%%%%%%%%%%%%%%%%%%%%%%%%
\begin{abstract}
-In this paper we revisit the krylov multisplitting algorithm presented in
-\cite{huang1993krylov} which uses a scalar method to minimize the krylov
+In this paper we revisit the Krylov multisplitting algorithm presented in
+\cite{huang1993krylov} which uses a scalar method to minimize the Krylov
iterations computed by a multisplitting algorithm. Our new algorithm is based on
a parallel multisplitting algorithm with few blocks of large size using a
-parallel GMRES method inside each block and on a parallel krylov minimization in
+parallel GMRES method inside each block and on a parallel Krylov minimization in
order to improve the convergence. Some large scale experiments with a 3D Poisson
problem are presented. They show the obtained improvements compared to a
classical GMRES both in terms of number of iterations and execution times.
the form $Ax=b$ because they are easier to parallelize than direct ones. Many
iterative methods have been proposed and adapted by many researchers. For
example, the GMRES method and the Conjugate Gradient method are very well known
-and used by many researchers~\cite{S96}. Both the method are based on the
-Krylov subspace which consists in forming a basis of the sequence of successive
+and used by many researchers~\cite{S96}. Both methods are based on the
+Krylov subspace which consists in forming a basis of a sequence of successive
matrix powers times the initial residual.
When solving large linear systems with many cores, iterative methods often
suffer from scalability problems. This is due to their need for collective
communications to perform matrix-vector products and reduction operations.
-Preconditionners can be used in order to increase the convergence of iterative
-solvers. However, most of the good preconditionners are not sclalable when
+Preconditioners can be used in order to increase the convergence of iterative
+solvers. However, most of the good preconditioners are not scalable when
thousands of cores are used.
Traditional iterative solvers have global synchronizations that penalize the
scalability. Two possible solutions consists either in using asynchronous
iterative methods~\cite{ref18} or to use multisplitting algorithms. In this
paper, we will reconsider the use of a multisplitting method. In opposition to
-traditionnal multisplitting method that suffer from slow convergence, as
+traditional multisplitting method that suffer from slow convergence, as
proposed in~\cite{huang1993krylov}, the use of a minimization process can
drastically improve the convergence.
In~\cite{huang1993krylov}, the authors proposed a parallel multisplitting
algorithm in which all the tasks except one are devoted to solve a sub-block of
-the splitting and to send their local solution to the first task which is in
+the splitting and to send their local solutions to the first task which is in
charge to combine the vectors at each iteration. These vectors form a Krylov
basis for which the first task minimizes the error function over the basis to
increase the convergence, then the other tasks receive the updated solution until
convergence of the global system.
In~\cite{couturier2008gremlins}, the authors proposed practical implementations
-of multisplitting algorithms that take benefit from multisplitting algorithms to
+of multisplitting algorithms that take benefit from multisplitting algorithms {\bf ???} to
solve large scale linear systems. Inner solvers could be based on scalar direct
method with the LU method or scalar iterative one with GMRES.
algorithm in which large blocks are solved using a GMRES solver. The authors have
performed large scale experiments up-to 32,768 cores and they conclude that
asynchronous multisplitting algorithm could be more efficient than traditional
-solvers on exascale architecture with hundreds of thousands of cores.
+solvers on an exascale architecture with hundreds of thousands of cores.
The key idea of a multisplitting method to solve a large system of linear equations $Ax=b$ is defined as follows. The first step consists in partitioning the matrix $A$ in $L$ several ways
\begin{equation}
-A = M_l - N_l,
+A = M_\ell - N_\ell,
\label{eq01}
\end{equation}
-where for all $l\in\{1,\ldots,L\}$ $M_l$ are non-singular matrices. Then the linear system is solved by iteration based on the obtained splittings as follows
+where for all $\ell\in\{1,\ldots,L\}$ $M_\ell$ are non-singular matrices. Then the linear system is solved by iteration based on the obtained splittings as follows
\begin{equation}
-x^{k+1}=\displaystyle\sum^L_{l=1} E_l M^{-1}_l (N_l x^k + b),~k=1,2,3,\ldots
+x^{k+1}=\displaystyle\sum^L_{\ell=1} E_\ell M^{-1}_\ell (N_\ell x^k + b),~k=1,2,3,\ldots
\label{eq02}
\end{equation}
-where $E_l$ are non-negative and diagonal weighting matrices and their sum is an identity matrix $I$. The convergence of such a method is dependent on the condition
+where $E_\ell$ are non-negative and diagonal weighting matrices and their sum is an identity matrix $I$. The convergence of such a method is dependent on the condition
\begin{equation}
-\rho(\displaystyle\sum^L_{l=1}E_l M^{-1}_l N_l)<1.
