new

[Krylov_multi.git] / krylov_multi.tex

diff --git a/krylov_multi.tex b/krylov_multi.tex
index 036c190c5612c23ab87675492892d7cf45be5da0..7a7e809f2375277dff3ee8c0ba7bba161518dcb1 100644 (file)
--- a/krylov_multi.tex
+++ b/krylov_multi.tex
@@ -3,14 +3,18 @@
 \usepackage{amsfonts,amssymb}
 \usepackage{amsmath}
 \usepackage{graphicx}
 \usepackage{amsfonts,amssymb}
 \usepackage{amsmath}
 \usepackage{graphicx}

-%\usepackage{algorithmic}
-%\usepackage[ruled,english,boxed,linesnumbered]{algorithm2e}
-%\usepackage[english]{algorithme}

 \usepackage{algorithm}
 \usepackage{algpseudocode}
 
 \usepackage{algorithm}
 \usepackage{algpseudocode}
 

+\algnewcommand\algorithmicinput{\textbf{Input:}}
+\algnewcommand\Input{\item[\algorithmicinput]}
+
+\algnewcommand\algorithmicoutput{\textbf{Output:}}
+\algnewcommand\Output{\item[\algorithmicoutput]}
+

 
 \title{A scalable multisplitting algorithm for solving large sparse linear systems} 
 
 \title{A scalable multisplitting algorithm for solving large sparse linear systems} 

+\date{}

 
 
 
 
 
 


@@ -25,7 +29,7 @@
 
 
 \begin{abstract}
 
 
 \begin{abstract}

-In  this  paper we  revist  the  krylov  multisplitting algorithm  presented  in
+In  this paper  we  revisit  the krylov  multisplitting  algorithm presented  in

 \cite{huang1993krylov}  which  uses  a  scalar  method to  minimize  the  krylov
 iterations computed by a multisplitting algorithm. Our new algorithm is based on
 a  parallel multisplitting  algorithm  with few  blocks  of large  size using  a
 \cite{huang1993krylov}  which  uses  a  scalar  method to  minimize  the  krylov
 iterations computed by a multisplitting algorithm. Our new algorithm is based on
 a  parallel multisplitting  algorithm  with few  blocks  of large  size using  a


@@ -58,8 +62,13 @@ solvers.   However, most  of the  good preconditionners  are not  sclalable when
 thousands of cores are used.
 
 
 thousands of cores are used.
 
 

-A completer...
-On ne peut pas parler de tout...\\
+Traditionnal 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
+traditionnal  multisplitting  method  that  suffer  from  slow  convergence,  as
+proposed  in~\cite{huang1993krylov},  the  use  of a  minimization  process  can
+drastically improve the convergence.

 
 
 
 
 
 


@@ -198,9 +207,9 @@ is solved  independently by a cluster of  processors and communication
 are required  to update the  right-hand side vectors $Y_l$,  such that
 the  vectors  $X_i$  represent   the  data  dependencies  between  the
 clusters. In this work,  we use the parallel GMRES method~\cite{ref34}
 are required  to update the  right-hand side vectors $Y_l$,  such that
 the  vectors  $X_i$  represent   the  data  dependencies  between  the
 clusters. In this work,  we use the parallel GMRES method~\cite{ref34}

-as     an     inner    iteration     method     for    solving     the
+as     an     inner      iteration     method     to     solve     the

 sub-systems~(\ref{sec03:eq03}).  It  is a well-known  iterative method
 sub-systems~(\ref{sec03:eq03}).  It  is a well-known  iterative method

-which  gives good performances  for solving  sparse linear  systems in
+which  gives good performances  to solve  sparse linear  systems in

 parallel on a cluster of processors.
 
 It should be noted that  the convergence of the inner iterative solver
 parallel on a cluster of processors.
 
 It should be noted that  the convergence of the inner iterative solver


@@ -226,10 +235,10 @@ 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
 \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 such a method is that the Krylov subspace does not need to be spanned by an orthogonal basis.
-The advantage  of such a  Krylov subspace is  that we need  neither an
-orthogonal basis  nor synchronizations between  the different clusters
-to generate this basis.
+solution of the  global linear system. The advantage  of such a Krylov
+subspace   is  that   we  need   neither  an   orthogonal   basis  nor
+synchronizations  between  the  different  clusters to  generate  this
+basis.

