X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/58544a2932f7124415b708bec79740f1269f8b52..dbb7065c4ccb3cdeb06911258b62798d3caa624d:/paper.tex

diff --git a/paper.tex b/paper.tex
index f2e0621..ba4d0b0 100644
--- a/paper.tex
+++ b/paper.tex
@@ -439,7 +439,7 @@ can be around 7 times faster.
 \end{abstract}
 
 \begin{IEEEkeywords}
-Iterative Krylov methods; sparse linear systems; error minimization; PETSc; %à voir... 
+Iterative Krylov methods; sparse linear systems; residual minimization; PETSc; %à voir... 
 \end{IEEEkeywords}
 
 
@@ -649,7 +649,7 @@ appropriate than a direct method in a parallel context.
     \State retrieve error
     \State $S_{k~mod~s}=x^k$ \label{algo:store}
     \If {$k$ mod $s=0$ {\bf and} error$>\epsilon_{kryl}$}
-      \State $R=AS$ \Comment{compute dense matrix}
+      \State $R=AS$ \Comment{compute dense matrix} \label{algo:matrix_mul}
       \State Solve least-squares problem $\underset{\alpha\in\mathbb{R}^{s}}{min}\|b-R\alpha\|_2$ \label{algo:}
       \State $x^k=S\alpha$  \Comment{compute new solution}
     \EndIf
@@ -671,7 +671,7 @@ solve the minimization problem, an  iterative method is used. Two parameters are
 required for that: the maximum number of iteration and the threshold to stop the
 method.
 
-To summarize, the important parameters of are:
+To summarize, the important parameters of TSARM are:
 \begin{itemize}
 \item $\epsilon_{kryl}$ the threshold to stop the method of the krylov method
 \item $max\_iter_{kryl}$ the maximum number of iterations for the krylov method
@@ -693,6 +693,46 @@ To summarize, the important parameters of are:
 \section{Parallelization}
 \label{sec:05}
 
+The  parallelisation  of  TSARM  relies   on  the  parallelization  of  all  its
+parts. More  precisely, except  the least-square step,  all the other  parts are
+obvious to  achieve out in parallel. In  order to develop a  parallel version of
+our   code,   we   have   chosen  to   use   PETSc~\cite{petsc-web-page}.    For
+line~\ref{algo:matrix_mul} the  matrix-matrix multiplication is  implemented and
+efficient since the  matrix $A$ is sparse and since the  matrix $S$ contains few
+colums in  practice. As explained  previously, at least  two methods seem  to be
+interesting to solve the least-square minimization, CGLS and LSQR.
+
+In the following  we remind the CGLS algorithm. The LSQR  method follows more or
+less the same principle but it take more place, so we briefly explain the parallelization of CGLS which is similar to LSQR.
+
+\begin{algorithm}[t]
+\caption{CGLS}
+\begin{algorithmic}[1]
+  \Input $A$ (matrix), $b$ (right-hand side)
+  \Output $x$ (solution vector)\vspace{0.2cm}
+  \State $r=b-Ax$
+  \State $p=A'r$
+  \State $s=p$
+  \State $g=||s||^2_2$
+  \For {$k=1,2,3,\ldots$ until convergence (g$<\epsilon_{ls}$)} \label{algo2:conv}
+    \State $q=Ap$
+    \State $\alpha=g/||q||^2_2$
+    \State $x=x+alpha*p$
+    \State $r=r-alpha*q$
+    \State $s=A'*r$
+    \State $g_{old}=g$
+    \State $g=||s||^2_2$
+    \State $\beta=g/g_{old}$
+  \EndFor
+\end{algorithmic}
+\label{algo:02}
+\end{algorithm}
+
+
+In each iteration  of CGLS, there is two  matrix-vector multiplications and some
+classical operations:  dots, norm, multiplication  and addition on  vectors. All
+these operations are easy to implement in PETSc or similar environment.
+
 %%%*********************************************************
 %%%*********************************************************
 \section{Experiments using petsc}
@@ -839,6 +879,7 @@ Larger experiments ....
 
 future plan : \\
 - study other kinds of matrices, problems, inner solvers\\
+- test the influence of all the parameters\\
 - adaptative number of outer iterations to minimize\\
 - other methods to minimize the residuals?\\
 - implement our solver inside PETSc