+\rho(\displaystyle\sum^L_{\ell=1}E_\ell M^{-1}_\ell N_\ell)<1.
\label{eq03}
\end{equation}
where $\rho$ is the spectral radius of the square matrix.
The advantage of the multisplitting method is that at each iteration $k$ there are $L$ different linear sub-systems
\begin{equation}
-v_l^k=M^{-1}_l N_l x_l^{k-1} + M^{-1}_l b,~l\in\{1,\ldots,L\},
+v_\ell^k=M^{-1}_\ell N_\ell x_\ell^{k-1} + M^{-1}_\ell b,~\ell\in\{1,\ldots,L\},
\label{eq04}
\end{equation}
-to be solved independently by a direct or an iterative method, where $v_l^k$ is the solution of the local sub-system. Thus the computations of $\{v_l\}_{1\leq l\leq L}$ may be performed in parallel by a set of processors. A multisplitting method using an iterative method as an inner solver is called an inner-outer iterative method or a two-stage method. The results $v_l$ obtained from the different splittings~(\ref{eq04}) are combined to compute solution $x$ of the linear system by using the diagonal weighting matrices
+to be solved independently by a direct or an iterative method, where $v_\ell$ is the solution of the local sub-system. Thus the computations of $\{v_\ell\}_{1\leq \ell\leq L}$ may be performed in parallel by a set of processors. A multisplitting method using an iterative method as an inner solver is called an inner-outer iterative method or a two-stage method. The results $v_\ell$ obtained from the different splittings~(\ref{eq04}) are combined to compute solution $x$ of the linear system by using the diagonal weighting matrices
\begin{equation}
-x^k = \displaystyle\sum^L_{l=1} E_l v_l^k,
+x^k = \displaystyle\sum^L_{\ell=1} E_\ell v_\ell^k,
\label{eq05}
\end{equation}
-In the case where the diagonal weighting matrices $E_l$ have only zero and one factors (i.e. $v_l$ are disjoint vectors), the multisplitting method is non-overlapping and corresponds to the block Jacobi method.
+In the case where the diagonal weighting matrices $E_\ell$ have only zero and one factors (i.e. $v_\ell$ are disjoint vectors), the multisplitting method is non-overlapping and corresponds to the block Jacobi method.
%%%%%%%%%%%%%%%%%%%%%%%%
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\section{A two-stage method with a minimization}
-Let $Ax=b$ be a given large and sparse linear system of $n$ equations to solve in parallel on $L$ clusters of processors, physically adjacent or geographically distant, where $A\in\mathbb{R}^{n\times n}$ is a 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. The multisplitting of this linear system is defined as follows
+Let $Ax=b$ be a given large and sparse linear system of $n$ equations to solve in parallel on $L$ clusters of processors, physically adjacent or geographically distant, where $A\in\mathbb{R}^{n\times n}$ is a 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. The multisplitting of this linear system is defined as follows
\begin{equation}
\left\{
\begin{array}{lll}
\right.