 
 The  multisplitting   method  is  periodically   restarted  every  $s$
 iterations  with   a  new  initial   guess  $\tilde{x}=S\alpha$  which
 
 The  multisplitting   method  is  periodically   restarted  every  $s$
 iterations  with   a  new  initial   guess  $\tilde{x}=S\alpha$  which


@@ -251,26 +260,28 @@ which is associated with the least squares problem
 \text{minimize}~\|b-R\alpha\|_2,
 \label{sec03:eq07}
 \end{equation}  
 \text{minimize}~\|b-R\alpha\|_2,
 \label{sec03:eq07}
 \end{equation}  

-where  $R^T$  denotes the  transpose  of  the  matrix $R$.  Since  $R$
-(i.e.  $AS$) and  $b$  are  split among  $L$  clusters, the  symmetric
-positive    definite    system~(\ref{sec03:eq06})    is   solved    in
-parallel. Thus, an  iterative method would be more  appropriate than a
-direct  one for  solving this  system. We  use the  parallel conjugate
-gradient method for the normal equations CGNR~\cite{S96,refCGNR}.
+where $R^T$ denotes the transpose  of the matrix $R$.  Since $R$ (i.e.
+$AS$) and  $b$ are  split among $L$  clusters, the  symmetric positive
+definite  system~(\ref{sec03:eq06}) is  solved in  parallel.  Thus, an
+iterative method would be more  appropriate than a direct one to solve
+this system.  We use  the parallel conjugate  gradient method  for the
+normal equations CGNR~\cite{S96,refCGNR}.

 
 \begin{algorithm}[!t]
 \caption{A two-stage linear solver with inner iteration GMRES method}
 \begin{algorithmic}[1]
 
 \begin{algorithm}[!t]
 \caption{A two-stage linear solver with inner iteration GMRES method}
 \begin{algorithmic}[1]

-\State Load $A_l$, $B_l$, initial guess $x^0$
+\Input $A_l$ (local sparse matrix), $B_l$ (local right-hand side), $x^0$ (initial guess)
+\Output $X_l$ (local solution vector)\vspace{0.2cm}
+\State Load $A_l$, $B_l$, $x^0$

 \State Initialize the minimizer $\tilde{x}^0=x^0$
 \For {$k=1,2,3,\ldots$ until the global convergence}
 \State Restart with $x^0=\tilde{x}^{k-1}$: \textbf{for} $j=1,2,\ldots,s$ \textbf{do}
 \State\hspace{0.5cm} Inner iteration solver: \Call{InnerSolver}{$x^0$, $j$}
 \State Initialize the minimizer $\tilde{x}^0=x^0$
 \For {$k=1,2,3,\ldots$ until the global convergence}
 \State Restart with $x^0=\tilde{x}^{k-1}$: \textbf{for} $j=1,2,\ldots,s$ \textbf{do}
 \State\hspace{0.5cm} Inner iteration solver: \Call{InnerSolver}{$x^0$, $j$}

-\State\hspace{0.5cm} Construct the basis $S$: add the column vector $X_l^j$ to the matrix $S_l$
+\State\hspace{0.5cm} Construct the basis $S$: add the column vector $X_l^j$ to the matrix $S_l^k$

 \State\hspace{0.5cm} Exchange the local solution vector $X_l^j$ with the neighboring clusters
 \State\hspace{0.5cm} Exchange the local solution vector $X_l^j$ with the neighboring clusters

-\State\hspace{0.5cm} Compute the dense matrix $R$: $R_l^j=\sum^L_{i=1}A_{li}X_i^j$ 
+\State\hspace{0.5cm} Compute the dense matrix $R$: $R_l^{k,j}=\sum^L_{i=1}A_{li}X_i^j$ 

 \State\textbf{end for} 
 \State\textbf{end for} 

-\State Minimization $\|b-R\alpha\|_2$: \Call{UpdateMinimizer}{$S_l$, $R$, $b$}
+\State Minimization $\|b-R\alpha\|_2$: \Call{UpdateMinimizer}{$S_l$, $R$, $b$, $k$}

 \State Local solution of the linear system $Ax=b$: $X_l^k=\tilde{X}_l^k$
 \State Exchange the local minimizer $\tilde{X}_l^k$ with the neighboring clusters
 \EndFor
 \State Local solution of the linear system $Ax=b$: $X_l^k=\tilde{X}_l^k$
 \State Exchange the local minimizer $\tilde{X}_l^k$ with the neighboring clusters
 \EndFor


@@ -279,31 +290,31 @@ gradient method for the normal equations CGNR~\cite{S96,refCGNR}.
 
