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\title{A scalable multisplitting algorithm for solving large sparse linear systems}
\date{}
proposed in~\cite{huang1993krylov}, the use of a minimization process can
drastically improve the convergence.
-\LZK[]{Suite\dots}
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convergence of the global system.
In~\cite{couturier2008gremlins}, the authors proposed practical implementations
-of multisplitting algorithms that take benefit from multisplitting algorithms\LZK[]{répétition ???} to
+of multisplitting algorithms that take benefit from multisplitting algorithms to
solve large scale linear systems. Inner solvers could be based on scalar direct
-method with the LU method or scalar iterative one with GMRES.\LZK[]{lu et gmres par exemple}
+method with the LU method or scalar iterative one with GMRES.
In~\cite{prace-multi}, the authors have proposed a parallel multisplitting
algorithm in which large blocks are solved using a GMRES solver. The authors have
asynchronous multisplitting algorithm could be more efficient than traditional
solvers on exascale architecture with hundreds of thousands of cores.
-\LZK[]{Peut-être autres related works\ldots}\\
-
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
\begin{equation}
A = M_l - N_l,
limit the number of internal step to 10. So an internal iteration is finished
when the precision is reached or when the maximum internal number of iterations
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.
-\LZK{J'ai ajouté les paramètres concernant la résolution du problème de moindres carrés. Confirmer leur valeurs.}
\section{Conclusion and perspectives}
We have implemented a Krylov multisplitting method to solve sparse linear systems on large-scale computing platforms. We have developed a synchronous two-stage method based on the block Jacobi multisplitting and uses GMRES iterative method as an inner iteration. Our contribution in this paper is twofold. First we have constituted a multi-cluster environment based on processors of the large-scale computing platform on which each linear sub-system issued from the splitting is solved in parallel by a cluster of processors. Second, we have implemented the outer iteration of the multisplitting method as a Krylov subspace method which minimizes some error function. This increases the convergence and improves the scalability of the multisplitting method.
-We have tested our multisplitting method for solving the sparse linear system issued from the discretization of the 3D Poisson problem. We have compared its performances to GMRES method on a supercomputer composed of 2048 to 8192 cores. The experimental results showed that the multisplitting method is about 4 to 6 times faster than the GMRES method for different sizes of the problem split into 2 or 4 blocks when using the multisplitting method. Indeed, the GMRES method has difficulties to scale with many cores while the Krylov multisplitting method allows to hide latency and reduce the inter-cluster communications.
+We have tested our multisplitting method for solving the sparse linear system issued from the discretization of the 3D Poisson problem. We have compared its performances to those of GMRES method on a supercomputer composed of 2048 to 8192 cores. The experimental results showed that the multisplitting method is about 4 to 6 times faster than the GMRES method for different sizes of the problem split into 2 or 4 blocks when using multisplitting method. Indeed, the GMRES method has difficulties to scale with many cores while the Krylov multisplitting method allows to hide latency and reduce the inter-cluster communications.
-In future works, we plan to conduct experiments on larger number of cores and test the scalability of our Krylov multisplitting method. It would be interesting to validate its performances for solving other linear/nonlinear and symmetric/nonsymmetric problems. Moreover, we intend to develop multisplitting methods based on asynchronous iteration in which communications are overlapped by computations. These methods would be interesting for platforms composed of distant clusters interconnected by a high-latency network. In addition, we intend to investigate the convergence improvements by using preconditioning techniques and multisplitting methods with overlapping blocks.
+In future works, we plan to conduct experiments on larger number of cores and test the scalability of our Krylov multisplitting method. It would be interesting to validate its performances for solving other linear/nonlinear and symmetric/nonsymmetric problems. Moreover, we intend to develop multisplitting methods based on asynchronous iteration in which communications are overlapped by computations. These methods would be interesting for platforms composed of distant clusters interconnected by a high-latency network. In addition, we intend to investigate the convergence improvements of our method by using preconditioning techniques for Krylov iterative methods and multisplitting methods with overlapping blocks.
%Other applications (=> other matrices)\\
