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25 \title{A scalable multisplitting algorithm to solve large sparse linear systems}
28 \author[1]{Raphaël Couturier}
29 \author[2]{ Lilia Ziane Khodja}
30 \affil[1]{ Femto-ST Institute\\
31 University of Franche Comte\\
33 email: raphael.couturier@univ-fcomte.fr}
34 \affil[2]{Inria Bordeaux Sud-Ouest\\
36 email: lilia.ziane@inria.fr}
43 %%%%%%%%%%%%%%%%%%%%%%%%
44 %%%%%%%%%%%%%%%%%%%%%%%%
47 In this paper we revisit the Krylov multisplitting algorithm presented in
48 \cite{huang1993krylov} which uses a sequential method to minimize the Krylov
49 iterations computed by a multisplitting algorithm. Our new algorithm is based on
50 a parallel multisplitting algorithm with few blocks of large size using a
51 parallel GMRES method inside each block and on a parallel Krylov minimization in
52 order to improve the convergence. Some large scale experiments with a 3D Poisson
53 problem are presented with up to 8,192 cores. They show the obtained
54 improvements compared to a classical GMRES both in terms of number of iterations
55 and in terms of execution times.
58 %%%%%%%%%%%%%%%%%%%%%%%%
59 %%%%%%%%%%%%%%%%%%%%%%%%
61 \section{Introduction}
62 Iterative methods are used to solve large sparse linear systems of equations of
63 the form $Ax=b$ because they are easier to parallelize than direct ones. Many
64 iterative methods have been proposed and adapted by different researchers. For
65 example, the GMRES method and the Conjugate Gradient method are very well known
66 and used~\cite{S96}. Both methods are based on the
67 Krylov subspace which consists in forming a basis of a sequence of successive
68 matrix powers times the initial residual.
70 When solving large linear systems with many cores, iterative methods often
71 suffer from scalability problems. This is due to their need for collective
72 communications to perform matrix-vector products and reduction operations.
73 Preconditioners can be used in order to increase the convergence of iterative
74 solvers. However, most of the good preconditioners are not scalable when
75 thousands of cores are used.
77 %Traditional iterative solvers have global synchronizations that penalize the
78 %scalability. Two possible solutions consists either in using asynchronous
79 %iterative methods~\cite{ref18} or to use multisplitting algorithms. In this
80 %paper, we will reconsider the use of a multisplitting method. In opposition to
81 %traditional multisplitting method that suffer from slow convergence, as
82 %proposed in~\cite{huang1993krylov}, the use of a minimization process can
83 %drastically improve the convergence.
85 Traditional parallel iterative solvers are based on fine-grain computations that
86 frequently require data exchanges between computing nodes and have global
87 synchronizations that penalize the scalability~\cite{zkcgb+14:ij}. Particularly,
88 they are more penalized on large scale architectures or on distributed platforms
89 composed of distant clusters interconnected by a high-latency network. It is
90 therefore imperative to develop coarse-grain based algorithms to reduce the
91 communications in the parallel iterative solvers. Two possible solutions
92 consists either in using asynchronous iterative methods~\cite{ref18} or in using
93 multisplitting algorithms. In this paper, we will reconsider the use of a
94 multisplitting method. In opposition to traditional multisplitting method that
95 suffer from slow convergence, as proposed in~\cite{huang1993krylov}, the use of
96 a minimization process can drastically improve the convergence.\\
99 %%% AJOUTE************************
100 %%%*******************************
101 \noindent {\bf Contributions:}\\ In this work we develop a new parallel
102 two-stage algorithm for large-scale clusters. Our objective is to create a mix
103 between Krylov based iterative methods and the multisplitting method to improve
104 scalability. In fact Krylov subspace methods are well-known for their good
105 convergence compared to other iterative methods. So, our main contribution is
106 to use the multisplitting method which splits the problem to solve into
107 different blocks in order to reduce the large amount of communications and, to
108 implement both inner and outer iterations as Krylov subspace iterations in order
109 to improve the convergence of the multisplitting algorithm.\\
110 %%%*******************************
111 %%%*******************************
113 The present paper is organized as follows. First, Section~\ref{sec:02} presents
114 some related works and the principle of multisplitting methods. Then, in
115 Section~\ref{sec:03} the algorithm of our Krylov multisplitting
116 method, based on inner-outer iterations, is presented. Finally, in Section~\ref{sec:04}, the
117 parallel experiments on Hector architecture show the performances of the Krylov
118 multisplitting algorithm compared to the classical GMRES algorithm to solve a 3D
122 %%%%%%%%%%%%%%%%%%%%%%%%
123 %%%%%%%%%%%%%%%%%%%%%%%%
125 \section{Related works and presentation of the multisplitting method}
127 A general framework to study parallel multisplitting methods has been presented in~\cite{o1985multi}
128 by O'Leary and White. Convergence conditions are given for the
129 most general cases. Many authors have improved multisplitting algorithms by proposing,
130 for example, an asynchronous version~\cite{bru1995parallel} or convergence
131 conditions~\cite{bai1999block,bahi2000asynchronous} or other
132 two-stage algorithms~\cite{frommer1992h,bru1995parallel}.