 \Function {InnerSolver}{$x^0$, $j$}
 \State Compute the local right-hand side: $Y_l = B_l - \sum^L_{i=1,i\neq l}A_{li}X_i^0$
 
 \Function {InnerSolver}{$x^0$, $j$}
 \State Compute the local right-hand side: $Y_l = B_l - \sum^L_{i=1,i\neq l}A_{li}X_i^0$

-\State Solving the local splitting $A_{ll}X_l^j=Y_l$ with the parallel GMRES method, such that $X_l^0$ is the initial guess.
+\State Solving the local splitting $A_{ll}X_l^j=Y_l$ using the parallel GMRES method, such that $X_l^0$ is the initial guess

 \State \Return $X_l^j$
 \EndFunction
 
 \Statex
 
 \Function {UpdateMinimizer}{$S_l$, $R$, $b$, $k$}
 \State \Return $X_l^j$
 \EndFunction
 
 \Statex
 
 \Function {UpdateMinimizer}{$S_l$, $R$, $b$, $k$}

-\State Solving the normal equations $R^TR\alpha=R^Tb$ in parallel by $L$ clusters using the parallel CGNR method
-\State Compute the local minimizer: $\tilde{X}_l^k=S_l\alpha$
+\State Solving the normal equations $(R^k)^TR^k\alpha^k=(R^k)^Tb$ in parallel by $L$ clusters using the parallel CGNR method
+\State Compute the local minimizer: $\tilde{X}_l^k=S_l^k\alpha^k$

 \State \Return $\tilde{X}_l^k$
 \EndFunction
 \end{algorithmic}
 \label{algo:01}
 \end{algorithm}
 
 \State \Return $\tilde{X}_l^k$
 \EndFunction
 \end{algorithmic}
 \label{algo:01}
 \end{algorithm}
 

-The main  key points  of the multisplitting  method for  solving large
-sparse  linear  systems are  given  in Algorithm~\ref{algo:01}.   This
+The  main key points  of the  multisplitting method  to solve  a large
+sparse  linear  system  are  given in  Algorithm~\ref{algo:01}.   This

 algorithm is based on a two-stage method with a minimization using the
 GMRES iterative method as an  inner solver. It is executed in parallel
 by  each cluster  of processors.   The matrices  and vectors  with the
 subscript  $l$ represent  the local  data for  the cluster  $l$, where
 $l\in\{1,\ldots,L\}$. The two-stage solver uses two different parallel
 algorithm is based on a two-stage method with a minimization using the
 GMRES iterative method as an  inner solver. It is executed in parallel
 by  each cluster  of processors.   The matrices  and vectors  with the
 subscript  $l$ represent  the local  data for  the cluster  $l$, where
 $l\in\{1,\ldots,L\}$. The two-stage solver uses two different parallel

-iterative algorithms: the GMRES method for solving each splitting on a
+iterative algorithms:  the GMRES method  to solve each splitting  on a

 cluster of processors, and the CGNR method executed in parallel by all
 cluster of processors, and the CGNR method executed in parallel by all

-clusters for  minimizing the function  error over the  Krylov subspace
+clusters  to minimize  the  function error  over  the Krylov  subspace

 spanned by  $S$.  The  algorithm requires two  global synchronizations
 between the $L$  clusters. The first one is  performed at line~$12$ in
 Algorithm~\ref{algo:01}  to exchange  the local  values of  the vector
 spanned by  $S$.  The  algorithm requires two  global synchronizations
 between the $L$  clusters. The first one is  performed at line~$12$ in
 Algorithm~\ref{algo:01}  to exchange  the local  values of  the vector


@@ -317,6 +328,36 @@ synchronizations   by   using   the   MPI   collective   communication
 subroutines.
 
 
 subroutines.
 
 

+\section{Experiments}
+
+In order  to illustrate  the interest  of our algorithm.   We have  compared our
+algorithm  with  the  GMRES  method  which  a very  well  used  method  in  many
+situations.  We have chosen to focus on only one problem which is very simple to
+implement: a 3 dimension Poisson problem.
+
+\begin{equation}
+\left\{
+                \begin{array}{ll}
+                  \nabla u&=f \mbox{~in~} \omega\\
+                  u &=0 \mbox{~on~}  \Gamma=\partial \omega
+                \end{array}
+              \right.
+\end{equation}
+
+After discretization, with a finite  difference scheme, a seven point stencil is
+used. It  is well-known that the  spectral radius of  matrices representing such
+problems are very close to 1.  Moreover, the larger the number of discretization
+points is,  the closer to 1  the spectral radius  is.  Hence, to solve  a matrix
+obtained for  a 3D Poisson  problem, the number  of iterations is high.  Using a
+preconditioner  it  is   possible  to  reduce  the  number   of  iterations  but
+preconditioners are not scalable when using many cores.
+
+\section{Conclusion and perspectives}
+
+Other applications (=> other matrices)\\
+Larger experiments\\
+Async\\
+Overlapping

 
 
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