N-1)))$ \\\\\n\n\\begin{displaymath}\nx_1(t) = \\left\\{\n\\begin{array}{lr}\nf(\\alpha f(\\alpha f(\\beta u(t-2N-3)) + \\beta u(t-N-2))) & ,~ t\\geq 2N+3 \\\\\nf(\\alpha f(\\beta u(t-N-2)) + \\beta u(t-1)) & ,~ t = 2N+2\n\\end{array}\n\\right.\n\\end{displaymath}\n\\\\\n\n\\begin{displaymath}\nx_2(t) = \\left\\{\n\\begin{array}{lr}\nf(\\alpha f(\\alpha f(\\beta u(t-2N-4)) + \\beta u(t-N-3))) & ,~ t \\geq 2N+4 \\\\\nf(\\alpha f(\\beta u(t-N-3)) + \\beta u(t-2)) & ,~ t = 2N+2, 2N+3\n\\end{array}\n\\right.\n\\end{displaymath}\n\\vdots \\\\\n\\begin{displaymath}\nx_i(t) = \\left\\{\n\\begin{array}{lr}\nf(\\alpha f(\\alpha f(\\beta u(t-2N-i-2)) + \\beta u(t-N-i-1))) & ,~ t \\geq 2N+i+2 \\\\\nf(\\alpha f(\\beta u(t-N-i-1)) + \\beta u(t-i)) & ,~ 2N+2 \\leq t \\leq 2N+i+1\n\\end{array}\n\\right.\n\\end{displaymath}\n\\end{center}\nLet $t = 2N+3+ i_0,~ i_0 \\in \\{0,\\dots,N\\}.$  By the same argument used to obtain \\eqref{10}, we see that the quantity $|y_u(t) - y_v(t)|$ is bounded above by\n\\begin{equation}\n\\begin{split}\\label{11}\n& \\sum_{0\\leq i\\leq i_0} [~~ |w_i| \\cdot |f(\\alpha f(\\alpha f(\\beta u(t-2N-i-2)) + \\beta u(t-N-i-1))) \\\\\n&\\hspace{1cm} - f(\\alpha f(\\alpha f(\\beta v(t-2N-i-2)) + \\beta v(t-N-i-1)))| ~~] \\\\\n&+ \\sum_{i_0 < i \\leq N} |w_i| \\cdot |f(\\alpha f(\\beta u(t-N-i-1)) + \\beta u(t-i)) - f(\\alpha f(\\beta v(t-N-i-1)) + \\beta v(t-i))|.\n\\end{split}\n\\end{equation}\nBy the same argument as in \\eqref{10}, the first summand in \\eqref{11} has the property\n\\begin{equation}\n\\begin{split}\n& |f(\\alpha f(\\alpha f(\\beta u(t-2N-i-2)) + \\beta u(t-N-i-1))) \\\\\n&\\hspace{1cm} - f(\\alpha f(\\alpha f(\\beta v(t-2N-i-2)) + \\beta v(t-N-i-1)))| \\\\\n&\\leq \\alpha L\\left[ \\alpha L^2 \\beta |u(t-2N-i-2)-v(t-2N-i-2)| + L\\beta|u(t-N-i-1)-v(t-N-i-1)| \\right].\n\\end{split}\n\\end{equation}\nThe second summand of \\eqref{11} is simply the first summand of \\eqref{10}, so similarly\n\\begin{equation}\n\\begin{split}\n& |f(\\alpha f(\\beta u(t-N-i-1)) + \\beta u(t-i)) - f(\\alpha f(\\beta v(t-N-i-1)) + \\beta v(t-i))| \\\\\n&\\leq \\alpha L^2 \\beta |u(t-N-i-1)-v(t-N-i-1)| + L\\beta|u(t-i)-v(t-i)|,\n\\end{split}\n\\end{equation}\nso that the quantity \\eqref{11} is bounded above by\n\\begin{equation}\n\\begin{split}\\label{12}\n& \\alpha^2 L^3 \\beta\\sum_{0\\leq i\\leq i_0} |w_i| \\cdot |u(t-2N-i-2)-v(t-2N-i-2)| \\\\\n&+ \\alpha L^2 \\beta \\sum_{i=0}^N |w_i| \\cdot |u(t-N-i-1)-v(t-N-i-1)| + L\\beta\\sum_{i_0 < i \\leq N} |w_i| \\cdot |u(t-i)-v(t-i)|.\n\\end{split}\n\\end{equation}\nLet $C_1 = L\\beta.$  For each pair $i,j$ write $z_j^i(t) = t-i-j(N+1)$. Then the square of \\eqref{12} becomes\n\\begin{equation}\n\\begin{split}\\label{13}\n& C_1^2 [~~ \\alpha^2 L^2 \\sum_{0\\leq i\\leq i_0} |w_i| \\cdot |u(z_2^i(t))-v(z_2^i(t))| + \\alpha L \\sum_i |w_i| \\cdot  |u(z_1^i(t))-v(z_1^i(t))| \\\\\n&\\hspace{1.5cm} + \\sum_{i_0< i\\leq N} |w_i| \\cdot |u(z_0^i(t))-v(z_0^i(t))| ~~]^2 \\\\\n&\\leq 2 C_1^2 [~~ 2\\left[\\alpha^2 L^2 \\sum_{0\\leq i\\leq i_0} |w_i| \\cdot |u(z_2^i(t))-v(z_2^i(t))| \\right]^2 + 2 \\left[\\alpha L \\sum_i |w_i| \\cdot |u(z_1^i(t))-v(z_1^i(t))| \\right]^2 \\\\\n&\\hspace{1.5cm} + \\left[ \\sum_{i_0< i\\leq N} |w_i| \\cdot |u(z_0^i(t))-v(z_0^i(t))| \\right]^2 ~~] \\\\\n&= 2 C_1^2 [~~ 2 (\\alpha L)^4 \\left[ \\sum_{0\\leq i\\leq i_0} |w_i| \\cdot |u(z_2^i(t))-v(z_2^i(t))| \\right]^2 + 2(\\alpha L)^2 \\left[ \\sum_i |w_i| \\cdot |u(z_1^i(t))-v(z_1^i(t))| \\right]^2 \\\\\n&\\hspace{1.5cm} + \\left[ \\sum_{i_0< i\\leq N} |w_i| \\cdot |u(z_0^i(t))-v(z_0^i(t))| \\right]^2 ~~] \\\\\n&\\leq 2 C_1^2 (N+1)^2 [~~ 2(\\alpha L)^4 \\sum_i |w_i|^2 \\cdot |u(z_2^i(t))-v(z_2^i(t))|^2 + 2(\\alpha L)^2 \\sum_i |w_i|^2 \\cdot |u(z_1^i(t))-v(z_1^i(t))|^2 \\\\\n&\\hspace{2.5cm} + \\sum_i |w_i|^2 \\cdot |u(z_0^i(t))-v(z_0^i(t))|^2 ~~]\n\\end{split}\n\\end{equation}\nBy \\eqref{13}\n\\begin{equation*}\n\\|y_u - y_v\\|_{\\ell^2(\\mathbb{R}^{N+2})}^2 \\leq (L\\beta )^2 (N+1) \\left[ 2^2(\\alpha L)^4 + 2^2(\\alpha L)^2 + 2 \\right] |w|^2 \\|u-v\\|_{\\ell^2(\\mathbb{R}^M)}^2.\n\\end{equation*}\n\n\\item\nIn general, on the interval $I_j = [j(N+1),(j+1)(N+1)]$ for $j\\in \\mathbb{Z}_{\\geq 0},$ we can iterate this construction to see that for $t\\in I_j,$\n\n\\begin{displaymath}\nx_i(t) = \\left\\{\n\\begin{array}{lr}\nf \\left( (\\alpha f)^{j-1} \\left[ \\alpha f(\\beta u(z_j^i(t))+\\beta u(z_{j-1}^i) \\right] \\right) & ,~ t\\geq jN+i+2 \\\\\nf \\left( (\\alpha f)^{j-2} \\left[ \\alpha f(\\beta u(z_{j-1}^i(t))+\\beta u(z_{j-2}^i) \\right] \\right) & ,~ t < jN+i+2\n\\end{array}\n\\right.\n\\end{displaymath}\nwhere $(\\alpha f)^j$ is the $j$th iterate of the function $(\\alpha f)(x)=\\alpha f(x).$ Note that when the convexity property is used iteratively, it follows the pattern\n\\begin{equation*}\n\\left(\\sum_{n=1}^N a_n + a_0 \\right)^2 \\leq 2^N a_1^2 + 2^N a_2^2 + 2^{N-1} a_3^2 + 2^{N-2} a_4^2 + \\cdots + 2^3 a_2^2 + 2^2 a_1^2 + 2a_0^2,\n\\end{equation*}\nso the highest power of $2$ is used twice, while the other powers of $2$ descend to $1.$\n\nIteration of the above argument using the Schwarz inequality and convexity of $x\\mapsto x^2$ shows that for $t\\in I_j = [j(N+1),(j+1)(N+1)],$\n\\begin{equation*}\n\\begin{split}\n\\|y_u - y_v\\|_{\\ell^2(\\mathbb{R}^{N+2})}^2\n&\\leq (L\\beta)^2(N+1) |w|^2 [ 2 + 2^2(\\alpha L)^2 + 2^3(\\alpha L)^4 + \\cdots \\\\\n&+ 2^{j-1}(\\alpha L)^{2(j-2)} + 2^j(\\alpha L)^{2(j-1)} + 2^j(\\alpha L)^{2j} ] \\|u-v\\|^2 \\\\\n&\\leq (L\\beta)^2(N+1) |w|^2 \\left[ 1 + \\sum_{k=1}^{\\infty} 2^k (\\alpha L)^{2(k-1)} \\right] \\|u-v\\|^2 \\\\\n&= (L\\beta)^2(N+1) |w|^2 \\left(1+\\frac{2}{1-2\\alpha^2 L^2} \\right)\\|u-v\\|_{\\ell^2(\\mathbb{R}^M)}^2.\n\\end{split}\n\\end{equation*}\n\nLet $j'= \\min \\{j : M\\leq (j+1)(N+1)\\}.$  Finally, the desired result is attainable since\n\\begin{equation*}\n\\begin{split}\n\\|y_u - y_v\\|_{\\ell^2(\\mathbb{R}^M)}^2 &= \\sum_{t=0}^M |y_u(t) - y_v(t)|^2 \\\\\n&\\leq \\sum_{j=0}^{j'} \\sum_{t\\in I_j} |y_u(t) - y_v(t)|^2 \\\\\n&= \\sum_{j=0}^{j'} \\|y_u - y_v\\|_{\\ell^2(\\mathbb{R}^{N+2})}^2~~~(\\text{where}~y_u,y_v:I_j \\rightarrow \\mathbb{R}) \\\\\n&\\leq \\sum_{j=0}^{j'} (L\\beta)^2 (N+1) |w|^2 \\left(1+\\frac{2}{1-2\\alpha^2 L^2} \\right)\\|u-v\\|_{\\ell^2(\\mathbb{R}^M)}^2 \\\\\n&\\leq M (L\\beta)^2 (N+1) |w|^2 \\left(1+\\frac{2}{1-2\\alpha^2 L^2} \\right)\\|u-v\\|_{\\ell^2(\\mathbb{R}^M)}^2. \\qedhere\n\\end{split} \n\\end{equation*} \n\\end{enumerate}\n\\end{proof}\n\nWhile aesthetically pleasing, the upper bound given by Theorem \\ref{main_result} is quite crude.  Moreover, this upper bound considers the entire time interval $[0,M],$ but gives no comparison of the two instantaneous reservoir states at time $t\\in [0,M]$ with respect to the inputs $u,v.$  It would be ideal to have an upper bound that applies at any time $t.$\n\\begin{question}\nIs there a constant $C=C(N,\\alpha,\\beta)$ such that $|y_u(t)-y_v(t)| \\leq C |u(t)-v(t)|$ for all $t$?\n\\end{question}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Separation and classification}\\label{classification_section}\nThere are several methods used to determine reservoir quality of ESNs and Liquid State Machines.  In \\cite{gibbons} Gibbons provided an overview of the Separation method (see \\cite{goodman},\\cite{norton}) in great generality.  Our goal is to reduce that overview to the case of the single non-linear node with delay line.  A fundamental difference between Separation for ESNs and Separation for the delay line structure is that with an ESN, there are multiple input nodes.  So if $k$ is the number of input nodes of an ESN, then at a given time $t$ an entire sequence of $k$ data points can be fed into the reservoir simultaneously to yield an instantaneous reservoir state $x(t)$.  For the delay line structure, however, a single input node does not provide such latitude.  If one wishes to feed in a sequence of data (i.e. an input) of length $k$, then $k$ time steps are required instead of just 1.\n\nLet $\\mathcal{U}$ be a set of inputs as in Subsection \\ref{notation_subsection}.  We assume that at time $t,$ the set of all reservoir states $\\mathcal{X}(t)=\\{x^{(u)}(t) : u\\in\\mathcal{U}\\}$ is a disjoint union of $N(\\mathcal{U})$ prescribed classes $X_1(t), X_2(t),\\dots,X_{N(\\mathcal{U})}(t)$.  It is desirable for reservoir states within the same class to be close to each other, and for states of different classes to be far from each other.  To that end, it is useful employ several averaging processes that provide insight into the Separation within and between classes.\n\nSeveral technical definitions are required to introduce the concept of Separation, all of which have been adapted from those in \\cite{gibbons}.\n\n\\begin{definition} For each $n,$ the \\textit{class average (center of mass)} of $X_n(t)$ is the average of all the members of $X_n(t)$:\n\\begin{equation}\\label{class_average}\n\\mu(X_n(t))= \\frac{1}{|X_n(t)|} \\sum_{x^{(u)}(t)\\in X_n(t)} x^{(u)}(t).\n\\end{equation}\nThe \\textit{inter-class distance} of $\\mathcal{X}(t)$ is the average of all distances between centers of mass in $\\mathcal{X}(t)$:\n\\begin{equation}\\label{interclass_distance}\nC_d(t)= \\frac{1}{N(\\mathcal{U})^2}\\sum_{n=1}^{N(\\mathcal{U})}\\sum_{m=1}^{N(\\mathcal{U})} |\\mu(X_n(t)) - \\mu(X_m(t))|.\n\\end{equation}\n\\end{definition}\n\nWe need a measure of spread within classes in order to define Separation; it is obtained in the following way.  For each class $X_n(t)$, find the average distance from the elements of $X_n(t)$ to the class average $\\mu(X_n(t)).$  Then, compute the average of this quantity over all classes.\n\n\\begin{definition}\\label{intraclass_variance} The \\textit{intra-class variance} of $\\mathcal{X}(t)$ is\n\\begin{equation*}\nC_v(t)= \\frac{1}{N(\\mathcal{U})} \\sum_{n=1}^{N(\\mathcal{U})} \\frac{1}{|X_n(t)|} \\sum_{x^{(u)}(t)\\in X_n(t)} |\\mu(X_n(t))-x^{(u)}(t)|.\n\\end{equation*}\n\\end{definition}\n\nTo perform good classification, the class averages should be separated by an amount substantial enough to differentiate them.  Moreover, the members of a given class should not stray too far from their class average, or else they could be misclassified.  Therefore the following definition provides a tool for measuring the effectiveness of a single-node reservoir classification system.\n\\begin{definition}\\label{separation}\nThe \\textit{separation of $\\mathcal{X}(t)$} is\n\\begin{equation}\n\\text{Sep}_{\\mathcal{X}}(t)=\\frac{C_d(t)}{C_v(t)+1}.\n\\end{equation}\n\\end{definition}\n\nDefinition \\ref{separation} first appeared in \\cite{goodman}, in which Goodman obtained results yielding large positive correlations between separation and classification accuracy of the output function.  This was done for temporal pattern classification problems in the context of randomly generated reservoirs~\\cites{gibbons,goodman}.  So at time step $t$, if $C_d(t)$ is large while  $C_v(t)$ is relatively small, one expects high classification accuracy.  It is shown in \\cite{gibbons} that a hallmark of good classification for random reservoirs is the existence of a constant $C>0$ such that\n\\begin{equation}\\label{inverse_lip}\n|x^{(u)}(t)-x^{(v)}(t)| \\geq C|u(t)-v(t)|\n\\end{equation}\nfor all pairs $u,v\\in\\mathcal{U}$ such that $|u(t)-v(t)|$ is ``large.\"  For sake of simplicity, let us say that \\eqref{inverse_lip} should hold whenever $|u(t)-v(t)|\\geq 1.$\n\n\\subsection{Injectivity and periodicity of $f$}\nIf the chosen nonlinear function $f$ is injective (which is fairly typical considering the common usage of $f=\\tanh$), then an inequality like \\eqref{inverse_lip} is not outside the realm of possibility, since the function $u(t)\\mapsto x^{(u)}(t)$ is also injective.\n\n\\begin{fact}\nLet $X(N,f,\\beta,\\alpha)$ be a reservoir where $f$ is injective. If two inputs $u$ and $v$ are such that $u(t)\\neq v(t),$ then $x^{(u)}(t)\\neq x^{(v)}(t).$\n\\end{fact}\n\\begin{proof}\nSuppose $u(t_0)\\neq v(t_0)$ and assume $x^{(u)}(t_0)=x^{(v)}(t_0).$  Put $t_1=\\min\\{t:u(t)\\neq v(t)\\}$.  By definition \\ref{resdef},\n\\begin{equation}\nx_1^{(u)}(t_1) = f(\\alpha x_N^{(u)}(t_1 -1)+\\beta u(t_1)),\\\\\n\\end{equation}\nand since $u(t_1-1)=v(t_1-1),$ we have\n\\begin{equation}\n\\begin{split}\\label{22}\nx_1^{(v)}(t_1) &= f(\\alpha x_N^{(v)}(t_1 -1)+\\beta v(t_1)) \\\\\n&= f(\\alpha x_N^{(u)}(t_1 -1)+\\beta v(t_1)).\n\\end{split}\n\\end{equation}\nThen $x_1^{(u)}(t_1)=x_1^{(v)}(t_1)$ since $x^{(u)}(t_0)=x^{(v)}(t_0),$ so it follows from \\eqref{22} that\n\\begin{equation}\\label{23}\n\\alpha x_N^{(u)}(t_1 -1)+\\beta u(t_1) = \\alpha x_N^{(u)}(t_1 -1)+\\beta v(t_1).\n\\end{equation}\nBy \\eqref{23} it is clear that $u(t_1)=v(t_1)$ since $\\beta >0,$ which is a contradiction.\n\\end{proof}\n\nSuppose that we graph $u$ as a function of time $t,$ and that we desire a single-node reservoir $X(N,f,\\beta,\\alpha)$ whose state is not extremely sensitive to vertical translations.  Intuitively, it seems reasonable to use a nonlinear function $f$ that is periodic.  The following fact shows that periodic functions $f$ can, in some sense, yield periodic reservoir states.\n\n\\begin{fact}\\label{periodic_fact}\nLet $X(N,f,\\beta,\\alpha)$ be a reservoir with inputs $u$ and $v,$ where $f$ is $P$-periodic.  For all $t$, if $v(t)=u(t)-\\nicefrac{P}{\\beta}$, then $x^{(u)}(t)=x^{(v)}(t).$\n\\end{fact}\n\\begin{proof}\nThe following basic argument proceeds by induction on $t,$ but this statement could be verified directly from the equations for $x_i(t)$ given in Part IV of the proof of Theorem \\ref{main_result}.\n\nThe case $t=0$ is trivial.  For the base case, let $t=1$ and note that\n\\begin{equation*}\nx_1^{(u)}(t)=f(\\alpha x_N^{(u)}(0) + \\beta u(1)) = f(\\alpha x_N^{(v)}(0) + \\beta v(1)+P)=x_1^{(v)}(1).\n\\end{equation*}\nClearly $x_i^{(u)}(1)=0=x_i^{(v)}(1)$ for all $1< j \\leq N,$ so that $x^{(u)}(1)=x^{(v)}(1).$  Now assume $x^{(u)}(k)=x^{(v)}(k).$  Then\n\\begin{equation}\\label{24}\nx_1^{(u)}(k+1)=f(\\alpha x_N^{(u)}(k) + \\beta u(k+1)) = f(\\alpha x_N^{(v)}(k) + \\beta v(k+1) + P) = x_1^{(v)}(k+1),\n\\end{equation}\nso it remains to show that $x_i^{(u)}(k+1) = x_i^{(v)}(k+1)$ for $1<i\\leq N.$  Since $x^{(u)}(k)=x^{(v)}(k),$\n\\begin{equation*}\n\\begin{split}\nx_2^{(u)}(k+1) &= x_1^{(u)}(k) = x_1^{(v)}(k) = x_2^{(v)}(k+1) \\\\\nx_3^{(u)}(k+1) &= x_2^{(u)}(k) = x_2^{(v)}(k) = x_3^{(v)}(k+1) \\\\\n&\\vdots \\\\\nx_N^{(u)}(k+1) &= x_{N-1}^{(u)}(k) = x_{N-1}^{(v)}(k) = x_N^{(v)}(k+1). \\qedhere\n\\end{split}\n\\end{equation*}\n\\end{proof}\n\n\\begin{remark}\nIn the case $f=\\sin$, Fact \\ref{periodic_fact} demonstrates the importance of pre-conditioning reservoir inputs $u$ so that $u(t)\\in [-1,1]$ for all $t\\in\\mathbb{Z}_{\\geq 0}$ and for all $u\\in\\mathcal{U}$.  Depending on the task, it may or may not be desirable to put $(u(1),\\dots,u(M))$ and $v=(u(1)+ \\nicefrac{P}{\\beta}, u(2)+\\nicefrac{P}{\\beta},\\dots,u(M)+\\nicefrac{P}{\\beta})$ into the same class, since the period $P$ can be made arbitrarily large.  So choosing $f$ to be periodic can yield the undesirable pair of separation properties $|u(t)-v(t)|=\\nicefrac{P}{\\beta}$ and $|x^{(u)}(t) -x^{(v)}(t)|=0,$ which is a blatant violation of condition \\eqref{inverse_lip}.  Since $f=\\sin$ is injective on $[-\\nicefrac{\\pi}{2},\\nicefrac{\\pi}{2}],$ such difficulties can easily be avoided by normalizing each $u\\in\\mathcal{U}$, which gives $|u(t)|\\leq \\|u\\|_2\\leq 1.$\n\\end{remark}\n\nIt is not clear whether any or all of the aforementioned quality metrics are useful for the single-node with delay line reservoir model.  Since the reservoir states rapidly change over time while receiving a signal, it is quite difficult to produce a lower bound on the difference in reservoir states in terms of the difference in inputs such as \\eqref{inverse_lip}.\n\n\n\n\n\n\\subsection*{Acknowledgement}\nThis article is based on research performed during the summer of 2015 by the author at Air Force Research Laboratory in Rome, NY under the direction of Ashley Prater.