134 In~\cite{huang1993krylov}, the authors have proposed a parallel multisplitting
135 algorithm in which all the tasks except one are devoted to solve a sub-block of
136 the splitting and to send their local solutions to the first task which is in
137 charge of combining the vectors at each iteration. These vectors form a Krylov
138 basis for which the first task minimizes the error function over the basis to
139 increase the convergence, then the other tasks receive the updated solution until the
140 convergence of the global system.
142 In~\cite{couturier2008gremlins}, the authors have developed practical implementations
143 of multisplitting algorithms to solve large scale linear systems. Inner solvers
144 could be based on sequential direct method with the LU method or sequential iterative
147 In~\cite{prace-multi}, the authors have designed a parallel multisplitting
148 algorithm in which large blocks are solved using a GMRES solver. The authors have
149 performed large scale experiments up-to 32,768 cores and they conclude that
150 an asynchronous multisplitting algorithm could be more efficient than traditional
151 solvers on an exascale architecture with hundreds of thousands of cores.
153 So, compared to these works, we propose in this paper a practical multisplitting method based on parallel iterative blocks which gives better results than classical GMRES method for the 3D Poisson problem we considered.
156 The key idea of a multisplitting method to solve a large system of linear equations $Ax=b$ is defined as follows. The first step consists in partitioning the matrix $A$ in $L$ several ways
161 where for all $\ell\in\{1,\ldots,L\}$ $M_\ell$ are non-singular matrices. Then the linear system is solved by an iteration based on the obtained splittings as follows
163 x^{k+1}=\displaystyle\sum^L_{\ell=1} E_\ell M^{-1}_\ell (N_\ell x^k + b),~k=1,2,3,\ldots
166 where $E_\ell$ are non-negative and diagonal weighting matrices and their sum is an identity matrix $I$. The convergence of such a method is dependent on the condition
168 \rho(\displaystyle\sum^L_{\ell=1}E_\ell M^{-1}_\ell N_\ell)<1.
171 where $\rho$ is the spectral radius of the square matrix.
173 The advantage of the multisplitting method is that at each iteration $k$ there are $L$ different linear sub-systems
175 v_\ell^k=M^{-1}_\ell N_\ell x_\ell^{k-1} + M^{-1}_\ell b,~\ell\in\{1,\ldots,L\},
178 to be solved independently by a direct or an iterative method, where $v_\ell$ is the solution of the local sub-system. Thus the computations of $\{v_\ell\}_{1\leq \ell\leq L}$ may be performed in parallel by a set of processors. A multisplitting method using an iterative method as an inner solver is called an inner-outer iterative method or a two-stage method. The results $v_\ell$ obtained from the different splittings~(\ref{eq04}) are combined to compute solution $x$ of the linear system by using the diagonal weighting matrices
180 x^k = \displaystyle\sum^L_{\ell=1} E_\ell v_\ell^k,
183 In the case where the diagonal weighting matrices $E_\ell$ have only zero and one factors (i.e. $v_\ell$ are disjoint vectors), the multisplitting method is non-overlapping and corresponds to the block Jacobi method.
