X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/acead508c1f820b919aa203c78668e5e645cc9b8..d40f73fb29cd99acadda6620fc198fc09066584e:/paper.tex diff --git a/paper.tex b/paper.tex index bf9e767..16beac7 100644 --- a/paper.tex +++ b/paper.tex @@ -380,8 +380,8 @@ % use a multiple column layout for up to two different % affiliations -\author{\IEEEauthorblockN{Rapha\"el Couturier\IEEEauthorrefmark{1}, Lilia Ziane Khodja \IEEEauthorrefmark{2}, and Christophe Guyeux\IEEEauthorrefmark{1}} -\IEEEauthorblockA{\IEEEauthorrefmark{1} Femto-ST Institute, University of Franche Comte, France\\ +\author{\IEEEauthorblockN{Rapha\"el Couturier\IEEEauthorrefmark{1}, Lilia Ziane Khodja\IEEEauthorrefmark{2}, and Christophe Guyeux\IEEEauthorrefmark{1}} +\IEEEauthorblockA{\IEEEauthorrefmark{1} Femto-ST Institute, University of Franche-Comt\'e, France\\ Email: \{raphael.couturier,christophe.guyeux\}@univ-fcomte.fr} \IEEEauthorblockA{\IEEEauthorrefmark{2} INRIA Bordeaux Sud-Ouest, France\\ Email: lilia.ziane@inria.fr} @@ -564,7 +564,7 @@ gradient and GMRES ones (Generalized Minimal RESidual). However, iterative methods suffer from scalability problems on parallel computing platforms with many processors, due to their need of reduction -operations, and to collective communications to achive matrix-vector +operations, and to collective communications to achieve matrix-vector multiplications. The communications on large clusters with thousands of cores and large sizes of messages can significantly affect the performances of these iterative methods. As a consequence, Krylov subspace iteration methods are often used @@ -618,22 +618,23 @@ It can be summarized as follows: the inner solver is a Krylov based one. In order to accelerate its convergence, the outer solver periodically applies a least-squares minimization on the residuals computed by the inner one. %Tsolver which does not required to be changed. -At each outer iteration, the sparse linear system $Ax=b$ is partially -solved using only $m$ -iterations of an iterative method, this latter being initialized with the -best known approximation previously obtained. -GMRES method~\cite{Saad86}, or any of its variants, can be used for instance as an -inner solver. The current approximation of the Krylov method is then stored inside a matrix -$S$ composed by the successive solutions that are computed during inner iterations. +At each outer iteration, the sparse linear system $Ax=b$ is partially solved +using only $m$ iterations of an iterative method, this latter being initialized +with the last obtained approximation. GMRES method~\cite{Saad86}, or any of its +variants, can potentially be used as inner solver. The current approximation of +the Krylov method is then stored inside a $n \times s$ matrix $S$, which is +composed by the $s$ last solutions that have been computed during the inner +iterations phase. In the remainder, the $i$-th column vector of $S$ will be +denoted by $S_i$. -At each $s$ iterations, the minimization step is applied in order to +At each $s$ iterations, another kind of minimization step is applied in order to 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} \end{equation} -with $R=AS$. Then the new solution $x$ is computed with $x=S\alpha$. +with $R=AS$. The new solution $x$ is then computed with $x=S\alpha$. In practice, $R$ is a dense rectangular matrix belonging in $\mathbb{R}^{n\times s}$, @@ -650,8 +651,7 @@ appropriate than a single direct method in a parallel context. \Output $x$ (solution vector)\vspace{0.2cm} \State Set the initial guess $x_0$ \For {$k=1,2,3,\ldots$ until convergence (error$<\epsilon_{tsirm}$)} \label{algo:conv} - \State $x_k=Solve(A,b,x_{k-1},max\_iter_{kryl})$ \label{algo:solve} - \State retrieve error + \State $[x_k,error]=Solve(A,b,x_{k-1},max\_iter_{kryl})$ \label{algo:solve} \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} @@ -663,18 +663,22 @@ appropriate than a single direct method in a parallel context. \label{algo:01} \end{algorithm} -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 $max\_iter_{kryl}$ 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 TSIRM algorithm (\emph{i.e.