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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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+In fact, the machine we have used, almost one year ago, is not accessible anymore, it has 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.
-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 convergence 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 takes a very long time and this is out of the scope of this paper.
-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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A paragraph is added in the introduction to show our main contribution of this work.
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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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+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 its good performances have been proven in many research works and it is used by many other researchers and tools. So we have compared our multisplitting method with it by using weak scaling which allows to have broadly a constant amount of computations on each core.
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Reviewer #5: In this paper, the authors have implemented a Krylov multisplitting method to solve sparse linear systems on large-scale computing platforms. The technical approach and analysis of this paper is reasonable and the paper is clear, logical, and understandable.
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However, the paper does not take into considerate account relevant current and past research on the topic.
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+%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.-
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Reviewer #6: In this paper it says that the Krylov GMRES method is compared with a new parallel muti-splitting method of the authors. The paper also says that this new method is an adaptation of another method based on references [11] and [9].
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-It is unclear from the paper whether the analysis includes the a comparison of their new method to the method of reference [9]. Does the new method do better than that one or is it similar or worse.
+It is unclear from the paper whether the analysis includes the a comparison of their new method to the method of reference [9]. Does the new method do better than that one or is it similar or worse.
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+The experiments in Section 4 show a comparison between the performances of our Krylov multisplitting algorithm and those of GMRES method. As said previously, we consider that GMRES is one of the most used method to solve large-scale sparse linear systems. The method of reference [9] is semi-parallel. In fact the task of the minimization is decoupled from the resolution of the different splittings, such as we could fall on a situation where the minimization cannot be performed until all splittings are solved. In addition, the minimization task of reference [9] is performed in sequential.
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The paper should be rewritten to clearly explain what is being compared. It seems as if the method in [9] is not included in the comparison.
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Was the method of reference [9] implemented by the authors of [9]? How did they do against GMRES?
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+Authors of [9] have not implemented the method of reference [9]. They have mainly focused on the convergence analysis of various forms of the algorithm [9] and presented results of numerical examples on a sequential computer.
