GMRES and the multisplitting methods. It should be noted that we took only the
inner number of iterations (i.e. the GMRES iterations) for the multisplitting
method. Iterations of CGNR are not taken into account. From this figure, it can
-be seen that the number of iteration per second is higher with GMRES but it is
+be seen that the number of iterations per second is higher with GMRES but it is
not so different with the multisplitting method. For the case with $8,192$
cores, the number of iterations per second with 4 clusters is approximately
equals to 115. So it is not different from GMRES.
analyze the convergence of this method compared to other method. In this work,
we have focused on the description of this method and its performance with a
typical application. Many other investigations are required for this method as explained in the next section.
-
-
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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.
,----
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.
,----
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.
-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".
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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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