\label{sec03:eq01}
\end{equation}
-where for $l\in\{1,\ldots,L\}$, $A_l$ is a rectangular block of size $n_l\times n$ and $X_l$ and $B_l$ are sub-vectors of size $n_l$ each, such that $\sum_ln_l=n$. In this work, we use a row-by-row splitting without overlapping in such a way that successive rows of sparse matrix $A$ and both vectors $x$ and $b$ are assigned to one cluster. So, the multisplitting format of the linear system is defined 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 $\sum_\ell n_\ell=n$. In this work, we use a row-by-row splitting without overlapping in such a way that successive rows of sparse matrix $A$ and both vectors $x$ and $b$ are assigned to one cluster. So, the multisplitting format of the linear system is defined as follows
\begin{equation}
-\forall l\in\{1,\ldots,L\} \mbox{,~} \displaystyle\sum_{m=1}^{l-1}A_{lm}X_m + A_{ll}X_l + \displaystyle\sum_{m=l+1}^{L}A_{lm}X_m = B_l,
+\forall \ell\in\{1,\ldots,L\} \mbox{,~} A_{\ell \ell}X_\ell + \displaystyle\sum_{\substack{m=1\\m\neq\ell}}^L A_{\ell m}X_m = B_\ell,
\label{sec03:eq02}
\end{equation}
-where $A_{lm}$ is a sub-block of size $n_l\times n_m$ of the rectangular matrix $A_l$, $X_m\neq X_l$ is a sub-vector of size $n_m$ of the solution vector $x$ and $\sum_{m\neq l}n_m+n_l=n$, for all $m\in\{1,\ldots,L\}$.
+where $A_{\ell m}$ is a sub-block of size $n_\ell\times n_m$ of the rectangular matrix $A_\ell$, $X_m\neq X_\ell$ is a sub-vector of size $n_m$ of the solution vector $x$ and $\sum_{m\neq \ell}n_m+n_\ell=n$, for all $m\in\{1,\ldots,L\}$.
Our multisplitting method proceeds by iteration for solving the linear system in such a way each sub-system
\begin{equation}
\left\{
\begin{array}{l}
-A_{ll}X_l = Y_l \mbox{,~such that}\\
-Y_l = B_l - \displaystyle\sum_{\substack{m=1\\m\neq l}}^{L}A_{lm}X_m,
+A_{\ell \ell}X_\ell = Y_\ell \mbox{,~such that}\\
+Y_\ell = B_\ell - \displaystyle\sum_{\substack{m=1\\m\neq \ell}}^{L}A_{\ell m}X_m,
\end{array}
\right.
\label{sec03:eq03}
\end{equation}
-is solved independently by a {\it cluster of processors} and communication are required to update the right-hand side vectors $Y_l$, such that the vectors $X_m$ represent the data dependencies between the clusters. In this work, we use the parallel restarted GMRES method~\cite{ref34} as an inner iteration method to solve sub-systems~(\ref{sec03:eq03}). GMRES is one of the most used Krylov iterative methods to solve sparse linear systems in parallel on clusters of processors. %In practice, GMRES is used with a preconditioner to improve its convergence. In this work, we used a preconditioning matrix equivalent to the main diagonal of sparse sub-matrix $A_{ll}$. This preconditioner is straightforward to implement in parallel and gives good performances in many situations.
+is solved independently by a {\it cluster of processors} and communications are required to update the right-hand side vectors $Y_\ell$, such that the vectors $X_m$ represent the data dependencies between the clusters. In this work, we use the parallel restarted GMRES method~\cite{ref34} as an inner iteration method to solve sub-systems~(\ref{sec03:eq03}). GMRES is one of the most used Krylov iterative methods to solve sparse linear systems. %In practice, GMRES is used with a preconditioner to improve its convergence. In this work, we used a preconditioning matrix equivalent to the main diagonal of sparse sub-matrix $A_{ll}$. This preconditioner is straightforward to implement in parallel and gives good performances in many situations.
-It should be noted that the convergence of the inner iterative solver for the different sub-systems~(\ref{sec03:eq03}) does not necessarily involve the convergence of the multisplitting method. It strongly depends on the properties of the global sparse linear system to be solved and the computing environment~\cite{o1985multi,ref18}. Furthermore, the multisplitting
-of the linear system among several clusters of processors increases the spectral radius of the iteration matrix, thereby slowing the convergence. In this work, we based on the work presented in~\cite{huang1993krylov} to increase the convergence and improve the scalability of the multisplitting methods.
