X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/Krylov_multi.git/blobdiff_plain/e82d3db730093a1ae5fd3f4ecb5442ee0eaef981..6d8fffd4d9fc58efe4a9ff5d7cd81767782ba572:/krylov_multi.tex?ds=sidebyside diff --git a/krylov_multi.tex b/krylov_multi.tex index d687ad8..87ee227 100644 --- a/krylov_multi.tex +++ b/krylov_multi.tex @@ -67,19 +67,24 @@ Preconditioners can be used in order to increase the convergence of iterative solvers. However, most of the good preconditioners are not scalable when thousands of cores are used. -Traditional 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 -traditional multisplitting method that suffer from slow convergence, as -proposed in~\cite{huang1993krylov}, the use of a minimization process can -drastically improve the convergence. +%Traditional 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 +%traditional multisplitting method that suffer from slow convergence, as +%proposed in~\cite{huang1993krylov}, the use of a minimization process can +%drastically improve the convergence. + +Traditional parallel iterative solvers are based on fine-grain computations that frequently require data exchanges between computing nodes and have global synchronizations that penalize the scalability. Particularly, they are more penalized on large scale architectures or on distributed platforms composed of distant clusters interconnected by a high-latency network. It is therefore imperative to develop coarse-grain based algorithms to reduce the communications in the parallel iterative solvers. 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 traditional multisplitting method that suffer from slow convergence, as proposed in~\cite{huang1993krylov}, the use of a minimization process can drastically improve the convergence. + +The present paper is organized as follows. First in Section~\ref{sec:02} is given some related works and the main principle of multisplitting methods. Then, in Section~\ref{sec:03} is presented the algorithm of our Krylov multisplitting method based on inner-outer iterations. Finally, in Section~\ref{sec:04}, the parallel experiments on Hector architecture show the performances of the Krylov multisplitting algorithm compared to the classical GMRES algorithm to solve a 3D Poisson problem. %%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%% -\section{Related works and presention of the multisplitting method} +\section{Related works and presentation of the multisplitting method} +\label{sec:02} 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 @@ -142,6 +147,7 @@ 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 to solve in parallel on $L$ clusters of processors, physically adjacent or geographically distant, where $A\in\mathbb{R}^{n\times n}$ is a 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. The multisplitting of this linear system is defined as follows \begin{equation} \left\{ @@ -253,6 +259,7 @@ The main key points of our Krylov multisplitting method to solve a large sparse %%%%%%%%%%%%%%%%%%%%%%%% \section{Experiments} +\label{sec:04} In order to illustrate the interest of our algorithm. We have compared our algorithm with the GMRES method which is a very well used method in many situations. We have chosen to focus on only one problem which is very simple to @@ -277,10 +284,10 @@ preconditioners are not scalable when using many cores. %Doing many experiments with many cores is not easy and requires to access to a supercomputer with several hours for developing a code and then improving it. In the following we present some experiments we could achieved out on the Hector -architecture, a UK's high-end computing resource, funded by the UK -Research Councils. This is a Cray XE6 supercomputer, equipped with two 16-core -AMD Opteron 2.3 Ghz and 32 GB of memory. Machines are interconnected with a 3D -torus. +architecture, a UK's high-end computing resource, funded by the UK Research +Councils~\cite{hector}. This is a Cray XE6 supercomputer, equipped with two +16-core AMD Opteron 2.3 Ghz and 32 GB of memory. Machines are interconnected +with a 3D torus. Table~\ref{tab1} shows the result of the experiments. The first column shows the size of the 3D Poisson problem. The size is chosen in order to have @@ -366,6 +373,8 @@ intend to investigate the convergence improvements of our method by using preconditioning techniques for Krylov iterative methods and multisplitting methods with overlapping blocks. +\section{Acknowledgement} +The authors would like to thank Mark Bull of the EPCC his fruitful remarks and the facilities of HECToR. %Other applications (=> other matrices)\\ %Larger experiments\\