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\title{A scalable multisplitting algorithm to solve large sparse linear systems}     
\author{}        
\date{}

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\section*{Reviewer \#2:} 
\begin{enumerate}
\item ``It is better to clearly state the major contributions of this paper in the introduction.''

\medskip
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 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.

\item ``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.''

\medskip
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. 

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. 


\item ``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.''

\medskip
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.

\item ``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.''

\medskip
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 splittings is, the larger the spectral radius is, thereby slowing the 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.
\end{enumerate}

\section*{Reviewer \#3:} 
\begin{enumerate}
\item ``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?''

\medskip
A paragraph is added in the introduction to show our main contribution of this work. 

\item ``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.''

\medskip
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.

We have added some experiments performed on a small cluster which compare our method to the GMRES method and the classical block Jacobi multisplitting method. 

\item ``In the last line on the page 7, there is apparent error "multi-saplitting".''

\medskip
The error is corrected.
\end{enumerate}

\section*{Reviewer \#5:} 
\begin{enumerate}
\item ``However, the paper does not take into considerate account relevant current and past research on the topic.''

\medskip
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 those of the GMRES method which is a very used method. In addition, the machine we have used is not accessible anymore, it has been reformed.
\end{enumerate}

\section*{Reviewer \#6:} 
\begin{enumerate}
\item ``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.''

\medskip
The experiments in Section 4 show a comparison between the performances of our Krylov multisplitting algorithm and those of the 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.   

\item ``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.''

\medskip
Section 4 has been rewritten in order to explain our choice 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.

\item ``Was the method of reference [9] implemented by the authors of [9]? How did they do against GMRES?''

\medskip
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 simulations of numerical examples on a sequential computer. 
\end{enumerate}
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