X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/Krylov_multi.git/blobdiff_plain/5e3a7342021a720e17be4f147b4e73e5ad5396b9..0d162d0a192a8e0e159e831ba451862a05260ee5:/krylov_multi.tex?ds=inline diff --git a/krylov_multi.tex b/krylov_multi.tex index 91e4745..ea6cfe9 100644 --- a/krylov_multi.tex +++ b/krylov_multi.tex @@ -3,6 +3,15 @@ \usepackage{amsfonts,amssymb} \usepackage{amsmath} \usepackage{graphicx} +\usepackage{algorithm} +\usepackage{algpseudocode} + +\algnewcommand\algorithmicinput{\textbf{Input:}} +\algnewcommand\Input{\item[\algorithmicinput]} + +\algnewcommand\algorithmicoutput{\textbf{Output:}} +\algnewcommand\Output{\item[\algorithmicoutput]} + \title{A scalable multisplitting algorithm for solving large sparse linear systems} @@ -89,12 +98,13 @@ v_l^k=M^{-1}_l N_l x_l^{k-1} + M^{-1}_l b,~l\in\{1,\ldots,L\}, \label{eq04} \end{equation} to be solved independently by a direct or an iterative method, where -$v_l^k$ is the solution of the local sub-system. A multisplitting -method using an iterative method for solving the $L$ linear -sub-systems is called an inner-outer iterative method or a two-stage -method. The results $v_l^k$ obtained from the different -splittings~(\ref{eq04}) are combined to compute the solution $x^k$ of -the linear system by using the diagonal weighting matrices +$v_l^k$ is the solution of the local sub-system. Thus, the +calculations of $v_l^k$ may be performed in parallel by a set of +processors. A multisplitting method using an iterative method for +solving the $L$ linear sub-systems is called an inner-outer iterative +method or a two-stage method. The results $v_l^k$ obtained from the +different splittings~(\ref{eq04}) are combined to compute the solution +$x^k$ of the linear system by using the diagonal weighting matrices \begin{equation} x^k = \displaystyle\sum^L_{l=1} E_l v_l^k, \label{eq05} @@ -144,12 +154,12 @@ solvers on exascale architecture with hunders of thousands of cores. \section{A two-stage method with a minimization} -Let $Ax=b$ be a given sparse and large linear system of $n$ equations -to solve in parallel on $L$ clusters, physically adjacent or geographically -distant, where $A\in\mathbb{R}^{n\times n}$ is a square and nonsingular -matrix, $x\in\mathbb{R}^{n}$ is the solution vector and $b\in\mathbb{R}^{n}$ -is the right-hand side vector. The multisplitting of this linear system -is defined as follows: +Let $Ax=b$ be a given sparse and large linear system of $n$ equations +to solve in parallel on $L$ clusters, physically adjacent or +geographically distant, where $A\in\mathbb{R}^{n\times n}$ is a square +and nonsingular matrix, $x\in\mathbb{R}^{n}$ is the solution vector +and $b\in\mathbb{R}^{n}$ is the right-hand side vector. The +multisplitting of this linear system is defined as follows: \begin{equation} \left\{ \begin{array}{lll} @@ -160,21 +170,24 @@ b & = & [B_{1}, \ldots, B_{L}] \right. \label{sec03:eq01} \end{equation} -where for all $l\in\{1,\ldots,L\}$ $A_l$ is a rectangular block of size $n_l\times n$ -and $X_l$ and $B_l$ are sub-vectors of size $n_l$, such that $\sum_ln_l=n$. In this -case, we use a row-by-row splitting without overlapping in such a way that successive -rows of the sparse matrix $A$ and both vectors $x$ and $b$ are assigned to one cluster. -So, the multisplitting format of the linear system is defined as follows: +where for $l\in\{1,\ldots,L\}$, $A_l$ is a rectangular block of size +$n_l\times n$ and $X_l$ and $B_l$ are sub-vectors of size $n_l$, such +that $\sum_ln_l=n$. In this case, we use a row-by-row splitting +without overlapping in such a way that successive rows of the sparse +matrix $A$ and both vectors $x$ and $b$ are assigned to one cluster. +So, the multisplitting format of the linear system is defined as +follows: \begin{equation} \forall l\in\{1,\ldots,L\} \mbox{,~} \displaystyle\sum_{i=1}^{l-1}A_{li}X_i + A_{ll}X_l + \displaystyle\sum_{i=l+1}^{L}A_{li}X_i = B_l, \label{sec03:eq02} \end{equation} -where $A_{li}$ is a block of size $n_l\times n_i$ of the rectangular matrix $A_l$, $X_i\neq X_l$ -is a sub-vector of size $n_i$ of the solution vector $x$ and $\sum_{il}n_i+n_l=n$, -for all $i\in\{1,\ldots,l-1,l+1,\ldots,L\}$. +where $A_{li}$ is a block of size $n_l\times n_i$ of the rectangular +matrix $A_l$, $X_i\neq X_l$ is a sub-vector of size $n_i$ of the +solution vector $x$ and $\sum_{il}n_i+n_l=n$, for all +$i\in\{1,\ldots,l-1,l+1,\ldots,L\}$. -The multisplitting method proceeds by iteration for solving the linear system in such a -way each sub-system +The multisplitting method proceeds by iteration for solving the linear +system in such a way each sub-system \begin{equation} \left\{ \begin{array}{l} @@ -184,14 +197,126 @@ Y_l = B_l - \displaystyle\sum_{i=1,i\neq l}^{L}A_{li}X_i, \right. \label{sec03:eq03} \end{equation} -is solved independently by a cluster of processors and communication are required to -update the right-hand side vectors $Y_l$, such that the vectors $X_i$ represent the data -dependencies between the clusters. In this case, the parallel GMRES method is used -as an inner iteration method for solving the linear sub-systems~(\ref{sec03:eq03}). - - - - +is solved independently by a cluster of processors and communication +are required to update the right-hand side vectors $Y_l$, such that +the vectors $X_i$ represent the data dependencies between the +clusters. In this work, we use the parallel GMRES method~\cite{ref34} +as an inner iteration method for solving the +sub-systems~(\ref{sec03:eq03}). It is a well-known iterative method +which gives good performances for solving sparse linear systems in +parallel on a cluster of processors. + +It should be noted that the convergence of the inner iterative solver +for the different linear sub-systems~(\ref{sec03:eq03}) does not +necessarily involve the convergence of the multisplitting method. It +strongly depends on the properties of the sparse linear system to be +solved and the computing +environment~\cite{o1985multi,ref18}. Furthermore, the multisplitting +of the linear system among several clusters of processors increases +the spectral radius of the iteration matrix, thereby slowing the +convergence. In this paper, we based on the work presented +in~\cite{huang1993krylov} to increase the convergence and improve the +scalability of the multisplitting methods. + +In order to accelerate the convergence, we implement the outer +iteration of the multisplitting solver as a Krylov subspace method +which minimizes some error function over a Krylov subspace~\cite{S96}. +The Krylov space of the method that we used is spanned by a basis +composed of successive solutions issued from solving the $L$ +splittings~(\ref{sec03:eq03}) +\begin{equation} +S=\{x^1,x^2,\ldots,x^s\},~s\leq n, +\label{sec03:eq04} +\end{equation} +where for $j\in\{1,\ldots,s\}$, $x^j=[X_1^j,\ldots,X_L^j]$ is a +solution of the global linear system. The advantage of such a Krylov subspace is that we need neither an orthogonal basis nor synchronizations between the different clusters to generate this basis. + +The multisplitting method is periodically restarted every $s$ +iterations with a new initial guess $\tilde{x}=S\alpha$ which +minimizes the error function $\|b-Ax\|_2$ over the Krylov subspace +spanned by the vectors of $S$. So, $\alpha$ is defined as the +solution of the large overdetermined linear system +\begin{equation} +R\alpha=b, +\label{sec03:eq05} +\end{equation} +where $R=AS$ is a dense rectangular matrix of size $n\times s$ and +$s\ll n$. This leads us to solve the system of