+It should be noted that the convergence of the inner iterative solver for the different sub-systems~(\ref{sec03:eq03}) does not necessarily involve the convergence of the multisplitting method. It strongly depends on the properties of the global sparse linear system to be solved and the computing environment~\cite{o1985multi,ref18}. Furthermore, the multisplitting of the linear system among several clusters of processors increases the spectral radius of the iteration matrix, thereby slowing the convergence. In this paper, we based on the work presented in~\cite{huang1993krylov} to increase the convergence and improve the scalability of the multisplitting methods.
-In order to accelerate the convergence, we implemented the outer iteration of the multisplitting solver as a Krylov subspace method which minimizes some error function over a Krylov subspace~\cite{S96}. The Krylov subspace that we used is spanned by a basis composed of successive solutions issued from solving the $L$ splittings~(\ref{sec03:eq03})
+In order to accelerate the convergence, we implemented the outer iteration of the multisplitting solver as a Krylov iterative method which minimizes some error function over a Krylov subspace~\cite{S96}. The Krylov subspace that we used is spanned by a basis composed of successive solutions issued from solving the $L$ splittings~(\ref{sec03:eq03})
\begin{equation}
S=\{x^1,x^2,\ldots,x^s\},~s\leq n,
\label{sec03:eq04}
\end{equation}
where for $j\in\{1,\ldots,s\}$, $x^j=[X_1^j,\ldots,X_L^j]$ is a solution of the global linear system. The advantage of such a Krylov subspace is that we need neither an orthogonal basis nor synchronizations between clusters to generate this basis.
-The multisplitting method is periodically restarted every $s$ iterations with a new initial guess $\tilde{x}=S\alpha$ which minimizes the error function $\|b-Ax\|_2$ 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 the error function $\|b-Ax\|_2$ over the Krylov subspace spanned by vectors of $S$. So $\alpha$ is defined as the solution of the large overdetermined linear system
\begin{equation}
R\alpha=b,
\label{sec03:eq05}
\text{minimize}~\|b-R\alpha\|_2,
\label{sec03:eq07}
\end{equation}
-where $R^T$ denotes the transpose of the matrix $R$. Since $R$ (i.e. $AS$) and $b$ are split among $L$ clusters, the symmetric positive definite system~(\ref{sec03:eq06}) is solved in parallel. Thus an iterative method would be more appropriate than a direct one to solve this system. We use the parallel conjugate gradient method for the normal equations CGNR~\cite{S96,refCGNR}.
+where $R^T$ denotes the transpose of matrix $R$. Since $R$ (i.e. $AS$) and $b$ are split among $L$ clusters, the symmetric positive definite system~(\ref{sec03:eq06}) is solved in parallel. Thus an iterative method would be more appropriate than a direct one to solve this system. We use the parallel Conjugate Gradient method for the normal equations CGNR~\cite{S96,refCGNR}.
\begin{algorithm}[!t]
\caption{A two-stage linear solver with inner iteration GMRES method}
\begin{algorithmic}[1]
-\Input $A_l$ (sparse sub-matrix), $B_l$ (right-hand side sub-vector)
-\Output $X_l$ (solution sub-vector)\vspace{0.2cm}
-\State Load $A_l$, $B_l$
-\State Initialize the initial guess $x^0$
+\Input $A_\ell$ (sparse sub-matrix), $B_\ell$ (right-hand side sub-vector)
+\Output $X_\ell$ (solution sub-vector)\vspace{0.2cm}
+\State Load $A_\ell$, $B_\ell$
+\State Set the initial guess $x^0$
\State Set the minimizer $\tilde{x}^0=x^0$
\For {$k=1,2,3,\ldots$ until the global convergence}
\State Restart with $x^0=\tilde{x}^{k-1}$:
\For {$j=1,2,\ldots,s$}
-\State Inner iteration solver: \Call{InnerSolver}{$x^0$, $j$}
-\State Construct basis $S$: add column vector $X_l^j$ to the matrix $S_l^k$
-\State Exchange local values of $X_l^j$ with the neighboring clusters
-\State Compute dense matrix $R$: $R_l^{k,j}=\sum^L_{i=1}A_{li}X_i^j$
+\State \label{line7}Inner iteration solver: \Call{InnerSolver}{$x^0$, $j$}
+\State Construct basis $S$: add column vector $X_\ell^j$ to the matrix $S_\ell^k$
+\State Exchange local values of $X_\ell^j$ with the neighboring clusters
+\State Compute dense matrix $R$: $R_\ell^{k,j}=\sum^L_{i=1}A_{\ell i}X_i^j$
\EndFor
-\State Minimization $\|b-R\alpha\|_2$: \Call{UpdateMinimizer}{$S_l$, $R$, $b$, $k$}
-\State Local solution of the linear system $Ax=b$: $X_l^k=\tilde{X}_l^k$
-\State Exchange the local minimizer $\tilde{X}_l^k$ with the neighboring clusters
+\State \label{line12}Minimization $\|b-R\alpha\|_2$: \Call{UpdateMinimizer}{$S_\ell$, $R$, $b$, $k$}
+\State Local solution of linear system $Ax=b$: $X_\ell^k=\tilde{X}_\ell^k$
+\State Exchange the local minimizer $\tilde{X}_\ell^k$ with the neighboring clusters
\EndFor
\Statex
\Function {InnerSolver}{$x^0$, $j$}
-\State Compute local right-hand side $Y_l = B_l - \sum^L_{\substack{m=1\\m\neq l}}A_{lm}X_m^0$
-\State Solving local splitting $A_{ll}X_l^j=Y_l$ using parallel GMRES method, such that $X_l^0$ is the initial guess
-\State \Return $X_l^j$
+\State Compute local right-hand side $Y_\ell = B_\ell - \sum^L_{\substack{m=1\\m\neq \ell}}A_{\ell m}X_m^0$
+\State Solving local splitting $A_{\ell \ell}X_\ell^j=Y_\ell$ using parallel GMRES method, such that $X_\ell^0$ is the initial guess
+\State \Return $X_\ell^j$
\EndFunction
\Statex
-\Function {UpdateMinimizer}{$S_l$, $R$, $b$, $k$}
+\Function {UpdateMinimizer}{$S_\ell$, $R$, $b$, $k$}
\State Solving normal equations $(R^k)^TR^k\alpha^k=(R^k)^Tb$ in parallel by $L$ clusters using parallel CGNR method
-\State Compute local minimizer $\tilde{X}_l^k=S_l^k\alpha^k$
-\State \Return $\tilde{X}_l^k$
+\State Compute local minimizer $\tilde{X}_\ell^k=S_\ell^k\alpha^k$
+\State \Return $\tilde{X}_\ell^k$
\EndFunction
\end{algorithmic}
\label{algo:01}
\end{algorithm}
-The main key points of our multisplitting method to solve a large sparse linear system are given in Algorithm~\ref{algo:01}. This algorithm is based on a two-stage method with a minimization using restarted GMRES iterative method as an inner solver. It is executed in parallel by each cluster of processors. Matrices and vectors with the subscript $l$ represent the local data for cluster $l$, where $l\in\{1,\ldots,L\}$. The two-stage solver uses two different parallel iterative algorithms: GMRES method to solve each splitting~(\ref{sec03:eq03}) on a cluster of processors, and CGNR method executed in parallel by all clusters to minimize the function error~(\ref{sec03:eq07}) over the Krylov subspace spanned by $S$. The algorithm requires two global synchronizations between $L$ clusters. The first one is performed at line~$12$ in Algorithm~\ref{algo:01} to exchange local values of vector solution $x$ (i.e. the minimizer $\tilde{x}$) required to restart the multisplitting solver. The second one is needed to construct the matrix $R$ of the Krylov subspace. We chose to perform this latter synchronization $s$ times in every outer iteration $k$ (line~$7$ in Algorithm~\ref{algo:01}). This is a straightforward way to compute the sparse matrix-dense matrix multiplication $R=AS$. We implemented all synchronizations by using message passing collective communications of MPI library.