185 %%%%%%%%%%%%%%%%%%%%%%%%
186 %%%%%%%%%%%%%%%%%%%%%%%%
188 \section{A two-stage method with a minimization}
191 %%% MODIFIE ************************
192 %%%*********************************
193 Let $Ax=b$ be a given large and sparse linear system of $n$ equations where $A\in\mathbb{R}^{n\times n}$ is a sparse square and non-singular matrix, $x\in\mathbb{R}^{n}$ is the solution vector and $b\in\mathbb{R}^{n}$ is the right-hand side vector. We use a multisplitting method to solve the linear system on a large computing platform in order to reduce communications. Let the computing platform be composed of $L$ blocks of processors physically adjacent or geographically distant. In this work we apply the block Jacobi multisplitting method to the linear system as follows
194 %%%*********************************
195 %%%*********************************
201 A & = & [A_{1}, \ldots, A_{L}]\\
202 x & = & [X_{1}, \ldots, X_{L}]\\
203 b & = & [B_{1}, \ldots, B_{L}]
208 where for $\ell\in\{1,\ldots,L\}$, $A_\ell$ is a rectangular block of size $n_\ell\times n$ and $X_\ell$ and $B_\ell$ are sub-vectors of size $n_\ell$ each, such that $\sum_\ell n_\ell=n$.
209 %%% MODIFIE ***********************
210 %%%********************************
211 The splitting is performed row-by-row without overlapping in such a way that successive rows of sparse matrix $A$ and both vectors $x$ and $b$ are assigned to a block of processors.
212 %%%********************************
213 %%%********************************
214 So, the multisplitting format of the linear system is defined as follows
216 \forall \ell\in\{1,\ldots,L\} \mbox{,~} A_{\ell \ell}X_\ell + \displaystyle\sum_{\substack{m=1\\m\neq\ell}}^L A_{\ell m}X_m = B_\ell,
219 where $A_{\ell m}$ is a sub-block of size $n_\ell\times n_m$ of the rectangular matrix $A_\ell$, $X_m\neq X_\ell$ is a sub-vector of size $n_m$ of the solution vector $x$ and $\sum_{m\neq \ell}n_m+n_\ell=n$, for all $m\in\{1,\ldots,L\}$.
221 Our multisplitting method proceeds by iteration to solve the linear system in such a way that each sub-system
225 A_{\ell \ell}X_\ell = Y_\ell \mbox{,~such that}\\
226 Y_\ell = B_\ell - \displaystyle\sum_{\substack{m=1\\m\neq \ell}}^{L}A_{\ell m}X_m,
231 is solved independently by a {\it block of processors} and communications are required to update the right-hand side vectors $Y_\ell$, such that the vectors $X_m$ represent the data dependencies between the blocks. In this work, we use the parallel restarted GMRES method~\cite{ref34} as an inner iteration method to solve sub-systems~(\ref{sec03:eq03}).
232 %%% MODIFIE ***********************
233 %%%********************************
234 GMRES is one of the most used Krylov iterative methods to solve sparse linear systems by minimizing the residuals over an orthonormal basis of a Krylov subspace.
235 %%%********************************
236 %%%********************************
237 %In practice, GMRES is used with a preconditioner to improve its convergence. In this work, we used a preconditioning matrix equivalent to the main diagonal of sparse sub-matrix $A_{ll}$. This preconditioner is straightforward to implement in parallel and gives good performances in many situations.
239 It should be noted that the convergence of the inner iterative solver for the
240 different sub-systems~(\ref{sec03:eq03}) does not necessarily involve the
241 convergence of the multisplitting algorithm. It strongly depends on the properties
242 of the global sparse linear system to be
243 solved~\cite{o1985multi,ref18}. Furthermore, the splitting of the linear system
244 among several blocks of processors increases the spectral radius of the
245 iteration matrix, thereby slowing the convergence. In fact, the larger the
246 number of splittings is, the larger the spectral radius is. In this paper, our
247 work is based on the work presented in~\cite{huang1993krylov} to increase the
248 convergence and improve the scalability of the multisplitting methods.
