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23 \title{A scalable multisplitting algorithm to solve large sparse linear systems}
29 \author{Raphaël Couturier \and Lilia Ziane Khodja}
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38 In this paper we revisit the Krylov multisplitting algorithm presented in
39 \cite{huang1993krylov} which uses a sequential method to minimize the Krylov
40 iterations computed by a multisplitting algorithm. Our new algorithm is based on
41 a parallel multisplitting algorithm with few blocks of large size using a
42 parallel GMRES method inside each block and on a parallel Krylov minimization in
43 order to improve the convergence. Some large scale experiments with a 3D Poisson
44 problem are presented with up to 8,192 cores. They show the obtained
45 improvements compared to a classical GMRES both in terms of number of iterations
46 and in terms of execution times.
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52 \section{Introduction}
53 Iterative methods are used to solve large sparse linear systems of equations of
54 the form $Ax=b$ because they are easier to parallelize than direct ones. Many
55 iterative methods have been proposed and adapted by different researchers. For
56 example, the GMRES method and the Conjugate Gradient method are very well known
57 and used~\cite{S96}. Both methods are based on the
58 Krylov subspace which consists in forming a basis of a sequence of successive
59 matrix powers times the initial residual.
61 When solving large linear systems with many cores, iterative methods often
62 suffer from scalability problems. This is due to their need for collective
63 communications to perform matrix-vector products and reduction operations.
64 Preconditioners can be used in order to increase the convergence of iterative
65 solvers. However, most of the good preconditioners are not scalable when
66 thousands of cores are used.
68 %Traditional iterative solvers have global synchronizations that penalize the
69 %scalability. Two possible solutions consists either in using asynchronous
70 %iterative methods~\cite{ref18} or to use multisplitting algorithms. In this
71 %paper, we will reconsider the use of a multisplitting method. In opposition to
72 %traditional multisplitting method that suffer from slow convergence, as
73 %proposed in~\cite{huang1993krylov}, the use of a minimization process can
74 %drastically improve the convergence.
76 Traditional parallel iterative solvers are based on fine-grain computations that
77 frequently require data exchanges between computing nodes and have global
78 synchronizations that penalize the scalability. Particularly, they are more
79 penalized on large scale architectures or on distributed platforms composed of
80 distant clusters interconnected by a high-latency network. It is therefore
81 imperative to develop coarse-grain based algorithms to reduce the communications
82 in the parallel iterative solvers. Two possible solutions consists either in
83 using asynchronous iterative methods~\cite{ref18} or in using multisplitting
84 algorithmss. In this paper, we will reconsider the use of a multisplitting
85 method. In opposition to traditional multisplitting method that suffer from slow
86 convergence, as proposed in~\cite{huang1993krylov}, the use of a minimization
87 process can drastically improve the convergence.
89 The present paper is organized as follows. First, Section~\ref{sec:02} presents
90 some related works and the principle of multisplitting methods. Then, in
91 Section~\ref{sec:03} the algorithm of our Krylov multisplitting
92 method, based on inner-outer iterations, is presented. Finally, in Section~\ref{sec:04}, the
93 parallel experiments on Hector architecture show the performances of the Krylov
94 multisplitting algorithm compared to the classical GMRES algorithm to solve a 3D
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101 \section{Related works and presentation of the multisplitting method}
103 A general framework to study parallel multisplitting methods has been presented in~\cite{o1985multi}
104 by O'Leary and White. Convergence conditions are given for the
105 most general cases. Many authors have improved multisplitting algorithms by proposing,
106 for example, an asynchronous version~\cite{bru1995parallel} or convergence
107 conditions~\cite{bai1999block,bahi2000asynchronous} or other
108 two-stage algorithms~\cite{frommer1992h,bru1995parallel}.
110 In~\cite{huang1993krylov}, the authors have proposed a parallel multisplitting
111 algorithm in which all the tasks except one are devoted to solve a sub-block of
112 the splitting and to send their local solutions to the first task which is in
113 charge of combining the vectors at each iteration. These vectors form a Krylov
114 basis for which the first task minimizes the error function over the basis to
115 increase the convergence, then the other tasks receive the updated solution until the
116 convergence of the global system.
