X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/b458d2ac5114a8e5c165514271869193a6c9388e..4f4061fdadd89f690136f185d51a8c9e6fb2f50c:/paper.tex

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+++ b/paper.tex
@@ -348,6 +348,21 @@
 \hyphenation{op-tical net-works semi-conduc-tor}
 
 
+
+\usepackage{algorithm}
+\usepackage{algpseudocode}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage{multirow}
+
+\algnewcommand\algorithmicinput{\textbf{Input:}}
+\algnewcommand\Input{\item[\algorithmicinput]}
+
+\algnewcommand\algorithmicoutput{\textbf{Output:}}
+\algnewcommand\Output{\item[\algorithmicoutput]}
+
+
+
 \begin{document}
 %
 % paper title
@@ -417,7 +432,7 @@ Email: lilia.ziane@inria.fr}
 \end{abstract}
 
 \begin{IEEEkeywords}
-Iterative Krylov methods; sparse linear systems; error minimization; PETSC; %Ã  voir... 
+Iterative Krylov methods; sparse linear systems; error minimization; PETSc; %Ã  voir... 
 \end{IEEEkeywords}
 
 
@@ -524,6 +539,42 @@ Iterative Krylov methods; sparse linear systems; error minimization; PETSC; %Ã 
 % no \IEEEPARstart
 % You must have at least 2 lines in the paragraph with the drop letter
 % (should never be an issue)
+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.
+
+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
+powers times an initial vector for example the residual. These methods are based
+on  orthogonality  of vectors  of  the Krylov  subspace  basis  to solve  linear
+systems.  The  most well-known iterative  Krylov subspace methods  are Conjugate
+Gradient method and GMRES method (generalized minimal residual).
+
+However,  iterative  methods suffer  from scalability  problems  on parallel
+computing  platforms  with many  processors  due  to  their need  for  reduction
+operations    and   collective    communications   to    perform   matrix-vector
+multiplications. The  communications on large  clusters with thousands  of cores
+and  large  sizes of  messages  can  significantly  affect the  performances  of
+iterative methods. In practice, Krylov subspace iteration methods are often used
+with preconditioners in order to increase their convergence and accelerate their
+performances.  However, most  of the  good preconditioners  are not  scalable on
+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
+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 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
+experimental results obtained on large  clusters of our algorithm using routines
+of PETSc toolkit.
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@@ -532,6 +583,7 @@ Iterative Krylov methods; sparse linear systems; error minimization; PETSC; %Ã 
 %%%*********************************************************
 %%%*********************************************************
 \section{Related works}
+\label{sec:02} 
 %Wherever Times is specified, Times Roman or Times New Roman may be used. If neither is available on your system, please use the font closest in appearance to Times. Avoid using bit-mapped fonts if possible. True-Type 1 or Open Type fonts are preferred. Please embed symbol fonts, as well, for math, etc.
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@@ -541,6 +593,67 @@ Iterative Krylov methods; sparse linear systems; error minimization; PETSC; %Ã 