250 %%% AJOUTE ************************
251 %%%********************************
252 Krylov subspace methods are well-known for their good convergence compared to other iterative methods.
253 %%%********************************
254 %%%********************************
255 In order to accelerate the convergence, we implemented the outer iteration of our multisplitting solver as a Krylov iterative method which minimizes some error function over a Krylov subspace~\cite{S96}. The Krylov subspace that we used is spanned by a basis composed of successive solutions issued from solving the $L$ splittings~(\ref{sec03:eq03})
257 S=\{x^1,x^2,\ldots,x^s\},~s\leq n,
260 where for $j\in\{1,\ldots,s\}$, $x^j=[X_1^j,\ldots,X_L^j]$ is a solution of the global linear system.
261 %%% MODIFIE ***********************
262 %%%********************************
263 The advantage of such a Krylov subspace is that we neither need an orthonormal basis nor any synchronization between the different blocks to orthogonalize the generated basis. This avoids to perform other synchronizations between the blocks of processors.
265 The multisplitting method is periodically restarted every $s$ iterations with a new initial guess $\tilde{x}=S\alpha$ which minimizes an error function, in our case it minimizes the residuals $\|b-Ax\|_2$ over the Krylov subspace spanned by vectors of $S$. So $\alpha$ is defined as the solution of the large overdetermined linear system.
266 %%%********************************
267 %%%********************************
273 where $R=AS$ is a dense rectangular matrix of size $n\times s$ and $s\ll n$. This leads us to solve a system of normal equations
278 which is associated with the least squares problem
280 \text{minimize}~\|b-R\alpha\|_2,
283 where $R^T$ denotes the transpose of matrix $R$. Since $R$ (i.e. $AS$) and $b$ are split among $L$ blocks, the symmetric positive definite system~(\ref{sec03:eq06}) is solved in parallel. Thus an iterative method would be more appropriate than a direct one to solve this system. We use the parallel Conjugate Gradient method for the normal equations CGNR~\cite{S96,refCGNR}.
285 \begin{algorithm}[!t]
286 \caption{A two-stage linear solver with inner iteration GMRES method}
287 \begin{algorithmic}[1]
288 \Input $A_\ell$ (sparse sub-matrix), $B_\ell$ (right-hand side sub-vector)
289 \Output $X_\ell$ (solution sub-vector)\vspace{0.2cm}
290 \State Load $A_\ell$, $B_\ell$
291 \State Set the initial guess $x^0$
292 \State Set the minimizer $\tilde{x}^0=x^0$
293 \For {$k=1,2,3,\ldots$ until the global convergence}
294 \State Restart with $x^0=\tilde{x}^{k-1}$:
295 \For {$j=1,2,\ldots,s$}
296 \State \label{line7}Inner iteration solver: \Call{InnerSolver}{$x^0$, $j$}
297 \State Construct basis $S$: add column vector $X_\ell^j$ to the matrix $S_\ell^k$
298 \State Exchange local values of $X_\ell^j$ with the neighboring blocks
299 \State Compute dense matrix $R$: $R_\ell^{k,j}=\sum^L_{i=1}A_{\ell i}X_i^j$
301 \State \label{line12}Minimization $\|b-R\alpha\|_2$: \Call{UpdateMinimizer}{$S_\ell$, $R$, $b$, $k$}
302 \State Local solution of linear system $Ax=b$: $X_\ell^k=\tilde{X}_\ell^k$
303 \State Exchange the local minimizer $\tilde{X}_\ell^k$ with the neighboring blocks
308 \Function {InnerSolver}{$x^0$, $j$}
309 \State Compute local right-hand side $Y_\ell = B_\ell - \sum^L_{\substack{m=1\\m\neq \ell}}A_{\ell m}X_m^0$
310 \State Solving local splitting $A_{\ell \ell}X_\ell^j=Y_\ell$ using parallel GMRES method, such that $X_\ell^0$ is the initial guess
311 \State \Return $X_\ell^j$
316 \Function {UpdateMinimizer}{$S_\ell$, $R$, $b$, $k$}
317 \State Solving normal equations $(R^k)^TR^k\alpha^k=(R^k)^Tb$ in parallel by $L$ blocks using parallel CGNR method
318 \State Compute local minimizer $\tilde{X}_\ell^k=S_\ell^k\alpha^k$
