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|1. It is better to clearly state the major contributions of this paper in the introduction.
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-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 sub-problems 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.
+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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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 multisplitting methods are well known to be more adapted to large-scale clusters of processors by minimizing the synchronizations but 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. We have used the parallel algorithm of the well known GMRES method to solve locally each block. In addition we have also implemented the outer iteration as a Krylov subspace iteration minimizing some error function which allows to improve the global 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 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.
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+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.
+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.
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Reviewer #3: In this paper the authors proposed a practical multi-splitting method based on parallel iterative blocks which gives better results than classical GMRES method for the 3D Poisson problem. The paper is well-organized, written smoothly, and provide solid theoretical analysis, detailed algorithm presentation and concrete experiment results.
-There are three problems/questions the reviewer is concerned with: i) what is the main contribution of this paper, i.e. the key advantage of the new algorithm compared to other multi-splitting methods, why not provide some experiments for comparison between them, rather than with only the classical GMRES?
+There are three problems/questions the reviewer is concerned with:
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+i) what is the main contribution of this paper, i.e. the key advantage of the new algorithm compared to other multi-splitting methods, why not provide some experiments for comparison between them, rather than with only the classical GMRES?
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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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iii) In the last line on the page 7, there is apparent error "multi-saplitting".
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+The error is corrected.
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* Reviewer #5 *
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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. However, the paper does not take into considerate account relevant current and past research on the topic.
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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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* Reviewer #6 *
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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]. 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.
+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.
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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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