 %%%*********************************************************
 %%%*********************************************************
 \section{A Krylov two-stage algorithm}
+\label{sec:03}
+A two-stage algorithm is proposed  to solve large  sparse linear systems  of the
+form  $Ax=b$,  where  $A\in\mathbb{R}^{n\times   n}$  is  a  sparse  and  square
+nonsingular   matrix,   $x\in\mathbb{R}^n$    is   the   solution   vector   and
+$b\in\mathbb{R}^n$ is  the right-hand side.  The algorithm is implemented  as an
+inner-outer iteration  solver based  on iterative Krylov  methods. The  main key
+points of our solver are given in Algorithm~\ref{algo:01}.
+
+In order to accelerate the convergence, the outer iteration is implemented as an
+iterative  Krylov method  which minimizes  some  error functions  over a  Krylov
+subspace~\cite{saad96}. At  each iteration, the  sparse linear system  $Ax=b$ is
+solved   iteratively    with   an   iterative   method,    for   example   GMRES
+method~\cite{saad86} or  some of its variants,  and the Krylov  subspace that we
+used is spanned by a basis  $S$ composed of successive solutions issued from the
+inner iteration
+\begin{equation}
+  S = \{x^1, x^2, \ldots, x^s\} \text{,~} s\leq n.
+\end{equation} 
+The advantage  of such a Krylov subspace  is that we neither  need an orthogonal
+basis nor  any synchronization  between processors to  generate this  basis. The
+algorithm  is periodically  restarted every  $s$ iterations  with a  new initial
+guess $x=S\alpha$ which minimizes the residual norm $\|b-Ax\|_2$ over the Krylov
+subspace spanned by  vectors of $S$, where $\alpha$ is a  solution of the normal
+equations
+\begin{equation}
+  R^TR\alpha = R^Tb,
+\end{equation}
+which is associated with the least-squares problem
+\begin{equation}
+   \underset{\alpha\in\mathbb{R}^{s}}{min}\|b-R\alpha\|_2
+\label{eq:01}
+\end{equation}
+such  that $R=AS$  is a  dense rectangular  matrix in  $\mathbb{R}^{n\times s}$,
+$s\ll n$,  and $R^T$ denotes  the transpose of  matrix $R$. We use  an iterative
+method   to  solve   the  least-squares   problem~(\ref{eq:01})  such   as  CGLS
+~\cite{hestenes52}  or LSQR~\cite{paige82}  which  are more  appropriate than  a
+direct method in the parallel context.
+
+\begin{algorithm}[t]
+\caption{A Krylov two-stage algorithm}
+\begin{algorithmic}[1]
+  \Input $A$ (sparse matrix), $b$ (right-hand side)
+  \Output $x$ (solution vector)\vspace{0.2cm}
+  \State Set the initial guess $x^0$
+  \For {$k=1,2,3,\ldots$ until convergence}
+    \State Solve iteratively $Ax^k=b$
+    \State $S_{k~mod~s}=x^k$ 
+    \If {$k$ mod $s=0$ {\bf and} not convergence}
+      \State Compute dense matrix $R=AS$
+      \State Solve least-squares problem $\underset{\alpha\in\mathbb{R}^{s}}{min}\|b-R\alpha\|_2$
+      \State Compute minimizer $x^k=S\alpha$
+    \EndIf
+  \EndFor
+\end{algorithmic}
+\label{algo:01}
+\end{algorithm}
+
+Operation $S_{k~  mod~ s}=x^k$ consists in  copying the residual  $x_k$ into the
+column $k~ mod~Â s$ of the matrix  $S$. After the minimization, the matrix $S$ is
+reused with the new values of the residuals.
+
 %%%*********************************************************
 %%%*********************************************************
 
