X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/Krylov_multi.git/blobdiff_plain/07677631c1d2e2d7aa7b88b82f62e12c72b11327..09702354d347f9bf651fba24d04f262c757e2cc5:/krylov_multi.tex diff --git a/krylov_multi.tex b/krylov_multi.tex index 036c190..7a7e809 100644 --- a/krylov_multi.tex +++ b/krylov_multi.tex @@ -3,14 +3,18 @@ \usepackage{amsfonts,amssymb} \usepackage{amsmath} \usepackage{graphicx} -%\usepackage{algorithmic} -%\usepackage[ruled,english,boxed,linesnumbered]{algorithm2e} -%\usepackage[english]{algorithme} \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} +\date{} @@ -25,7 +29,7 @@ \begin{abstract} -In this paper we revist the krylov multisplitting algorithm presented in +In this paper we revisit the krylov multisplitting algorithm presented in \cite{huang1993krylov} which uses a scalar method to minimize the krylov iterations computed by a multisplitting algorithm. Our new algorithm is based on a parallel multisplitting algorithm with few blocks of large size using a @@ -58,8 +62,13 @@ solvers. However, most of the good preconditionners are not sclalable when thousands of cores are used. -A completer... -On ne peut pas parler de tout...\\ +Traditionnal iterative solvers have global synchronizations that penalize the +scalability. Two possible solutions consists either in using asynchronous +iterative methods~\cite{ref18} or to use multisplitting algorithms. In this +paper, we will reconsider the use of a multisplitting method. In opposition to +traditionnal multisplitting method that suffer from slow convergence, as +proposed in~\cite{huang1993krylov}, the use of a minimization process can +drastically improve the convergence. @@ -198,9 +207,9 @@ 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 +as an inner iteration method to solve the sub-systems~(\ref{sec03:eq03}). It is a well-known iterative method -which gives good performances for solving sparse linear systems in +which gives good performances to solve sparse linear systems in parallel on a cluster of processors. It should be noted that the convergence of the inner iterative solver @@ -226,10 +235,10 @@ 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 such a method is that the Krylov subspace does not need to be spanned by an orthogonal basis. -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. +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 @@ -251,26 +260,28 @@ which is associated with the least squares problem \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}. +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 to solve +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] -\State Load $A_l$, $B_l$, initial guess $x^0$ +\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 $S_l$ +\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 $R$: $R_l^j=\sum^L_{i=1}A_{li}X_i^j$ +\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}{$S_l$, $R$, $b$} +\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 @@ -279,31 +290,31 @@ gradient method for the normal equations CGNR~\cite{S96,refCGNR}. \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 $A_{ll}X_l^j=Y_l$ with the parallel GMRES method, such that $X_l^0$ is the initial guess. +\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 $R^TR\alpha=R^Tb$ in parallel by $L$ clusters using the parallel CGNR method -\State Compute the local minimizer: $\tilde{X}_l^k=S_l\alpha$ +\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 +The main key points of the multisplitting method to solve a large +sparse linear system 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 +iterative algorithms: the GMRES method to solve 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 +clusters to minimize 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 @@ -317,6 +328,36 @@ synchronizations by using the MPI collective communication subroutines. +\section{Experiments} + +In order to illustrate the interest of our algorithm. We have compared our +algorithm with the GMRES method which a very well used method in many +situations. We have chosen to focus on only one problem which is very simple to +implement: a 3 dimension Poisson problem. + +\begin{equation} +\left\{ + \begin{array}{ll} + \nabla u&=f \mbox{~in~} \omega\\ + u &=0 \mbox{~on~} \Gamma=\partial \omega + \end{array} + \right. +\end{equation} + +After discretization, with a finite difference scheme, a seven point stencil is +used. It is well-known that the spectral radius of matrices representing such +problems are very close to 1. Moreover, the larger the number of discretization +points is, the closer to 1 the spectral radius is. Hence, to solve a matrix +obtained for a 3D Poisson problem, the number of iterations is high. Using a +preconditioner it is possible to reduce the number of iterations but +preconditioners are not scalable when using many cores. + +\section{Conclusion and perspectives} + +Other applications (=> other matrices)\\ +Larger experiments\\ +Async\\ +Overlapping %%%%%%%%%%%%%%%%%%%%%%%%