1 Dear raphael Couturier,
3 We have received the reports from our advisors on your manuscript, "A scalable multisplitting algorithm to solve large sparse linear systems", which you submitted to Journal of Supercomputing.
5 Based on the advice received, the Editor feels that your manuscript could be accepted for publication should you be prepared to incorporate minor revisions. When preparing your revised manuscript, you are asked to carefully consider the reviewer comments which are attached, and submit a list of responses to the comments. Your list of responses should be uploaded as a file in addition to your revised manuscript.
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20 Springer Journals Editorial Office
21 Journal of Supercomputing
25 COMMENTS FOR THE AUTHOR:
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30 Reviewer #2: This work focus on an better algorithm that solves very large sparse linear systems. The presentation of this paper is clear and within the scope of the journal. However, the paper can be improved in the following ways:
32 |1. It is better to clearly state the major contributions of this paper in the introduction.
34 The following paragraph is added in the introduction:
35 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 other 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.
39 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.
41 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.
43 We have added some experimental results obtained on a small cluster comparing the performances of our Krylov multisplitting method with those of the well-known block Jacobi multisplitting method and the GMRES method. These experiments clearly show that our method is better than the other two methods and the classical multisplitting method is the worst one. For this reason in the rest of the work we have compared the performances of our method only to those of the GMRES method.
47 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.
49 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.
53 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.
55 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 that 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.
57 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.
59 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.
62 RAPH : on peut modifier des trucs pour répondre dans le papier? ca serait bien :-)
63 Lilia: J'ai un peu modifié la section 3, mais on peut toujours l'améliorer. As-tu d'autres idées à ajouter, Mr...? :)
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71 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.
73 There are three problems/questions the reviewer is concerned with:
75 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?
77 A paragraph is added in the introduction to show our main contribution of this work.
81 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.
83 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.
85 As said previously, we have added some experiments performed on a small cluster comparing our method to the GMRES method and the classical block Jacobi multisplitting method.
89 iii) In the last line on the page 7, there is apparent error "multi-saplitting".
91 The error is corrected.
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99 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.
101 However, the paper does not take into considerate account relevant current and past research on the topic.
103 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. This is why in our work we have focused on experiments to solve one well-known sparse linear equations which is the 3D Poisson problem and to compare the performances of our Krylov multisplitting method to the GMRES method which is a very used method. In addition, the machine we have used is not accessible anymore, it has been reformed.
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111 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].
113 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.
115 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.
119 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.
121 Section 4 has been rewritten in order to explain our choice and to compare our Krylov multisplitting method with only the GMRES method. We have added in the paper some experimental results obtained on a small cluster which clearly show that our method is more efficient than GMRES and block Jacobi multisplitting methods.
125 Was the method of reference [9] implemented by the authors of [9]? How did they do against GMRES?
127 As explained in the paper, 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.