\n\n\n\n\n\\begin{bibdiv}\n\\begin{biblist}\n\n\\bib{appeltant}{article}{\n  title={Reservoir computing based on delay-dynamical systems},\n  author={Appeltant, Lennert},\n  journal={These de Doctorat, Vrije Universiteit Brussel\/Universitat de les Illes Balears},\n  year={2012}\n}\n\n\\bib{appeltantetal}{article}{,\n  title={Information processing using a single dynamical node as complex system},\n  author={Appeltant, Lennert},\n  author={Soriano, Miguel Cornelles},\n  author={Van der Sande, Guy},\n  author={Danckaert, Jan},\n  author={Massar, Serge},\n  author={Dambre, Joni},\n  author={Schrauwen, Benjamin},\n  author={Mirasso, Claudio R},\n  author={Fischer, Ingo},\n  journal={Nature communications},\n  volume={2},\n  pages={468},\n  year={2011},\n  publisher={Nature Publishing Group}\n}\n\n\\bib{brunner}{article}{\n  title={Parallel photonic information processing at gigabyte per second data rates using transient states},\n  author={Brunner, Daniel},\n  author={Soriano, Miguel C},\n  author={Mirasso, Claudio R},\n  author={Fischer, Ingo},\n  journal={Nature communications},\n  volume={4},\n  pages={1364},\n  year={2013},\n  publisher={Nature Publishing Group}\n}\n\n\\bib{candes}{article}{\nauthor ={Candes, Emmanuel},\nauthor={Tao, Terence},\ndoi = {10.1214\/009053606000001523},\njournal = {The Annals of Statistics},\njournal = {Ann. Statist.},\nnumber = {6},\npages = {2313--2351},\npublisher = {The Institute of Mathematical Statistics},\ntitle = {The Dantzig selector: Statistical estimation when p is much larger than n},\nurl = {http:\/\/dx.doi.org\/10.1214\/009053606000001523},\nvolume = {35},\nyear = {2007}\n}\n\n\\bib{duport}{article}{\n  title={All-optical reservoir computing},\n  author={Duport, Fran{\\c{c}}ois},\n  author={Schneider, Bendix},\n  author={Smerieri, Anteo},\n  author={Haelterman, Marc},\n  author={Massar, Serge},\n  journal={Optics express},\n  volume={20},\n  number={20},\n  pages={22783--22795},\n  year={2012},\n  publisher={Optical Society of America}\n}\n\n\\bib{gibbons}{article}{\n  title={Unifying quality metrics for reservoir networks},\n  author={Gibbons, Thomas E},\n  booktitle={Neural Networks (IJCNN), The 2010 International Joint Conference on},\n  pages={1--7},\n  year={2010},\n  organization={IEEE}\n}\n\n\\bib{goodman}{article}{\n  title={Spatiotemporal pattern recognition via liquid state machines},\n  author={Goodman, Eric},\n  author={Ventura, Dan A},\n  year={2006},\n  publisher={IEEE}\n}\n\n\n\\bib{jaeger}{article}{\n  title={The \"echo state\" approach to analysing and training recurrent neural networks-with an erratum note},\n  author={Jaeger, Herbert},\n  journal={Bonn, Germany: German National Research Center for Information Technology GMD Technical Report},\n  volume={148},\n  pages={34},\n  year={2001}\n}\n\n\\bib{norton}{article}{\n  title={Improving liquid state machines through iterative refinement of the reservoir},\n  author={Norton, David},\n  author={Ventura, Dan},\n  journal={Neurocomputing},\n  volume={73},\n  number={16},\n  pages={2893--2904},\n  year={2010},\n  publisher={Elsevier}\n}\n\n\\bib{paquot}{article}{\n  title={Optoelectronic reservoir computing},\n  author={Paquot, Yvan},\n  author={Duport, Francois},\n  author={Smerieri, Antoneo},\n  author={Dambre, Joni},\n  author={Schrauwen, Benjamin},\n  author={Haelterman, Marc},\n  author={Massar, Serge},\n  journal={Scientific reports},\n  volume={2},\n  year={2012},\n  publisher={Nature Publishing Group}\n}\n\n\\bib{prater}{article}{\n   author={Prater, Ashley},\n   author={Shen, Lixin},\n   author={Suter, Bruce W.},\n   title={Finding Dantzig selectors with a proximity operator based\n   fixed-point algorithm},\n   journal={Comput. Statist. Data Anal.},\n   volume={90},\n   date={2015},\n   pages={36--46},\n   issn={0167-9473},\n   review={\\MR{3354827}},\n   doi={10.1016\/j.csda.2015.04.005},\n}\n\n\n\\end{biblist}\n\\end{bibdiv}\n\n\\end{document}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nHidden Valley models of dark matter allow for a strong interaction in the dark sector \\cite{hidden_valley}. Such a strong interaction would cause showering of the dark particles, producing both invisible dark matter and Standard Model (SM) hadrons. As a result, so-called semivisible (SV) jets would be formed, containing some fraction $r_{inv}$ of invisible particles \\cite{sv_jets}. Depending on the mass of the mediator \\ensuremath{m_{\\PZprime}}\\xspace, SV jets could be produced at the Large Hadron Collider (LHC) energies and registered in detectors \\cite{CMS-PAS-EXO-19-020}. \n\nEvents containing such jets would be characterised by missing transverse energy \\ensuremath{\\cancel{E}_{T}}\\xspace being aligned with a jet, which is a scenario difficult to distinguish from detector issues and therefore usually neglected in LHC data analyses. Moreover, the details of the Hidden Valley physics depend on a number of parameters. There are four parameters that most influence the kinematic behavior in the final state:\n\n\\begin{itemize}\n\\item \\ensuremath{\\alpha_{\\text{dark}}}\\xspace -- the coupling constant of the dark sector strong interaction equivalent,\n\\item \\ensuremath{m_{\\PZprime}}\\xspace -- the mass of the mediator,\n\\item \\ensuremath{m_{\\text{dark}}}\\xspace -- the mass of the dark hadrons,\n\\item \\ensuremath{r_{\\text{inv}}}\\xspace -- the fraction of stable, invisible dark hadrons.\n\\end{itemize}\n\n\nThese parameters could be a function of other parameters in a fully-specified Hidden Valley model, as well as parameters that impact the modeling of strong dynamics in the event generator (hadronization, fragmentation). This leads to an extremely large model space with a huge number of possible scenarios that can easily evade any constraints from e.g. cosmological measurements. It is clearly impractical to perform dedicated searches for all possible model variations.\n\nIn order to gain robustness against detector effects, as well as details of the model implementation, anomaly detection techniques such as unsupervised deep learning can be used. Those are designed to detect objects significantly different from the training sample, without prior knowledge of signal characteristics. The presented approach is suitable to search for a wide class of new physics models containing anomalous jets, including the SV jets stemming from Hidden Valley models, which are the main focus of this work.\n\nHere, an autoencoder-based analysis for SV jets detection will be presented. Unlike previous attempts \\cite{qcd_or_what, new_physics_deep_autoencoders, lhc_olympics, dark_machines} that use jet images (four-momenta of jet constituents), we incorporate high-level jet features and substructure variables computed based on the four-momenta of the jet constituents: Energy Flow Polynomials (EFPs)~\\cite{eflow}, Energy Correlation Functions (ECFs) \\cite{ecf} and their ratios: C\\textsubscript{2} and D\\textsubscript{2}, as well as the jet \\ensuremath{p_{T}}\\xspace dispersion \\ensuremath{p_{T}D}\\xspace~\\cite{ptd} and jet axes \\cite{jets_at_lhc}. These prove to be highly useful in discriminating between the QCD background and the SV jet signal \\cite{svj_substructure}.\n\nThe trained model can then be used to tag anomalous jets, as the autoencoder has been trained to recognize QCD jets in the signal region, but not SV signal jets (or any other anomalous jets, such as emerging jets \\cite{emerging_jets}).\n\nIt is worth noting that, while the performance is assessed using SV jet models as a benchmark, the training process is independent of the signal implementation, unlike in case of the supervised learning. Given that the landscape of possible Hidden Valley scenarios is so vast, using one classifier (even parametric \\cite{parametric_ml}) per signal model becomes impractical.\n\n\n\\section{Data samples}\n\\label{sec:data}\n\n\\subsection{Generation}\nBoth multi-jet QCD and dark sector samples are generated using \\texttt{PYTHIA8}\\xspace~\\cite{pythia} and reconstructed with \\texttt{DELPHES}\\xspace~\\cite{delphes} providing a detector response approximating that of CMS. Proton-proton collisions at a centre-of-mass energy of 13 TeV are considered, with approximately 50 collisions per bunch crossing. The particle flow reconstruction algorithm distributed with \\texttt{DELPHES}\\xspace is used. In particular, jets are clustered from reconstructed particles and pileup reduction is applied.\n\nTo examine the robustness of autoencoder models to varying signal parameters, multiple SV jet samples were generated with values of the fraction of stable jet hadrons $\\ensuremath{r_{\\text{inv}}}\\xspace \\in \\{0.3, 0.5, 0.7\\}$ and the \\ensuremath{{\\PZ}^{\\prime}}\\xspace boson mass $\\ensuremath{m_{\\PZprime}}\\xspace \\in \\{1.5, 2.0, 2.5, 3.0, 3.5, 4.0 \\}$. The number of events generated varies depending on the sample, accounting for the different selection efficiency (see Sec.~\\ref{subsec:preselection}):\n\n\\begin{itemize}\n    \\item QCD: $3.7 \\cdot 10^7$ events,\n    \\item SV jets, $\\ensuremath{m_{\\PZprime}}\\xspace > 1.5$ TeV: 50k events per sample,\n    \\item SV jets, $\\ensuremath{m_{\\PZprime}}\\xspace = 1.5$ TeV, $\\ensuremath{r_{\\text{inv}}}\\xspace \\in \\{0.3, 0.5\\}$: 100k events per sample,\n    \\item SV jets, $\\ensuremath{m_{\\PZprime}}\\xspace = 1.5$ TeV, $\\ensuremath{r_{\\text{inv}}}\\xspace = 0.7$: 140k events per sample.\n\\end{itemize}\n\n\n\\subsection{Preselection}\n\\label{subsec:preselection}\n\nA set of preselection criteria (based on those outlined in Ref.~\\cite{sv_jets}) is applied to both training and testing samples on an event-by-event basis. These requirements are aimed at isolating the SV jet signal, as well as reproducing the realistic triggering capabilities of LHC experiments. The signal region is defined to include events passing the following criteria:\n\\begin{itemize}\n    \\item at least 2 jets with $|\\eta| < 2.4$ and $\\ensuremath{p_{T}}\\xspace > 200$ GeV\n    \\item then, for the two leading jets:\n    \\begin{itemize}\n        \\item $|\\Delta\\eta| < 1.5$,\n        \\item $\\ensuremath{m_{T}}\\xspace > 1500$ GeV,\n        \\item $\\ensuremath{\\cancel{E}_{T}}\\xspace\/\\ensuremath{m_{T}}\\xspace > 0.25$,\n    \\end{itemize}    \n\\end{itemize}\nwhere $\\eta$ is the pseudorapidity, \\ensuremath{p_{T}}\\xspace is the transverse momentum for each jet, $\\Delta\\eta$ is the absolute dijet $\\eta$ difference, \\ensuremath{\\cancel{E}_{T}}\\xspace is the missing momentum in the transverse plane for the event, and \\ensuremath{m_{T}}\\xspace is the transverse mass of the leading dijet system and the \\ensuremath{\\cancel{E}_{T}}\\xspace.\n\nThe selection efficiency was found to be at the level of $0.13 \\%$ for the QCD events and between $0.2$ and $15.3 \\%$ for SV jets, depending on the signal model parameters. The number of QCD jets after applying selections is $\\approx$100k (split between training, validation, and testing) and varies between 500 and 15k for SV jets, depending on the model parameters.\n\n\n\\subsection{Feature Selection}\n\nSince the goal of this study is to find anomalies on the basis of tagging individual anomalous jets rather than anomalous events, a set of jet-level and jet substructure variables is determined for the training.\n\nThe coordinates of each jet ($\\eta$ and $\\phi$) are included in the training to allow the network to learn about problematic regions of the detector and avoid tagging noise or other detector failures as anomalous signals.\n\nIn order to avoid bias, the transverse momentum of the jet was not included in the training. Moreover, the jet \\ensuremath{p_{T}}\\xspace distribution was flattened (by applying appropriate weights) for the training.\n\nA number of other jet-level and substructure variables have been considered for the training. A fraction of them were rejected due to poor signal-background discrimination. The correlations between remaining variables were tested, as autoencoders tend to perform better with uncorrelated inputs. It was found that for QCD and SV jets, the EFPs are fully correlated with each other, as well as with the girth \\cite{jet_substructure}, Les-Houches Angularity (LHA) \\cite{lha}, and normalized ECFs: e\\textsubscript{2} and e\\textsubscript{3}. Because of that, it was decided to only keep one of those variables for the training, namely EFP\\textsubscript{1} \\cite{eflow} which exploits pairwise relations between jet constituents.\n\nThe jet constituent four-momenta have also been considered; however, they proved not to bring any significant gain in performance, while increasing the complexity (${\\approx}100$ instead of ${\\approx}10$ input features) and training time of the autoencoder. This can be understood since EFPs and ECFs are included in the training and provide a linear basis for characterizing jet constituent substructure, making up for the discrimination that is lost when using high-level jet features rather than jet constituents.\n\nUltimately, the following set of jet-related input variables is selected: $\\eta$, $\\phi$, invariant mass \\ensuremath{m_{j}}\\xspace, jet fragmentation function \\ensuremath{p_{T}D}\\xspace, jet ellipse minor and major axes, EFP\\textsubscript{1}, and ECF ratios: C\\textsubscript{2} and D\\textsubscript{2}. The distributions of those variables for the QCD background and selected signals is presented in Fig.~\\ref{fig:inputs}.\n\nThe first three of these features are provided directly by the \\texttt{DELPHES}\\xspace reconstruction algorithm. The remaining ones are calculated using a list of jet constituents, which consists of all particles in the event which lie within a cone of radius $R = 0.8$ from the jet axis.\n\nAll of the selected high level features discussed here provide some meaningful discrimination power, with the exception of the angular coordinates, $\\eta$ and $\\phi$. Since those are randomly distributed, they provide no meaningful information about QCD jet structure. However, given the ability of autoencoders to learn a compressed representation of input signals, it is feasible that they might also learn the locations and behaviors of defective detector components. For instance, they could learn location of energy spikes in certain detector locations as normal behavior that does not indicate an anomaly associated with a potential signal. Such detector malfunctions are not included in the \\texttt{DELPHES}\\xspace reconstruction; however, they may be present in the collision data, on which the autoencoder can be trained.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=1.0\\textwidth]{images\/inputReco.pdf}\n    \\caption{Distributions of input variables for QCD background and selected signal models.}\n    \\label{fig:inputs}\n\\end{figure}\n\n\n\\section{Models}\\label{sec:models}\n\nAll unsupervised models presented in the following sections are trained using the QCD data sample described in Sec.~\\ref{sec:data} as background, with a 70:15:15 split between training, validation, and testing, respectively. \n\n\n\\subsection{Autoencoders}\\label{sec:models:ae}\n\nAn autoencoder is a neural network which approximates the identity transformation as the application of a function $f$ from the input space to a latent space (encoding) and a function $f^{-1}$ from the latent space back to the input space (decoding), which provides an approximation of the input as the output. The constraint forces the autoencoder to find hidden relationships between features in the input data set. In the case of a neural autoencoder, this constraint takes the form of a bottleneck in the network architecture: an input vector $\\vec{x} \\in \\mathbb{R}^n$ is compressed into a latent space vector $\\vec{l} \\in \\mathbb{R}^m$ with $m < n$, enforced by an $m$-node layer within the architecture of an autoencoder with input dimension $n$. This latent space vector is then reconstructed by a decoding network in an attempt to match the input values.