} -$\epsilon_{tsirm}$). 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 iterations and the threshold to stop the -method. +Algorithm~\ref{algo:01} summarizes the principle of the proposed method. The +outer iteration is inside the \emph{for} loop. Line~\ref{algo:solve}, the Krylov +method is called for a maximum of $max\_iter_{kryl}$ iterations. In practice, +we suggest to set this parameter equal to the restart number in 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 TSIRM algorithm (\emph{i.e.}, $\epsilon_{tsirm}$). We also consider that +after the call of the $Solve$ function, we obtain the vector $x_k$ and the error +which is defined by $||Ax^k-b||_2$. + + Line~\ref{algo:store}, +$S_{k \mod s}=x^k$ consists in copying the solution $x_k$ into the column $k +\mod s$ of $S$. 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 iterations +and the threshold to stop the method. Let us summarize the most important parameters of TSIRM: \begin{itemize} @@ -686,13 +690,13 @@ Let us summarize the most important parameters of TSIRM: \end{itemize} -The parallelisation of TSIRM relies on the parallelization of all its +The parallelization of TSIRM relies on the parallelization of all its parts. More precisely, except the least-squares 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 +columns in practice. As explained previously, at least two methods seem to be interesting to solve the least-squares minimization, CGLS and LSQR. In the following we remind the CGLS algorithm. The LSQR method follows more or @@ -735,17 +739,59 @@ these operations are easy to implement in PETSc or similar environment. \section{Convergence results} \label{sec:04} -Let us recall the following result, see~\cite{Saad86}. +Let us recall the following result, see~\cite{Saad86} for further readings. \begin{proposition} +\label{prop:saad} Suppose that $A$ is a positive real matrix with symmetric part $M$. Then the residual norm provided at the $m$-th step of GMRES satisfies: \begin{equation} ||r_m|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{m}{2}} ||r_0|| , \end{equation} -where $\alpha = \lambda_min(M)^2$ and $\beta = \lambda_max(A^T A)$, which proves +where $\alpha = \lambda_{min}(M)^2$ and $\beta = \lambda_{max}(A^T A)$, which proves the convergence of GMRES($m$) for all $m$ under that assumption regarding $A$. \end{proposition} +We can now claim that, +\begin{proposition} +If $A$ is a positive real matrix and GMRES($m$) is used as solver, then the TSIRM algorithm is convergent. Furthermore, +let $r_k$ be the +$k$-th residue of TSIRM, then +we still have: +\begin{equation} +||r_k|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0|| , +\end{equation} +where $\alpha$ and $\beta$ are defined as in Proposition~\ref{prop:saad}. +\end{proposition} + +\begin{proof} +We will prove by a mathematical induction that, for each $k \in \mathbb{N}^\ast$, +$||r_k|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{mk}{2}} ||r_0||.$ + +The base case is obvious, as for $k=1$, the TSIRM algorithm simply consists in applying GMRES($m$) once, leading to a new residual $r_1$ which follows the inductive hypothesis due to Proposition~\ref{prop:saad}. + +Suppose now that the claim holds for all $m=1, 2, \hdots, k-1$, that is, $\forall m \in \{1,2,\hdots, k-1\}$, $||r_m|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0||$. +We will show that the statement holds too for $r_k$. Two situations can occur: +\begin{itemize} +\item If $k \mod m \neq 0$, then the TSIRM algorithm consists in executing GMRES once. In that case, we obtain $||r_k|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{m}{2}} ||r_{k-1}||\leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0||$ by the inductive hypothesis. +\item Else, the TSIRM algorithm consists in two stages: a first GMRES($m$) execution leads to a temporary $x_k$ whose residue satisfies $||r_k|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{m}{2}} ||r_{k-1}||\leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0||$, and a least squares resolution. +Let $\operatorname{span}(S) = \left \{ {\sum_{i=1}^k \lambda_i v_i \Big| k \in \mathbb{N}, v_i \in S, \lambda _i \in \mathbb{R}} \right \}$ be the linear span of a set of real vectors $S$. So,\\ +$\min_{\alpha \in \mathbb{R}^s} ||b-R\alpha ||_2 = \min_{\alpha \in \mathbb{R}^s} ||b-AS\alpha ||_2$ + +$\begin{array}{ll} +& = \min_{x \in span\left(S_{k-s+1}, S_{k-s+2}, \hdots, S_{k} \right)} ||b-AS\alpha ||_2\\ +& = \min_{x \in span\left(x_{k-s+1}, x_{k-s}+2, \hdots, x_{k} \right)} ||b-AS\alpha ||_2\\ +& \leqslant \min_{x \in span\left( x_{k} \right)} ||b-Ax ||_2\\ +& \leqslant \min_{\lambda \in \mathbb{R}} ||b-\lambda Ax_{k} ||_2\\ +& \leqslant ||b-Ax_{k}||_2\\ +& = ||r_k||_2\\ +& \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0||, +\end{array}$ +\end{itemize} +which concludes the induction and the proof. +\end{proof} + +We can remark that, at each iterate, the residue of the TSIRM algorithm is lower +than the one of the GMRES method. %%%********************************************************* %%%********************************************************* @@ -817,7 +863,7 @@ torso3 & fgmres / sor & 37.70 & 565 & 34.97 & 510 \\ \hline \end{tabular} -\caption{Comparison of (F)GMRES and 2 stage (F)GMRES algorithms in sequential with some matrices, time is expressed in seconds.} +\caption{Comparison of (F)GMRES and TSIRM with (F)GMRES in sequential with some matrices, time is expressed in seconds.} \label{tab:02} \end{center} \end{table} @@ -826,7 +872,7 @@ torso3 & fgmres / sor & 37.70 & 565 & 34.97 & 510 \\ -In order to perform larger experiments, we have tested some example applications +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} @@ -839,11 +885,23 @@ scalable linear equations solvers: finite elements. For this example, the user can define the scaling of material coefficient in embedded circle called $\alpha$. \end{itemize} -For more technical details on these applications, interested readers are invited -to read the codes available in the PETSc sources. Those problems have been -chosen because they are scalable with many cores which is not the case of other problems that we have tested. +For more technical details on these applications, interested readers are invited +to read the codes available in the PETSc sources. Those problems have been +chosen because they are scalable with many cores which is not the case of other +problems that we have tested. + +In the following larger experiments are described on two large scale +architectures: Curie and Juqeen. Both these architectures are supercomputer +composed of 80,640 cores for Curie and 458,752 cores for Juqueen. Those machines +are respectively hosted by GENCI in France and Jülich Supercomputing Centre in +Germany. They belongs with other similar architectures of the PRACE initiative ( +Partnership for Advanced Computing in Europe) which aims at proposing high +performance supercomputing architecture to enhance research in Europe. The Curie +architecture is composed of Intel E5-2680 processors at 2.7 GHz with 2Gb memory +by core. The Juqueen architecture is composed of IBM PowerPC A2 at 1.6 GHz with +1Gb memory per core. Both those architecture are equiped with a dedicated high +speed network. -In the following larger experiments are described on two large scale architectures: Curie and Juqeen... {\bf description...}\\ {\bf Description of preconditioners}\\ @@ -876,7 +934,7 @@ Table~\ref{tab:03} shows the execution times and the number of iterations of example ex15 of PETSc on the Juqueen architecture. Different numbers of cores are studied ranging from 2,048 up-to 16,383. Two preconditioners have been tested: {\it mg} and {\it sor}. For those experiments, the number of components (or unknowns of the -problems) per processor is fixed to 25,000, also called weak scaling. This +problems) per core is fixed to 25,000, also called weak scaling. This number can seem relatively small. In fact, for some applications that need a lot of memory, the number of components per processor requires sometimes to be small. @@ -905,7 +963,7 @@ corresponds to 30*12, there are $max\_iter_{ls}$ which corresponds to 15. In Figure~\ref{fig:01}, the