X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/58544a2932f7124415b708bec79740f1269f8b52..99819ede8e00abfefaff7e709c42952c94e96f4c:/paper.tex diff --git a/paper.tex b/paper.tex index f2e0621..deff6f3 100644 --- a/paper.tex +++ b/paper.tex @@ -431,15 +431,15 @@ convergence of Krylov iterative methods, typically those of GMRES variants. The principle of our approach is to build an external iteration over the Krylov method and to save the current residual frequently (for example, for each restart of GMRES). Then after a given number of outer iterations, a minimization -step is applied on the matrix composed of the save residuals in order to compute -a better solution and make a new iteration if necessary. We prove that our -method has the same convergence property than the inner method used. Some +step is applied on the matrix composed of the saved residuals in order to +compute a better solution and make a new iteration if necessary. We prove that +our method has the same convergence property than the inner method used. Some experiments using up to 16,394 cores show that compared to GMRES our algorithm 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} @@ -753,7 +793,7 @@ minimization. \begin{tabular}{|c|c|r|r|r|r|} \hline - \multirow{2}{*}{Matrix name} & Solver / & \multicolumn{2}{c|}{gmres variant} & \multicolumn{2}{c|}{2 stage CGLS} \\ + \multirow{2}{*}{Matrix name} & Solver / & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSARM CGLS} \\ \cline{3-6} & precond & Time & \# Iter. & Time & \# Iter. \\\hline \hline @@ -774,14 +814,16 @@ torso3 & fgmres / sor & 37.70 & 565 & 34.97 & 510 \\ -Larger experiments .... +Larger experiments ....\\ + +Describe the problems ex15 and ex54 \begin{table*} \begin{center} \begin{tabular}{|r|r|r|r|r|r|r|r|r|} \hline - nb. cores & precond & \multicolumn{2}{c|}{gmres variant} & \multicolumn{2}{c|}{2 stage CGLS} & \multicolumn{2}{c|}{2 stage LSQR} & best gain \\ + nb. cores & precond & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSARM CGLS} & \multicolumn{2}{c|}{TSARM LSQR} & best gain \\ \cline{3-8} & & Time & \# Iter. & Time & \# Iter. & Time & \# Iter. & \\\hline \hline 2,048 & mg & 403.49 & 18,210 & 73.89 & 3,060 & 77.84 & 3,270 & 5.46 \\ @@ -806,7 +848,7 @@ Larger experiments .... \begin{tabular}{|r|r|r|r|r|r|r|r|r|} \hline - nb. cores & threshold & \multicolumn{2}{c|}{gmres variant} & \multicolumn{2}{c|}{2 stage CGLS} & \multicolumn{2}{c|}{2 stage LSQR} & best gain \\ + nb. cores & threshold & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSARM CGLS} & \multicolumn{2}{c|}{TSARM LSQR} & best gain \\ \cline{3-8} & & Time & \# Iter. & Time & \# Iter. & Time & \# Iter. & \\\hline \hline 2,048 & 8e-5 & 108.88 & 16,560 & 23.06 & 3,630 & 22.79 & 3,630 & 4.77 \\ @@ -814,7 +856,7 @@ Larger experiments .... 4,096 & 7e-5 & 160.59 & 22,530 & 35.15 & 5,130 & 29.21 & 4,350 & 5.49 \\ 4,096 & 6e-5 & 249.27 & 35,520 & 52.13 & 7,950 & 39.24 & 5,790 & 6.35 \\ 8,192 & 6e-5 & 149.54 & 17,280 & 28.68 & 3,810 & 29.05 & 3,990 & 5.21 \\ - 8,192 & 5e-5 & 792.11 & 109,590 & 76.83 & 10,470 & 65.20 & 9,030 & 12.14 \\ + 8,192 & 5e-5 & 785.04 & 109,590 & 76.07 & 10,470 & 69.42 & 9,030 & 11.30 \\ 16,384 & 4e-5 & 718.61 & 86,400 & 98.98 & 10,830 & 131.86 & 14,790 & 7.26 \\ \hline @@ -823,6 +865,33 @@ Larger experiments .... \label{tab:04} \end{center} \end{table*} + + + + + +\begin{table*} +\begin{center} +\begin{tabular}{|r|r|r|r|r|r|r|r|r|r|r|} +\hline + + nb. cores & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSARM CGLS} & \multicolumn{2}{c|}{TSARM LSQR} & best gain & \multicolumn{3}{c|}{efficiency} \\ +\cline{2-7} \cline{9-11} + & Time & \# Iter. & Time & \# Iter. & Time & \# Iter. & & GMRES & TS CGLS & TS LSQR\\\hline \hline + 512 & 3,969.69 & 33,120 & 709.57 & 5,790 & 622.76 & 5,070 & 6.37 & 1 & 1 & 1 \\ + 1024 & 1,530.06 & 25,860 & 290.95 & 4,830 & 307.71 & 5,070 & 5.25 & 1.30 & 1.21 & 1.01 \\ + 2048 & 919.62 & 31,470 & 237.52 & 8,040 & 194.22 & 6,510 & 4.73 & 1.08 & .75 & .80\\ + 4096 & 405.60 & 28,380 & 111.67 & 7,590 & 91.72 & 6,510 & 4.42 & 1.22 & .79 & .84 \\ + 8192 & 785.04 & 109,590 & 76.07 & 10,470 & 69.42 & 9,030 & 11.30 & .32 & .58 & .56 \\ + +\hline + +\end{tabular} +\caption{Comparison of FGMRES and 2 stage FGMRES algorithms for ex54 of Petsc (both with the MG preconditioner) with 204,919,225 components on Curie with different number of cores (restart=30, s=12, threshol 5e-5), time is expressed in seconds.} +\label{tab:05} +\end{center} +\end{table*} + %%%********************************************************* %%%********************************************************* @@ -839,6 +908,7 @@ 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