\n\n\\begin{figure}\\centering\n\\includegraphics[scale=0.5]{images\/autoencoder_diagram.pdf}\n\\caption{Neural Network architecture.}\n\\label{nn_arch}\n\\end{figure}\n\nThe autoencoder is trained to minimize the reconstruction error of each sample, defined for an $n$-dimensional input feature vector $\\vec{x}$ with components $x_1, ..., x_n$. A number of loss functions were considered and the mean absolute error was selected, as it had the best performance on the considered benchmark data sets:\n\\begin{equation}\n\\label{eq:mae}\n    \\text{MAE} = \\frac{1}{N}\\sum_i^N \\left|x^\\prime_i - x_i\\right|\n\\end{equation}\n\n\\noindent where $x_i$ is the original value and $x^\\prime_i$ is the reconstructed value for the $i^{th}$ feature of a jet.\n\nThe training data consists of the simulated QCD background events that fall within our signal region as per the cuts described in Sec.~\\ref{sec:data}. \n\nOnce a model is trained to recognize the background, one can evaluate it on unlabeled data and use the reconstruction errors as a measure of similarity of a given sample to the training data set. Typically, a model trained on QCD will return low reconstruction loss for samples similar to those it was trained on, and high loss for samples that are more complex \\cite{ae_anomaly_hep}. In case of a search for new physics, an autoencoder would be trained and evaluated, with the largest 10\\%, 1\\%, and 0.1\\% of its reconstruction losses isolated and analyzed.\n\n\n\\subsection{Alternative Models}\n\\label{sec:models:others}\n\nAlong with neural autoencoders - the simplest form of autoencoder - other types of autoencoders were considered: \n\\begin{itemize}\n    \\item Variational Autoencoders~\\cite{vae} (VAE): provide probabilistic representation of the input data, which can give higher performance in some applications.\n   \n    \\item Principal Component Analysis (PCA) \\cite{pca}: can be viewed as a linear autoencoder with no hidden layers, which allows determining the weights analytically.\n\\end{itemize}\n\nThe results obtained with the VAE and PCA will be reported in this work.\nSparse Autoencoders (SAE)~\\cite{sae:eth} were investigated in the early stages of this study and found to give worse results than the dense autoencoder; therefore, they will not be included.\n\n\n\\subsection{Model Optimization}\n\nIn order to determine the optimal architecture, normalization, and hyper-parameters for the SV jet search, a large number of models were trained and evaluated. The optimization was focused on maximizing the Area Under Receiver Operating Characteristic (ROC) Curve (AUC). For each set of hyper-parameters and each choice of architecture, 80\\% of best models were selected and the average AUC over all signal samples and all models was used to choose the highest performing one.\n\nIt was found that the best results are obtained by standardizing the data, i.e., by normalizing them such that the mean of the distribution is at zero and the standard deviation is unity. An extensive architecture scan was performed and the architecture with two hidden layers of size 10, followed by the bottleneck of size 6 was found to be optimal. The activation function of the hidden layers and the bottleneck is `elu' \\cite{elu} and the output layer uses linear activation. \n\nThe autoencoder models were implemented in Keras\/TensorFlow ~\\cite{keras,tensorflow}. Each model was trained for 200 epochs (with early stopping enabled), with a batch size of 256. The learning rate was set to $10^{-6}$ and the optimizer found to yield the best performance was `Nadam'~\\cite{nadam}. \n\n\\section{Results}\n\nIn this section, the results from the best autoencoder will be presented. The best autoencoder was found by training 150 models with the optimal hyper-parameters and varying random seed and selecting the best performing one in terms of AUC. No signs of overfitting were observed, and the loss evolution was smooth and saturated after around 100 epochs.\n\n\\subsection{Reconstructed Distributions}\n\nThe differences between input and reconstructed jet features are shown for both SVJ and QCD data in Fig.~\\ref{fig:input_and_reco}. One can observe that, in almost all cases, the difference is larger for the signals than for the background. What is more, the distributions for the signal do not change with varying mass; however there is a slight difference in shape when moving from \\ensuremath{r_{\\text{inv}}}\\xspace = 0.3 to 0.7.\n\n\\begin{figure}\\centering\n\\includegraphics[width=1.0\\linewidth]{images\/inputRecoDifferences.pdf}\n\\caption{Distributions of difference between input and reconstructed jet features for QCD and a few example SVJ signals.}\n\\label{fig:input_and_reco}\n\\end{figure}\n\n\\subsection{Reconstruction Loss}\n\nAn important figure of merit is the distribution of losses for the background and signal samples. As shown in Fig.~\\ref{fig:losses}, QCD tends to have lower loss values than SVJ, providing good discrimination.\n\nIn order to achieve the best possible performance, alternative approaches to autoencoder training and evaluation were studied. For those, reference distributions were built for each node of the network from the training data set, using the reconstructed variables, MAE values, or even values from the latent space nodes. The performance was then evaluated by comparing values from the tested jets to those reference distributions. \n\nThe comparison was done using the negative log-likelihood, as well as the Mahalanobis distance \\cite{mahalanobis_distance}. It was found that none of the six combinations performed better than the MAE loss function.\n\n\\begin{figure}\\centering\n\\includegraphics[width=1.0\\textwidth]{images\/lossesComparison.pdf}\n\\caption{Reconstruction loss for QCD and different SVJ signals.}\n\\label{fig:losses}\n\\end{figure}\n\n\\clearpage\n\\newpage\n\n\\subsection{Discrimination Power}\n    \nIn order to compare discrimination power of different architectures, one can examine the ROC curves and AUC values for each generated signal. As a reference for the best-case performance, we considered a Boosted Decision Trees (BDT), providing a strong baseline for supervised algorithms. The BDT was implemented using SKLearn's AdaBoostClassifier, trained on a mixture of all signals, and had its hyper-parameters optimized for maximum AUC. The input features were the same as for the autoencoder. For the final comparison, 10 models were trained and the best one was selected (although the performance spread between different models was negligible).\n\nA comparison of ROC curves for the autoencoder, BDT, PCA and VAE is shown in Fig. \\ref{fig:rocs}. As can be seen, the BDT classifier provides the best performance regardless of the signal considered, although it is rather closely followed by the autoencoder. What is more, as will be shown in the further part of this section, the AE can become more powerful than the BDT if the latter was trained on a wrong signal hypothesis. The VAE has been trained predominantly on the reconstruction loss and evaluated on reconstruction loss exclusively, neglecting the KL divergence \\cite{kl_divergence}; therefore, it could be further improved with a dedicated study. The PCA performance is significantly worse than those of autoencoders and BDT.\n\n\\begin{figure}\\centering\n\\includegraphics[width=1.0\\textwidth]{images\/rocCurvesComparison.pdf}\n\\caption{Comparison of the ROC curves for the autoencoder, PCA, BDT and VAE for a few selected signals.}\n\\label{fig:rocs}\n\\end{figure}\n\nFigure \\ref{fig:aucs} presents a comparison of the AUC values between the autoencoder and BDT for each of the signals. It is clear from this plot that signals with lower values of \\ensuremath{r_{\\text{inv}}}\\xspace are much more efficiently identified, although all signals can be tagged at relatively high rates. Moreover, even though the autoencoder performs worse than the BDT in all signal cases, it is still quite good, with the added benefit of requiring no particular signal to train against.\n\n\n\\subsection{Robustness}\n\nAs an additional test, the autoencoder performance was compared to that of a BDT trained on the wrong signal hypothesis. This allows to study the generalization power of the autoencoder against that of a supervised model and assess their ability to discover ``by accident'' a signal other than the targeted one. As can be seen in the rightmost panel of Fig.\\ref{fig:aucs}, a clear decrease in the performance can be observed in two cases, up to $19\\%$ worse when trained on $\\ensuremath{m_{\\PZprime}}\\xspace = 2$ TeV, $\\ensuremath{r_{\\text{inv}}}\\xspace = 0.7$, but tested on $\\ensuremath{m_{\\PZprime}}\\xspace = 4$ TeV, $\\ensuremath{r_{\\text{inv}}}\\xspace = 0.3$. What is remarkable is that in this particular case, the BDT performs worse than the autoencoder, which was trained without any information about the signal characteristics.\n\nAnother robustness test was to evaluate both the BDT and the AE on signal samples with varying \\ensuremath{m_{\\text{dark}}}\\xspace values, with \\ensuremath{m_{\\PZprime}}\\xspace fixed to 3~TeV. The results of this study are presented in Fig.\\ref{fig:aucs_mdark}. Here, one can see again that in certain cases the BDT trained on a wrong signal hypothesis can be outperformed by the AE. In this scenario, the AE becomes more powerful than a classifier for $\\ensuremath{m_{\\text{dark}}}\\xspace = 100$~GeV, when the BDT training was performed on a mixture of 18 signal samples (with varying \\ensuremath{m_{\\PZprime}}\\xspace and \\ensuremath{r_{\\text{inv}}}\\xspace) with $\\ensuremath{m_{\\text{dark}}}\\xspace = 20$~GeV. More details on the signal samples with varying \\ensuremath{m_{\\text{dark}}}\\xspace values can be found in Appendix~\\ref{app:mdark}.\n\nWe noticed that training the BDT on a specific signal point with low \\ensuremath{r_{\\text{inv}}}\\xspace leads to higher efficiency for signals with higher \\ensuremath{r_{\\text{inv}}}\\xspace with respect to training on a mixture of all signal points. This result comes from the fact that low \\ensuremath{r_{\\text{inv}}}\\xspace jets are more similar to background jets. Such conclusions suggest that it might be a better strategy to train a classifier on low \\ensuremath{r_{\\text{inv}}}\\xspace signals rather than on a mixture of low and high parameter values.\n\nFrom this test, we conclude that it would be preferable for an LHC signal-specific search to have a signal-agnostic equivalent exploiting autoencoders to complement model-dependent searches, as opposed to relying on accidental generalization properties of the signal-specific selection algorithm. The synergy of the two approaches would enhance the chances of discovering a signal.\n\n\\begin{figure}\\centering\n\\includegraphics[width=1.0\\textwidth]{images\/aucsTablesComparison.pdf}\n\\caption{Left and middle panels: comparison of AUC values of the autoencoder and BDT. Right panel: AUC values for a BDT trained on a signal with parameters different from those it was tested on. E.g., the AUC value presented in the top left corner of the table comes from a model trained on the lower right corner sample.}\n\\label{fig:aucs}\n\\end{figure}\n\n\\begin{figure}\\centering\n\\includegraphics[width=0.6\\textwidth]{images\/aucsTablesComparisonMDarkExt.pdf}\n\\caption{Comparison of AUC values of the autoencoder and BDT with varying \\ensuremath{m_{\\text{dark}}}\\xspace values. The BDT was trained on a mixture of all signals with $\\ensuremath{m_{\\text{dark}}}\\xspace = 20$~GeV.}\n\\label{fig:aucs_mdark}\n\\end{figure}\n\n\\subsection{Sensitivity to Semivisible Jet Models}\n\nAs a last step, a threshold was optimized for the SVJ signal samples to tag jets with loss > 0.06 as anomalous. Doing so, one can mimic the impact of the AE algorithm on a real analysis, estimating expected upper limits. Events passing the preselection were grouped in categories with 0, 1 or 2 SV jets, and dijet transverse mass distributions for the signal and background in these categories were used to set approximate exclusion limits on the cross section, depending on the \\ensuremath{{\\PZ}^{\\prime}}\\xspace mass. The results, which take into account basic uncertainties (luminosity and trigger), are presented in Figs.~\\ref{fig:limits_0p3} and \\ref{fig:limits_0p7} and compared with the example theoretical cross section. The results without applying an autoencoder-based tagger are also shown.\n\nAs expected, the limits are stronger for lower values of \\ensuremath{r_{\\text{inv}}}\\xspace and the improvement from using the tagger is more pronounced in this region. The presented results show that one can expect an approach based on the autoencoder to be able to cover a large area of the phase space.\n\n\\begin{figure}[!tbp]\n  \\centering\n  \\subfloat[\\ensuremath{r_{\\text{inv}}}\\xspace = 0.3]{\n    \\includegraphics[width=0.47\\textwidth]{images\/limits_rinv0p3_Run2.pdf}\n    \\label{fig:limits_0p3}\n  }\n  \\hfill\n  \\subfloat[\\ensuremath{r_{\\text{inv}}}\\xspace = 0.7] {\n    \\includegraphics[width=0.47\\textwidth]{images\/limits_rinv0p7_Run2.pdf}\n    \\label{fig:limits_0p7}\n  }\n  \\hfill\n  \\caption{The cross section limits vs. \\ensuremath{m_{\\PZprime}}\\xspace for different values of \\ensuremath{r_{\\text{inv}}}\\xspace, scaled to the luminosity of the full Run 2 of the LHC.}\n\\end{figure}\n\n\\section{Conclusion}\n\nThis study shows that the usage of a neural autoencoder based on physics-motivated high-level features, despite its simplicity, is extremely effective as an anomaly detection model. This is especially true in cases of a signal with widely varying parameters, such as semivisible jets, as the model is trained on the background sample only and is therefore highly model independent. The projections of the exclusion limits for this particular class of models show a 40-60\\% improvement in the search reach compared to a study without such an anomalous jet tagger. What is more, the autoencoder could be trained on the real data, accounting for any effects of an imperfect detector simulation.\n\nThis result demonstrates the usefulness of the autoencoder network and jet substructure variables such as Energy Flow Polynomials, Energy Correlation Functions, the jet \\ensuremath{p_{T}}\\xspace dispersion \\ensuremath{p_{T}D}\\xspace, and the jet ellipse axes for the detection of anomalous jets. It is also worthwhile to acknowledge certain aspects that should be further investigated in the future.\n\nThe assumption that the network should be insensitive to detector effects could be tested by simulating such issues and contaminating the training sample with affected events. Next, in order to train autoencoder on the collision data, one has to verify that a small admixture of signal events in the training sample does not influence the performance. Finally, one could try to further exploit the four-momenta of the jet constituents using convolutional or graph autoencoders, which should preserve information about the spatial distribution of particles within the jet.\n\n\\section*{Acknowledgements}\n\nM.~Pierini is supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (Grant Agreement No. 772369). A.~de~Cosa and J.~Niedziela are supported by the Swiss National Science Fundation (SNFS) under the SNSF Eccellenza program (PCEFP2\\_186878). L.