X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/Krylov_multi.git/blobdiff_plain/05dd9db495c67be95f59c5d072cce9df954f114e..b185b4f9f3ee16bc9c59ac9a91edb7554dd525fe:/krylov_multi.tex?ds=sidebyside

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@@ -14,25 +14,171 @@
 \maketitle
 
 
+%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%
+
+
 \begin{abstract}
 In  this  paper we  revist  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 simply a
-parallel multisplitting algorithm  with few blocks of large  size and a parallel
-krylov  minimization  is used  to  improve  the  convergence. Some  large  scale
-experiments  with a 3D  Poisson problem  are presented.  They show  the obtained
-improvements compared to a classical GMRES both in terms of number of iterations
-and execution times.
+iterations computed by a multisplitting algorithm. Our new algorithm is based on
+a  parallel multisplitting  algorithm  with few  blocks  of large  size using  a
+parallel GMRES method inside each block and on a parallel krylov minimization in
+order to improve the convergence. Some large scale experiments with a 3D Poisson
+problem  are presented.   They  show  the obtained  improvements  compared to  a
+classical GMRES both in terms of number of iterations and execution times.
 \end{abstract}
 
+
+%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%
+
+
 \section{Introduction}
 
 Iterative methods are used to solve  large sparse linear systems of equations of
 the form  $Ax=b$ because they are  easier to parallelize than  direct ones. Many
-iterative  methods have  been proposed  and adpated  by many  researchers.  When
+iterative  methods have  been proposed  and  adapted by  many researchers.  When
 solving large  linear systems  with many cores,  iterative methods  often suffer
 from  scalability  problems.    This  is  due  to  their   need  for  collective
-communications to perform matrix-vector products and reduction operations.
+communications  to  perform  matrix-vector  products and  reduction  operations.
+Preconditionners can be  used in order to increase  the convergence of iterative
+solvers.   However, most  of the  good preconditionners  are not  sclalable when
+thousands of cores are used.
+
+
+A completer...
+On ne peut pas parler de tout...\\
+
+
+
+
+%%%%%%%%%%%%%%%%%%%%%%%
+%% BEGIN
+%%%%%%%%%%%%%%%%%%%%%%%
+The key idea  of the multisplitting method for  solving a large system
+of linear equations $Ax=b$ consists  in partitioning the matrix $A$ in
+$L$ several ways
+\begin{equation}
+A = M_l - N_l,~l\in\{1,\ldots,L\},
+\label{eq01}
+\end{equation}
+where $M_l$ are nonsingular matrices. Then the linear system is solved
+by iteration based on the multisplittings as follows
+\begin{equation}
+x^{k+1}=\displaystyle\sum^L_{l=1} E_l M^{-1}_l (N_l x^k + b),~k=1,2,3,\ldots
+\label{eq02}
+\end{equation}
+where $E_l$ are non-negative and diagonal weighting matrices such that
+$\sum^L_{l=1}E_l=I$ ($I$ is an identity matrix).  Thus the convergence
+of such a method is dependent on the condition
+\begin{equation}
+\rho(\displaystyle\sum^L_{l=1}E_l M^{-1}_l N_l)<1.
+\label{eq03}
+\end{equation}
+
+The advantage of  the multisplitting method is that  at each iteration
+$k$ there are $L$ different linear sub-systems
+\begin{equation}
+y_l^k=M^{-1}_l N_l x_l^{k-1} + M^{-1}_l b,~l\in\{1,\ldots,L\},
+\label{eq04}
+\end{equation}
+to be solved  independently by a direct or  an iterative method, where
+$y_l^k$  is the solution  of the  local sub-system.   A multisplitting
+method  using   an  iterative  method  for  solving   the  $L$  linear
+sub-systems is  called an inner-outer iterative method  or a two-stage
