X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/23e0645a70bedc9a87517c244a7500dc07db0638..dbb7065c4ccb3cdeb06911258b62798d3caa624d:/paper.tex

diff --git a/paper.tex b/paper.tex
index 23bb18b..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}
 
 
@@ -583,10 +583,10 @@ performances.
 The present paper is organized  as follows. In Section~\ref{sec:02} some related
 works are presented. Section~\ref{sec:03} presents our two-stage algorithm using
 a  least-square  residual  minimization.   Section~\ref{sec:04}  describes  some
-convergence   results   on  this   method.    Section~\ref{sec:05}  shows   some
-experimental results obtained on large  clusters of our algorithm using routines
-of  PETSc  toolkit.  Finally   Section~\ref{sec:06}  concludes  and  gives  some
-perspectives.
+convergence  results  on this  method.   In Section~\ref{sec:05},  parallization
+details  of  TSARM  are  given.  Section~\ref{sec:06}  shows  some  experimental
+results  obtained on large  clusters of  our algorithm  using routines  of PETSc
+toolkit.  Finally Section~\ref{sec:06} concludes and gives some perspectives.
 %%%*********************************************************
 %%%*********************************************************
 
@@ -604,7 +604,7 @@ perspectives.
 
 %%%*********************************************************
 %%%*********************************************************
-\section{A Krylov two-stage algorithm}
+\section{Two-stage algorithm with least-square residuals minimization}
 \label{sec:03}
 A two-stage algorithm is proposed  to solve large  sparse linear systems  of the
 form  $Ax=b$,  where  $A\in\mathbb{R}^{n\times   n}$  is  a  sparse  and  square
@@ -644,12 +644,13 @@ appropriate than a direct method in a parallel context.
   \Input $A$ (sparse matrix), $b$ (right-hand side)
   \Output $x$ (solution vector)\vspace{0.2cm}
   \State Set the initial guess $x^0$
-  \For {$k=1,2,3,\ldots$ until convergence} \label{algo:conv}
-    \State  $x^k=Solve(A,b,x^{k-1},m)$   \label{algo:solve}
+  \For {$k=1,2,3,\ldots$ until convergence (error$<\epsilon_{kryl}$)} \label{algo:conv}
+    \State  $x^k=Solve(A,b,x^{k-1},max\_iter_{kryl})$   \label{algo:solve}
+    \State retrieve error
     \State $S_{k~mod~s}=x^k$ \label{algo:store}
-    \If {$k$ mod $s=0$ {\bf and} not convergence}
-      \State $R=AS$ \Comment{compute dense matrix}
-      \State Solve least-squares problem $\underset{\alpha\in\mathbb{R}^{s}}{min}\|b-R\alpha\|_2$
+    \If {$k$ mod $s=0$ {\bf and} error$>\epsilon_{kryl}$}
+      \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
   \EndFor
@@ -659,11 +660,25 @@ appropriate than a direct method in a parallel context.
 
 Algorithm~\ref{algo:01}  summarizes  the principle  of  our  method.  The  outer
 iteration is  inside the for  loop. Line~\ref{algo:solve}, the Krylov  method is
-called for a  maximum of $m$ iterations.  In practice, we  suggest to choose $m$
-equals to  the restart number  of the GMRES like  method. Line~\ref{algo:store},
-$S_{k~ mod~ s}=x^k$  consists in copying the solution $x_k$  into the column $k~
-mod~ s$ of the matrix $S$. After the minimization, the matrix $S$ is reused with
-the new values of the residuals.
+called for a  maximum of $max\_iter_{kryl}$ iterations.  In practice, we  suggest to set this parameter
+equals to  the restart  number of the  GMRES-like method. Moreover,  a tolerance
+threshold must be specified for the  solver. In practice, this threshold must be
+much  smaller  than the  convergence  threshold  of  the TSARM  algorithm  (i.e.
+$\epsilon$).  Line~\ref{algo:store}, $S_{k~ mod~ s}=x^k$ consists in copying the
+solution  $x_k$  into the  column  $k~  mod~ s$ of  the  matrix  $S$. After  the
+minimization, the matrix $S$ is reused with the new values of the residuals.  To
+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 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
+\item $s$ the number of outer iterations before applying the minimization step
+\item $max\_iter_{ls}$ the maximum number of iterations for the iterative least-square method
+\item $\epsilon_{ls}$ the threshold to stop the least-square method
+\end{itemize}
 
 %%%*********************************************************
 %%%*********************************************************
@@ -671,11 +686,58 @@ the new values of the residuals.
 \section{Convergence results}
 \label{sec:04}
 
+
+
 %%%*********************************************************
 %%%*********************************************************
-\section{Experiments using petsc}
+\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}
+\label{sec:06}
+
 
 In order to see the influence of our algorithm with only one processor, we first
 show  a comparison  with the  standard version  of GMRES  and our  algorithm. In
@@ -809,7 +871,7 @@ Larger experiments ....
 %%%*********************************************************
 %%%*********************************************************
 \section{Conclusion}
-\label{sec:06}
+\label{sec:07}
 %The conclusion goes here. this is more of the conclusion
 %%%*********************************************************
 %%%*********************************************************
@@ -817,6 +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