~Le~Pottier is supported by the  Swiss State Secretariat for Education, Research and Innovation (SERI) under the ThinkSwiss program. K. Pedro is supported by the Fermi National Accelerator Laboratory, managed and operated by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction\\label{sec:intro}}\n\n\nThe understanding about how stars rotate is essential to describe and modelling many aspect of stellar evolution. \nFrom spectroscopy observations we can only get the projected velocity, $v \\sin i$, where $i$ is the inclination angle with respect to the line of sight. Furthermore, in order to deconvolve (disentangle or unfold) the rotational velocity distribution function, an assumption on the distribution of\nrotational axes is required. The standard choice is that the distribution of stellar axes is uniformly (randomly) \ndistributed over the sphere. Using this assumption Chandrasekhar \\&  M\\\"unch (1950) studied the integral equation that describe the distribution of 'true'  ($v$) and  apparent ($v \\sin i$) rotational velocities, deriving a formal solution, which is proportional to a derivative of an Abel's Integral. \nChandrasekhar \\& M\\\"unch (1950) method is not usually applied, because the differentiation of the formal \nsolution can lead to misleading results due to intrinsic numerical problems associated to the derivative of the Abel's integral. \n\nCur\\'e et al. (2014) extended the work of Chandrasekhar \\& M\\\"unch (1950), integrating the formal solution and obtained \nthe cumulative distribution function (CDF) for the rotational velocities. This CDF is attained in one step demonstrating the robustness to this method.\n\nWhile the CDF identifies the distribution of the speed of rotation it is sometimes useful to have the probability density function (PDF) for easy handling and to appreciate directly certain properties of the distribution (e.g., the maximum, its  symmetry, variability, etc). It is known also that the observed values of the projected rotational velocities are provided with measurement error. The goal of this work is to propose a methodology that provides straightforward the PDF, taking into account the measurement errors and avoiding numerical problems arising from the derivative of the CDF. \nRegularization methods are a technique widely used to deconvolve inverse problems. Image processing, geophysics and  machine learning are some of the areas where they are usually applied (Bouhamidi $2007$, Deng et al. $2013$, Fomel $2007$). Among the regularization methods we find: Truncated Singular Value Decomposition (TSVD), Selective Singular Value Decomposition (SSVD) and Tikhonov Regularization Method (Hansen 2010).\n\nIn this article we obtain the estimated probability distribution function directly from the Fredholm integral by means of the Tikhonov regularization method.\n\nAfter its introduction by Tikhonov (1943) to solve integral equation problems, this method (known as Ridge Regression in statistics) has been developed and extensively used since then (see, e.g., Tikhonov 1963, Tikhonov and Arsenin 1977, Tikhonov et al. 1995, Eggermont 1993, Hansen 2010). It allows an increase in the numerical stability and dealing with errors of measurement.\n\nThis article is structured as follows: In section 2  we briefly present the mathematical description of the method and describe a procedure to calculate the Tikhonov factor. In section 3, we perform Monte Carlo simulations to show the robustness of this method. In section 4,  a real sample of cluster stars are deconvolved by Tikhonov regularization,  confidence intervals are calculated using bootstrap method and a comparison between our PDF results with the one obtained with the Lucy (1974) method and CDF results from the work of Cur\\'e et al. (2014) are performed. Last section presents our conclusions and future work.\n\n\\section{Tikhonov Regularization Method\\label{sec:method}}\n\nMany inverse problems in physics and astronomy are given in terms of the Fredholm integral of the first kind (Lucy 1994, Hansen 2010), namely:\n\\begin{equation}\n\\label{eq1}\nf_{Y}(y)=\\int p(y\\,|\\,x)\\,f_{X}(x)dx,\n\\end{equation}\nhere $f_{Y}$ is a function accessible to observation and $f_{X}$ is the function of interest. The kernel $p(y\\,|\\,x)$ of this integral is related to the remoteness of the measurement process, in this case, the projection of the distribution of stellar axes.\n\nChandrasekhar \\& M\\\"unch (1950) were the first in considering the integral equation governing the distribution of 'true' an the apparent (projected)  rotational velocities of stars, $y=x\\sin i$, where $x=v$ is the rotational speed and $i$ is the inclination angle with respect to the line of sight.\nAssuming an  uniform distribution of stellar axes over the sphere (see Cur\\'e et al. 2014 for details), this integral equation (Eq. \\ref{eq1}) reads as  follows:\n\n\n\\begin{equation}\n\\label{eq-problema}\nf_{Y}(y)=\\int_{y}^{\\infty}\\dfrac{y}{x\\sqrt{x^{2}-y^{2}}}\\,f_{X}(x)dx.\n\\end{equation}\n\n\nExpressing Eq. (\\ref{eq-problema}) in matrix form (by a quadrature discretization of the problem), we get:\n\n\\begin{equation} \\label{eq-ill}\nY\\,=\\, A \\, X\n\\end{equation}\n\n\n\nNow, $A$ is a matrix representing the kernel $p(y|x)$, $Y$ is a vector representing the density of projected rotational velocities $f_Y(y)$ and $X$ is the unknown vector representing the density of 'true' rotational velocities $f_X(x)$.\\\\\n\nSince the observed data are measured with error, last equation is an example of a discrete ill--posed problem, i.e., small errors in the measured data can produce large \nvariations in the recovered function which make the solution unstable (Ivanov et al. 2002 and references therein).\nNevertheless, in the decades after the work of Chandrasekhar \\&  M\\\"unch (1950), much mathematical work on this kind of problems has been developed. Among them, one of the most common methods is the Tikhonov regularisation method \n(Tikhonov \\& Arsenin 1977, Tikhonov et al. 1995, Hansen 2010). \n\nThe standard method to solve Eq. (\\ref{eq-ill}) is to apply ordinary least squares (OLS), i.e., $\\min\\{||A\\,X-Y||^2\\}$, where $||\\cdot||$ represents the euclidean norm, but for ill--posed problems this method fails in the sense that can produce unstable estimators. In order to avoid this problem Tikhonov regularization method\nimposes a regularization term to be included in the minimization process, namely: \n\\begin{equation}\\label{eq-regterm}\n\\min\\{||A\\,X-Y||^2\\} \\rightarrow \\min\\{||A\\,X-Y||^2 + \\lambda^2 \\,|| L \\,(X-X_0) ||^2\\},\n\\end{equation}\nwhere $\\lambda$ is the Tikhonov factor. The standard definition for the $L$ matrix is $L=I$, where $I$ is the identity matrix and $X_0$ is an initial estimation, setting $X_0 = 0$, when there is no previous information. There exist different quantitative approaches to obtain Tikhonov factor, e.g.,  Generalized Cross-Validation (GCV), L-curve Method, Discrepancy Principle, Restricted Maximum Likelihood. More details of these are explained in, e.g., Press et al. (2007), Hansen (2010), Tikhonov \\& Arsenin (1977).\nOnce the $\\lambda$-value is attained, the solution $X_\\lambda$ of the regularized problem by Tikhonov method is given by:\n\\begin{equation}\nX_\\lambda =(A^{T} A+\\lambda^2 I)^{-1} A^{T} Y.\n\\label{solTik}\n\\end{equation}\n\nIn this article we use the Tikhonov regularization method using singular value decomposition \n(see appendix A for details) to deconvolve the distribution of the rotational stellar velocities. \n\nIn the data analysed in this article the L-curve method failed, i. e., we do not obtain the ``L\" shape in the L-curve plot, but only the horizontal part of it (see details in Appendix B). For this reason we propose the method described below to chose the Tikhonov factor based on the fact that, when $\\lambda \\to 0$, $X_{\\lambda}$ tends to the exact solution $X$, whereby the difference between two regularized solutions tends to $0$. In Monte Carlo runs (sect. \\ref{sec:mcsim}) the Tikhonov factor has been calculated with our proposed method (see below). We proved (sect. \\ref{sec:mcsim}) empirically that, Tikhonov estimator we obtained, is unbiased and consistent, both desirables properties of any statistical estimator.\n\n\nWe determine the value of Tikhonov factor, $\\lambda$, using the following iterative procedure, which turned out to be faster and efficient to obtain the regularization parameter in case of smooth solutions:\n\\begin{itemize} \n\t\\item[i)]We start with an initial value of $\\lambda$ ($\\lambda=\\lambda_0$). \n\t\\item[ii)] In each following iteration we reduce the value of $\\lambda$ by a factor $f$, ($\\lambda_j=\\,f^j \\,\\lambda_0$), we use typically $f=0.99$. \n\t\\item[iii)] At iteration step $j$ we calculate the difference between the correspondent regularization solutions:  $\\phi = ||X_{\\lambda_j}-X_{\\lambda_{j-1}}||$.\n\t\\item[iv)] If $\\phi$ is small enough, that is, $\\phi<\\epsilon$, we stop the iterative process and get the value of $\\lambda$. Typically a value of $\\epsilon= 10^{-7}$ has been used in this procedure.\n\\end{itemize}\nIn appendix B, we show the criteria for selecting $\\lambda_0$ and  factor $f$.\n\n\\section{Monte Carlo Simulation\\label{sec:mcsim}}\n\nIn this section we present the results of Monte Carlo numerical simulations, to assess the performance of Tikhonov regularization method when applying to deconvolve rotational velocities distribution from Fredhoml integral. Our Monte Carlo runs consist in $n_{MC}= 1000$ independent replications for each of chosen scenarios, where we considered two specific distributions of rotational velocities. Therefore, we simulate 30 different cases described as follows:\n\n\\begin{itemize}\n\\item[a)]Unimodal Distribution: We choose a Maxwellian distribution\n\\begin{equation}\nf_M(x)= \\sqrt{\\frac{2}{\\pi }}\\, \\frac{1}{\\sigma ^3}\\, x^2 e^{-\\frac{x^2}{2 \\sigma ^2}}, \\quad  x>0,\n\\label{eq1M}\n\\end{equation}\nwith parameter $\\sigma=8$, which is the same distribution used in Cur\\'e et al. ($2014$). Furthermore, we consider \nthree different cases, each one including an additive error from a uniform distribution $U[-\\sigma_{\\epsilon}, \\sigma_{\\epsilon}]$, with PDF given by $f_{U}(x)\\,=\\,1\/(2\\sigma_{\\epsilon})$ for $-\\sigma_{\\epsilon}\\leq x\\leq\\sigma_{\\epsilon}$. \nThe chosen values of $\\sigma_{\\epsilon}$ are: $\\sigma_{\\epsilon} = 0.5, 1, 2\\,(km\/ s)$.\n\n\n\\item[b)] Bimodal Distribution: For a mixed of two Maxwellian distributions\n\\begin{equation}\nf_{2M}(x)= \\sqrt{\\frac{2}{\\pi }}\\,\\frac{x^2}{A+B}  \\left(\\frac{A}{\\sigma_1^3} e^{-\\frac{x^2}{2 \\sigma_1^2}} + \\frac{B}{\\sigma_2^3} e^{-\\frac{x^2}{2 \\sigma_2^2}} \\right) , \\quad  x>0,\n\\label{eq2M}\n\\end{equation}\n dispersion parameters are: $\\sigma_1=5$ and $\\sigma_2=15$, and amplitudes: $A=0.3$ and $B=0.7$. We consider the same  additive error cases as the unimodal distribution.\n\\end{itemize}\n\nFurthermore, for both (uni and bimodal) cases, we consider five sample lengths $n_s$: $n_s=30, 100, 300, 1\\,000, 10\\,000$.\n\n\n\nFor each independent Monte Carlo sample we need to simulate two samples, one from the distribution of the rotational velocities (uni or bimodal) and other for the kernel,  $p(y|x)$, representing the distribution of the inclination angles in the Fredholm integral (Eq. \\ref{eq-problema}). Then, we multiply each element of the first sample with the correspondent of the second sample and add the error term. This gives the final sample of $vsin i$ of each scenario. \nThe following step is to estimate the PDF of the projected rotational velocities with a Kernel Density Estimator (KDE, Silverman 1986). Using a grid of $n_g$ points we discretized the Fredholm integral obtaining the linear system (Eq. \\ref{eq-regterm}). \nWith this data we calculate the Tikhonov factor $\\lambda$ using the procedure described above and obtained the Tikhonov regularization solution, $X_\\lambda$, which is the estimated PDF of rotational speeds.\n\n\n\\begin{figure*}[ht]\n\t\\centering\n\t\\includegraphics[scale=0.55]{fig1.eps}\n\t\\caption{\n\t\tUpper panels: Univariate Maxwellian distribution, with parameter $\\sigma=8$, is shown in solid line in all upper panels, black squares connected by dashed line represents the mean of the $n_{MC}=1000$ samples of Tikhonov regularization. Results are for:  $n_s=30$ with $\\sigma_{\\epsilon}=0.5$ (upper left), $n_s=100$ with $\\sigma_{\\epsilon}=1$ (upper center) and $n_s=1000$ with $\\sigma_{\\epsilon}=2$ (upper right).\n\t\tLower panels: Bivariate Maxwellian distributions is shown in solid line in all lower panels, with parameters $\\sigma_1=5$ and $\\sigma_2=15$ and amplitudes $A=0.7$ and $B=0.3$. Black squares connected by dashed line show the estimated PDFs obtained by Tikhonov regularization. Results are for: $n_s=30$ with $\\sigma_{\\epsilon}=0.5$ (lower left), $n_s=300$ with $\\sigma_{\\epsilon}=1$ (lower center) and $n_s=1000$ with $\\sigma_{\\epsilon}=2$ (lower right).}\n\t\\label{Fig1}\n\\end{figure*}\n\nFigure \\ref{Fig1}  upper panels show, in solid line, the original Maxwellian distribution (Eq. \\ref{eq1M}) together with the mean estimated PDF of all Monte Carlo simulations (black squares connected by dashed line) for different values of $\\sigma_{\\epsilon}$ and $n_s$. It is clearly shown that sample lengths of order $n_s \\sim 30$ gives acceptable results when compared with the original sample. For larger sample lengths, $n_s \\gtrsim 100$, the agreement between original distribution and mean of the estimated PDF is almost exact. Although the mean estimated distribution are slightly shifted to lower velocities. Lower panels of Fig. \\ref{Fig1} show the original bimodal mixed Maxwellian distributions (in solid line) together with the mean estimated PDF (black squares connected by dashed line). When a sample length is of order $n_s \\sim 30$, a difference between the estimated PDF and the original PDF is observed. Nevertheless, Tikhonov regularized solution retrieves the bimodality and deliver approximately the position of maximum of both components, but gives a wrong estimate of the tail of the original distribution.\\\\\nIn the other cases ($n_s \\gtrsim 100$) the mean of the Tikhonov regularization solutions is very close to the original mixture of Maxwellian distributions, although the estimated value of the amplitudes is slightly lower (first distribution) and slightly higher (second distribution) than the original one.\n\n\nIn order to quantify the error of the estimated PDF, we calculate (following Cur\\'e et al. 2014) the Mean Integrated Square Error (MISE), that is:\n\n\\begin{equation}\n\\mathrm{MISE}=\\frac{1}{n_{MC}} \\sum_{j=1}^{n_{MC}}\\left(\\frac{1}{n_{g}}\\sum_{i=1}^{n_{g}} (\\hat{f_{j}}(x_i)-f(x_i))^2\\right).\n\\end{equation}\nwhere $f(x)$ represent the original distribution function of rotational speeds and $\\hat{f}_{j}(x)$ represent the estimated Tikhonov regularization density of the $j$-run in Monte Carlo simulations.\n\nIn the left panel of Fig. \\ref{Fig2}  we plotted the MISE values as function of sample length for $\\sigma_{\\epsilon}=0.5$.In the other cases ($\\sigma_{\\epsilon}=1, \\, 2$) the MISE value is very similar with values $\\mathrm{MISE} \\lesssim 10^{-4}$.\nAlso it can be seen that as increasing the sample size, MISE tends to zero, that is, $\\mathrm{MISE}(\\hat{f})\\tiny{\\rightarrow} 0$ when $n_s\\tiny{\\rightarrow} \\infty$. \n\n\\begin{figure*}[ht]\n\t\\centering\n\t\\includegraphics[scale=0.55]{fig2.eps}\n\t\\caption{Left panel: The value of MISE (black dots) from the estimated PDF for univariate distributions (solid line) and  bivariate distributions (dashed line) both using  $\\sigma_{\\epsilon}=0.5$. Right panel show the value of Tikhonov factors $\\lambda$ as function of sample size for the cases where $\\sigma_{\\epsilon}=2$. See text for details.}\n\t\\label{Fig2}\n\\end{figure*}\n\n\nThe right panel in Fig. \\ref{Fig2} shows Tikhonov factors as function of sample size, these factors are of the same order of magnitude for both types of distributions (unimodal and bimodal). Our simulations confirm for all sample lengths and different $\\sigma_{\\epsilon}$ values that, Tikhonov factor ($\\lambda$) is almost independent of the magnitude of the error $\\sigma_\\epsilon$. Furthermore, since Tikhonov parameter changes slightly as function of sample size,  we can consider the Tikhonov factor is almost independent of the sample length, $n_s$. \n\nTo confirm this result, we have performed MC simulations with a fixed value of $\\lambda$. The range of $\\lambda$ was from $0.002$ to $0.01$ with a step of $\\Delta_ \\lambda=0.001$. We calculate the MISE from $n_{MC}=1000$ samples, each with a size of $n_s=1000$ for each value of $\\lambda$. For the unimodal\nMaxwellian distribution the values of the MISE vary increasing from $7.319 \\times 10^{-5}$ to  $7.320 \\times 10^{-5}$ for\nthis range of $\\lambda$, a difference almost negligible. In the case of a bimodal Maxwellian distribution the scenario is\nvery similar using the same range of  $\\lambda$, the MISE values vary increasing from $4.717 \\times 10^{-5}$ until \n$4.718 \\times 10^{-5}$. Similar behaviour is found when $n_s=30, 100, 300, 10000$,  supporting our claim about Tikhonov factor ($\\lambda$) is almost independent of the sample size.