From: lilia Date: Mon, 8 Dec 2014 10:44:06 +0000 (+0100) Subject: 08-12-2014 v00 X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/Krylov_multi.git/commitdiff_plain/c9242f9fee68ee32282c6b18679048eb4c3056b4 08-12-2014 v00 --- diff --git a/krylov_multi_reviewed.tex b/krylov_multi_reviewed.tex index 98b4433..73881ba 100644 --- a/krylov_multi_reviewed.tex +++ b/krylov_multi_reviewed.tex @@ -94,7 +94,12 @@ method. In opposition to traditional multisplitting method that suffer from slow convergence, as proposed in~\cite{huang1993krylov}, the use of a minimization process can drastically improve the convergence. + +%%% AJOUTE************************ +%%%******************************* In this work we develop a new parallel two-stage algorithm for large-scale clusters. Our objective is to mix between Krylov based iterative methods and the multisplitting method to improve the scalability. In fact Krylov subspace methods are well-known for their good convergence compared to others iterative methods. So our main contribution is to use the multisplitting method which splits the problem to solve into different blocks in order to reduce the large amount of communications and, to implement both inner and outer iterations as Krylov subspace iterations improving the convergence of the multisplitting algorithm. +%%%******************************* +%%%******************************* The present paper is organized as follows. First, Section~\ref{sec:02} presents some related works and the principle of multisplitting methods. Then, in @@ -173,7 +178,14 @@ In the case where the diagonal weighting matrices $E_\ell$ have only zero and on \section{A two-stage method with a minimization} \label{sec:03} -Let $Ax=b$ be a given large and sparse linear system of $n$ equations where $A\in\mathbb{R}^{n\times n}$ is a sparse square and non-singular matrix, $x\in\mathbb{R}^{n}$ is the solution vector and $b\in\mathbb{R}^{n}$ is the right-hand side vector. We use a multisplittig method to solve the linear system on a large computing platform in order to reduce the communications. Let the computing platform be composed of $L$ clusters of processors physically adjacent or geographically distant. In this work we apply the block Jacobi multisplitting to the linear system as follows + +%%% MODIFIE ************************ +%%%********************************* +Let $Ax=b$ be a given large and sparse linear system of $n$ equations where $A\in\mathbb{R}^{n\times n}$ is a sparse square and non-singular matrix, $x\in\mathbb{R}^{n}$ is the solution vector and $b\in\mathbb{R}^{n}$ is the right-hand side vector. We use a multisplitting method to solve the linear system on a large computing platform in order to reduce the communications. Let the computing platform be composed of $L$ clusters of processors physically adjacent or geographically distant. In this work we apply the block Jacobi multisplitting to the linear system as follows +%%%********************************* +%%%********************************* + + \begin{equation} \left\{ \begin{array}{lll} @@ -184,7 +196,13 @@ b & = & [B_{1}, \ldots, B_{L}] \right. \label{sec03:eq01} \end{equation} -where for $\ell\in\{1,\ldots,L\}$, $A_\ell$ is a rectangular block of size $n_\ell\times n$ and $X_\ell$ and $B_\ell$ are sub-vectors of size $n_\ell$ each, such that $\sum_\ell n_\ell=n$. The splitting is performed row-by-row without overlapping in such a way that successive rows of 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 $\ell\in\{1,\ldots,L\}$, $A_\ell$ is a rectangular block of size $n_\ell\times n$ and $X_\ell$ and $B_\ell$ are sub-vectors of size $n_\ell$ each, such that $\sum_\ell n_\ell=n$. +%%% MODIFIE *********************** +%%%******************************** +The splitting is performed row-by-row without overlapping in such a way that successive rows of 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 \ell\in\{1,\ldots,L\} \mbox{,~} A_{\ell \ell}X_\ell + \displaystyle\sum_{\substack{m=1\\m\neq\ell}}^L A_{\ell m}X_m = B_\ell, \label{sec03:eq02} @@ -201,7 +219,13 @@ Y_\ell = B_\ell - \displaystyle\sum_{\substack{m=1\\m\neq \ell}}^{L}A_{\ell m}X_ \right. \label{sec03:eq03} \end{equation} -is solved independently by a {\it cluster of processors} and communications are required to update the right-hand side vectors $Y_\ell$, such that the vectors $X_m$ represent the data dependencies between the clusters. In this work, we use the parallel restarted GMRES method~\cite{ref34} as an inner iteration method to solve sub-systems~(\ref{sec03:eq03}). GMRES is one of the most used Krylov iterative methods to solve sparse linear systems by minimizing the residuals over an orthonormal basis of a Krylov subspace. %In practice, GMRES is used with a preconditioner to improve its convergence. In this work, we used a preconditioning matrix equivalent to the main diagonal of sparse sub-matrix $A_{ll}$. This preconditioner is straightforward to implement in parallel and gives good performances in many situations. +is solved independently by a {\it cluster of processors} and communications are required to update the right-hand side vectors $Y_\ell$, such that the vectors $X_m$ represent the data dependencies between the clusters. In this work, we use the parallel restarted GMRES method~\cite{ref34} as an inner iteration method to solve sub-systems~(\ref{sec03:eq03}). +%%% MODIFIE *********************** +%%%******************************** +GMRES is one of the most used Krylov iterative