+method.   The   results    $y_l^k$   obtained   from   the   different
+splittings~(\ref{eq04}) are combined to  compute the solution $x^k$ of
+the linear system by using the diagonal weighting matrices
+\begin{equation}
+x^k = \displaystyle\sum^L_{l=1} E_l y_l^k,
+\label{eq05}
+\end{equation}    
+In the case where the diagonal weighting matrices $E_l$ have only zero
+and   one   factors  (i.e.   $y_l^k$   are   disjoint  vectors),   the
+multisplitting method is non-overlapping  and corresponds to the block
+Jacobi method.
+%%%%%%%%%%%%%%%%%%%%%%%
+%% END
+%%%%%%%%%%%%%%%%%%%%%%%
+
+\section{Related works}
+
+
+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
+for  example  a  asynchronous  version  \cite{bru1995parallel}  and  convergence
+condition  \cite{bai1999block,bahi2000asynchronous}   in  this  case   or  other
+two-stage algorithms~\cite{frommer1992h,bru1995parallel}
+
+In  \cite{huang1993krylov},  the  authors  proposed  a  parallel  multisplitting
+algorithm in which all the tasks except  one are devoted to solve a sub-block of
+the splitting  and to send their  local solution to  the first task which  is in
+charge to  combine the vectors at  each iteration.  These vectors  form a Krylov
+basis for  which the first tasks minimize  the error function over  the basis to
+increase the convergence, then the other tasks receive the update solution until
+convergence of the global system. 
+
+
+
+In \cite{couturier2008gremlins}, the  authors proposed practical implementations
+of multisplitting algorithms that take benefit from multisplitting algorithms to
+solve large scale linear systems. Inner  solvers could be based on scalar direct
+method with the LU method or scalar iterative one with GMRES.
+
+
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+\section{A two-stage method with a minimization}
+Let $Ax=b$ be a given sparse and large linear system of $n$ equations
+to solve in parallel on $L$ clusters, physically adjacent or geographically
+distant, where $A\in\mathbb{R}^{n\times n}$ is a square and nonsingular
+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\{
+\begin{array}{lll}
+A & = & [A_{1}, \ldots, A_{L}]\\
+x & = & [X_{1}, \ldots, X_{L}]\\
+b & = & [B_{1}, \ldots, B_{L}]
+\end{array}
+\right.
+\label{sec03:eq01}
+\end{equation}  
+where for all $l\in\{1,\ldots,L\}$ $A_l$ is a rectangular block of size $n_l\times n$
+and $X_l$ and $B_l$ are sub-vectors of size $n_l$, such that $\sum_ln_l=n$. In this
+case, we use a row-by-row splitting without overlapping in such a way that successive
+rows of the sparse matrix $A$ and both vectors $x$ and $b$ are assigned to one cluster.
+So, the multisplitting format of the linear system is defined as follows:
+\begin{equation}
+\forall l\in\{1,\ldots,L\} \mbox{,~} \displaystyle\sum_{i=1}^{l-1}A_{li}X_i + A_{ll}X_l + \displaystyle\sum_{i=l+1}^{L}A_{li}X_i = B_l, 
+\label{sec03:eq02}
+\end{equation} 
+where $A_{li}$ is a block of size $n_l\times n_i$ of the rectangular matrix $A_l$, $X_i\neq X_l$
+is a sub-vector of size $n_i$ of the solution vector $x$ and $\sum_{i<l}n_i+\sum_{i>l}n_i+n_l=n$,
+for all $i\in\{1,\ldots,l-1,l+1,\ldots,L\}$. Therefore, each cluster $l$ is in charge of solving
+the following spare sub-linear system: 
+\begin{equation}
+\left\{
+\begin{array}{l}
+A_{ll}X_l = Y_l \mbox{,~such that}\\
+Y_l = B_l - \displaystyle\sum_{i=1,i\neq l}^{L}A_{li}X_i,
+\end{array}
+\right.
+\label{sec03:eq03}
+\end{equation}
+where the sub-vectors $X_i$ define the data dependencies between the cluster $l$ and other clusters.
+
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%
 
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