\n\n\nBy means of the average of the estimated PDFs we can estimate the expected value for the Tikhonov regularization solution. In all cases, the mean of the estimated PDFs is very close to the original unimodal or bimodal distributions, and this mean probability density function is closer to the true PDF when increasing the sample size. \n\nThis fact shows, empirically, that the studied estimator is asymptotically unbiased. Therefore, since MISE tends to zero when $n_s$ tends to infinity, it implies that the variance of the Tikhonov regularization estimator tends to zero as well and hence it is a consistent estimator.\n\n\n\\section{Deconvolving a Real Sample \\label{sec:realsample}}\nIn this section, we perform the following steps: i) Apply Tikhonov regularization method to a sample of measured $v\\sin i$ data of cluster stars in order to estimate the rotational velocity probability density distribution, ii) Compare  the application of different methods to deconvolve the velocity distribution together with previous non-parametric results from the literature.\n\n\\subsection{Tarantula Sample \\label{sec:tarantula}}\nWe select the Tarantula sample for single O-type stars from the VLT Flames Tarantula Survey, where Ram\\'irez-Agudelo et al. (2013) deconvolved the rotational velocity distribution using the Lucy (1974) method (see also Richardson 1972). This sample contains 216 stars with $v \\sin i$ data from $40  \\, km\/s$ up to $610 \\, km\/s$. Following Ram\\'irez-Agudelo et al. (2013), for comparison purposes, we also omitted the two largest values of the sample (outliers). To build the $Y$ vector, we used the KDE method with the following bandwidths (Silverman, 1986, pages 45 and 47):\n\\begin{eqnarray}\nh_1&=&0.79 \\, IQR\\,  n_s^{-1\/5} \\\\\nh_2&=& 0.9 \\,\\min\\{\\Sigma,IQR\/1.34\\} \\,n_s^{-1\/5},\n\\end{eqnarray}\nhere, $IQR$ is the interquartile range and $\\Sigma$ is the standard deviation of the sample and $n_s$ is the sample length.\\\\\n\nFigure \\ref{Fig5} shows, in solid line, the rotational velocity distribution after Tikhonov regularization. Our procedure for Tikhonov factor determination gives a value of $\\lambda=0.0174$ for a step of $\\Delta x= 2\\, km\/s$. Left panel uses a bandwidth $h_1=35.676$ and right panel a bandwidth $h_2=30.313$. In Fig. \\ref{Fig5} we also plotted in light gray the confidence intervals calculated using bootstrap method ($n_{BS}=3000$). The lower is the bandwidth, the wider is the confidence interval. The bump around $ 400 - 450  \\, km\/s$ is wider in our case ranging from  $\\sim 340 \\, km\/s$ to $  \\sim 480  \\, km\/s$. This discrepancy is probably due to the use of the KDE method with a Gaussian kernel in $Y$.\n\n\\begin{figure*}[ht]\n\t\\centering\n\t\\includegraphics[scale=0.6]{fig3.eps}\n\t\\caption{The estimated PDF from Tarantula sample in solid lines. Both panels with $\\lambda=0.0174$ and $\\Delta x=2\\, km\/s$, Left panel with bandwidth $h_1=35.676$ and right panel with bandwidth $h_2=30.313$. Gray--shaded regions represent the $2.5\\%$ (lower) and $97.5\\%$ (upper) confidence intervals calculated by bootstrap method. Dashed lines show the PDF (from Ram\\'irez-Agudelo et al. 2013) obtained using Lucy (1974) method.}\n\t\\label{Fig5}\n\\end{figure*}\n\n\\subsection{Comparing Results \\label{sec:compresults}}\n\\vspace{-.1cm}\nFor the Tarantula sample, we calculate the CDF by direct integration of the PDF obtained by Tikhonov regularization method  and compare with the CDF calculated by the method described in Cur\\'e et al (2014). Figure \\ref{Fig10}  shows both CDFs, the agreement between both CDFs is remarkable.In addition to our results for the PDF, Fig. \\ref{Fig5} also show in dashed lines the PDF obtained from Ram\\'irez-Agudelo et al. (2013, see their Fig. [17]) calculated using Lucy (1974) method. It can be clearly seen that Lucy-PDF lies inside our confidence interval.\\\\ \nIn order to evaluate if both estimated PDFs correspond to the same distribution, we obtained the q--q plot, calculating the respective quantiles. Figure \\ref{Fig11} shows the q--q plot of these densities, confirming that both coming from the same probability distribution.\n\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[scale=0.38]{fig10.eps}\n\t\\caption{The estimated cumulative rotational velocity distribution function for Tarantula sample (solid line) obtained using Tikhonov regularization using a spacing of $\\Delta x=2\\, km\\, s^{-1}$ for the velocities. Dots connected by dashed line shows the CDF calculated using Cur\\'e et al. (2014) method with a spacing of $\\Delta x=10\\, km\\, s^{-1}$.}\n\t\\label{Fig10}\n\\end{figure}\n\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[scale=0.38]{figqq.eps}\n\t\\caption{q-q Plot from Tarantula sample, black dots represent the quantiles of each distribution, one calculated using Tikhonov regularization method and the other calculated using the Lucy method (data from Ram\\'irez-Agudelo et al., 2013).}\n\t\\label{Fig11}\n\\end{figure}\n\n\\section{Conclusions}\n\nIn this work we have obtained the estimated probability distribution function of 'true' rotational velocities using Tikhonov regularization method.\nFurthermore, this estimated PDF uses a Tikhonov parameter $\\lambda$ obtained by means of an iterative method with a specific stopping criterion in comparison with the widely used  iterative method of Lucy (1974).\\\\\nThrough Monte Carlo numerical simulations we assess the proposed method in two cases: when the rotational velocity distribution is described by a Maxwell distribution and for a mixture of two Maxwell distributions. For each situation different scenarios were evaluated obtaining good results for all of them except for $n_s=30$, when the velocities are described by a mixture of two Maxwellian distributions.\\\\\nThis method retrieve the typical rotational velocities distribution for uni- and bimodal distribution. \nWe showed, empirically, that the studied estimator is asymptotically unbiased and its variance tends to zero. Furthermore, as measure of goodness of fit, the MISE $\\lesssim 10^{-4}$ for all sample sizes and tends to zero when $n_s$ tends to infinity.\\\\\n\nWe apply this method to a set of observed data from Tarantula cluster (Ram\\'irez-Agudelo et al. 2013). The estimated PDF from Tikhonov regularization method agreed very well with the PDF obtained using Lucy method, as the q-q plot shows, demonstrating a very good performance to deconvolve rotational velocity distribution (PDF).\\\\\nIn comparison with the method that delivers the CDF described in Cure et al. (2014), Tikhonov regularization solution gives, by direct integration of the PDF, almost the same non--parametric estimation of the true underlying cumulative distribution function of  rotational velocities.\\\\\n\n\n\n\nSummarizing, in  Cur\\'e et al. (2014) we developed a method to obtain the CDF of  'true' rotational velocities and in this work we present Tikhonov regularization method to obtain the corresponding PDF directly from Fredhoml integral, both methods calculate in a simple and straightforward way, the PDF or CDF, without any assumptions of the underlying distribution. \\\\\n\n\n\nFuture work: We want to develop a general function of the kernel of Fredholm integral, $p(y|x)$, in order to describe an arbitrary orientation of rotational axes. Thus, we can \nstudy the distribution of rotational speeds relaxing the standard assumption of uniformity of stellar axes.\n\n\n\n\\begin{acknowledgements}\nAC thanks the support from Instituto de Estad\\'{i}stica, Pontificia Universidad Cat\\'{o}lica de Valpara\\'{i}so. PE Thanks the support from Advanced Center for Electrical and Electronic Engineering, AC3E, Basal Fund Conicyt FB0008.\nMC thanks the support  Centro de Astrof\\'isica de Valpara\\'iso and Centro Interdiciplinario de Estudios Atmosf\\'ericos y Astroestad\\'istica. JC thanks the financial support from project: \"Ecuaciones Diferenciales y An\\'alisis Num\\'erico\", Instituto de Ciencias, Instituto de Desarrollo Humano e Instituto de Industria, Universidad Nacional de General Sarmiento. DR acknowledge the support of project PIP11420090100165, CONICET.\n\n\\end{acknowledgements}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nPhotogrammetry is a technique to determine the geometric properties of an object from a sequence of photographic images, which has a long history since 1950s. Photogram\u00acmetry can be classified into two main categories as Aerial Photogrammetry and Close Range Photogrammetry (CRP) based on the camera location during the image acquisition process. In CRP, the camera is closed to the subject and 3D models, measurements and point clouds are the most common outputs. This technique is used in different application areas, such as modeling and measuring buildings and large engineering structures, accident scenes inspection, mine inspection, quality controlling and film industry.\n\n\\begin{figure}[!b]\n\\begin{center}\n\\subfloat[]{\\label{binary}\\includegraphics[width=2.7cm]{Fig1a}}\n\\hspace{0.1em}\n\\subfloat[]{\\label{binary}\\includegraphics[width=2.7cm]{Fig1b}}\n\\hspace{0.1em}\n\\end{center}\n\\caption{Two types of CTs. (a) Dot distributions (b) Concentric rings.}\n\\label{fig:fig1}\n\\end{figure}\n\n\\section{Coded Targets}\nIndustrial photogrammetric measurement techniques utilize retro-reflective coded targets (CTs) encoded with a unique identifier to signalize feature points. Two main categories of CTs; concentric rings [1, 2] and dot distributions [3-5] have been introduced so far and two examples are shown in Fig. \\ref{fig:fig1}.\n\nConcentric rings are relatively easy to recognize and decode, but gives a limited number of code possibilities. Therefore, it is not suitable for measuring large scale structures which need several hundreds of markers to cover the whole structure. Dot distribution targets solve this problem by allowing a large number of code possibilities, but have difficulties in recognition due to the complexity. \n\nDifferent dots distribution CTs have been introduced so far and some examples are shown in Fig. \\ref{fig:fig2}. CRP systems work with a large number of CTs within an image and 30 to 100 images for one network. Therefore, it is really important to achieve fully automatic target detection to make the system cost and time effective. In this paper, we introduce a fully automatic coded target detection and decoding method based on the CT shown in Fig. \\ref{fig:fig2}(b).\n\n\\begin{figure}[!b]\n\\centering\n\\includegraphics[width=0.4\\textwidth]{Fig2}\n\\caption{Different types of dot distribution CTs [3-5].}\n\\label{fig:fig2}\n\\end{figure} \n\n\\section{Coded Target Detection}\nThe coded target detection process starts by converting the captured image to gray-scale and removing the image noise using the Gaussian smoothing filter, explained in the Eq. \\ref{Eq1}. Then the adaptive inverse thresholding with a Gaussian kernel is employed to convert the smoothed image to a binary image. Adaptive thresholding defines different threshold values for each pixel in the image by examining the surrounding neighborhood pixels as explained in Eq. \\ref{Eq2} where $T(x,y)$ is the weighted sum of the neighborhood of $(x,y)$ and $c$ is a user defined constant.\n\n\\begin{equation}\ng(x,y) = \\frac{1}{{2\\pi {\\sigma ^2}}}{e^{ - \\frac{{{x^2} + {y^2}}}{{2{\\sigma ^2}}}}}\n\\label{Eq1}\n\\end{equation}\n\n\\begin{equation}\ndst(x,y) = \\left\\{ {\\begin{array}{*{20}{c}}\n0\\\\\n{255}\n\\end{array}} \\right.\\begin{array}{*{20}{c}}\n{if\\;src(x,y) > T(x,y) - c}\\\\\n{otherwise}\n\\end{array}\n\\label{Eq2}\n\\end{equation}\n\nContour detection is applied to the binary image for finding possible circular dots belong to the CTs. After detecting contours, first we remove very large or small contours which are possibly not circular dots by examining the contour size. Then each contour is approximate to a polygon using Douglas-Peucker algorithm and analyze the shape to remove the contours which are not convex and have fewer number of sides. Algorithm introduced in [6] is used to fit each contour to an ellipse and find the center coordinates of circular dots. Finally, these detected circular dots are grouped into possible markers (CTs) using a nearest neighbor search algorithm based on a k-d tree data structure.\n\n\\begin{figure}[!b]\n\\centering\n\\includegraphics[width=0.4\\textwidth]{Fig3}\n\\caption{Flow chart of the decoding process.}\n\\label{fig:fig3}\n\\end{figure} \n\n\\section{Coded Target Decoding}\nEach CT use in this research has eight circular dots where five of them are common for all the CTs and define the axis (orientation) of the CT. Remaining three circles can be chosen from 22 possibilities and they define the code-word of the CT. The first step of the decoding process is to identify the five common circles from the detected possible CTs.\n\nWe measure the total distance from each circle to every other circles within a CT and select the circle with the lowest total as center circle $x_0$. Then we identify the two corner circles $x_1$ and $x_3$ by finding the two circles which make a straight line with $x_0$. From all the eight circles, only $x_1$, $x_0$ and $x_3$ make a straight line within a CT. $x_1$ and $x_3$ can be differentiated by finding the distance to the $x_0$. To find the $x_4$ and $x_5$, we find the two closest circles to the $x_1$ from the remaining circles. Those two circles should reside on the two sides of the line created by $x_1$ and $x_3$ and can be differentiated as $x_4$ and $x_5$ according to the side they reside. \n\nAfter finding the five common circles, the perspective transformation between the original projected CT and the detected one can be calculated. Then the remaining three circles can be projected onto the original marker and the code-word of the CT can be identified using a mask created by original positions of the code circles. Flow chart in Fig. \\ref{fig:fig3} summarizes this decoding process.\n\nWe test our proposed method with different structures (Fig. \\ref{fig:fig4}) and achieve nearly 100\\% detection rate when the CTs are nearly orthogonal to the camera view and about 85\\% for other cases. For every successful detection of a CT, decoding accuracy is almost 100\\% and most of the errors come from the target detection process.\n\n\\section{Conclusions}\nIn an effort to enhance the performance of photogrammetric measurement techniques, in this research, we proposed a fully automated coded target detection and decoding method by using various image processing and vision techniques. As a future work we are planning to integrate this target detection technique with natural feature tracking techniques to develop a more accurate and enhanced photogrammetric system.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{Fig4}\n\\caption{Results of the experiments with different structures.}\n\\label{fig:fig4}\n\\end{figure} \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section*{Acknowledgment}\n\nThis work was supported by the Industry Core Technology Program granted financial resource from the Ministry of Trade, Industry \\& Energy, Republic of Korea (No. 10043897) and Samsung Heavy Industries Co.