X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/Krylov_multi.git/blobdiff_plain/e82d3db730093a1ae5fd3f4ecb5442ee0eaef981..b7730f90b7bda6c50247f984c438b3fb1fde97bf:/krylov_multi.tex

diff --git a/krylov_multi.tex b/krylov_multi.tex
index d687ad8..2d0bc69 100644
--- a/krylov_multi.tex
+++ b/krylov_multi.tex
@@ -75,11 +75,14 @@ traditional  multisplitting  method  that  suffer  from  slow  convergence,  as
 proposed  in~\cite{huang1993krylov},  the  use  of a  minimization  process  can
 drastically improve the convergence.
 
+The paper is organized as follows. First in Section~\ref{sec:02} is given some related works and the main principle of multisplitting methods. The, in Section~\ref{sec:03} is presented the algorithm of our Krylov multisplitting method based on inner-outer iterations. Finally, in Section~\ref{sec:04}, the parallel experiments on Hector architecture show the performances of the Krylov multisplitting algorithm compared to the classical GMRES algorithm to solve a 3D Poisson problem.
+
 
 %%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%
 
-\section{Related works and presention of the multisplitting method}
+\section{Related works and presentation of the multisplitting method}
+\label{sec:02}
 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
@@ -142,6 +145,7 @@ In the case where the diagonal weighting matrices $E_\ell$ have only zero and on
 %%%%%%%%%%%%%%%%%%%%%%%%
 
 \section{A two-stage method with a minimization}
+\label{sec:03}
 Let $Ax=b$ be a given large and sparse linear system of $n$ equations to solve in parallel on $L$ clusters of processors, physically adjacent or geographically distant, where $A\in\mathbb{R}^{n\times n}$ is a square and non-singular 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\{
@@ -253,6 +257,7 @@ The main key points of our Krylov multisplitting method to solve a large sparse
 %%%%%%%%%%%%%%%%%%%%%%%%
 
 \section{Experiments}
+\label{sec:04}
 In order to illustrate  the interest  of our algorithm. We have  compared our
 algorithm  with  the  GMRES  method  which  is a very  well  used  method  in  many
 situations.  We have chosen to focus on only one problem which is very simple to
@@ -277,10 +282,10 @@ preconditioners are not scalable when using many cores.
 
 %Doing many experiments  with many cores is  not easy and requires to  access to a supercomputer  with several  hours for  developing  a code  and then  improving it. 
 In the following we present some experiments we could achieved out on the Hector
-architecture, a UK's high-end  computing resource, funded by  the UK
-Research Councils. This  is a Cray XE6 supercomputer,  equipped with two 16-core
-AMD Opteron 2.3 Ghz  and 32 GB of memory. Machines are  interconnected with a 3D
-torus.
+architecture,  a UK's  high-end computing  resource, funded  by the  UK Research
+Councils~\cite{hector}.  This is  a Cray  XE6 supercomputer,  equipped  with two
+16-core AMD  Opteron 2.3 Ghz  and 32 GB  of memory. Machines  are interconnected
+with a 3D torus.
 
 Table~\ref{tab1} shows  the result of  the experiments.  The first  column shows
 the  size of  the  3D Poisson  problem.  The size  is chosen  in  order to  have
@@ -366,6 +371,8 @@ intend  to investigate  the  convergence  improvements of  our  method by  using
 preconditioning  techniques  for  Krylov  iterative methods  and  multisplitting
 methods with overlapping blocks.
 
+\section{Acknowledgement}
+The authors would like to thank Mark Bull of the EPCC his fruitful remarks and the facilities of HECToR.
 
 %Other applications (=> other matrices)\\
 %Larger experiments\\