update ex49

[GMRES2stage.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index 312270cdad324e70a9c5c6777fcc5b056fd86898..0b8d05cdefb5b1e044029d5e67561878a01a77bf 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -351,6 +351,9 @@
 
 \usepackage{algorithm}
 \usepackage{algpseudocode}
 
 \usepackage{algorithm}
 \usepackage{algpseudocode}

+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage{multirow}

 
 \algnewcommand\algorithmicinput{\textbf{Input:}}
 \algnewcommand\Input{\item[\algorithmicinput]}
 
 \algnewcommand\algorithmicinput{\textbf{Input:}}
 \algnewcommand\Input{\item[\algorithmicinput]}


@@ -429,7 +432,7 @@ Email: lilia.ziane@inria.fr}
 \end{abstract}
 
 \begin{IEEEkeywords}
 \end{abstract}
 
 \begin{IEEEkeywords}

-Iterative Krylov methods; sparse linear systems; error minimization; PETSC; %à voir... 
+Iterative Krylov methods; sparse linear systems; error minimization; PETSc; %à voir... 

 \end{IEEEkeywords}
 
 
 \end{IEEEkeywords}
 
 


@@ -536,6 +539,42 @@ 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.
+
+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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@@ -544,6 +583,7 @@ Iterative Krylov methods; sparse linear systems; error minimization; PETSC; %à
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 \section{Related works}
 %%%*********************************************************
 %%%*********************************************************
 \section{Related works}

+\label{sec:02} 

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@@ -553,47 +593,67 @@ Iterative Krylov methods; sparse linear systems; error minimization; PETSC; %à
 %%%*********************************************************
 %%%*********************************************************
 \section{A Krylov two-stage algorithm}
 %%%*********************************************************
 %%%*********************************************************
 \section{A Krylov two-stage algorithm}

-
-
-\begin{algorithm}[!t]
+\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]
 \caption{A Krylov two-stage algorithm}
 \begin{algorithmic}[1]

-\Input $A_\ell$ (sparse sub-matrix), $B_\ell$ (right-hand side sub-vector)
-\Output $X_\ell$ (solution sub-vector)\vspace{0.2cm}
-\State Load $A_\ell$, $B_\ell$
-\State Set the initial guess $x^0$
-\State Set the minimizer $\tilde{x}^0=x^0$
-\For {$k=1,2,3,\ldots$ until the global convergence}
-\State Restart with $x^0=\tilde{x}^{k-1}$:
-\For {$j=1,2,\ldots,s$}
-\State \label{line7}Inner iteration solver: \Call{InnerSolver}{$x^0$, $j$}
-\State Construct basis $S$: add column vector $X_\ell^j$ to the matrix $S_\ell^k$
-\State Exchange local values of $X_\ell^j$ with the neighboring clusters
-\State Compute dense matrix $R$: $R_\ell^{k,j}=\sum^L_{i=1}A_{\ell i}X_i^j$ 
-\EndFor 
-\State \label{line12}Minimization $\|b-R\alpha\|_2$: \Call{UpdateMinimizer}{$S_\ell$, $R$, $b$, $k$}
-\State Local solution of linear system $Ax=b$: $X_\ell^k=\tilde{X}_\ell^k$
-\State Exchange the local minimizer $\tilde{X}_\ell^k$ with the neighboring clusters
-\EndFor
-
-\Statex
-
-\Function {InnerSolver}{$x^0$, $j$}
-\State Compute local right-hand side $Y_\ell = B_\ell - \sum^L_{\substack{m=1\\m\neq \ell}}A_{\ell m}X_m^0$
-\State Solving local splitting $A_{\ell \ell}X_\ell^j=Y_\ell$ using parallel GMRES method, such that $X_\ell^0$ is the initial guess
-\State \Return $X_\ell^j$
-\EndFunction
-
-\Statex
-
-\Function {UpdateMinimizer}{$S_\ell$, $R$, $b$, $k$}
-\State Solving normal equations $(R^k)^TR^k\alpha^k=(R^k)^Tb$ in parallel by $L$ clusters using parallel CGNR method
-\State Compute local minimizer $\tilde{X}_\ell^k=S_\ell^k\alpha^k$
-\State \Return $\tilde{X}_\ell^k$
-\EndFunction
+  \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} \label{algo:conv}
+    \State Solve iteratively $Ax^k=b$  \label{algo:solve}
+    \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}
 \end{algorithmic}
 \label{algo:01}
 \end{algorithm}

