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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  create a  mix between  Krylov  based iterative
methods and  the multisplitting method  to improve 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 in order to  improve  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 used, almost one year ago, is not accessible anymore, it has been replaced and we do not have access to the new one. 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 compare 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 results 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
Iterative  algorithms  suffer  from  scalability  problems  on  large  computing
platforms due  to the  large amount of  communications and  synchronizations. In
this  context, multisplitting  methods  are  well-known to  be  more adapted  to
large-scale clusters of processors.   The main feature of multisplitting methods
is to split large problems 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
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 multisplitting methods is defined in Section 2. Section 3, presenting our two-stage algorithm, has been slightly  modified to show our motivations to create a mix between 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 in 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 no longer available 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 access a supercomputer for several hours to develop a code and then improve it. This is why, in our work, we have focused on experiments to solve a well-known sparse linear equations system 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, so that 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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