\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.
+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 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.
+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 have compared the performances of our method only to those of the GMRES 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 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 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
-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.
+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 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.
+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:}
\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.
+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 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 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 ``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.
+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:}
\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.
+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.''
%%% AJOUTE************************
%%%*******************************
-\noindent {\bf Contributions:}\\
-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.\\
+\noindent {\bf Contributions:}\\ 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.\\
%%%*******************************
%%%*******************************
%%% MODIFIE ************************
%%%*********************************
-Let $Ax=b$ be a given large and sparse linear system of $n$ equations where $A\in\mathbb{R}^{n\times n}$ is a sparse square and non-singular matrix, $x\in\mathbb{R}^{n}$ is the solution vector and $b\in\mathbb{R}^{n}$ is the right-hand side vector. We use a multisplitting method to solve the linear system on a large computing platform in order to reduce the communications. Let the computing platform be composed of $L$ blocks of processors physically adjacent or geographically distant. In this work we apply the block Jacobi multisplitting to the linear system as follows
+Let $Ax=b$ be a given large and sparse linear system of $n$ equations where $A\in\mathbb{R}^{n\times n}$ is a sparse square and non-singular matrix, $x\in\mathbb{R}^{n}$ is the solution vector and $b\in\mathbb{R}^{n}$ is the right-hand side vector. We use a multisplitting method to solve the linear system on a large computing platform in order to reduce communications. Let the computing platform be composed of $L$ blocks of processors physically adjacent or geographically distant. In this work we apply the block Jacobi multisplitting method to the linear system as follows
%%%*********************************
%%%*********************************
where for $\ell\in\{1,\ldots,L\}$, $A_\ell$ is a rectangular block of size $n_\ell\times n$ and $X_\ell$ and $B_\ell$ are sub-vectors of size $n_\ell$ each, such that $\sum_\ell n_\ell=n$.
%%% MODIFIE ***********************
%%%********************************
-The splitting is performed row-by-row without overlapping in such a way that successive rows of sparse matrix $A$ and both vectors $x$ and $b$ are assigned to one block of processors.
+The splitting is performed row-by-row without overlapping in such a way that successive rows of sparse matrix $A$ and both vectors $x$ and $b$ are assigned to a block of processors.
%%%********************************
%%%********************************
So, the multisplitting format of the linear system is defined as follows
where for $j\in\{1,\ldots,s\}$, $x^j=[X_1^j,\ldots,X_L^j]$ is a solution of the global linear system.
%%% MODIFIE ***********************
%%%********************************
-The advantage of such a Krylov subspace is that we neither need an orthonormal basis nor any synchronization between the different blocks is necessary to orthogonalize the generated basis. This avoids to perform other synchronizations between the blocks of processors.
+The advantage of such a Krylov subspace is that we neither need an orthonormal basis nor any synchronization between the different blocks to orthogonalize the generated basis. This avoids to perform other synchronizations between the blocks of processors.
-The multisplitting method is periodically restarted every $s$ iterations with a new initial guess $\tilde{x}=S\alpha$ which minimizes an error function, in our case it minimizes the residuals $\|b-Ax\|_2$ over the Krylov subspace spanned by vectors of $S$. So $\alpha$ is defined as the solution of the large overdetermined linear system
+The multisplitting method is periodically restarted every $s$ iterations with a new initial guess $\tilde{x}=S\alpha$ which minimizes an error function, in our case it minimizes the residuals $\|b-Ax\|_2$ over the Krylov subspace spanned by vectors of $S$. So $\alpha$ is defined as the solution of the large overdetermined linear system.
%%%********************************
%%%********************************
%%% AJOUTE ************************
%%%********************************
-We have performed some experiments on an infiniband cluster of three Intel Xeon quad-core E5620 CPUs of 2.40 GHz and 12 GB of memory. For the GMRES code (alone and in both multisplitting versions) the restart parameter is fixed to 16. The precision of the GMRES version is fixed to 1e-6. For the multisplitting versions, there are two precisions, one for the external solver which is fixed to 1e-6 and another one for the inner solver (GMRES) which is fixed to 1e-10. It should be noted that a high precision is used but we also fixed a maximum number of iterations for each internal step. In practice, we limit the number of iterations in the internal step to 10. So an internal iteration is finished when the precision is reached or when the maximum internal number of iterations is reached. The precision and the maximum number of iterations of CGNR method used by our Krylov multisplitting algorithm are fixed to 1e-25 and 20 respectively. The size of the Krylov subspace basis $S$ is fixed to 10 vectors.
+We have performed some experiments on an infiniband cluster of three Intel Xeon
+quad-core E5620 CPUs of 2.40 GHz and 12 GB of memory. For the GMRES code (alone
+and in both multisplitting versions) the restart parameter is fixed to 16. The
+precision of the GMRES version is fixed to 1e-6. For the multisplitting
+versions, there are two precisions, one for the external solver which is fixed
+to 1e-6 and another one for the inner solver (GMRES) which is fixed to 1e-10. It
+should be noted that a high precision is used but we also fixed a maximum number
+of iterations for each internal step. In practice, we limit the number of
+iterations in the internal step to 10. So an internal iteration is finished when
+the precision is reached or when the maximum internal number of iterations is
+reached. The precision and the maximum number of iterations of CGNR method used
+by our Krylov multisplitting algorithm are fixed to 1e-25 and 20
+respectively. The size of the Krylov subspace basis $S$ is fixed to 10 vectors.
\begin{figure}[htbp]
\centering
\end{figure}
%%The experiments are performed on 3 different clusters of cores interconnected by an infiniband network (each cluster is a quad-core CPU).
