X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/Krylov_multi.git/blobdiff_plain/03b1aa97afc150a187a7ec819386a3b5768a2f9e..d9105f3cc518086be070a250845823a6ec0f7b4a:/krylov_multi.tex

diff --git a/krylov_multi.tex b/krylov_multi.tex
index 1010f07..0ff9175 100644
--- a/krylov_multi.tex
+++ b/krylov_multi.tex
@@ -6,6 +6,12 @@
 \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} 
 
@@ -55,8 +61,13 @@ 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...\\
+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.
 
 
 
@@ -255,7 +266,9 @@ 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]
-\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}