\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{amsfonts,amssymb}
\usepackage{amsmath}
\usepackage{graphicx}

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



\begin{document}
\author{Raphaël Couturier \and Lilia Ziane Khodja}

\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 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  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.
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
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 system.
A multisplitting method using an iterative method for solving the $L$ linear systems is called an inner-outer
iterative method or a two-stage method. The solution of the global linear system at the iteration $k$ is computed
as follows
\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.



%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%

\bibliographystyle{plain}
\bibliography{biblio}

\end{document}