319 \State \Return $\tilde{X}_\ell^k$
325 The main key points of our Krylov multisplitting method to solve a large sparse linear system are given in Algorithm~\ref{algo:01}. This algorithm is based on a two-stage method with a minimization using restarted GMRES iterative method as an inner solver. It is executed in parallel by each block of processors. Matrices and vectors with the subscript $\ell$ represent the local data for block $\ell$, where $\ell\in\{1,\ldots,L\}$. The two-stage solver uses two different parallel iterative algorithms: the GMRES method to solve each splitting~(\ref{sec03:eq03}) on a block of processors, and the CGNR method, executed periodically in parallel by all blocks to minimize the function error~(\ref{sec03:eq07}) over the Krylov subspace spanned by $S$. The algorithm requires two global synchronizations between $L$ blocks. The first one is performed line~\ref{line12} in Algorithm~\ref{algo:01} to exchange local values of vector solution $x$ (i.e. the minimizer $\tilde{x}$) required to restart the multisplitting solver. The second one is needed to construct the matrix $R$. We chose to perform this latter synchronization $s$ times in every outer iteration $k$ (line~\ref{line7} in Algorithm~\ref{algo:01}). This is a straightforward way to compute the sparse matrix-dense matrix multiplication $R=AS$. We implemented all synchronizations by using message passing collective communications of MPI library.
327 %%%%%%%%%%%%%%%%%%%%%%%%
328 %%%%%%%%%%%%%%%%%%%%%%%%
330 \section{Experiments}
332 %%% MODIFIE ***********************
333 %%%********************************
334 In order to illustrate the interest of our Krylov multisplitting algorithm, we have compared its performances with those of a classical block Jacobi multisplitting method and those of the GMRES method which is a commonly used method in many situations.
335 %%%********************************
336 %%%********************************
337 We have chosen to focus on only one problem which is very simple to implement: a 3 dimension Poisson problem.
342 \nabla u&=f \mbox{~in~} \omega\\
343 u &=0 \mbox{~on~} \Gamma=\partial \omega
348 After discretization, with a finite difference scheme, a seven point stencil is
349 used. It is well-known that the spectral radius of matrices representing such
350 problems are very close to 1. Moreover, the larger the number of discretization
351 points is, the closer to 1 the spectral radius is. Hence, to solve a matrix
352 obtained for a 3D Poisson problem, the number of iterations is high. Using a
353 preconditioner it is possible to reduce the number of iterations but
354 preconditioners are not scalable when using many cores.
357 %%% AJOUTE ************************
358 %%%********************************
359 We have performed some experiments on an infiniband cluster of three Intel Xeon
360 quad-core E5620 CPUs of 2.40 GHz and 12 GB of memory. For the GMRES code (alone
361 and in both multisplitting versions) the restart parameter is fixed to 16. The
362 precision of the GMRES version is fixed to 1e-6. For the multisplitting
363 versions, there are two precisions, one for the external solver which is fixed
364 to 1e-6 and another one for the inner solver (GMRES) which is fixed to 1e-10. It
365 should be noted that a high precision is used but we also fixed a maximum number
366 of iterations for each internal step. In practice, we limit the number of
367 iterations in the internal step to 10. So an internal iteration is finished when
368 the precision is reached or when the maximum internal number of iterations is
369 reached. The precision and the maximum number of iterations of CGNR method used
370 by our Krylov multisplitting algorithm are fixed to 1e-25 and 20
371 respectively. The size of the Krylov subspace basis $S$ is fixed to 10 vectors.
