The following paragraph is added in the introduction:
In this work we develop a new parallel two-stage algorithm for large-scale clusters. Our objective is to mix between Krylov based iterative methods and the multisplitting method to improve the scalability. In fact Krylov subspace methods are well-known for their good convergence compared to others iterative methods. So our main contribution is to use the multisplitting method which splits the problem to solve into different blocks in order to reduce the large amount of communications and, to implement both inner and outer iterations as Krylov subspace iterations improving the convergence of the multisplitting algorithm.
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2. Given that the focus of the paper is to provide a better solution on a well known problem with several well studied approaches. It is essential for the author to provide extensive comparison studies with those approaches. In Section 4 the paper provides some experiments with very limited scope (solving one simple problem and comparing with one well known problems). This seems not enough. Another way is to provide a qualitative comparison against other proposed approaches and explain why the proposed approach is better. But this is also not found.
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+tentative de réponse : In fact, the machine we have used, almost one year ago, is not accessible anymore, it is been reformed. In this paper, we show that, for a very well known problem, the 3D Poisson problem that is used in many simulations, our method is more efficient than the GMRES method which is a very well known method.
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3. It is better if the paper can provide a quantitative study on the speed-up achieved by the proposed algorithm so that the reader can get insights on how much is the performance improvement in theory.
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+With all numerical methods, the converge is a very difficult problem. In this study, we show that a very simple method can provide faster result than the GMRES method. Of course, many theoretical works need to be added, but it take a very long time and this is out of the scope of this paper.
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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.
+RAPH : on peut modifier des trucs pour répondre dans le papier? ca serait bien :-)
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* Reviewer #3 *
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ii) The authors supposed a good scalability of the new algorithm, but the experiment's proof seems not enough, as it just gave the weak scalability comparison, which just could lead to a conclusion of improved execution time, while a strong scalability curve might be more persuasive.
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-Some figures showing the strong scalability of the two tested algorithms are added in Section 4.
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+As said previously, the machine we have used is reformed and currently we have no access to make other large scale tests. In fact, we consider that GMRES is quite scalable because it is used by many other researchers and tools. So we have compared our method with it using weak scalability.
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iii) In the last line on the page 7, there is apparent error "multi-saplitting".