normal equations +\begin{equation} +R^TR\alpha=R^Tb, +\label{sec03:eq06} +\end{equation} +which is associated with the least squares problem +\begin{equation} +\text{minimize}~\|b-R\alpha\|_2, +\label{sec03:eq07} +\end{equation} +where $R^T$ denotes the transpose of the matrix $R$. Since $R$ +(i.e. $AS$) and $b$ are split among $L$ clusters, the symmetric +positive definite system~(\ref{sec03:eq06}) is solved in +parallel. Thus, an iterative method would be more appropriate than a +direct one for solving this system. We use the parallel conjugate +gradient method for the normal equations CGNR~\cite{S96,refCGNR}. + +\begin{algorithm}[!t] +\caption{A two-stage linear solver with inner iteration GMRES method} +\begin{algorithmic}[1] +\Input $A_l$ (local sparse matrix), $B_l$ (local right-hand side), $x^0$ (initial guess) +\Output $X_l$ (local solution vector)\vspace{0.2cm} +\State Load $A_l$, $B_l$, $x^0$ +\State Initialize the minimizer $\tilde{x}^0=x^0$ +\For {$k=1,2,3,\ldots$ until the global convergence} +\State Restart with $x^0=\tilde{x}^{k-1}$: \textbf{for} $j=1,2,\ldots,s$ \textbf{do} +\State\hspace{0.5cm} Inner iteration solver: \Call{InnerSolver}{$x^0$, $j$} +\State\hspace{0.5cm} Construct the basis $S$: add the column vector $X_l^j$ to the matrix $S_l^k$ +\State\hspace{0.5cm} Exchange the local solution vector $X_l^j$ with the neighboring clusters +\State\hspace{0.5cm} Compute the dense matrix $R$: $R_l^{k,j}=\sum^L_{i=1}A_{li}X_i^j$ +\State\textbf{end for} +\State Minimization $\|b-R\alpha\|_2$: \Call{UpdateMinimizer}{$S_l$, $R$, $b$, $k$} +\State Local solution of the linear system $Ax=b$: $X_l^k=\tilde{X}_l^k$ +\State Exchange the local minimizer $\tilde{X}_l^k$ with the neighboring clusters +\EndFor + +\Statex + +\Function {InnerSolver}{$x^0$, $j$} +\State Compute the local right-hand side: $Y_l = B_l - \sum^L_{i=1,i\neq l}A_{li}X_i^0$ +\State Solving the local splitting $A_{ll}X_l^j=Y_l$ using the parallel GMRES method, such that $X_l^0$ is the initial guess +\State \Return $X_l^j$ +\EndFunction + +\Statex + +\Function {UpdateMinimizer}{$S_l$, $R$, $b$, $k$} +\State Solving the normal equations $(R^k)^TR^k\alpha^k=(R^k)^Tb$ in parallel by $L$ clusters using the parallel CGNR method +\State Compute the local minimizer: $\tilde{X}_l^k=S_l^k\alpha^k$ +\State \Return $\tilde{X}_l^k$ +\EndFunction +\end{algorithmic} +\label{algo:01} +\end{algorithm} + +The main key points of the multisplitting method for solving large +sparse linear systems are given in Algorithm~\ref{algo:01}. This +algorithm is based on a two-stage method with a minimization using the +GMRES iterative method as an inner solver. It is executed in parallel +by each cluster of processors. The matrices and vectors with the +subscript $l$ represent the local data for the cluster $l$, where +$l\in\{1,\ldots,L\}$. The two-stage solver uses two different parallel +iterative algorithms: the GMRES method for solving each splitting on a +cluster of processors, and the CGNR method executed in parallel by all +clusters for minimizing the function error over the Krylov subspace +spanned by $S$. The algorithm requires two global synchronizations +between the $L$ clusters. The first one is performed at line~$12$ in +Algorithm~\ref{algo:01} to exchange the local values of the vector +solution $x$ (i.e. the minimizer $\tilde{x}$) required to restart the +multisplitting solver. The second one is needed to construct the +matrix $R$ of the Krylov subspace. We choose to perform this latter +synchronization $s$ times in every outer iteration $k$ (line~$7$ in +Algorithm~\ref{algo:01}). This is a straightforward way to compute the +matrix-matrix multiplication $R=AS$. We implement all +synchronizations by using the MPI collective communication +subroutines.