375 \includegraphics[width=0.8\textwidth]{strong_scaling_150x150x150}
376 \caption{Strong scaling with 3 blocks of 4 cores each to solve a 3D Poisson problem of size $150^3$ components}
383 \includegraphics[width=0.8\textwidth]{weak_scaling_280k} \\ \includegraphics[width=0.8\textwidth]{weak_scaling_280K2}\\
385 \caption{Weak scaling with 3 blocks of 4 cores each to solve a 3D Poisson problem with approximately 280K components per core}
389 %%The experiments are performed on 3 different clusters of cores interconnected by an infiniband network (each cluster is a quad-core CPU).
390 Figures~\ref{fig:001} and~\ref{fig:002} show the scalability performances of
391 GMRES, classical multisplitting and Krylov multisplitting methods: strong and
392 weak scaling are presented respectively. We can remark from these figures that
393 the performances of our Krylov multisplitting method are better than those of
394 GMRES and classical multisplitting methods. In the experiments conducted in this
395 work, our method is approximately twice faster than the GMRES method and about 9
396 times faster than the classical multisplitting method. Our multisplitting method
397 uses a minimization step over a Krylov subspace which reduces the number of
398 iterations and accelerates the convergence. We can also remark that the
399 performances of the classical block Jacobi multisplitting method are the worst
400 compared with those of the other two methods. This is why in the following
401 experiments we compare the performances of our Krylov multisplitting method with
402 only those of the GMRES method.
403 %%%********************************
404 %%%********************************
407 %%% MODIFIE ************************
408 %%%*********************************
409 %Doing many experiments with many cores is not easy and requires to access to a supercomputer with several hours for developing a code and then improving it.
410 In the following we present some experiments we could achieve out on the Hector
411 architecture, a UK high-end computing resource, funded by the UK Research
412 Councils~\cite{hector}. This is a Cray XE6 supercomputer, equipped with two
413 16-core AMD Opteron 2.3 GHz and 32 GB of memory. Machines are interconnected
414 with a 3D torus. The different parameters used by the GMRES and the Krylov multisplitting codes are as those previously mentioned.
416 Table~\ref{tab1} shows the result of the experiments. The first column shows
417 the size of the 3D Poisson problem. The size is chosen in order to have
418 approximately 50,000 components per core. The second column represents the
419 number of cores used. Between brackets, one can find the decomposition used for
420 the Krylov multisplitting. The third column and the sixth column respectively
421 show the execution time for the GMRES and the Krylov multisplitting codes. The
422 fourth and the seventh column describe the number of iterations. For the
423 multisplitting code, the total number of inner iterations is represented between
425 %%%********************************
426 %%%********************************
430 \begin{changemargin}{-1.8cm}{0cm}
432 \begin{tabular}{|c|c||c|c|c||c|c|c||c|}
434 \multirow{2}{*}{Pb size}&\multirow{2}{*}{Nb. cores} & \multicolumn{3}{c||}{GMRES} & \multicolumn{3}{c||}{Krylov Multisplitting} & \multirow{2}{*}{Ratio}\\
436 & & Time (s) & nb Iter. & $\Delta$ & Time (s)& nb Iter. & $\Delta$ & \\
438 $468^3$ & 2,048 (2x1,024) & 299.7 & 41,028 & 5.02e-8 & 48.4 & 691(6,146) & 8.24e-08 & 6.19 \\
440 $590^3$ & 4,096 (2x2,048) & 433.1 & 55,494 & 4.92e-7 & 74.1 & 1,101(8,211) & 6.62e-08 & 5.84 \\
442 $743^3$ & 8,192 (2x4,096) & 704.4 & 87,822 & 4.80e-07 & 151.2 & 3,061(14,914) & 5.87e-08 & 4.65 \\
444 $743^3$ & 8,192 (4x2,048) & 704.4 & 87,822 & 4.80e-07 & 110.3 & 1,531(12,721) & 1.47e-07& 6.39 \\
455 From these experiments, it can be observed that the multisplitting version is
456 always faster than the GMRES version. The acceleration gain of the
457 multisplitting version ranges between 4 and 6. It can be noticed that the number of
458 iterations is drastically reduced with the multisplitting version even it is not
459 negligible. Moreover, with 8,192 cores, we can see that using 4 blocks of cores gives a
460 better performance than simply using 2 blocks. In fact, we can notice that the
461 precision with 2 blocks is slightly better but in both cases the precision is
462 under the specified threshold.