\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\label{sec:introduction}The last two decades in the development of iterative\nsubstructuring class of the domain decomposition methods has been marked by a\nsignificant progress of the two most advanced methods:\\ \\emph{Finite Element\nTearing and Interconnecting - Dual, Primal} (FETI-DP) and\\ \\emph{Balancing\nDomain Decomposition by Constraints} (BDDC). The FETI-DP method by Farhat et\nal.~\\cite{Farhat-2001-FDP,Farhat-2000-SDP} was developed from an earlier\nversion, called FETI (or FETI-1) by Farhat and\nRoux~\\cite{Farhat-1994-OCP,Farhat-1991-MFE}. The methods from the FETI\\ family\nare \\emph{dual} because they iterate on a system of Lagrange multipliers that\nenforce continuity on the interfaces. However, in the FETI-DP some of the\n\\textquotedblleft coarse\\textquotedblright\\ variables are treated as\n\\emph{primal}. The BDDC method was developed by\nDohrmann~\\cite{Dohrmann-2003-PSC,Mandel-2003-CBD}, as a fully \\emph{primal}\ncounterpart of the FETI-DP, from an earlier\\ BDD\\ method by\nMandel~\\cite{Mandel-1993-BDD}. Interestingly, two methods that turned out to\nbe equivalent to BDDC\\ were developed independently by\nCros~\\cite{Cros-2003-PSC} and by Fragakis and\nPapadrakakis~\\cite{Fragakis-2003-MHP},\nsee~\\cite{Mandel-2007-BFM,Sousedik-2008-EPD} for a proof. The methods are\nclosely related, and the essential role here is played by the coarse problem.\nThe proof that the eigenvalues of the FETI-DP\\ and BDDC\\ are the same except\npossibly for the multiplicity of eigenvalue equal to one, provided that the\ncoarse components of both methods are the same, was obtained by Mandel et\nal.~\\cite{Mandel-2005-ATP}, and simpler proofs were derived soon afterwards by\nLi and Widlund~\\cite{Li-2006-FBB}, and also by Brenner and\nSung~\\cite{Brenner-2007-BFW}.\n\n\nEven though both methods were originally developed for elliptic problems, they\nwere later extended to other models see, e.g., the\nmonographs~\\cite{Kruis-2006-DDM,Pechstein-2013-FBE,Toselli-2005-DDM} for an\noverview. From our current perspective, we find the most important extensions\nbeyond the original two-level algorithms. The possibility for a multilevel\nextension of the BDDC was already mentioned by\nDohrmann~\\cite{Dohrmann-2003-PSC}. The theory of three-level BDDC was\ndeveloped by Tu~\\cite{Tu-2007-TBT3D,Tu-2007-TBT} and extended into a general\nmultispace BDDC framework (with multilevel BDDC\\ as a particular instance)\\ by\nMandel et al.~\\cite{Mandel-2008-MMB}. For further developments of multilevel\nBDDC\\ see,\ne.g.,~\\cite{Kim-2009-TLB,Peng-2018-ABA,Sistek-2013-PIM,Sousedik-2013-NBS,Sousedik-2013-AMB,Tu-2011-TBA,Zampini-2017-MBD}%\n. Extensions of the FETI-DP\\ are less straightforward because the inverse of\nthe coarse problem is built directly into the system matrix, and to the best\nof our knowledge the literature on this topic is limited. For example, a\nhybrid approach to three-level FETI was proposed by Klawonn and\nRheinbach~\\cite{Klawonn-2008-HA3}, and a recursive application of the FETI-DP\nwas recently proposed by Toivanen et al.~\\cite{Toivanen-2018-MFM}, see also\nHor\\'{a}k~\\cite[Chapter~11]{Horak-2007-FBD}.\n\nThe FETI-DP\\ method is, due to the use of dual variables, suitable in\nparticular for numerical solution of certain classes of coercive variational\ninequalities such as the contact problems see,\ne.g.,~\\cite{Dostal-2009-OQP,Dostal-2009-SFA,Jarosova-2012-PPT}. However,\nfactorization of \\textcolor{black}{the} coarse problem may become a computational bottleneck when\nthe number of substructures is large. Our main goal here is to develop a\nvariant of the FETI-DP method that would allow for an approximate solution of\nthe coarse problem using a multilevel strategy, but we do not apply the method\nrecursively as~\\cite{Toivanen-2018-MFM}. Instead, we keep the coarse variables\nprimal. Klawonn and Rheinbach~\\cite{Klawonn-2007-IFM} studied inexact FETI-DP\nmethods based on the saddle-point formulation of the FETI-DP which allows to\nmove (the approximation of) the inverse of the coarse problem into the\npreconditioner. We also use the saddle-point formulation, and the\npreconditioner based on an approximation of the block inverse of the system\nmatrix. However, while~\\cite{Klawonn-2007-IFM} applied algebraic multigrid\n(AMG) to the coarse problem, our main focus here is the multilevel BDDC and\nalso our formulation of the algorithms is different since we do not rely on\npartial subassembly and a change of variables. This strategy then naturally\nleads to a version of multilevel FETI-DP method, which we refer to as\n\\emph{primal}. We show that the spectra of the preconditioned multilevel\nFETI-DP and BDDC\\ operators are essentially the same, in the same way as in\nthe two-level case, and we present numerical experiments that confirm the\ntheory. We note that an alternative point of view is that our method is a\nvariant of the multilevel BDDC that allows to use the dual, FETI-DP\nformulation\\ on the finite element level.\n\nThe paper is organized as follows. In Section~\\ref{sec:model} we introduce the\nmodel problem and substructuring components, in Section~\\ref{sec:methods} we\nrecall the FETI-DP and BDDC\\ methods, in Section~\\ref{sec:bddc-dp} we\nformulate the inexact and multilevel FETI-DP\\ method, in\nSection~\\ref{sec:analysis} we relate it to the multilevel BDDC, and in\nSection~\\ref{sec:numerical} we present results of numerical experiments.\n\n\n\\section{Model problem and substructuring components}\n\n\\label{sec:model}We introduce a model problem and some standard substructuring\nconcepts. So, let us assume we are interested in solving a system of\nlinear\\ equations\n\\begin{equation}\nAu=f, \\label{eq:Au=f}%\n\\end{equation}\nwhere $u\\in U$, and $A$ is a symmetric, positive definite matrix obtained by a\ndiscretization of an elliptic partial differential equation subject to\nappropriate boundary conditions in a bounded domain $\\Omega\\subset%\n\\mathbb{R}\n^{d}$, $d=2$ or $3$.\nLet $\\Omega$ be decomposed into $N$ nonoverlapping subdomains$~\\Omega^{s}$,\n$s=1,\\dots,N$, equally called substructures.\\ Let each substructure be\nobtained as a union of \\textcolor{black}{Lagrangian} P1 or Q1 finite elements with mesh size$~h$,\nand let$~H$ denote the characteristic size of a subdomain. The first step in\nsubstructuring is elimination of degrees of freedom interior to each\nsubstrure, $u_{I}\\in U_{I}$, by so-called \\emph{static condensation}, and the\nproblem is reduced to substructure interfaces. To this end, we denote by\n$W_{i}$ the space of degrees of freedom on the substructure$~i$\\ interface, by\n$w_{i}$ the vector of substructure interface degrees of freedom, and let\n\\[\nW=W^{1}\\times\\cdots W^{N}.\n\\]\nNext, we denote by $S^{i}:W^{i}\\rightarrow W^{i}$ the Schur complement of\nsubdomain$~i$ after eliminating the interior degrees of freedom, and by\n$g^{i}$ the substructure force vector. Also, we introduce $U_{\\Gamma}$ as the\nspace of the global interface degrees of freedom, and denote by $\\widehat{W}$\nthe space of degrees of freedom continuous across the substructure interfaces.\nFinally, we introduce~$R^{i}:U_{\\Gamma}\\rightarrow W^{i}$ as the restriction\nmap (a zero-one matrix) of global to local degrees of freedom. Then, we\ndenote\n\\begin{equation}\nS=\\left[\n\\begin{array}\n[c]{ccc}%\nS^{1} &  & \\\\\n& \\ddots & \\\\\n&  & S^{N}%\n\\end{array}\n\\right]  \\quad w=\\left[\n\\begin{array}\n[c]{c}%\nw^{1}\\\\\n\\vdots\\\\\nw^{N}%\n\\end{array}\n\\right]  ,\\quad g=\\left[\n\\begin{array}\n[c]{c}%\ng^{1}\\\\\n\\vdots\\\\\ng^{N}%\n\\end{array}\n\\right]  ,\\quad R=\\left[\n\\begin{array}\n[c]{c}%\nR^{1}\\\\\n\\vdots\\\\\nR^{N}%\n\\end{array}\n\\right]  , \\label{eq:blocks}%\n\\end{equation}\nwhere $R$ has a full column rank such that $R^{i}R^{iT}=I$ , and\n$\\widehat{W}=\\operatorname*{range}R$. Then, problem~(\\ref{eq:Au=f}) reduced to\ninterfaces may be written as\n\\begin{equation}\n\\widehat{S}w=f_{\\Gamma},\\qquad\\widehat{S}=R^{T}SR,\\quad f_{\\Gamma}=R^{T}g.\n\\label{eq:Sw=f}%\n\\end{equation}\nNext ingredient is the averaging operator $E=R^{T}D_{P}$, where $D_{P}%\n:W\\rightarrow W$ is a given weight matrix such that $ER=I.$ Also, let\n$B:W\\rightarrow\\Lambda$ denote the matrix enforcing the condition\n$w\\in\\widehat{W}$ by $Bw=0$, that is $\\operatorname*{range}%\nR=\\operatorname*{null}B$. In agreement with the block\nstructure~(\\ref{eq:blocks}), we consider\n\\[\nB=\\left[\n\\begin{array}\n[c]{ccc}%\nB^{1} & \\cdots & B^{N}%\n\\end{array}\n\\right]  ,\\qquad B^{i}=W^{i}\\rightarrow\\Lambda,\\quad\\operatorname*{range}%\nB=\\Lambda.\n\\]\nLet $B_{D}^{T}$ denote a generalized inverse of $B$, that is $B_{D}%\n^{T}:\\Lambda\\rightarrow W$ and $BB_{D}^{T}=I_{\\Lambda}$. In computations, the\nmatrix $B_{D}$ is constructed from $B$\\ and $D_{P}$,\nsee~\\cite{Klawonn-2002-DPF,Mandel-2005-ATP}, and, in particular, we will\nassume that\n\\[\nB_{D}^{T}B+RE=I.\n\\]\n\n\nThe FETI-DP and BDDC\\ methods are characterized by a selection of \\emph{coarse\ndegrees of freedom }(resp. \\emph{constraints}). These can be values at corners\nor averages over edges or faces. Let $\\widetilde{W}\\subset W$ be the subspace\nof vectors with continuous coarse degrees of freedom across substructure\ninterface. Specifically, suppose we are given a space$~X$ and a matrix\n$C:W\\rightarrow X$, and define\n\\begin{equation}\n\\widetilde{W}=\\left\\{  w\\in W:C\\left(  I-RE\\right)  w=0\\right\\}  ,\n\\label{eq:W-tilde-1}%\n\\end{equation}\nwhere each row of $C$ defines one constraint. The values $Cw$ are called local\ncoarse degrees of freedom and by~(\\ref{eq:W-tilde-1}) they have zero jumps on\nadjacent substructures. To represent the global coarse degrees of freedom, we\nuse a space$~U_{c}$ and a matrix$~R_{c}$ with full column rank, such that\n\\[\n\\widetilde{W}=\\left\\{  w\\in W:\\exists u_{c}\\in U_{c}:Cw=R_{c}u_{c}\\right\\}  .\n\\]\nNext, let $\\Phi$ be a matrix that defines the basis for the coarse problem,\ngiven as\n\\[\n\\Phi=\\left[\n\\begin{array}\n[c]{c}%\n\\Phi^{1}\\\\\n\\vdots\\\\\n\\Phi^{N}%\n\\end{array}\n\\right]  ,\\qquad C\\Phi=R_{c}.\n\\]\n\\textcolor{black}{Here we use energy minimal coarse basis functions$~\\Psi$ that minimize \nenergy of the form $x^T \\Phi^{T}S\\Phi x$ subject to constraints $Cx=e$ for some vector $e$ given for each basis function as a column of matrix $R_c$.\nSpecifically, the basis functions are found by solving saddle-point problems}\n\\begin{equation}\n\\left[\n\\begin{array}\n[c]{cc}%\nS & C^{T}\\\\\nC & 0\n\\end{array}\n\\right]  \\left[\n\\begin{array}\n[c]{c}%\n\\Psi\\\\\n\\Lambda\n\\end{array}\n\\right]  =\\left[\n\\begin{array}\n[c]{c}%\n0\\\\\nR_{c}%\n\\end{array}\n\\right]  . \\label{eq:energy-minimization}%\n\\end{equation}\nIn implementation, we use a decomposition of the space $\\widetilde{W}$,\nsee~\\cite[Lemma~8]{Mandel-2005-ATP},\n\\begin{equation}\n\\widetilde{W}=\\widetilde{W}_{\\Pi}\\oplus\\widetilde{W}_{\\Delta},\n\\label{eq:dharm-dec-interface}%\n\\end{equation}\nwhere $\\widetilde{W}_{\\Pi}=\\operatorname*{range}\\Phi$ is the \\emph{primal}\nspace, which gives rise to the (global) coarse problem, and $\\widetilde{W}%\n_{\\Delta}=\\operatorname*{null}C$ is the \\emph{dual} space with coarse degrees\nof freedom equal to zero, which gives rise to independent subdomain problems.\n\n\n\\section{FETI-DP and BDDC\\ methods}\n\n\\label{sec:methods}\n\n\\subsection{FETI-DP}\n\nProblem~(\\ref{eq:Sw=f}) may be equivalently\\ written as\n\\begin{equation}\n\\min_{w\\in\\widetilde{W}}\\max_{\\lambda\\in\\Lambda}\\mathcal{L}(w,\\lambda),\n\\label{eq:nested-minimization}%\n\\end{equation}\nwhere $\\mathcal{L}\\left(  w,\\lambda\\right)  $ is the \\textcolor{black}{Lagrangian} \n\\[\n\\mathcal{L}\\left(  w,\\lambda\\right)  =\\frac{1}{2}w^{T}Sw-w^{T}E^{T}f_{\\Gamma\n}+w^{T}B^{T}\\lambda.\n\\]\nProblem~(\\ref{eq:nested-minimization}) is further equivalent to the dual\nproblem $\\max_{\\lambda\\in\\Lambda}\\mathcal{F}(\\lambda)$, where\n\\begin{equation}\n\\mathcal{F}(\\lambda)=\\min_{w\\in\\widetilde{W}}\\mathcal{L}(w,\\lambda)\n\\label{eq:dual-functional}%\n\\end{equation}\nis the dual functional.\n\nNext, we use the splitting~(\\ref{eq:dharm-dec-interface}) and introduce\nLagrange multipliers$~\\mu$ to enforce the zero values of coarse degrees of\nfreedom in$~\\widetilde{W}_{\\Delta}$. Then,\nfunctional~(\\ref{eq:dual-functional}) can be written with minimization\nover$~\\widetilde{W}$ instead of$~\\widetilde{W}_{\\Delta}$ as\n\\begin{equation}\n\\mathcal{F}(\\lambda)=\\min_{w_{\\Delta}\\in\\widetilde{W}}\\min_{w_{\\Pi}%\n\\in\\widetilde{W}_{\\Pi}}\\sup_{\\mu}\\mathcal{L}(w_{\\Delta}+w_{\\Pi},\\lambda\n)+w_{\\Delta}^{T}C^{T}\\mu.\\label{eq:augmented-nested-minimization}%\n\\end{equation}\nWriting $w_{\\Pi}=\\Phi u_{c}$,\nand differentiating with respect to $w_{\\Delta}$, $\\mu$, $u_{c}$, $\\lambda$,\nwe get a linear system, with a solution equvalent to solving the dual problem,\nwhich is\n\\begin{equation}\n\\left[\n\\begin{array}\n[c]{cccc}%\nS & C^{T} & S\\Phi & B^{T}\\\\\nC & 0 & 0 & 0\\\\\n\\Phi^{T}S^{T} & 0 & \\Phi^{T}S\\Phi & \\Phi^{T}B^{T}\\\\\nB & 0 & B\\Phi & 0\n\\end{array}\n\\right]  \\left[\n\\begin{array}\n[c]{c}%\nw_{\\Delta}\\\\\n\\mu\\\\\nu_{c}\\\\\n\\lambda\n\\end{array}\n\\right]  =\\left[\n\\begin{array}\n[c]{c}%\nE^{T}f_{\\Gamma}\\\\\n0\\\\\n\\Phi^{T}E^{T}f_{\\Gamma}\\\\\n0\n\\end{array}\n\\right]  .\\label{eq:feti-dp-full}%\n\\end{equation}\nEliminating $w_{\\Delta}$ and $\\mu$ from the first two equations, we derive the\nso-called \\emph{saddle-point }FETI-DP formulation, which can be written as\n\\begin{equation}\n\\left[\n\\begin{array}\n[c]{cc}%\nS_{c} & \\Psi^{T}B^{T}\\\\\nB\\Psi & -BS_{\\Delta}^{-1}B^{T}%\n\\end{array}\n\\right]  \\left[\n\\begin{array}\n[c]{c}%\nu_{c}\\\\\n\\lambda\n\\end{array}\n\\right]  =\\left[\n\\begin{array}\n[c]{c}%\n\\Psi^{T}E^{T}f_{\\Gamma}\\\\\n-BS_{\\Delta}^{-1}E^{T}f_{\\Gamma}%\n\\end{array}\n\\right]  ,\\label{eq:bddc-dp}%\n\\end{equation}\nwhere $S_{\\Delta}^{-1}$ corresponds to independent subdomain solves\n\\begin{equation}\nS_{\\Delta}^{-1}=\\left[\n\\begin{array}\n[c]{c}%\nI\\\\\n0\n\\end{array}\n\\right]  ^{T}\\left[\n\\begin{array}\n[c]{cc}%\nS & C^{T}\\\\\nC & 0\n\\end{array}\n\\right]  ^{-1}\\left[\n\\begin{array}\n[c]{c}%\nI\\\\\n0\n\\end{array}\n\\right]  ,\\label{eq:sub_solve}%\n\\end{equation}\nso that $\\Psi=\\left(  I-S_{\\Delta}^{-1}S\\right)  \\Phi$ are the energy\nminimizing coarse basis functions, the same as\nin~(\\ref{eq:energy-minimization}), cf.~\\cite[eqs.~($22$) and ($25$%\n)]{Mandel-2005-ATP}, and $S_{c}$ is the coarse matrix\n\\begin{equation}\nS_{c}=\\Phi^{T}S\\Phi-\\left[\n\\begin{array}\n[c]{c}%\nS\\Phi\\\\\n0\n\\end{array}\n\\right]  ^{T}\\left[\n\\begin{array}\n[c]{cc}%\nS & C^{T}\\\\\nC & 0\n\\end{array}\n\\right]  ^{-1}\\left[\n\\begin{array}\n[c]{c}%\nS\\Phi\\\\\n0\n\\end{array}\n\\right]  =\\Psi^{T}S\\Psi,\\label{eq:coarse_matrix}%\n\\end{equation}\nwhere the second equality follows from~\\cite[Theorem~$3$]{Mandel-2005-ATP}.\n\nWe note that eliminating$~u_{c}$ from (\\ref{eq:bddc-dp}) yields the\n\\emph{original} FETI-DP\\ system\n\\begin{equation}\nB\\left(  \\Psi S_{c}^{-1}\\Psi^{T}+S_{\\Delta}^{-1}\\right)  B^{T}\\lambda=B\\left(\n\\Psi S_{c}^{-1}\\Psi^{T}+S_{\\Delta}^{-1}\\right)  E^{T}f_{\\Gamma}%\n.\\label{eq:feti-dp-orig}%\n\\end{equation}\nThe original FETI-DP is the method of preconditioned conjugate gradients\napplied to~(\\ref{eq:feti-dp-orig}), used typically with the so-called\nDirichlet preconditioner\n\\begin{equation}\nM_{D}=B_{D}SB_{D}^{T}.\\label{eq:Dirichlet-preconditioner}%\n\\end{equation}\n\n\n\\subsection{BDDC}\n\nThe BDDC preconditioner was originally formulated by\nDohrmann~\\cite{Dohrmann-2003-PSC} for problem~(\\ref{eq:Au=f}). Formulations\nfor the reduced problem~(\\ref{eq:Sw=f}) followed\nin~\\cite{Mandel-2003-CBD,Mandel-2005-ATP}. The BDDC is the method of\npreconditioned conjugate gradients, in which the (two-level)\nBDDC\\ preconditioner for problem~(\\ref{eq:Sw=f}) is given as,\ncf.