+
+Operation $S_{k~  mod~ s}=x^k$ consists in  copying the residual  $x_k$ into the
+column $k~ mod~ s$ of the matrix  $S$. After the minimization, the matrix $S$ is
+reused with the new values of the residuals.
+

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@@ -602,6 +662,112 @@ Iterative Krylov methods; sparse linear systems; error minimization; PETSC; %à
 %%%*********************************************************
 %%%*********************************************************
 \section{Experiments using petsc}
 %%%*********************************************************
 %%%*********************************************************
 \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}
+
+The following  parameters have been chosen  for our experiments.   As by default
+the restart  of GMRES is performed every  30 iterations, we have  chosen to stop
+the     GMRES    every     30    iterations     (line     \ref{algo:solve}    in
+Algorithm~\ref{algo:01}).   $s$ is  set to  8. CGLS  is chosen  to  minimize the
+least-squares  problem.  Two  conditions  are  used to  stop  CGLS,  either  the
+precision is under $1e-40$ or the  number of iterations is greater to $20$.  The
+external   precision    is   set    to   $1e-10$   (line    \ref{algo:conv}   in
+Algorithm~\ref{algo:01}).  Those  experiments have been performed  on a Intel(R)
+Core(TM) i7-3630QM CPU @ 2.40GHz with the version 3.5.1 of PETSc.
+
+
+In  Table~\ref{tab:02}, some  experiments comparing  the solving  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.   This  Table shows  that  the 2  stage
+algorithm  can  drastically  reduce  the  number  of  iterations  to  reach  the
+convergence when the  number of iterations for the normal GMRES  is more or less
+greater than  500. In fact  this also depends  on tow parameters: the  number of
+iterations  to  stop  GMRES  and   the  number  of  iterations  to  perform  the
+minimization.
+
+
+\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 CGLS} \\ 
+\cline{3-6}
+       &  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  & 37.70 & 565 & 34.97 & 510 \\
+\hline
+
+\end{tabular}
+\caption{Comparison of (F)GMRES and 2 stage (F)GMRES algorithms in sequential with some matrices, time is expressed in seconds.}
+\label{tab:02}
+\end{center}
+\end{table}
+
+
+
+
+Larger experiments ....
+
+\begin{table*}
+\begin{center}
+\begin{tabular}{|r|r|r|r|r|r|r|r|r|} 
+\hline
+
+  nb. cores & precond   & \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      & mg                    & 403.49   & 18,210    & 73.89  & 3,060   & 77.84  & 3,270  & 5.46 \\
+  2,048      & sor                   & 745.37   & 57,060    & 87.31  & 6,150   & 104.21 & 7,230  & 8.53 \\
+  4,096      & mg                    & 562.25   & 25,170    & 97.23  & 3,990   & 89.71  & 3,630  & 6.27 \\
+  4,096      & sor                   & 912.12   & 70,194    & 145.57 & 9,750   & 168.97 & 10,980 & 6.26 \\
+  8,192      & mg                    & 917.02   & 40,290    & 148.81 & 5,730   & 143.03 & 5,280  & 6.41 \\
+  8,192      & sor                   & 1,404.53 & 106,530   & 212.55 & 12,990  & 180.97 & 10,470 & 7.76 \\
+  16,384     & mg                    & 1,430.56 & 63,930    & 237.17 & 8,310   & 244.26 & 7,950  & 6.03 \\
+  16,384     & sor                   & 2,852.14 & 216,240   & 418.46 & 21,690  & 505.26 & 23,970 & 6.82 \\
+\hline
+
+\end{tabular}
+\caption{Comparison of FGMRES and 2 stage FGMRES algorithms for ex15 of Petsc with 25000 components per core on Juqueen (threshold 1e-3, restart=30, s=12),  time is expressed in seconds.}
+\label{tab:03}
+\end{center}
+\end{table*}
+
+
+

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@@ -610,6 +776,7 @@ Iterative Krylov methods; sparse linear systems; error minimization; PETSC; %à
 %%%*********************************************************
 %%%*********************************************************
 \section{Conclusion}
 %%%*********************************************************
 %%%*********************************************************
 \section{Conclusion}

+\label{sec:05}

 %The conclusion goes here. this is more of the conclusion
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 %The conclusion goes here. this is more of the conclusion
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@@ -654,10 +821,13 @@ Iterative Krylov methods; sparse linear systems; error minimization; PETSC; %à
 % (used to reserve space for the reference number labels box)
 \begin{thebibliography}{1}
 
 % (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}
 
 
 \end{thebibliography}