118 In~\cite{couturier2008gremlins}, the authors have developed practical implementations
119 of multisplitting algorithms to solve large scale linear systems. Inner solvers
120 could be based on sequential direct method with the LU method or sequential iterative
123 In~\cite{prace-multi}, the authors have designed a parallel multisplitting
124 algorithm in which large blocks are solved using a GMRES solver. The authors have
125 performed large scale experiments up-to 32,768 cores and they conclude that
126 an asynchronous multisplitting algorithm could be more efficient than traditional
127 solvers on an exascale architecture with hundreds of thousands of cores.
129 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.
132 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
137 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
139 x^{k+1}=\displaystyle\sum^L_{\ell=1} E_\ell M^{-1}_\ell (N_\ell x^k + b),~k=1,2,3,\ldots
142 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
144 \rho(\displaystyle\sum^L_{\ell=1}E_\ell M^{-1}_\ell N_\ell)<1.
147 where $\rho$ is the spectral radius of the square matrix.
149 The advantage of the multisplitting method is that at each iteration $k$ there are $L$ different linear sub-systems
151 v_\ell^k=M^{-1}_\ell N_\ell x_\ell^{k-1} + M^{-1}_\ell b,~\ell\in\{1,\ldots,L\},
154 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
156 x^k = \displaystyle\sum^L_{\ell=1} E_\ell v_\ell^k,
159 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.
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164 \section{A two-stage method with a minimization}
166 Let $Ax=b$ be a given large and sparse linear system of $n$ equations to solve in parallel on $L$ clusters of processors, physically adjacent or geographically distant, where $A\in\mathbb{R}^{n\times n}$ is a 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. The multisplitting of this linear system is defined as follows
170 A & = & [A_{1}, \ldots, A_{L}]\\
171 x & = & [X_{1}, \ldots, X_{L}]\\
172 b & = & [B_{1}, \ldots, B_{L}]
177 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 $\sum_\ell n_\ell=n$. In this work, we use a row-by-row splitting without overlapping in such a way that successive rows of sparse matrix $A$ and both vectors $x$ and $b$ are assigned to one cluster. So, the multisplitting format of the linear system is defined as follows
179 \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,
182 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 $m\in\{1,\ldots,L\}$.
184 Our multisplitting method proceeds by iteration to solve the linear system in such a way that each sub-system
188 A_{\ell \ell}X_\ell = Y_\ell \mbox{,~such that}\\
189 Y_\ell = B_\ell - \displaystyle\sum_{\substack{m=1\\m\neq \ell}}^{L}A_{\ell m}X_m,
194 is solved independently by a {\it cluster 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 clusters. In this work, we use the parallel restarted GMRES method~\cite{ref34} as an inner iteration method to solve sub-systems~(\ref{sec03:eq03}). GMRES is one of the most used Krylov iterative methods to solve sparse linear systems. %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.
196 It should be noted that the convergence of the inner iterative solver for the
197 different sub-systems~(\ref{sec03:eq03}) does not necessarily involve the
198 convergence of the multisplitting method. It strongly depends on the properties
199 of the global sparse linear system to be
200 solved~\cite{o1985multi,ref18}. Furthermore, the splitting of the linear system
201 among several clusters of processors increases the spectral radius of the
202 iteration matrix, thereby slowing the convergence. In fact, the larger the
203 number of splitting is, the larger the spectral radius is. In this paper, our
204 work is based on the work presented in~\cite{huang1993krylov} to increase the
205 convergence and improve the scalability of the multisplitting methods.
207 In order to accelerate the convergence, we implemented the outer iteration of the 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})
209 S=\{x^1,x^2,\ldots,x^s\},~s\leq n,
212 where for $j\in\{1,\ldots,s\}$, $x^j=[X_1^j,\ldots,X_L^j]$ is a solution of the global linear system. The advantage of such a Krylov subspace is that we neither need an orthogonal basis nor any synchronization between clusters to generate this basis.
214 The multisplitting method is periodically restarted every $s$ iterations with a new initial guess $\tilde{x}=S\alpha$ which minimizes the error function $\|b-Ax\|_2$ over the Krylov subspace spanned by vectors of $S$. So $\alpha$ is defined as the solution of the large overdetermined linear system
219 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
224 which is associated with the least squares problem
226 \text{minimize}~\|b-R\alpha\|_2,
229 where $R^T$ denotes the transpose of matrix $R$. Since $R$ (i.e. $AS$) and $b$ are split among $L$ clusters, 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}.