@@ -549,6 +662,73 @@ Iterative Krylov methods; sparse linear systems; error minimization; PETSC; %Ã 
 %%%*********************************************************
 %%%*********************************************************
 \section{Experiments using petsc}
+\label{sec:04}
+
+
+In order to see the influence of our algorithm with only one processor, we first
+show  a comparison  with the  standard version  of GMRES  and our  algorithm. In
+table~\ref{tab:01},  we  show  the  matrices  we  have used  and  some  of  them
+characteristics. For all  the matrices, the name, the field,  the number of rows
+and the number of nonzero elements is given.
+
+\begin{table}
+\begin{center}
+\begin{tabular}{|c|c|r|r|r|} 
+\hline
+Matrix name              & Field             &\# Rows   & \# Nonzeros   \\\hline \hline
+crashbasis         & Optimization      & 160,000  &  1,750,416  \\
+parabolic\_fem     & Computational fluid dynamics  & 525,825 & 2,100,225 \\
+epb3               & Thermal problem   & 84,617  & 463,625  \\
+atmosmodj          & Computational fluid dynamics  & 1,270,432 & 8,814,880 \\
+bfwa398            & Electromagnetics problem & 398 & 3,678 \\
+torso3             & 2D/3D problem & 259,156 & 4,429,042 \\
+\hline
+
+\end{tabular}
+\caption{Main characteristics of the sparse matrices chosen from the Davis collection}
+\label{tab:01}
+\end{center}
+\end{table}
+
+In  table~\ref{tab:02}, some  experiments comparing  the sovling  of  the linear
+systems obtained with the previous matrices  with a GMRES variant and with out 2
+stage algorithm are  given. In the second column, it can  be noticed that either
+gmres or fgmres is used to  solve the linear system.  According to the matrices,
+different preconditioner is used.  With the 2  stage algorithm, the same
+solver and the same preconditionner is used.
+
+
+\begin{table}
+\begin{center}
+\begin{tabular}{|c|c|r|r|r|r|} 
+\hline
+
+ \multirow{2}{*}{Matrix name}  & Solver /   & \multicolumn{2}{c|}{gmres variant} & \multicolumn{2}{c|}{2 stage} \\
+       &  precond             & Time  & \# Iter.  & Time  & \# Iter.  \\\hline \hline
+
+crashbasis         & gmres / none             &  15.65     & 518  &  14.12 & 450  \\
+parabolic\_fem     & gmres / ilu           & 1009.94   & 7573 & 401.52 & 2970 \\
+epb3               & fgmres / sor             &  8.67     & 600  &  8.21 & 540  \\
+atmosmodj          &  fgmres / sor & 104.23  & 451 & 88.97 & 366  \\
+bfwa398            & gmres / none  & 1.42 & 9612 & 0.28 & 1650 \\
+torso3             & fgmres/sor  & 565  & 37.70 & 34.97 & 510 \\
+\hline
+
+\end{tabular}
+\caption{Comparison of GMRES and 2 stage GMRES algorithms in sequential with some matrices, time is expressed in seconds.}
+\label{tab:02}
+\end{center}
+\end{table}
+
+
+Param : retart 30 iters
+cols = 8
+iter cgls = 20
+cgls prec = 1e-40
+prec = 1e-10
+Intel(R) Core(TM) i7-3630QM CPU @ 2.40GHz
+
+
 %%%*********************************************************
 %%%*********************************************************
 
@@ -557,6 +737,7 @@ Iterative Krylov methods; sparse linear systems; error minimization; PETSC; %Ã 
 %%%*********************************************************
 %%%*********************************************************
 \section{Conclusion}
+\label{sec:05}
 %The conclusion goes here. this is more of the conclusion
 %%%*********************************************************
 %%%*********************************************************
@@ -601,10 +782,13 @@ Iterative Krylov methods; sparse linear systems; error minimization; PETSC; %Ã 
 % (used to reserve space for the reference number labels box)
 \begin{thebibliography}{1}
 
-\bibitem{IEEEhowto:kopka}
-%H.~Kopka and P.~W. Daly, \emph{A Guide to \LaTeX}, 3rd~ed.\hskip 1em plus
-%  0.5em minus 0.4em\relax Harlow, England: Addison-Wesley, 1999.
+\bibitem{saad86} Y.~Saad and M.~H.~Schultz, \emph{GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems}, SIAM Journal on Scientific and Statistical Computing, 7(3):856--869, 1986.
+
+\bibitem{saad96} Y.~Saad, \emph{Iterative Methods for Sparse Linear Systems}, PWS Publishing, New York, 1996.
+
+\bibitem{hestenes52} M.~R.~Hestenes and E.~Stiefel, \emph{Methods of conjugate gradients for solving linear system}, Journal of Research of National Bureau of Standards, B49:409--436, 1952.
 
+\bibitem{paige82} C.~C.~Paige and A.~M.~Saunders, \emph{LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares}, ACM Trans. Math. Softw. 8(1):43--71, 1982.
 \end{thebibliography}
 
 