methods to solve sparse linear systems by minimizing the residuals over an orthonormal basis of a Krylov subspace. +%%%******************************** +%%%******************************** +%In practice, GMRES is used with a preconditioner to improve its convergence. In this work, we used a preconditioning matrix equivalent to the main diagonal of sparse sub-matrix $A_{ll}$. This preconditioner is straightforward to implement in parallel and gives good performances in many situations. It should be noted that the convergence of the inner iterative solver for the different sub-systems~(\ref{sec03:eq03}) does not necessarily involve the @@ -214,12 +238,22 @@ number of splitting is, the larger the spectral radius is. In this paper, ou work is based on the work presented in~\cite{huang1993krylov} to increase the convergence and improve the scalability of the multisplitting methods. -Krylov subspace methods are well-known for their good convergence compared to other iterative methods. In order to accelerate the convergence, we implemented the outer iteration of our multisplitting solver as a Krylov iterative method which minimizes some error function over a Krylov subspace~\cite{S96}. The Krylov subspace that we used is spanned by a basis composed of successive solutions issued from solving the $L$ splittings~(\ref{sec03:eq03}) +%%% AJOUTE ************************ +%%%******************************** +Krylov subspace methods are well-known for their good convergence compared to other iterative methods. +%%%******************************** +%%%******************************** +In order to accelerate the convergence, we implemented the outer iteration of our multisplitting solver as a Krylov iterative method which minimizes some error function over a Krylov subspace~\cite{S96}. The Krylov subspace 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 neither need an orthonormal basis nor any synchronization between clusters is necessary to orthogonalize the generated basis. This avoids to perform other synchronizations between the blocks of processors. +where for $j\in\{1,\ldots,s\}$, $x^j=[X_1^j,\ldots,X_L^j]$ is a solution of the global linear system. +%%% MODIFIE *********************** +%%%******************************** +The advantage of such a Krylov subspace is that we neither need an orthonormal basis nor any synchronization between clusters is necessary to orthogonalize the generated basis. This avoids to perform other synchronizations between the blocks of processors. +%%%******************************** +%%%******************************** 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 vectors of $S$. So $\alpha$ is defined as the solution of the large overdetermined linear system \begin{equation} diff --git a/review.txt b/review.txt index fc1b382..06e7a8c 100644 --- a/review.txt +++ b/review.txt @@ -45,9 +45,9 @@ In this work we develop a new parallel two-stage algorithm for large-scale clust ,---- 4. In Section 3. it is better if the paper can explain the intuition of multi-splitting. Currently it is more like "Here is what I did" presentation but "why do we do this" is left for the reader to guess. `---- -The iterative algorithms suffer from the scalability problem on large computing platforms due to the large amount of communications and synchronisations. In this context, the multisplitting methods are well-known to be more adapted to large-scale clusters of processors. The main principle of the multispliting methods is to split the large problem to solve in different blocks in such a way each block can be solved by a processor or a set of processors and thus to minimize by this way the synchronizations over the large cluster. However they suffer from slow convergence. In fact, the larger the number of splitting is, the larger the spectral radius is, thereby slowing the convergence of the multisplitting algorithm. +The iterative algorithms suffer from the scalability problem on large computing platforms due to the large amount of communications and synchronizations. In this context, the multisplitting methods are well-known to be more adapted to large-scale clusters of processors. The main principle of the multisplitting methods is to split the large problem to solve in different blocks in such a way each block can be solved by a processor or a set of processors and thus to minimize by this way the synchronizations over the large cluster. However these methods suffer from slow convergence. In fact, the larger the number of splitting is, the larger the spectral radius is, thereby slowing the convergence of the multisplitting algorithm. -We have used the parallel algorithm of the well-known GMRES method to solve locally each block by a set of processors. In addition we have also implemented the outer iteration as a Krylov subspace iteration minimizing some error function which allows to accelerate the global convergence of the multisplitting algorithm. +We have used the well-known GMRES method to solve locally in parallel each block by a set of processors. In addition we have also implemented the outer iteration as a Krylov subspace iteration minimizing some error function which allows to accelerate the global convergence of the multisplitting algorithm. The main principle of the multisplitting methods is defined in Section 2. Section 3 presenting our two-stage algorithm is little modified to show our motivations to mix between the multisplitting methods and Krylov iterative methods.