~\\cite[Lemma~27]{Mandel-2005-ATP},\n\\begin{equation}\nM_{\\widehat{S}}=EHE^{T}, \\label{eq:bddc}%\n\\end{equation}\nwhere%\n\\begin{equation}\nH=\\left[\n\\begin{array}\n[c]{c}%\n\\Psi^{T}\\\\\nI\n\\end{array}\n\\right]  ^{T}\\left[\n\\begin{array}\n[c]{cc}%\nS_{c} & 0\\\\\n0 & S_{\\Delta}%\n\\end{array}\n\\right]  ^{-1}\\left[\n\\begin{array}\n[c]{c}%\n\\Psi^{T}\\\\\nI\n\\end{array}\n\\right]  =\\Psi S_{c}^{-1}\\Psi^{T}+S_{\\Delta}^{-1}. \\label{eq:H}%\n\\end{equation}\nThe preconditioner formulated for problem~(\\ref{eq:Au=f}) includes in addition\na pre-processing step with a static condensation of the residual and\npost-processing step with recovery of the approximate solution corresponding\nto the subdomain interiors. Both formulations are equivalent, and in\nparticular by~\\cite[Theorem~14]{Mandel-2008-MMB}, the eigenvalues of the two\npreconditioned operators are the same except possibly for multiplicity of\neigenvalue equal to one.\n\n\n\\subsubsection{Multilevel BDDC}\n\nIn case when the number of subdomains in large, solving the \\emph{coarse\nproblem}\\ becomes a bottleneck. This motivated the introduction of the\nmultilevel extensions~\\cite{Mandel-2008-MMB,Tu-2007-TBT3D,Tu-2007-TBT} that\nsolve the coarse problem only approximately by applying the two-level BDDC\nrecursively. Since we will apply this idea in the FETI-DP\\ framework, we\nbriefly recall the Multilevel BDDC from~\\cite{Mandel-2008-MMB}.\nThe substructuring components\\ will be denoted by an additional\nsubscript$~_{1}$ as $U_{1}$, $\\widetilde{W}_{\\Pi1}$\\ and $\\Omega_{1}^{s}$,\n$s=1,\\dots,N_{1}$, etc., and called level$~1$. Level$~1$ coarse problem will\nbe called level$~2$ problem. Because it has an analogous structure as the\noriginal problem on level$~1$, we put $U_{2}=\\widetilde{W}_{\\Pi1}$, and we\nintroduce the substructuring components for level$~2$ in the same way as we\nintroduced level$~1$ components.\nGenerally, in a design of $L$-level method, we repeat this process recursively\nfor levels $\\ell=1,\\dots,L-1$. On each decomposition level$~\\ell$, we assume\nthat the subdomains$~\\Omega_{\\ell}^{s}$, $s=1,\\dots,N_{\\ell}$, have\ncharacteristic size $H_{\\ell}$ and form a conforming triangulation of the\ndomain$~\\Omega$. \\ Level$~\\ell-1$ substructures become level$~\\ell$ elements,\nlevel$~\\ell-1$ coarse degrees of freedom become level$~\\ell$ degrees of\nfreedom, and $\\Gamma_{\\ell}\\subset\\Gamma_{\\ell-1}$. The finite element level\nwill be denoted as level$~0$\\ and $H_{0}=h$. \n\\textcolor{black}{An example of a decomposition is shown in Figure~\\ref{fig:decomposition}.}\n\n\\begin{figure}[tbh]\n\\begin{center}%\n\\begin{tikzpicture}[scale=.14] \n\\foreach \\i in {0,...,9} { \\draw [very thin,gray] (\\i,0) -- (\\i,9);} \\foreach \\i in {0,...,9} { \\draw [very thin,gray] (0,\\i) -- (9,\\i); } \n\\foreach \\i in {0,...,9} { \\draw [thick,red] (3*\\i,0) -- (3*\\i,18);} \\foreach \\i in {0,...,6} { \\draw [thick,red] (0,3*\\i) -- (27,3*\\i); } \n\\foreach \\i in {0,...,3} { \\draw [very thick,blue] (9*\\i,0) -- (9*\\i,27);} \\foreach \\i in {0,...,3} { \\draw [very thick,blue] (0,9*\\i) -- (27,9*\\i); } \n\\draw (-21,0) -- (0,0); \\draw (-2.8,1) -- (0,1);  \\draw [->] (-2,-1.5) -- (-2,0); \\draw (-2,0) -- (-2,1); \\draw [->] (-2,2.5) -- (-2,1); \\node at (-6.5,1.5) {$H_0=h$};\n\\draw (-15,3) -- (0,3); \\draw [->] (-14,-2) -- (-14,0); \\draw (-14,0) -- (-14,3); \\draw [->] (-14,5) -- (-14,3); \\node at (-16,1.5) {$H_1$};\n\\draw (-21,9) -- (0,9); \\draw [->] (-20,-2) -- (-20,0); \\draw (-20,0) -- (-20,9); \\draw [->] (-20,12) -- (-20,9); \\node at (-22,4.5) {$H_2$};\n\\draw [{Circle[color=red]}-] (25,1.5) -- (30,1.5);  \\node[font = {\\large}, red] at (33,1.5) {$\\Omega_1^9$}; \n\\draw [{Circle[color=blue]}-] (22,22.5) -- (30,22.5);  \\node[font = {\\large}, blue] at (33,22.5) {$\\large \\Omega_2^9$}; \n\\end{tikzpicture} \n\\end{center}\n\\caption{An example of a uniform decomposition for a three-level method with $H_\\ell\/H_{\\ell-1}=3$, where $\\ell=1,2$. \nLevel~$1$ subdomains are shown in red and level~$2$ subdomains in blue.}%\n\\label{fig:decomposition}%\n\\end{figure}\n\nThe shape functions on\nlevel$~\\ell$ are determined by minimization of energy with respect to\nlevel$~\\ell-1$ shape functions, subject to the value of exactly one\nlevel$~\\ell$ degree of freedom being one and others level $\\ell$ degrees of\nfreedom being zero. The minimization is done on each level$~\\ell$ element\n(level$~\\ell-1$ substructure) separately, so the values of level$~\\ell-1$\ndegrees of freedom are in general discontinuous between level$~\\ell-1$\nsubstructures, and only the values of level$~\\ell$ degrees of freedom between\nneighboring level$~\\ell$ elements coincide.\nThe hierarchy of the spaces for $L$-level BDDC\\ method can be written as\n\\[%\n\\begin{array}\n[c]{ccccccccccccccccccc}\n&  & U_{I1} &  &  &  &  &  &  &  &  &  &  &  & U_{IL-1} &  &  &  & \\\\\nU_{1} & = & \\oplus &  &  &  & \\widetilde{W}_{\\Pi1} & = & U_{2} &  & \\cdots &\n& U_{L-1} & = & \\oplus &  &  &  & \\widetilde{W}_{\\Pi L-1}\\\\\n&  & \\widehat{W}_{1} &\n\\genfrac{}{}{0pt}{}{\\genfrac{}{}{0pt}{}{E_{1}}{\\leftarrow}}{\\subset}%\n& \\widetilde{W_{1}} & = & \\oplus &  &  &  &  &  &  &  & \\widehat{W}_{L-1} &\n\\genfrac{}{}{0pt}{}{\\genfrac{}{}{0pt}{}{E_{L-1}}{\\leftarrow}}{\\subset}%\n& \\widetilde{W}_{L-1} & = & \\oplus\\\\\n&  &  &  &  &  & \\widetilde{W}_{\\Delta1} &  &  &  &  &  &  &  &  &  &  &  &\n\\widetilde{W}_{\\Delta L-1}%\n\\end{array}\n.\n\\]\nThe two-level BDDC\\ preconditioner for problem~(\\ref{eq:Sw=f}) entails finding\ncorrections in the spaces$~\\widetilde{W}_{\\Delta1}$ and$~\\widetilde{W}_{\\Pi1}%\n$, and the multilevel BDDC\\ for the same problem entails computing corrections\nin the spaces as above, that is\n\\begin{equation}\nV_{1}=\\widetilde{W}_{\\Delta1},\\quad V_{2}=U_{I2},\\quad V_{3}=\\widetilde{W}%\n_{\\Delta2},\\quad\\dots\\quad V_{2(L-1)}=\\widetilde{W}_{\\Pi L-1},\n\\label{eq:ml-spaces}%\n\\end{equation}\nsuch that\n\\begin{equation}\n\\widetilde{W}_{1}=\\left(  \\widetilde{W}_{\\Delta1}\\oplus\\widetilde{W}_{\\Pi\n1}\\right)  \\subset\\widetilde{V}=\\sum\\nolimits_{k=1}^{2(L-1)}V_{k}\\subset W.\n\\label{eq:ml-spaces-embedding}%\n\\end{equation}\nThe precise formulation of the multilevel BDDC preconditioner can be found\nin~\\cite[Algorithm 17]{Mandel-2008-MMB}. More generally, as an analogy\nto~(\\ref{eq:bddc})--(\\ref{eq:H}), we may consider an \\emph{inexact}\nBDDC\\ preconditioner given as\n\\begin{equation}\n\\widetilde{M}_{\\widehat{S}}=E\\widetilde{H}E^{T}, \\label{eq:mBDDC}%\n\\end{equation}\nwhere\n\\begin{equation}\n\\widetilde{H}=\\left[\n\\begin{array}\n[c]{c}%\n\\Psi^{T}\\\\\nI\n\\end{array}\n\\right]  ^{T}\\left[\n\\begin{array}\n[c]{cc}%\nM_{c} & 0\\\\\n0 & S_{\\Delta}^{-1}%\n\\end{array}\n\\right]  \\left[\n\\begin{array}\n[c]{c}%\n\\Psi^{T}\\\\\nI\n\\end{array}\n\\right]  =\\Psi M_{c}\\Psi^{T}+S_{\\Delta}^{-1}, \\label{eq:H-tilde}%\n\\end{equation}\nand $M_{c}$ is a preconditioner for the coarse matrix$~S_{c}$. The multilevel\nBDDC\\ applied to the coarse problem is then a particular example of$~M_{c}$.\n\nFinally, we note that the derivation of the FETI-DP~(\\ref{eq:feti-dp-orig})\nprovides an alternative proof to~\\cite[Lemma~$27$]{Mandel-2005-ATP}, which\nstates that the preconditioned BDDC\\ and FETI-DP\\ operators can be written,\nrespectively, as\n\\begin{align}\nM_{\\widehat{S}}\\widehat{S}  &  =\\left(  EHE^{T}\\right)  \\left(  R^{T}%\nSR\\right)  ,\\label{eq:PA}\\\\\nM_{D}F  &  =\\left(  B_{D}SB_{D}^{T}\\right)  \\left(  BHB^{T}\\right)  ,\n\\label{eq:MF}%\n\\end{align}\nwhich is used in the proof of equivalence of the spectra of the two operators.\n\n\\begin{theorem}\n[{\\cite[Theorem~$26$]{Mandel-2005-ATP}}]\n\\label{thm:equiv}\nThe eigenvalues of the preconditioned\noperators of the BDDC and FETI-DP\\ methods given, respectively, by~(\\ref{eq:PA}) and~(\\ref{eq:MF})\nare the same except possibly for the eigenvalues equal to zero and one.\n\\end{theorem}\n\n\n\\begin{proof}\nSee Appendix.\n\\end{proof}\n\n\nOur next goal is to design a multilevel version of the FETI-DP\\ method, based\non formulation~(\\ref{eq:bddc-dp}), and relate its spectrum to that of\nMultilevel BDDC.\n\n\n\\section{Inexact and multilevel FETI-DP methods}\n\n\\label{sec:bddc-dp}Since the inverse of the coarse matrix$~S_{c}$ is built\ninto the original FETI-DP operator in (\\ref{eq:feti-dp-orig}), its application\nmay become a bottleneck when the number of substructures is large. Therefore,\nwe work with the saddle-point formulation~(\\ref{eq:bddc-dp}) that allows to\nsolve coarse problem inexactly, which is a strategy proposed by Klawonn and\nRheinbach~\\cite{Klawonn-2007-IFM}.\n\n\nThe proposed preconditioner is based on the block inverse\n\\begin{align*}\n\\left[\n\\begin{array}\n[c]{cc}%\nX & Y^{T}\\\\\nY & -Z\n\\end{array}\n\\right]  ^{-1}  &  =\\left[\n\\begin{array}\n[c]{cc}%\nI & -X^{-1}Y^{T}\\\\\n0 & I\n\\end{array}\n\\right]  \\left[\n\\begin{array}\n[c]{cc}%\nX^{-1} & 0\\\\\n0 & -S_{X}^{-1}%\n\\end{array}\n\\right]  \\left[\n\\begin{array}\n[c]{cc}%\nI & 0\\\\\n-YX^{-1} & I\n\\end{array}\n\\right]  =\\\\\n&  =\\left[\n\\begin{array}\n[c]{cc}%\nI & -X^{-1}Y^{T}\\\\\n0 & I\n\\end{array}\n\\right]  \\left[\n\\begin{array}\n[c]{cc}%\nX^{-1} & 0\\\\\nS_{X}^{-1}YX^{-1} & -S_{X}^{-1}%\n\\end{array}\n\\right]  ,\n\\end{align*}\nand $S_{X}=Z+YX^{-1}Y^{T}$. For the inverse of matrix in~(\\ref{eq:bddc-dp}),\nwe then get\n\\begin{equation}\n\\left[\n\\begin{array}\n[c]{cc}%\nS_{c} & \\Psi^{T}B^{T}\\\\\nB\\Psi & -BS_{\\Delta}^{-1}B^{T}%\n\\end{array}\n\\right]  ^{-1}=\\left[\n\\begin{array}\n[c]{cc}%\nI & -S_{c}^{-1}\\Psi^{T}B^{T}\\\\\n0 & I\n\\end{array}\n\\right]  \\left[\n\\begin{array}\n[c]{cc}%\nS_{c}^{-1} & 0\\\\\nF^{-1}B\\Psi S_{c}^{-1} & -F^{-1}%\n\\end{array}\n\\right]  , \\label{eq:block-dec-bddc-dp}%\n\\end{equation}\nand $F=B\\left(  \\Psi S_{c}^{-1}\\Psi^{T}+S_{\\Delta}^{-1}\\right)  B^{T}$ is the\nsame as the FETI-DP matrix in~(\\ref{eq:feti-dp-orig}). The matrix on the right\nin~(\\ref{eq:block-dec-bddc-dp}) then motivates the choice of a preconditioner\n\\begin{equation}\nM_{F}=\\left[\n\\begin{array}\n[c]{cc}%\nM_{c} & 0\\\\\nM_{D}B\\Psi M_{c} & -M_{D}%\n\\end{array}\n\\right]  , \\label{eq:M_D}%\n\\end{equation}\nwhere$~M_{D}$ is the Dirichlet\npreconditioner~(\\ref{eq:Dirichlet-preconditioner}), and $M_{c}$ is the\npreconditioner for the coarse matrix$~S_{c}$. In general, we may consider any\nsuitable$~M_{c}$ giving rise to \\emph{inexact} FETI-DP. For example, Klawonn\nand Rheinbach~\\cite{Klawonn-2007-IFM} used algebraic multigrid (AMG). Here,\nour main focus is on application of multilevel BDDC in place of$~M_{c}$, which\nwe then refer to as \\emph{primal multilevel} FETI-DP method. In\nimplementation, the application of the preconditioner~$M_{F}$ entails one\napplication of both~$M_{c}$ and~$M_{D}$, which may be computed as follows.\n\n\\begin{algorithm}\n\\label{alg:M_D}The preconditioner$~M_{F}:$ $\\left(  r_{c},r_{\\lambda}\\right)\n\\mapsto\\left(  u_{c},\\lambda\\right)  $\nis applied as:\\newline First, apply the preconditioner $M_{c}$ as $u_{c}%\n=M_{c}r_{c}$, and compute%\n\\begin{align*}\nx  &  =B\\Psi u_{c},\\\\\ny  &  =x-r_{\\lambda}.\n\\end{align*}\nThen, apply the preconditioner $M_{D}$ as $\\lambda=M_{D}y$, and concatenate\nthe results\n\\[\n\\left[\n\\begin{array}\n[c]{c}%\nu_{c}\\\\\n\\lambda\n\\end{array}\n\\right]  .\n\\]\n\n\\end{algorithm}\n\n\\begin{remark}\nAlgorithm~\\ref{alg:M_D} is closely related to the second algorithm by\nKlawonn and Rheinbach~\\cite{Klawonn-2007-IFM}.\nHowever, our formulation of FETI-DP does not use partial\nsubassembly and a change of variables,\nand we consider multilevel BDDC besides algebraic multigrid as a preconditioner~$M_{c}$\nfor the coarse matrix~$S_c$.\n\\end{remark}\n\n\n\\section{Analysis of the methods}\n\n\\label{sec:analysis}The saddle-point FETI-DP matrix from~(\\ref{eq:bddc-dp})\npreconditioned by$~M_{F}$\\ from~(\\ref{eq:M_D})~is%\n\\begin{equation}\n\\left[\n\\begin{array}\n[c]{cc}%\nM_{c}S_{c} & M_{c}\\Psi^{T}B^{T}\\\\\nM_{D}B\\Psi\\left(  M_{c}S_{c}-I\\right)  & M_{D}B\\left(  \\Psi M_{c}\\Psi\n^{T}+S_{\\Delta}^{-1}\\right)  B^{T}%\n\\end{array}\n\\right]  . \\label{eq:M_D_bddc-dp}%\n\\end{equation}\nFirst, note that with the exact coarse solve, i.e., setting $M_{c}=S_{c}^{-1}%\n$, the matrix in position (1,1) is the identity so the matrix in position\n(2,1) vanishes, and the matrix in position (2,2) is the same as the\npreconditioned FETI-DP operator~(\\ref{eq:MF}). Finally, multiplying the\nright-hand side of~(\\ref{eq:bddc-dp}) by$~M_{F}$ we may check that solving the\nsecond set of equations therein is the same as solving the original\nFETI-DP\\ problem~(\\ref{eq:feti-dp-orig}) with the Dirichlet\npreconditioner~$M_{D}$.\n\nNext, let us observe that the inexact BDDC preconditioner~(\\ref{eq:mBDDC})\nwith $\\widetilde{H}$ from~(\\ref{eq:H-tilde})\\ and the entry in position (2,2)\nof matrix (\\ref{eq:M_D_bddc-dp}) with the Dirichlet preconditioner$~M_{D}$\ndefined by~(\\ref{eq:Dirichlet-preconditioner}) may be written, respectively,\nas\n\\begin{align}\n\\widetilde{M}_{\\widehat{S}}\\widehat{S}  &  =\\left(  E\\widetilde{H}%\nE^{T}\\right)  \\left(  R^{T}SR\\right)  ,\\label{eq:mBDDC-prec-op}\\\\\nM_{D}\\widetilde{F}  &  =\\left(  B_{D}SB_{D}^{T}\\right)  \\left(  B\\widetilde{H}%\nB^{T}\\right)  . \\label{eq:(2,2)}%\n\\end{align}\nThat is (\\ref{eq:mBDDC-prec-op})--(\\ref{eq:(2,2)}) is an analogy to\n(\\ref{eq:PA})--(\\ref{eq:MF}), just with $\\widetilde{H}$ in place of $H$.\n\nLet us now focus on the primal multilevel FETI-DP\\ method, which combines the\ndual (FETI-DP type of) approach to the solve in~$\\widetilde{W}_{\\Delta}$ and\napplies the (primal) multilevel BDDC preconditioner to the coarse solve\nin~$\\widetilde{W}_{\\Pi}$.\n\n\\begin{assumption}\n\\label{assum:coarse-split}We will assume that the spaces $V_{k}$,\n$k=1,\\dots,2(L-1)$, are constructed so that the associated linear systems are\nnonsingular, and the multilevel BDDC uses exact solves in these subspaces, so\nthat\n\\begin{equation}\n\\left.  \\widetilde{H}S\\right\\vert _{\\widetilde{V}\\subset W}=I.\\label{eq:HS}%\n\\end{equation}\n\n\\end{assumption}\n\nFor the two-level BDDC method the Assumption~\\ref{assum:coarse-split} holds\ndue to the exact coarse solve, see also~\\cite[Lemma~28]{Mandel-2005-ATP}. For\nmultispace and multilevel BDDC see \\cite[Corollary~4, Remark~5 and\nLemma~18]{Mandel-2008-MMB}. \n\nThen, a variant of Theorem~\\ref{thm:equiv} also holds for the two\noperators~(\\ref{eq:mBDDC-prec-op})--(\\ref{eq:(2,2)}).\n\n\\begin{lemma}\n\\label{lem:equiv-mlevel}\nLet $\\widetilde{H}$ be defined as in~(\\ref{eq:H-tilde}%\n) and so that assumption~(\\ref{eq:HS}) is satisfied. Then the eigenvalues\nof the preconditioned operators given by (\\ref{eq:mBDDC-prec-op}) and\n(\\ref{eq:(2,2)}) are the same except possibly for the eigenvalues equal to\nzero and one.\n\\end{lemma}\n\n\n\\begin{proof}\nSee Appendix.\n\\end{proof}\n\n\n\\textcolor{black}{Let us now use Lemma~\\ref{lem:equiv-mlevel} to examine the eigenvalues of the preconditioned operator~(\\ref{eq:M_D_bddc-dp}). \nThe corresponding eigenvalue problem can be written~as} \n\n\\begin{equation}%\n\\begin{array}\n[c]{ccccc}%\nM_{c}S_{c}u & + & M_{c}\\Psi^{T}B^{T}p & = & \\lambda u,\\\\\nM_{D}B\\Psi\\left(  M_{c}S_{c}-I\\right)  u & + & M_{D}B\\left(  \\Psi M_{c}%\n\\Psi^{T}+S_{\\Delta}^{-1}\\right)  B^{T}p & = & \\lambda p.