number of iterations per second corresponding to -Table~\ref{tab:01} is displayed. It can be noticed that the number of +Table~\ref{tab:03} is displayed. It can be noticed that the number of iterations per second of FMGRES is constant whereas it decreases with TSIRM with both preconditioners. This can be explained by the fact that when the number of cores increases the time for the least-squares minimization step also increases but, generally, @@ -923,7 +981,7 @@ the number of iterations. So, the overall benefit of using TSIRM is interesting. \begin{tabular}{|r|r|r|r|r|r|r|r|r|} \hline - nb. cores & threshold & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSIRM CGLS} & \multicolumn{2}{c|}{TSIRM LSQR} & best gain \\ + nb. cores & threshold & \multicolumn{2}{c|}{FGMRES} & \multicolumn{2}{c|}{TSIRM CGLS} & \multicolumn{2}{c|}{TSIRM 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 \\ @@ -936,7 +994,7 @@ the number of iterations. So, the overall benefit of using TSIRM is interesting. \hline \end{tabular} -\caption{Comparison of FGMRES and 2 stage FGMRES algorithms for ex54 of Petsc (both with the MG preconditioner) with 25000 components per core on Curie (restart=30, s=12), time is expressed in seconds.} +\caption{Comparison of FGMRES and TSIRM with FGMRES algorithms for ex54 of Petsc (both with the MG preconditioner) with 25,000 components per core on Curie (restart=30, s=12), time is expressed in seconds.} \label{tab:04} \end{center} \end{table*} @@ -950,9 +1008,9 @@ In Table~\ref{tab:04}, some experiments with example ex54 on the Curie architect \begin{tabular}{|r|r|r|r|r|r|r|r|r|r|r|} \hline - nb. cores & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSIRM CGLS} & \multicolumn{2}{c|}{TSIRM LSQR} & best gain & \multicolumn{3}{c|}{efficiency} \\ + nb. cores & \multicolumn{2}{c|}{FGMRES} & \multicolumn{2}{c|}{TSIRM CGLS} & \multicolumn{2}{c|}{TSIRM 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 + & Time & \# Iter. & Time & \# Iter. & Time & \# Iter. & & FGMRES & 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\\ @@ -962,7 +1020,7 @@ In Table~\ref{tab:04}, some experiments with example ex54 on the Curie architect \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.} +\caption{Comparison of FGMRES and TSIRM with FGMRES 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, threshold 5e-5), time is expressed in seconds.} \label{tab:05} \end{center} \end{table*} @@ -987,13 +1045,22 @@ In Table~\ref{tab:04}, some experiments with example ex54 on the Curie architect %%%********************************************************* %%%********************************************************* +A novel two-stage iterative algorithm has been proposed in this article, +in order to accelerate the convergence Krylov iterative methods. +Our TSIRM proposal acts as a merger between Krylov based solvers and +a least-squares minimization step. +The convergence of the method has been proven in some situations, while +experiments up to 16,394 cores have been led to verify that TSIRM runs +5 or 7 times faster than GMRES. -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 + +For future work, the authors' intention is to investigate +other kinds of matrices, problems, and inner solvers. The +influence of all parameters must be tested too, while +other methods to minimize the residuals must be regarded. +The number of outer iterations to minimize should become +adaptative to improve the overall performances of the proposal. +Finally, this solver will be implemented inside PETSc. % conference papers do not normally have an appendix @@ -1005,7 +1072,7 @@ future plan : \\ %%%********************************************************* \section*{Acknowledgment} This paper is partially funded by the Labex ACTION program (contract -ANR-11-LABX-01-01). We acknowledge PRACE for awarding us access to resource +ANR-11-LABX-01-01). We acknowledge PRACE for awarding us access to resources Curie and Juqueen respectively based in France and Germany. @@ -1048,5 +1115,3 @@ Curie and Juqueen respectively based in France and Germany. % that's all folks \end{document} - -