231 \begin{algorithm}[!t]
232 \caption{A two-stage linear solver with inner iteration GMRES method}
233 \begin{algorithmic}[1]
234 \Input $A_\ell$ (sparse sub-matrix), $B_\ell$ (right-hand side sub-vector)
235 \Output $X_\ell$ (solution sub-vector)\vspace{0.2cm}
236 \State Load $A_\ell$, $B_\ell$
237 \State Set the initial guess $x^0$
238 \State Set the minimizer $\tilde{x}^0=x^0$
239 \For {$k=1,2,3,\ldots$ until the global convergence}
240 \State Restart with $x^0=\tilde{x}^{k-1}$:
241 \For {$j=1,2,\ldots,s$}
242 \State \label{line7}Inner iteration solver: \Call{InnerSolver}{$x^0$, $j$}
243 \State Construct basis $S$: add column vector $X_\ell^j$ to the matrix $S_\ell^k$
244 \State Exchange local values of $X_\ell^j$ with the neighboring clusters
245 \State Compute dense matrix $R$: $R_\ell^{k,j}=\sum^L_{i=1}A_{\ell i}X_i^j$
247 \State \label{line12}Minimization $\|b-R\alpha\|_2$: \Call{UpdateMinimizer}{$S_\ell$, $R$, $b$, $k$}
248 \State Local solution of linear system $Ax=b$: $X_\ell^k=\tilde{X}_\ell^k$
249 \State Exchange the local minimizer $\tilde{X}_\ell^k$ with the neighboring clusters
254 \Function {InnerSolver}{$x^0$, $j$}
255 \State Compute local right-hand side $Y_\ell = B_\ell - \sum^L_{\substack{m=1\\m\neq \ell}}A_{\ell m}X_m^0$
256 \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
257 \State \Return $X_\ell^j$
262 \Function {UpdateMinimizer}{$S_\ell$, $R$, $b$, $k$}
263 \State Solving normal equations $(R^k)^TR^k\alpha^k=(R^k)^Tb$ in parallel by $L$ clusters using parallel CGNR method
264 \State Compute local minimizer $\tilde{X}_\ell^k=S_\ell^k\alpha^k$
265 \State \Return $\tilde{X}_\ell^k$
271 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 cluster of processors. Matrices and vectors with the subscript $\ell$ represent the local data for cluster $\ell$, 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 cluster of processors, and the CGNR method, executed in parallel by all clusters, to minimize the function error~(\ref{sec03:eq07}) over the Krylov subspace spanned by $S$. The algorithm requires two global synchronizations between $L$ clusters. 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.
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276 \section{Experiments}
278 In order to illustrate the interest of our algorithm, we have compared our
279 algorithm with the GMRES method which is a commonly used method in many
280 situations. We have chosen to focus on only one problem which is very simple to
281 implement: a 3 dimension Poisson problem.
286 \nabla u&=f \mbox{~in~} \omega\\
287 u &=0 \mbox{~on~} \Gamma=\partial \omega
292 After discretization, with a finite difference scheme, a seven point stencil is
293 used. It is well-known that the spectral radius of matrices representing such
294 problems are very close to 1. Moreover, the larger the number of discretization
295 points is, the closer to 1 the spectral radius is. Hence, to solve a matrix
296 obtained for a 3D Poisson problem, the number of iterations is high. Using a
297 preconditioner it is possible to reduce the number of iterations but
298 preconditioners are not scalable when using many cores.
300 %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.
301 In the following we present some experiments we could achieve out on the Hector
302 architecture, a UK's high-end computing resource, funded by the UK Research
303 Councils~\cite{hector}. This is a Cray XE6 supercomputer, equipped with two
304 16-core AMD Opteron 2.3 Ghz and 32 GB of memory. Machines are interconnected
307 Table~\ref{tab1} shows the result of the experiments. The first column shows
308 the size of the 3D Poisson problem. The size is chosen in order to have
309 approximately 50,000 components per core. The second column represents the
310 number of cores used. In brackets, one can find the decomposition used for the
311 Krylov multisplitting. The third column and the sixth column respectively show
312 the execution time for the GMRES and the Krylov multisplitting codes. The fourth
313 and the seventh column describe the number of iterations. For the
314 multisplitting code, the total number of inner iterations is represented in
315 brackets. For the GMRES code (alone and in the multisplitting version) the
316 restart parameter is fixed to 16. The precision of the GMRES version is fixed to
317 1e-6. For the multisplitting, there are two precisions, one for the external
318 solver which is fixed to 1e-6 and another one for the inner solver (GMRES) which
319 is fixed to 1e-10. It should be noted that a high precision is used but we also
320 fixed a maximum number of iterations for each internal step. In practice, we
321 limit the number of iterations in the internal step to 10. So an internal iteration is finished
322 when the precision is reached or when the maximum internal number of iterations
323 is reached. The precision and the maximum number of iterations of CGNR method are fixed to 1e-25 and 20 respectively. The size of the Krylov subspace basis $S$ is fixed to 10 vectors.
