X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/534d3695d610fd2e8face91f6516d39e2473580d..61de3472de119515c917449ea4aea0a3240c8cb0:/paper.tex diff --git a/paper.tex b/paper.tex index 463fe2c..4cd16c3 100644 --- a/paper.tex +++ b/paper.tex @@ -370,10 +370,7 @@ % paper title % can use linebreaks \\ within to get better formatting as desired \title{TSIRM: A Two-Stage Iteration with least-square Residual Minimization algorithm to solve large sparse linear systems} -%où -%\title{A two-stage algorithm with error minimization to solve large sparse linear systems} -%où -%\title{???} + @@ -607,7 +604,7 @@ is summarized while intended perspectives are provided. %%%********************************************************* %%%********************************************************* -\section{Two-stage algorithm with least-square residuals minimization} +\section{Two-stage iteration with least-square residuals minimization algorithm} \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 @@ -629,8 +626,8 @@ inner solver. The current approximation of the Krylov method is then stored insi $S$ composed by the successive solutions that are computed during inner iterations. At each $s$ iterations, the minimization step is applied in order to -compute a new solution $x$. For that, the previous residuals are computed with -$(b-AS)$. The minimization of the residuals is obtained by +compute a new solution $x$. For that, the previous residuals of $Ax=b$ are computed by +the inner iterations with $(b-AS)$. The minimization of the residuals is obtained by \begin{equation} \underset{\alpha\in\mathbb{R}^{s}}{min}\|b-R\alpha\|_2 \label{eq:01} @@ -639,7 +636,7 @@ with $R=AS$. Then the new solution $x$ is computed with $x=S\alpha$. In practice, $R$ is a dense rectangular matrix belonging in $\mathbb{R}^{n\times s}$, -with $s\ll n$. In order to minimize~(\eqref{eq:01}), a least-square method such as +with $s\ll n$. In order to minimize~\eqref{eq:01}, a least-square method such as CGLS ~\cite{Hestenes52} or LSQR~\cite{Paige82} is used. Remark that these methods are more appropriate than a single direct method in a parallel context. @@ -657,7 +654,7 @@ appropriate than a single direct method in a parallel context. \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} \label{algo:matrix_mul} - \State Solve least-square problem $\underset{\alpha\in\mathbb{R}^{s}}{min}\|b-R\alpha\|_2$ \label{algo:} + \State $\alpha=Solve\_Least\_Squares(R,b,max\_iter_{ls})$ \label{algo:} \State $x^k=S\alpha$ \Comment{compute new solution} \EndIf \EndFor @@ -675,7 +672,7 @@ $\epsilon_{tsirm}$). Line~\ref{algo:store}, $S_{k~ mod~ s}=x^k$ consists in cop 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 +required for that: the maximum number of iterations and the threshold to stop the method. Let us summarize the most important parameters of TSIRM: @@ -698,7 +695,7 @@ 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. +less the same principle but it takes more place, so we briefly explain the parallelization of CGLS which is similar to LSQR. \begin{algorithm}[t] \caption{CGLS} @@ -725,7 +722,7 @@ less the same principle but it take more place, so we briefly explain the parall In each iteration of CGLS, there is two matrix-vector multiplications and some -classical operations: dots, norm, multiplication and addition on vectors. All +classical operations: dot product, norm, multiplication and addition on vectors. All these operations are easy to implement in PETSc or similar environment. @@ -757,7 +754,7 @@ 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 Table~\ref{tab:01}, we show the matrices we have used and some of them characteristics. For all the matrices, the name, the field, the number of rows -and the number of nonzero elements is given. +and the number of nonzero elements are given. \begin{table}[htbp] \begin{center} @@ -780,7 +777,7 @@ torso3 & 2D/3D problem & 259,156 & 4,429,042 \\ The following parameters have been chosen for our experiments. As by default the restart of GMRES is performed every 30 iterations, we have chosen to stop -the GMRES every 30 iterations, $max\_iter_{kryl}=30$). $s$ is set to 8. CGLS is +the GMRES every 30 iterations (\emph{i.e.} $max\_iter_{kryl}=30$). $s$ is set to 8. CGLS is chosen to minimize the least-squares problem with the following parameters: $\epsilon_{ls}=1e-40$ and $max\_iter_{ls}=20$. The external precision is set to $\epsilon_{tsirm}=1e-10$. Those experiments have been performed on a Intel(R) @@ -792,7 +789,7 @@ systems obtained with the previous matrices with a GMRES variant and with out 2 stage algorithm are given. In the second column, it can be noticed that either gmres or fgmres is used to solve the linear system. According to the matrices, different preconditioner is used. With TSIRM, the same solver and the same -preconditionner is used. This Table shows that TSIRM can drastically reduce the +preconditionner are used. This Table shows that TSIRM can drastically reduce the number of iterations to reach the convergence when the number of iterations for the normal GMRES is more or less greater than 500. In fact this also depends on tow parameters: the number of iterations to stop GMRES and the number of @@ -826,12 +823,12 @@ torso3 & fgmres / sor & 37.70 & 565 & 34.97 & 510 \\ -In order to perform larger experiments, we have tested some example application +In order to perform larger experiments, we have tested some example applications of PETSc. Those applications are available in the ksp part which is suited for scalable linear equations solvers: \begin{itemize} \item ex15 is an example which solves in parallel an operator using a finite - difference scheme. The diagonal is equals to 4 and 4 extra-diagonals + difference scheme. The diagonal is equal to 4 and 4 extra-diagonals representing the neighbors in each directions is equal to -1. This example is used in many physical phenomena, for example, heat and fluid flow, wave propagation... @@ -943,7 +940,7 @@ the number of iterations. So, the overall benefit of using TSIRM is interesting. \end{table*} -In Table~\ref{tab:04}, some experiments with example ex54 on the Curie architecture are reported +In Table~\ref{tab:04}, some experiments with example ex54 on the Curie architecture are reported. \begin{table*}[htbp]