new results

[GMRES2stage.git] / paper.tex

diff --git a/paper.tex b/paper.tex
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@@ -367,13 +367,16 @@
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-\title{A Krylov two-stage algorithm to solve large sparse linear systems}
+\title{TSARM: A Two-Stage Algorithm with least-square Residual Minimization to solve large sparse linear systems}

 %où
 %\title{A two-stage algorithm with error minimization to solve large sparse linear systems}
 %où
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 %où
 %\title{A two-stage algorithm with error minimization to solve large sparse linear systems}
 %où
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+
+
+

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@@ -423,7 +426,16 @@ Email: lilia.ziane@inria.fr}
 
 
 \begin{abstract}
 
 
 \begin{abstract}

-%The abstract goes here. DO NOT USE SPECIAL CHARACTERS, SYMBOLS, OR MATH IN YOUR TITLE OR ABSTRACT.
+In  this paper  we propose  a  two stage  iterative method  which increases  the
+convergence of Krylov iterative methods,  typically those of GMRES variants. The
+principle of  our approach  is to  build an external  iteration over  the Krylov
+method  and to  save  the current  residual  frequently (for  example, for  each
+restart of GMRES). Then after a given number of outer iterations, a minimization
+step is applied on the matrix composed of the save residuals in order to compute
+a  better solution and  make a  new iteration  if necessary.  We prove  that our
+method  has the  same  convergence property  than  the inner  method used.  Some
+experiments using up  to 16,394 cores show that compared  to GMRES our algorithm
+can be around 7 times faster.

 \end{abstract}
 
 \begin{IEEEkeywords}
 \end{abstract}
 
 \begin{IEEEkeywords}


@@ -534,12 +546,14 @@ Iterative Krylov methods; sparse linear systems; error minimization; PETSc; %à
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-Iterative  methods are become  more attractive  than direct  ones to  solve very
-large sparse linear  systems. They are more effective in  a parallel context and
-require less memory  and arithmetic operations than direct  methods. A number of
-iterative methods are proposed and adapted by many researchers and the increased
-need for solving  very large sparse linear systems  triggered the development of
-efficient iterative techniques suitable for the parallel processing.
+
+Iterative methods  became more attractive than  direct ones to  solve very large
+sparse  linear systems.  Iterative  methods  are more  effecient  in a  parallel
+context,  with  thousands  of  cores,  and  require  less  memory  and  arithmetic
+operations than direct  methods. A number of iterative  methods are proposed and
+adapted by many researchers and the increased need for solving very large sparse
+linear  systems  triggered the  development  of  efficient iterative  techniques
+suitable for the parallel processing.

 
 Most of the successful iterative methods currently available are based on Krylov
 subspaces which  consist in forming a  basis of a sequence  of successive matrix
 
 Most of the successful iterative methods currently available are based on Krylov
 subspaces which  consist in forming a  basis of a sequence  of successive matrix


@@ -561,13 +575,16 @@ large clusters.
 In this  paper we propose a  two-stage algorithm based on  two nested iterations
 called inner-outer  iterations.  This algorithm  consists in solving  the sparse
 linear system iteratively  with a small number of  inner iterations and restarts
 In this  paper we propose a  two-stage algorithm based on  two nested iterations
 called inner-outer  iterations.  This algorithm  consists in solving  the sparse
 linear system iteratively  with a small number of  inner iterations and restarts

-the outer step with a new solution minimizing some error functions over a Krylov
-subspace. This algorithm is iterative  and easy to parallelize on large clusters
-and the minimization technique improves its convergence and performances.
+the outer  step with a  new solution minimizing  some error functions  over some
+previous residuals. This algorithm is iterative and easy to parallelize on large
+clusters   and  the   minimization  technique   improves  its   convergence  and
+performances.

 
 The present paper is organized  as follows. In Section~\ref{sec:02} some related
 
 The present paper is organized  as follows. In Section~\ref{sec:02} some related

-works are presented. Section~\ref{sec:03} presents our two-stage algorithm based
-on   Krylov  subspace   iteration  methods.   Section~\ref{sec:04}   shows  some
+works are presented. Section~\ref{sec:03} presents our two-stage algorithm using
+a  least-square  residual   minimization.  Section~\ref{sec:04}  describes  some
+convergence results on this method.
+   Section~\ref{sec:05}  shows   some

 experimental results obtained on large  clusters of our algorithm using routines
 of PETSc toolkit.
 %%%*********************************************************
 experimental results obtained on large  clusters of our algorithm using routines
 of PETSc toolkit.
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@@ -652,12 +669,13 @@ reused with the new values of the residuals.
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-
+\section{Convergence results}
+\label{sec:04}

 
 %%%*********************************************************
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 \section{Experiments using petsc}
 
 %%%*********************************************************
 %%%*********************************************************
 \section{Experiments using petsc}

-\label{sec:04}
+\label{sec:05}

 
 
 In order to see the influence of our algorithm with only one processor, we first
 
 
 In order to see the influence of our algorithm with only one processor, we first


@@ -762,7 +780,28 @@ Larger experiments ....
 \end{table*}
 
 
 \end{table*}
 
 

+\begin{table*}
+\begin{center}
+\begin{tabular}{|r|r|r|r|r|r|r|r|r|} 
+\hline
+
+  nb. cores & threshold   & \multicolumn{2}{c|}{gmres variant} & \multicolumn{2}{c|}{2 stage CGLS} &  \multicolumn{2}{c|}{2 stage LSQR} & best gain \\ 
+\cline{3-8}
+             &                       & Time  & \# Iter.  & Time  & \# Iter. & Time  & \# Iter. & \\\hline \hline
+  2,048      & 8e-5                  & 108.88 & 16,560  & 23.06  &  3,630  & 22.79  & 3,630   & 4.77 \\
+  2,048      & 6e-5                  & 194.01 & 30,270  & 35.50  &  5,430  & 27.74  & 4,350   & 6.99 \\
+  4,096      & 7e-5                  & 160.59 & 22,530  & 35.15  &  5,130  & 29.21  & 4,350   & 5.49 \\
+  4,096      & 6e-5                  & 249.27 & 35,520  & 52.13  &  7,950  & 39.24  & 5,790   & 6.35 \\
+  8,192      & 6e-5                  & 149.54 & 17,280  & 28.68  &  3,810  & 29.05  & 3,990  & 5.21 \\
+  8,192      & 5e-5                  & 792.11 & 109,590 & 76.83  &  10,470  & 65.20  & 9,030  & 12.14 \\
+  16,384     & 4e-5                  & 718.61 & 86,400 & 98.98  &  10,830  & 131.86  & 14,790  & 7.26 \\
+\hline

 
 

+\end{tabular}
+\caption{Comparison of FGMRES  and 2 stage FGMRES algorithms for ex54 of Petsc (both with the MG preconditioner) with 25000 components per core on Curie (restart=30, s=12),  time is expressed in seconds.}
+\label{tab:04}
+\end{center}
+\end{table*}

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@@ -777,6 +816,12 @@ Larger experiments ....
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+future plan : \\
+- study other kinds of matrices, problems, inner solvers\\
+- adaptative number of outer iterations to minimize\\
+- other methods to minimize the residuals?\\
+- implement our solver inside PETSc
+

 
 % conference papers do not normally have an appendix
 
 
 % conference papers do not normally have an appendix