327 \begin{changemargin}{-1.8cm}{0cm}
329 \begin{tabular}{|c|c||c|c|c||c|c|c||c|}
331 \multirow{2}{*}{Pb size}&\multirow{2}{*}{Nb. cores} & \multicolumn{3}{c||}{GMRES} & \multicolumn{3}{c||}{Krylov Multisplitting} & \multirow{2}{*}{Ratio}\\
333 & & Time (s) & nb Iter. & $\Delta$ & Time (s)& nb Iter. & $\Delta$ & \\
335 $468^3$ & 2,048 (2x1,024) & 299.7 & 41,028 & 5.02e-8 & 48.4 & 691(6,146) & 8.24e-08 & 6.19 \\
337 $590^3$ & 4,096 (2x2,048) & 433.1 & 55,494 & 4.92e-7 & 74.1 & 1,101(8,211) & 6.62e-08 & 5.84 \\
339 $743^3$ & 8,192 (2x4,096) & 704.4 & 87,822 & 4.80e-07 & 151.2 & 3,061(14,914) & 5.87e-08 & 4.65 \\
341 $743^3$ & 8,192 (4x2,048) & 704.4 & 87,822 & 4.80e-07 & 110.3 & 1,531(12,721) & 1.47e-07& 6.39 \\
353 From these experiments, it can be observed that the multisplitting version is
354 always faster than the GMRES version. The acceleration gain of the
355 multisplitting version ranges between 4 and 6. It can be noticed that the number of
356 iterations is drastically reduced with the multisplitting version even it is not
357 negligible. Moreover, with 8,192 cores, we can see that using 4 clusters gives a
358 better performance than simply using 2 clusters. In fact, we can notice that the
359 precision with 2 clusters is slightly better but in both cases the precision is
360 under the specified threshold.
362 \section{Conclusion and perspectives}
363 We have implemented a Krylov multisplitting method to solve sparse linear
364 systems on large-scale computing platforms. We have developed a synchronous
365 two-stage method based on the block Jacobi multisaplitting which uses GMRES
366 iterative method as an inner iteration. Our contribution in this paper is
367 twofold. First we provide a multi cluster decomposition that allows us to choose
368 the appropriate size of the clusters according to the architecures of the
369 supercomputer. Second, we have implemented the outer iteration of the
370 multisplitting method as a Krylov subspace method which minimizes some error
371 function. This increases the convergence and improves the scalability of the
372 multisplitting method.
374 We have tested our multisplitting method to solve the sparse linear system
375 issued from the discretization of a 3D Poisson problem. We have compared its
376 performances to the classical GMRES method on a supercomputer composed of 2,048
377 to 8,192 cores. The experimental results showed that the multisplitting method is
378 about 4 to 6 times faster than the GMRES method for different sizes of the
379 problem split into 2 or 4 blocks when using the multisplitting method. Indeed, the
380 GMRES method has difficulties to scale with many cores while the Krylov
381 multisplitting method allows to hide latency and reduce the inter-cluster
384 In future works, we plan to conduct experiments on larger numbers of cores and
385 test the scalability of our Krylov multisplitting method. It would be
386 interesting to validate its performances to solve other linear/nonlinear and
387 symmetric/nonsymmetric problems. Moreover, we intend to develop multisplitting
388 methods based on asynchronous iterations in which communications are overlapped
389 by computations. These methods would be interesting for platforms composed of
390 distant clusters interconnected by a high-latency network. In addition, we
391 intend to investigate the convergence improvements of our method by using
392 preconditioning techniques for Krylov iterative methods and multisplitting
393 methods with overlapping blocks.
395 \section{Acknowledgement}
396 The authors would like to thank Mark Bull of the EPCC his fruitful remarks and the facilities of HECToR.
398 %Other applications (=> other matrices)\\
399 %Larger experiments\\
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