465 %%% AJOUTE************************
466 %%%*******************************
467 In Figure~\ref{fig:01}, the number of iterations per second is reported for both
468 GMRES and the multisplitting methods. It should be noted that we took only the
469 inner number of iterations (i.e. the GMRES iterations) for the multisplitting
470 method. Iterations of CGNR are not taken into account. From this figure, it can
471 be seen that the number of iterations per second is higher with GMRES but it is
472 not so different with the multisplitting method. For the case with $8,192$
473 cores, the number of iterations per second with 4 blocks is approximately equal
474 to 115. So it is not different from GMRES.
478 \includegraphics[width=0.7\textwidth]{nb_iter_sec}
479 \caption{Number of iterations per second with the same parameters as in Table~\ref{tab1} (weak scaling) with only blocks of cores}
483 \noindent {\bf Final remarks:}\\ It should be noted, on the one hand, that the
484 development of a complete new code usable with any kind of problem is a really
485 long and fastidious task if one is working from scratch. On the other hand,
486 using an existing tool for the inner solver is also quite difficult because it
487 requires to establish a link between the inner solver and the outer one. We
488 plan to do that later with engineers working specifically on that point.
489 Moreover, we think that it is very important to analyze the convergence of this
490 method compared to other methods. In this work, we have focused on the
491 description of this method and its performances with a typical application. Many
492 other investigations are required for this method as explained in the next
494 %%%*******************************
495 %%%*******************************
497 \section{Conclusion and perspectives}
498 We have implemented a Krylov multisplitting method to solve sparse linear
499 systems on large-scale computing platforms. We have developed a synchronous
500 two-stage method based on the block Jacobi multisplitting which uses GMRES
501 iterative method as an inner iteration. Our contribution in this paper is
502 twofold. First we provide a multi block decomposition that allows us to choose
503 the appropriate size of the blocks according to the architecures of the
504 supercomputer. Second, we have implemented the outer iteration of the
505 multisplitting method as a Krylov subspace method which minimizes some error
506 function. This increases the convergence and improves the scalability of the
507 multisplitting method.
509 We have tested our multisplitting method to solve the sparse linear system
510 issued from the discretization of a 3D Poisson problem. We have compared its
511 performances to the classical GMRES method on a supercomputer composed of 2,048
512 up-to 8,192 cores. The experimental results showed that the multisplitting method is
513 about 4 to 6 times faster than the GMRES method for different sizes of the
514 problem split into 2 or 4 blocks when using the multisplitting method. Indeed, the
515 GMRES method has difficulties to scale with many cores while the Krylov
516 multisplitting method allows to hide latency and reduce the inter-block
519 In future works, we plan to conduct experiments on larger numbers of cores and
520 test the scalability of our Krylov multisplitting method. It would be
521 interesting to validate its performances to solve other linear/nonlinear and
522 symmetric/nonsymmetric problems. Moreover, we intend to develop multisplitting
523 methods based on asynchronous iterations in which communications are overlapped
524 by computations. These methods would be interesting for platforms composed of
525 distant clusters interconnected by a high-latency network. In addition, we
526 intend to investigate the convergence improvements of our method by using
527 preconditioning techniques for Krylov iterative methods and multisplitting
528 methods with overlapping blocks.
530 \section{Acknowledgement}
531 The authors would like to thank Mark Bull of the EPCC his fruitful remarks and the facilities of HECToR. This work has been partially supported by the Labex
532 ACTION project (contract “ANR-11-LABX-01-01”).
535 %Other applications (=> other matrices)\\
536 %Larger experiments\\
542 %%%%%%%%%%%%%%%%%%%%%%%%
543 %%%%%%%%%%%%%%%%%%%%%%%%
545 \bibliographystyle{plain}
546 \bibliography{biblio}