-Figures~\ref{fig:001} and~\ref{fig:002} show the scalability performances of GMRES, classical multisplitting and Krylov multisplitting methods: strong and weak scaling are presented respectively. We can remark from these figures that the performances of our Krylov multisplitting method are better than those of GMRES and classical multisplitting methods. In the experiments conducted in this work, our method is about twice faster than the GMRES method and about 9 times faster than the classical multisplitting method. Our multisplitting method uses a minimization step over a Krylov subspace which reduces the number of iterations and accelerates the convergence. We can also remark that the performances of the classical block Jacobi multisplitting method are the worst compared with those of the other two methods. This is why in the following experiments we compare the performances of our Krylov multisplitting method with only those of the GMRES method.
+Figures~\ref{fig:001} and~\ref{fig:002} show the scalability performances of
+GMRES, classical multisplitting and Krylov multisplitting methods: strong and
+weak scaling are presented respectively. We can remark from these figures that
+the performances of our Krylov multisplitting method are better than those of
+GMRES and classical multisplitting methods. In the experiments conducted in this
+work, our method is approximately twice faster than the GMRES method and about 9
+times faster than the classical multisplitting method. Our multisplitting method
+uses a minimization step over a Krylov subspace which reduces the number of
+iterations and accelerates the convergence. We can also remark that the
+performances of the classical block Jacobi multisplitting method are the worst
+compared with those of the other two methods. This is why in the following
+experiments we compare the performances of our Krylov multisplitting method with
+only those of the GMRES method.
%%%********************************
%%%********************************
%%%*********************************
%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.
In the following we present some experiments we could achieve out on the Hector
-architecture, a UK's high-end computing resource, funded by the UK Research
+architecture, a UK high-end computing resource, funded by the UK Research
Councils~\cite{hector}. This is a Cray XE6 supercomputer, equipped with two
16-core AMD Opteron 2.3 GHz and 32 GB of memory. Machines are interconnected
with a 3D torus. The different parameters used by the GMRES and the Krylov multisplitting codes are as those previously mentioned.
Table~\ref{tab1} shows the result of the experiments. The first column shows
-the size of the 3D Poisson problem. The size is chosen in order to have
+the size of the 3D Poisson problem. The size is chosen in order to have
approximately 50,000 components per core. The second column represents the
-number of cores used. In brackets, one can find the decomposition used for the
-Krylov multisplitting. The third column and the sixth column respectively show
-the execution time for the GMRES and the Krylov multisplitting codes. The fourth
-and the seventh column describe the number of iterations. For the
-multisplitting code, the total number of inner iterations is represented in
+number of cores used. Between brackets, one can find the decomposition used for
+the Krylov multisplitting. The third column and the sixth column respectively
+show the execution time for the GMRES and the Krylov multisplitting codes. The
+fourth and the seventh column describe the number of iterations. For the
+multisplitting code, the total number of inner iterations is represented between
brackets.
%%%********************************
%%%********************************
%%%*******************************
In Figure~\ref{fig:01}, the number of iterations per second is reported for both
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
+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 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 blocks is approximately
-equals to 115. So it is not different from GMRES.
+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 blocks is approximately equal
+to 115. So it is not different from GMRES.
\begin{figure}[htbp]
\centering
\label{fig:01}
\end{figure}
-\noindent {\bf Final remarks:}\\
-It should be noted, on the one hand, that the development of a complete new
-method usable with any kind of problem is a really long and fastidious task if
-one is working from scratch. On the other hand, using an existing tool for the
-inner solver is also not easy because it requires to make link between the inner
-solver and the outer one. We plan to do that later with engineers working
-specifically on that point. Moreover, we think that it is very important to
-analyze the convergence of this method compared to other methods. 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.
+\noindent {\bf Final remarks:}\\ It should be noted, on the one hand, that the
+development of a complete new code usable with any kind of problem is a really
+long and fastidious task if one is working from scratch. On the other hand,
+using an existing tool for the inner solver is also quite difficult because it
+requires to establish a link between the inner solver and the outer one. We
+plan to do that later with engineers working specifically on that point.
+Moreover, we think that it is very important to analyze the convergence of this
+method compared to other methods. In this work, we have focused on the
+description of this method and its performances with a typical application. Many
+other investigations are required for this method as explained in the next
+section.
%%%*******************************
%%%*******************************
methods with overlapping blocks.
\section{Acknowledgement}
-The authors would like to thank Mark Bull of the EPCC his fruitful remarks and the facilities of HECToR.
+The authors would like to thank Mark Bull of the EPCC his fruitful remarks and the facilities of HECToR. This work has been partially supported by the Labex
+ACTION project (contract “ANR-11-LABX-01-01”).
+
%Other applications (=> other matrices)\\
%Larger experiments\\
