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@@ -324,9 +324,9 @@ iterations is drastically reduced with  the multisplitting version even it is no
 neglectable.
 
 \section{Conclusion and perspectives}
 neglectable.
 
 \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 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 using Krylov subspace method which minimizes some error function. This increases the convergence and improves the scalability of the multisplitting method.
+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 virtual multi-cluster environment based on processors of the 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 a 3D Poisson problem. We have compared its performances to those of classical 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.
+We have tested our multisplitting method for solving the sparse linear system issued from the discretization of a 3D Poisson problem. We have compared its performances to the classical 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 of our method by using preconditioning techniques for Krylov iterative methods 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.