X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/0c77825b587a6051bceb28a3a8762f660f3c6c52..5890d7827b348b48bcfa9f454f2a10f591deef14:/paper.tex diff --git a/paper.tex b/paper.tex index 3b19b2d..126ff34 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} @@ -439,7 +439,7 @@ GMRES. \end{abstract} \begin{IEEEkeywords} -Iterative Krylov methods; sparse linear systems; residual minimization; PETSc; %à voir... +Iterative Krylov methods; sparse linear systems; two stage iteration; least-squares residual minimization; PETSc \end{IEEEkeywords} @@ -547,38 +547,42 @@ Iterative Krylov methods; sparse linear systems; residual minimization; PETSc; % % You must have at least 2 lines in the paragraph with the drop letter % (should never be an issue) -Iterative methods have recently become more attractive than direct ones to solve very large -sparse linear systems. They are more efficient in a parallel -context, supporting thousands of cores, and they require less memory and arithmetic -operations than direct methods. This is why new iterative methods are frequently -proposed or adapted by researchers, and the increasing need to solve very large sparse -linear systems has triggered the development of such efficient iterative techniques -suitable for parallel processing. - -Most of the successful iterative methods currently available are based on so-called ``Krylov -subspaces''. They consist in forming a basis of successive matrix -powers multiplied by an initial vector, which can be for instance the residual. These methods use vectors orthogonality of the Krylov subspace basis in order to solve linear -systems. The most known iterative Krylov subspace methods are conjugate -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 +Iterative methods have recently become more attractive than direct ones to solve +very large sparse linear systems\cite{Saad2003}. They are more efficient in a +parallel context, supporting thousands of cores, and they require less memory +and arithmetic operations than direct methods~\cite{bahicontascoutu}. This is +why new iterative methods are frequently proposed or adapted by researchers, and +the increasing need to solve very large sparse linear systems has triggered the +development of such efficient iterative techniques suitable for parallel +processing. + +Most of the successful iterative methods currently available are based on +so-called ``Krylov subspaces''. They consist in forming a basis of successive +matrix powers multiplied by an initial vector, which can be for instance the +residual. These methods use vectors orthogonality of the Krylov subspace basis +in order to solve linear systems. The most known iterative Krylov subspace +methods are conjugate 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 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 -with preconditioners in practice, to increase their convergence and accelerate their -performances. However, most of the good preconditioners are not scalable on -large clusters. - -In this research work, a two-stage algorithm based on two nested iterations -called inner-outer iterations is proposed. This algorithm consists in solving the sparse -linear system iteratively with a small number of inner iterations, and restarting -the outer step with a new solution minimizing some error functions over some -previous residuals. This algorithm is iterative and easy to parallelize on large -clusters. Furthermore, the minimization technique improves its convergence and -performances. +and large sizes of messages can significantly affect the performances of these +iterative methods. As a consequence, Krylov subspace iteration methods are often +used with preconditioners in practice, to increase their convergence and +accelerate their performances. However, most of the good preconditioners are +not scalable on large clusters. + +In this research work, a two-stage algorithm based on two nested iterations +called inner-outer iterations is proposed. This algorithm consists in solving +the sparse linear system iteratively with a small number of inner iterations, +and restarting the outer step with a new solution minimizing some error +functions over some previous residuals. For further information on two-stage +iteration methods, interested readers are invited to +consult~\cite{Nichols:1973:CTS}. Two-stage algorithms are easy to parallelize on +large clusters. Furthermore, the least-squares minimization technique improves +its convergence and performances. The present article is organized as follows. Related works are presented in Section~\ref{sec:02}. Section~\ref{sec:03} details the two-stage algorithm using @@ -618,22 +622,23 @@ 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 $\mathbb{R}^{n\times s}$, @@ -650,9 +655,8 @@ 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 $S_{k \mod s}=x_k$ \label{algo:store} + \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} \Comment{update column (k mod s) of S} \If {$k \mod s=0$ {\bf and} error$>\epsilon_{kryl}$} \State $R=AS$ \Comment{compute dense matrix} \label{algo:matrix_mul} \State $\alpha=Least\_Squares(R,b,max\_iter_{ls})$ \label{algo:} @@ -663,18 +667,22 @@ 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 +694,13 @@ 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 @@ -735,32 +743,59 @@ 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} -<<<<<<< HEAD -======= 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. +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} -Let $r_k = b-Ax_k$, where $x_k$ is the approximation of the solution after the -$k$-th iterate of TSIRM. -We will prove that $r_k \rightarrow 0$ when $k \rightarrow +\infty$. +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}. -Each step of the TSIRM algorithm +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} ->>>>>>> 84e15020344b77e5497c4a516cc20b472b2914cd + +We can remark that, at each iterate, the residue of the TSIRM algorithm is lower +than the one of the GMRES method. %%%********************************************************* %%%********************************************************* @@ -769,10 +804,12 @@ Each step of the TSIRM algorithm 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 are given. +show a comparison with GMRES or FGMRES and our algorithm. In Table~\ref{tab:01}, +we show the matrices we have used and some of them characteristics. Those +matrices are chosen from the Davis collection of the University of +Florida~\cite{Dav97}. They are matrices arising in real-world applications. For +all the matrices, the name, the field, the number of rows and the number of +nonzero elements are given. \begin{table}[htbp] \begin{center} @@ -805,13 +842,14 @@ Core(TM) i7-3630QM CPU @ 2.40GHz with the version 3.5.1 of PETSc. In Table~\ref{tab:02}, some experiments comparing the solving of the linear 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 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 -iterations to perform the minimization. +GRMES or FGMRES (Flexible GMRES)~\cite{Saad:1993} is used to solve the linear +system. According to the matrices, different preconditioner is used. With +TSIRM, the same solver and the same 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 iterations to perform the +minimization. \begin{table}[htbp] @@ -832,7 +870,7 @@ 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} @@ -841,7 +879,7 @@ 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} @@ -854,11 +892,23 @@ 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}\\ @@ -891,7 +941,7 @@ 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. @@ -938,7 +988,7 @@ 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 \\ @@ -951,7 +1001,7 @@ 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*} @@ -965,9 +1015,9 @@ 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\\ @@ -977,7 +1027,7 @@ 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*} @@ -1002,13 +1052,22 @@ 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 @@ -1020,7 +1079,7 @@ 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. @@ -1063,5 +1122,3 @@ Curie and Juqueen respectively based in France and Germany. % that's all folks \end{document} - -