X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/Krylov_multi.git/blobdiff_plain/c0329df983b381995f844afef4536e2853e59e78..1b340b04ffddcee33acad4949f55a44d2eadf683:/krylov_multi.tex?ds=sidebyside diff --git a/krylov_multi.tex b/krylov_multi.tex index b96a5b9..0ea51d8 100644 --- a/krylov_multi.tex +++ b/krylov_multi.tex @@ -21,12 +21,12 @@ \begin{abstract} In this paper we revist 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 simply a -parallel multisplitting algorithm with few blocks of large size and a parallel -krylov minimization is used to improve the convergence. Some large scale -experiments with a 3D Poisson problem are presented. They show the obtained -improvements compared to a classical GMRES both in terms of number of iterations -and execution times. +iterations computed by a multisplitting algorithm. Our new algorithm is based on +a parallel multisplitting algorithm with few blocks of large size using a +parallel GMRES method inside each block and on a parallel krylov minimization in +order to improve the convergence. Some large scale experiments with a 3D Poisson +problem are presented. They show the obtained improvements compared to a +classical GMRES both in terms of number of iterations and execution times. \end{abstract} @@ -38,14 +38,106 @@ and execution times. Iterative methods are used to solve large sparse linear systems of equations of the form $Ax=b$ because they are easier to parallelize than direct ones. Many -iterative methods have been proposed and adpated by many researchers. When -solving large linear systems with many cores, iterative methods often suffer -from scalability problems. This is due to their need for collective -communications to perform matrix-vector products and reduction operations. - -%%%%% Lilia -% doit-on définir le principe et les préliminaires du multisplitting dans l'intro ou dans l'autre section? -% valides-tu le titre de la 2eme section? celle que je voudrai rédiger. +iterative methods have been proposed and adapted by many researchers. For +example, the GMRES method and the Conjugate Gradient method are very well known +and used by many researchers ~\cite{S96}. Both the method are based on the +Krylov subspace which consists in forming a basis of the sequence of successive +matrix powers times the initial residual. + +When solving large linear systems with many cores, iterative methods often +suffer from scalability problems. This is due to their need for collective +communications to perform matrix-vector products and reduction operations. +Preconditionners can be used in order to increase the convergence of iterative +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...\\ + + + + +%%%%%%%%%%%%%%%%%%%%%%% +%% BEGIN +%%%%%%%%%%%%%%%%%%%%%%% +The key idea of the multisplitting method for solving a large system +of linear equations $Ax=b$ consists in partitioning the matrix $A$ in +$L$ several ways +\begin{equation} +A = M_l - N_l,~l\in\{1,\ldots,L\}, +\label{eq01} +\end{equation} +where $M_l$ are nonsingular matrices. Then the linear system is solved +by iteration based on the multisplittings as follows +\begin{equation} +x^{k+1}=\displaystyle\sum^L_{l=1} E_l M^{-1}_l (N_l x^k + b),~k=1,2,3,\ldots +\label{eq02} +\end{equation} +where $E_l$ are non-negative and diagonal weighting matrices such that +$\sum^L_{l=1}E_l=I$ ($I$ is an identity matrix). Thus the convergence +of such a method is dependent on the condition +\begin{equation} +\rho(\displaystyle\sum^L_{l=1}E_l M^{-1}_l N_l)<1. +\label{eq03} +\end{equation} + +The advantage of the multisplitting method is that at each iteration +$k$ there are $L$ different linear sub-systems +\begin{equation} +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. 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} +\end{equation} +In the case where the diagonal weighting matrices $E_l$ have only zero +and one factors (i.e. $v_l^k$ are disjoint vectors), the +multisplitting method is non-overlapping and corresponds to the block +Jacobi method. +%%%%%%%%%%%%%%%%%%%%%%% +%% END +%%%%%%%%%%%%%%%%%%%%%%% + +\section{Related works} + + +A general framework for studying parallel multisplitting has been presented in +\cite{o1985multi} by O'Leary and White. Convergence conditions are given for the +most general case. Many authors improved multisplitting algorithms by proposing +for example an asynchronous version \cite{bru1995parallel} and convergence +conditions \cite{bai1999block,bahi2000asynchronous} in this case or other +two-stage algorithms~\cite{frommer1992h,bru1995parallel}. + +In \cite{huang1993krylov}, the authors proposed a parallel multisplitting +algorithm in which all the tasks except one are devoted to solve a sub-block of +the splitting and to send their local solution to the first task which is in +charge to combine the vectors at each iteration. These vectors form a Krylov +basis for which the first task minimizes the error function over the basis to +increase the convergence, then the other tasks receive the update solution until +convergence of the global system. + + + +In \cite{couturier2008gremlins}, the authors proposed practical implementations +of multisplitting algorithms that take benefit from multisplitting algorithms to +solve large scale linear systems. Inner solvers could be based on scalar direct +method with the LU method or scalar iterative one with GMRES. + +In~\cite{prace-multi}, the authors have proposed a parallel multisplitting +algorithm in which large block are solved using a GMRES solver. The authors have +performed large scale experimentations upto 32.768 cores and they conclude that +asynchronous multisplitting algorithm could more efficient than traditionnal +solvers on exascale architecture with hunders of thousands of cores. %%%%%%%%%%%%%%%%%%%%%%%% @@ -53,6 +145,79 @@ communications to perform matrix-vector products and reduction operations. \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: +\begin{equation} +\left\{ +\begin{array}{lll} +A & = & [A_{1}, \ldots, A_{L}]\\ +x & = & [X_{1}, \ldots, X_{L}]\\ +b & = & [B_{1}, \ldots, B_{L}] +\end{array} +\right. +\label{sec03:eq01} +\end{equation} +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 $i\in\{1,\ldots,l-1,l+1,\ldots,L\}$. + +The multisplitting method proceeds by iteration for solving the linear system in such a +way each sub-system +\begin{equation} +\left\{ +\begin{array}{l} +A_{ll}X_l = Y_l \mbox{,~such that}\\ +Y_l = B_l - \displaystyle\sum_{i=1,i\neq l}^{L}A_{li}X_i, +\end{array} +\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 work, we use the GMRES method 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 the solutions issued from +solving the $L$ splittings~(\ref{sec03:eq03}) +\begin{equation} +\{x^1,x^2,\ldots,x^s\},~s\ll n, +\label{sec03:eq04} +\end{equation} +where for $k\in\{1,\ldots,s\}$, $x^k=[X_1^k,\ldots,X_L^k]$ 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 method is that the Krylov subspace need neither to be spanned by an orthogonal +basis nor synchronizations between the different clusters to generate this basis. + + + + %%%%%%%%%%%%%%%%%%%%%%%%