\n\\end{array}\n\\label{eq:1}%\n\\end{equation}\nWe begin by considering possible eigenvalues of $M_{c}S_{c}$. There are two\npossible cases for $M_{c}S_{c}u=\\mu u$ where $(\\mu,u)$ is the corresponding\neigenpair: $\\mu=1$ or $\\mu\\neq1$. If $\\mu=1$, then $M_{c}S_{c}u=u$ and,\nassuming $\\lambda\\neq1$, we get from the first equation $u=1\/(\\lambda\n-1)M_{c}\\Psi^{T}B^{T}p$. Then, the second equation of~(\\ref{eq:1}) simplifies\nto the eigenvalue problem for~(\\ref{eq:(2,2)}), that is $M_{D}\\widetilde{F}%\np=\\lambda p$, which has essentially the same eigenvalues as the multilevel\nBDDC preconditioned operator $\\widetilde{M}_{\\widehat{S}}\\widehat{S}$ by\nLemma~\\ref{lem:equiv-mlevel}. If $\\mu\\neq1$, then $M_{c}S_{c}u=\\mu u$ and,\nassuming $\\lambda\\neq\\mu$, we get from the first equation $u=1\/(\\lambda\n-\\mu)M_{c}\\Psi^{T}B^{T}p$. Substituting this into the second equation\nof~(\\ref{eq:1}), we obtain\n\\[\nM_{D}B\\left(  \\frac{\\mu-1}{\\lambda-\\mu}\\Psi M_{c}\\Psi^{T}+\\Psi M_{c}\\Psi\n^{T}+S_{\\Delta}^{-1}\\right)  B^{T}p=\\lambda p,\n\\]\nwhich reduces with $\\lambda=1$ to $M_{D}BS_{\\Delta}^{-1}B^{T}p=\\lambda p$.\nTherefore, Theorem~\\ref{thm:equiv} for two-level methods translates to the\nmultilevel BDDC\\ and FETI-DP\\ methods.\n\n\\begin{theorem}\nThe eigenvalues of the preconditioned operators of the multilevel BDDC and primal multilevel FETI-DP methods\nare the same except possibly for the eigenvalues equal to zero and one.\n\\end{theorem}\n\nFor further discussion of multilevel BDDC\\ method we refer\nto~\\cite{Mandel-2008-MMB}. It is well known that the eigenvalues of the\noperators provide only asymptotic insight into the convergence of GMRES, which\nis the main iterative method used in the numerical experiments. The bound on\nthe error at every step involves the condition number of the operator\neigenvectors see, e.g.,~\\cite{Elman-2014-FEF,Saad-2003-IMS,Simoncini-2004-BTP}%\n, which is difficult to estimate. Nevertheless, the numerical experiments\npresented in the next section illustrate that the convergence of GMRES applied\nto either multilevel FETI-DP\\ or BDDC\\ method is quite comparable to that of\nthe preconditioned conjugate gradients applied to the multilevel BDDC. We also\nnote that if the preconditioned operator is symmetric, positive definite in\nsome inner product then it may be also used in conjugate gradient\nmethod~\\cite{Klawonn-1998-BTP,Klawonn-2007-IFM}.\n\n\\section{Numerical experiments}\n\n\\label{sec:numerical}The methods were implemented in \\textsc{Matlab}. We used\nthe right-preconditioned GMRES\\ method with no restarts and (for multilevel\nBDDC also) preconditioned conjugate gradients, with relative residual\ntolerance$~10^{-8}$ as the stopping criterion and zero initial guess. To\nestimate the largest eigenvalues of the preconditioned operators, we used the\n\\textsc{Matlab} function \\texttt{eigs}.\n\n\nFirst, we presents results of numerical experiments for the model problem\ncorresponding to the scalar second-order elliptic problem with zero Dirichlet\nboundary conditions on a square domain discretized by linear quadrilateral\nfinite elements. The domain was uniformly divided into substructures with\nfixed $H_{\\ell}\/H_{\\ell-1}$ ratio on each level$~\\ell$. Table~\\ref{tab:c}\nshows the largest eigenvalues of the preconditioned operators and the\niteration counts of the multilevel FETI-DP\\ and BDDC\\ methods set up using\ncorner constraints and varying numbers of levels$~L$ and coarsening ratios\n$H_{\\ell}\/H_{\\ell-1}$, $\\ell=1,\\dots,L-1$. Table~\\ref{tab:c+f} then shows\nresults for the same set of problems and the two methods set up using corner\nconstraints combined with arithmetic averages over edges. We used\nright-preconditioned GMRES for the FETI-DP\\ method~(\\ref{eq:bddc-dp})\npreconditioned by$~M_{F}$ from~(\\ref{eq:M_D}), in which multilevel\nBDDC\\ preconditioner was applied as$~M_{c}$, and for the multilevel\nBDDC\\ method we used also preconditioned conjugate gradients (PCG). In\naddition, for the FETI-DP\\ formulation, we studied performance of a\nblock-diagonal variant of the preconditioner$~M_{F}$, that is\n\\begin{equation}\n\\left[\n\\begin{array}\n[c]{cc}%\nM_{c} & 0\\\\\n0 & -M_{D}%\n\\end{array}\n\\right]  ,\\label{eq:M_BD}%\n\\end{equation}\nwhich from the theory in~\\cite{Murphy-2000-NPI} may be nearly as efficient.\nFrom the tables it can be seen the largest eigenvalues of both multilevel\nFETI-DP\\ and BDDC\\ preconditioned operators were the same. Also, the iteration\ncounts of GMRES\\ were essentially the same for the two methods, and the same\ntrend can be observed for the PCG\\ method. However, in case of the FETI-DP\nmethod\\ with the block-diagonal preconditioner the iteration counts of\nGMRES\\ are higher in all cases. Since the overhead associated with the\napplication of$~M_{F}$ is small, cf. Algorithm~\\ref{alg:M_D}, the\nblock-diagonal preconditioner appears to be less suitable. \n\nFigure~\\ref{fig} displays \\textcolor{black}{estimates of} the largest $150$ eigenvalues of the multilevel\nFETI-DP\\ and BDDC preconditioned operators in the setup with three levels and\ncoarsening ratio $H_{2}\/H_{1}=H_{1}\/H_{0}=3$. Specifically, the left panel\ncorresponds to the second row of Table~\\ref{tab:c} and the right panel to the\nsecond row of Table~\\ref{tab:c+f}. In both panels all eigenvalues match as\npredicted by the theory.\n\n\\begin{figure}[ptbh]\n\\begin{center}%\n\\begin{tabular}\n[c]{cc}%\n\\includegraphics[width=5.5cm]{laplace_square9x9Hh3_c.pdf} &\n\\includegraphics[width=5.5cm]{laplace_square9x9Hh3_c+f.pdf}\n\\end{tabular}\n\\end{center}\n\\caption{The largest $150$ eigenvalues of the three-level BDDC and FETI-DP\npreconditioned operators with coarsening ratio $H_{\\ell}\/H_{\\ell-1}=3$, where\n$\\ell=1,2$, with corner constraints (left), and corners and arithmetic\naverages over \\textcolor{black}{edges} (right).}%\n\\label{fig}%\n\\end{figure}\n\n\\begin{table}[ptbh]\n\\caption{Numbers of GMRES iterations (and GMRES\/PCG for BDDC) for the\nmultilevel FETI-DP and BDDC methods, and for the FETI-DP with the block\ndiagonal preconditioner~(BD) from~(\\ref{eq:M_BD}), with increasing number of\nlevels $L$ and coarsening ratios $H_{\\ell}\/H_{\\ell-1}$, $\\ell=1,\\dots,L-1$\nwith corner constraints. Here $nsub$ is the number of subdomains on each\nlevel, $ndof$ is the number of degrees of freedom, and $\\lambda_{\\max}$ is the\nlargest eigenvalue of the multilevel BDDC and FETI-DP preconditioned\noperators.}%\n\\label{tab:c}\n\\begin{center}%\n\\begin{tabular}\n[c]{|c|c|c|c|c|c|c|}\\hline\n$L$ & $nsub$ & $ndof$ & $\\lambda_{\\max}$ & BD & FETI-DP & BDDC\\\\\\hline\n\\multicolumn{7}{|c|}{$H_{\\ell}\/H_{\\ell-1}=3$}\\\\\\hline\n2 & 9 & 100 & 1.8781 & 14 & 9 & 10\/10\\\\\n3 & 81\/9 & 784 & 3.2636 & 23 & 14 & 15\/15\\\\\n4 & 729\/81\/9 & 6724 & 5.3709 & 29 & 20 & 21\/21\\\\\n5 & 6561\/729\/81\/9 & 59,536 & 8.8857 & 35 & 27 & 28\/29\\\\\\hline\n\\multicolumn{7}{|c|}{$H_{\\ell}\/H_{\\ell-1}=4$}\\\\\\hline\n2 & 16 & 289 & 2.1797 & 10 & 6 & 12\/11\\\\\n3 & 256\/16 & 4225 & 4.1758 & 29 & 18 & 19\/19\\\\\n4 & 4096\/256\/16 & 66,049 & 7.8472 & 37 & 27 & 28\/29\\\\\\hline\n\\multicolumn{7}{|c|}{$H_{\\ell}\/H_{\\ell-1}=6$}\\\\\\hline\n2 & 36 & 1369 & 2.7982 & 24 & 14 & 15\/15\\\\\n3 & 1296\/36 & 47,089 & 6.0284 & 36 & 24 & 25\/26\\\\\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\begin{table}[ptbh]\n\\caption{Numbers of GMRES (and GMRES\/PCG for BDDC) iterations for the same problems\nas in Table~\\ref{tab:c}, but here with corner constraints combined with\naverages over edges.}\n\\begin{center}%\n\\begin{tabular}\n[c]{|c|c|c|c|c|c|c|}\\hline\n$L$ & $nsub$ & $ndof$ & $\\lambda_{\\max}$ & BD & FETI-DP & BDDC\\\\\\hline\n\\multicolumn{7}{|c|}{$H_{\\ell}\/H_{\\ell-1}=3$}\\\\\\hline\n2 & 9 & 100 & 1.0550 & 8 & 4 & 4\/5\\\\\n3 & 81\/9 & 784 & 1.2866 & 11 & 8 & 8\/8\\\\\n4 & 729\/81\/9 & 6724 & 1.6980 & 14 & 11 & 11\/11\\\\\n5 & 6561\/729\/81\/9 & 59,536 & 2.1212 & 17 & 14 & 14\/14\\\\\\hline\n\\multicolumn{7}{|c|}{$H_{\\ell}\/H_{\\ell-1}=4$}\\\\\\hline\n2 & 16 & 289 & 1.1094 & 7 & 4 & 6\/6\\\\\n3 & 256\/16 & 4225 & 1.4779 & 13 & 9 & 9\/9\\\\\n4 & 4096\/256\/16 & 66,049 & 1.9393 & 17 & 13 & 13\/13\\\\\\hline\n\\multicolumn{7}{|c|}{$H_{\\ell}\/H_{\\ell-1}=6$}\\\\\\hline\n2 & 36 & 1369 & 1.2280 & 13 & 7 & 7\/7\\\\\n3 & 1296\/36 & 47,089 & 1.7775 & 17 & 11 & 11\/11\\\\\\hline\n\\end{tabular}\n\\end{center}\n\\label{tab:c+f}%\n\\end{table}\n\n\\begin{figure}[ptbh]\n\\begin{center}%\n\\begin{tabular}\n[c]{cc}%\n\\includegraphics[height=5.5cm]{dam2_400_back.png} &\n\\includegraphics[height=5.5cm]{dam2_400_front.png}\\\\\n\\includegraphics[height=5cm]{dam2_400+8_front.png} &\n\\includegraphics[height=5cm]{dam2_400_side.png}\n\n\\end{tabular}\n\\end{center}\n\\caption{Finite element discretization and substructuring of the dam\nconsisting of $3,800,080$ elements distributed into $400$ subdomains in the\nfirst level (top) and $8$ subdomains in the second level (bottom left). Model by\ncourtesy of Jaroslav Kruis, images reproduced\nfrom~\\cite{Sousedik-2010-AMB-thesis}.}%\n\\label{fig:dam}%\n\\end{figure}\n\n\\begin{figure}[ptbh]\n\\begin{center}%\n\\begin{tabular}\n[c]{cc}%\n\\includegraphics[width=5.5cm]{dam_400_3level_c_50eigs} &\n\\includegraphics[width=5.5cm]{dam_400_3level_c+e_50eigs}\n\\end{tabular}\n\\end{center}\n\\caption{The largest $50$ eigenvalues of the three-level BDDC and FETI-DP\npreconditioned operators for the dam problem using corner constraints (left),\nand corners combined with arithmetic averages over edges (right).\n\\label{fig:dam-eig}%\n\\end{figure}\n\n\\begin{table}[ptbh]\n\\caption{Numbers of GMRES iterations (and GMRES\/PCG for BDDC) for the dam\nproblem and the BDDC and FETI-DP methods with varying choices \\textcolor{black}{of} constraints:\ncorners (c), arithmetic averages over edges (e) and faces (f), and their total\nnumbers on level~1\/level~2, and also the largest eigenvalues~$\\lambda_{\\max}$\nof the preconditioned operators. The inexact methods apply algebraic multigrid\n(AMG) to the coarse problem. }\n\\begin{center}%\n\\begin{tabular}\n[c]{|c|c|c|c|c|c|c|c|}\\hline\n\\multicolumn{2}{|c|}{} & \\multicolumn{2}{|c|}{inexact methods} &\n\\multicolumn{4}{|c|}{three-level methods}\\\\\\hline\n\\multicolumn{2}{|c|}{constraints} & \\multicolumn{1}{|c|}{FETI-DP} &\n\\multicolumn{1}{|c|}{BDDC} & \\multicolumn{2}{|c|}{FETI-DP} &\n\\multicolumn{2}{|c|}{BDDC}\\\\\\hline\ntype & level~1\/level~2 & it & it & $\\lambda_{\\max}$ & it & $\\lambda_{\\max}$ &\nit\\\\\\hline\nc & 11,970\/99 & 65 & 71\/75 & 99.5 & 66 & 99.5 & 73\/87 \\\\\nc+e & 21,180\/144 & 105 & 107\/125 & 22.0 & 43 & 21.1 & 42\/45 \\\\\nc+e+f & 28,002\/198 & 134 & 127\/140 & 15.1 & 37 & 14.1 & 33\/35 \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\label{tab:dam}%\n\\end{table}\n\n\n\\textcolor{black}{As the second problem we used a real-world model of linear elasticity in a dam\ndiscretized using $3,800,080$ finite elements with $2,006,748$ degrees of\nfreedom, distributed into $400$ subdomains with $3990$ corners, $3070$\nedges and $2274$ faces. The subdomains in the first level were distributed\ninto $8$ subdomains for the three-level method.} The distribution of the\nelements into subdomains in both substructuring levels was obtained using\n\\textsc{Metis}~\\cite{Karypis-1998-FHQ,Karypis-1998-MSP}, see\nFigure~\\ref{fig:dam}. Table~\\ref{tab:dam} shows the iteration counts of GMRES\n(and PCG) for inexact and three-level FETI-DP\\ and BDDC\\ methods with varying\nchoices of constraints. The table also shows the largest eigenvalues of the\ntwo preconditioned operators for the three-level methods. For the inexact\nmethods we applied algebraic multigrid (AMG) as the preconditioner$~M_{c}$\nin~(\\ref{eq:M_D}) and~(\\ref{eq:mBDDC})--(\\ref{eq:H-tilde}). Specifically, we\nused a \\textsc{Matlab} version of the routine \\texttt{HSL\\_MI20}%\n~\\cite{Boyle-2010-HSL}, which performs grid coarsening using the classical\nalgorithm by Ruge and St\\\"{u}ben~\\cite{Ruge-1987-AMG}. We used the\nAMG\\ routine with its default settings. It can be seen that adding more\nconstraints \\textcolor{black}{as averages over the edges and faces} to the coarse problem (and thus increasing its size) leads to an\nincrease in the iteration count for the inexact methods. Nevertheless with the\ncorner constraints, the inexact methods converge in fewer iterations than the\nthree-level methods. On the other hand, it is expected that adding more constraints would improve convergence for the multilevel methods. \nWe see that this \\textcolor{black}{happens, in particular after combining the corners with the averages over edges,\nand the improvement is less dramatic after adding the averages over faces. \nWe also note that there} \nare adaptive techniques for construction of the the coarse spaces,\nin\\ particular the Adaptive-Multilevel\nBDDC~\\cite{Mandel-2007-ASF,Sousedik-2013-AMB}, which could be applied to the\nmultilevel methods presented here, but this is beyond the focus of our study.\nAlso, a thorough comparison of the proposed methods with AMG\\ would be of\nindependent interest. For example, in a recent study Klawonn et\nal.~\\cite{Klawonn-2019-PCP} found that, for the BDDC method, inexact\npreconditioning\\ using AMG\\ and the two-level BDDC applied to the coarse\nproblem show in most situations a very similar behavior. \n\n\\textcolor{black}{Figure~\\ref{fig:dam-eig} displays estimates of the $50$ largest eigenvalues for the\nthree-level FETI-DP\\ and BDDC\\ preconditioned operators for the setup using\ncorner constraints (left panel), and using corners combined with arithmetic\naverages over edges (right panel), for the dam problem. \nWe see that adding averages of the edges reduces the largest eigenvalue approximately four times,\nand even though the eigenvalues in the case of the corner constraints decrease at a somewhat faster rate,\ntheir values remain higher than those in the case when both corner and edges are used for constraints. \nIn both panels a (close) correspondence of the eigenvalue estimates can be observed for all eigenvalues.}\n\nFinally, we note that the iteration counts of GMRES\\ and PCG in\nTable~\\ref{tab:dam}\\ are close also for both inexact methods and each set of\nconstraints. This suggests that the eigenvalues of the inexact methods may be\nalso similar. A comparison of \\textcolor{black}{the eigenvalue estimates} for the inexact methods is provided\nby Figure~\\ref{fig:amg}. The left panel corresponds to the problem \\textcolor{black}{with the same} setup \\textcolor{black}{as} in\nthe second row of Table~\\ref{tab:c}, cf. also the left panel in\nFigure~\\ref{fig}. For this problem, the inexact FETI-DP\\ converged in $11$\niterations and the inexact BDDC\\ converged in $12$ iterations using both\nGMRES\\ and PCG. It can be seen that in this case, with geometric\ndiscretization and uniform partitioning of the domain into subdomains, the \\textcolor{black}{eigenvalue estimates} are the same (and real). \n\\textcolor{black}{An agreement can be observed also in the right panel, which\ndisplays $50$\neigenvalues of the largest magnitude\\ corresponding to the dam problem with\ncorner constraints.\nNevertheless, even though all eigenvalues in Figure~\\ref{fig:amg} are real,  \nwe note that the eigenvalues of the inexact FETI-DP\\ preconditioned operator are in general complex, \nand a strategy to make them real was provided in~\\cite{Klawonn-2007-IFM}.}\n\n\\begin{figure}[ptbh]\n\\begin{center}%\n\\begin{tabular}\n[c]{cc}%\n\\includegraphics[width=5.5cm]{laplace_square9x9Hh3_c_amg_large} &\n\\includegraphics[width=5.5cm]{dam_400_amg_c_50eigs}\n\\end{tabular}\n\\end{center}\n\\caption{The largest $150$ eigenvalues of the inexact BDDC and FETI-DP\npreconditioned operators with geometric discretization and partitioning of the\ndomain (left panel), and $50$ eigenvalues of the largest magnitude\nfor the dam problem (right).}%\n\\label{fig:amg}%\n\\end{figure}\n\n\n\n\\begin{acknowledgements}\nThe author is grateful to Jaroslav Kruis for providing the discretization of the dam problem, \nto Marian Brezina for visualizations using his own software package~\\emph{MyVis},\n\\textcolor{black}{and to Jan Mandel for continuing support and interest in the domain decomposition methods. \nHe would also like to thank the anonymous reviewers for comments and suggestions that helped to improve the quality of the manuscript.}\n\\end{acknowledgements}\n\n\n\\bibliographystyle{spmpsci}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}