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4. In Section 3. it is better if the paper can explain the intuition of multi-splitting. Currently it is more like "Here is what I did" presentation but "why do we do this" is left for the reader to guess.
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-The iterative algorithms suffer from the scalability problem on large computing platforms due to the large amount of communications and synchronisations. In this context, the multisplitting methods are well-known to be more adapted to large-scale clusters of processors. The main principle of the multispliting methods is to split the large problem to solve in different blocks in such a way each block can be solved by a processor or a set of processors and thus to minimize by this way the synchronizations over the large cluster. However they suffer from slow convergence. In fact, the larger the number of splitting is, the larger the spectral radius is, thereby slowing the convergence of the multisplitting algorithm.
+The iterative algorithms suffer from the scalability problem on large computing platforms due to the large amount of communications and synchronizations. In this context, the multisplitting methods are well-known to be more adapted to large-scale clusters of processors. The main principle of the multisplitting methods is to split the large problem to solve in different blocks in such a way each block can be solved by a processor or a set of processors and thus to minimize by this way the synchronizations over the large cluster. However these methods suffer from slow convergence. In fact, the larger the number of splitting is, the larger the spectral radius is, thereby slowing the convergence of the multisplitting algorithm.
-We have used the parallel algorithm of the well-known GMRES method to solve locally each block by a set of processors. In addition we have also implemented the outer iteration as a Krylov subspace iteration minimizing some error function which allows to accelerate the global convergence of the multisplitting algorithm.
+We have used the well-known GMRES method to solve locally in parallel each block by a set of processors. In addition we have also implemented the outer iteration as a Krylov subspace iteration minimizing some error function which allows to accelerate the global convergence of the multisplitting algorithm.
The main principle of the multisplitting methods is defined in Section 2. Section 3 presenting our two-stage algorithm is little modified to show our motivations to mix between the multisplitting methods and Krylov iterative methods.