+The main key points of our Krylov multisplitting method to solve a large sparse linear system are given in Algorithm~\ref{algo:01}. This algorithm is based on a two-stage method with a minimization using restarted GMRES iterative method as an inner solver. It is executed in parallel by each cluster of processors. Matrices and vectors with the subscript $\ell$ represent the local data for cluster $\ell$, where $\ell\in\{1,\ldots,L\}$. The two-stage solver uses two different parallel iterative algorithms: GMRES method to solve each splitting~(\ref{sec03:eq03}) on a cluster of processors, and CGNR method executed in parallel by all clusters to minimize the function error~(\ref{sec03:eq07}) over the Krylov subspace spanned by $S$. The algorithm requires two global synchronizations between $L$ clusters. The first one is performed at line~\ref{line12} in Algorithm~\ref{algo:01} to exchange local values of vector solution $x$ (i.e. the minimizer $\tilde{x}$) required to restart the multisplitting solver. The second one is needed to construct the matrix $R$. We chose to perform this latter synchronization $s$ times in every outer iteration $k$ (line~\ref{line7} in Algorithm~\ref{algo:01}). This is a straightforward way to compute the sparse matrix-dense matrix multiplication $R=AS$. We implemented all synchronizations by using message passing collective communications of MPI library.
%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%
approximately 50,000 components per core. The second column represents the
number of cores used. In parenthesis, there is the decomposition used for the
Krylov multisplitting. The third column and the sixth column respectively show
-the execution time for the GMRES and the Kyrlov multisplitting codes. The fourth
+the execution time for the GMRES and the Krylov multisplitting codes. The fourth
and the seventh column describes the number of iterations. For the
multisplitting code, the total number of inner iterations is represented in
parenthesis. For the GMRES code (alone and in the multisplitting version) the
1e-6. For the multisplitting, 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 practise, we
-limit the number of internal step to 10. So an internal iteration is finished
+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 are fixed to 1e-25 and 20, respectively. The size of the Krylov subspace basis $S$ is fixed to 10 vectors.
-
-
+is reached. The precision and the maximum number of iterations of CGNR method are fixed to 1e-25 and 20 respectively. The size of the Krylov subspace basis $S$ is fixed to 10 vectors.
\begin{table}[htbp]
\begin{center}
neglectable.
\section{Conclusion and perspectives}
-We have implemented a Krylov multisplitting method to solve sparse linear systems on large-scale computing platforms. We have developed a synchronous two-stage method based on the block Jacobi multisplitting and uses GMRES iterative method as an inner iteration. Our contribution in this paper is twofold. First we have constituted a multi-cluster environment based on processors of the large-scale computing platform on which each linear sub-system issued from the splitting is solved in parallel by a cluster of processors. Second, we have implemented the outer iteration of the multisplitting method as a Krylov subspace method which minimizes some error function. This increases the convergence and improves the scalability of the multisplitting method.
+We have implemented a Krylov multisplitting method to solve sparse linear systems on large-scale computing platforms. We have developed a synchronous two-stage method based on the block Jacobi multisplitting and uses GMRES iterative method as an inner iteration. Our contribution in this paper is twofold. First we have constituted a multi-cluster environment based on processors of the computing platform on which each linear sub-system issued from the splitting is solved in parallel by a cluster of processors. Second, we have implemented the outer iteration of the multisplitting method using Krylov subspace method which minimizes some error function. This increases the convergence and improves the scalability of the multisplitting method.
-We have tested our multisplitting method for solving the sparse linear system issued from the discretization of the 3D Poisson problem. We have compared its performances to those of GMRES method on a supercomputer composed of 2048 to 8192 cores. The experimental results showed that the multisplitting method is about 4 to 6 times faster than the GMRES method for different sizes of the problem split into 2 or 4 blocks when using multisplitting method. Indeed, the GMRES method has difficulties to scale with many cores while the Krylov multisplitting method allows to hide latency and reduce the inter-cluster communications.
+We have tested our multisplitting method for solving the sparse linear system issued from the discretization of a 3D Poisson problem. We have compared its performances to those of classical GMRES method on a supercomputer composed of 2048 to 8192 cores. The experimental results showed that the multisplitting method is about 4 to 6 times faster than the GMRES method for different sizes of the problem split into 2 or 4 blocks when using multisplitting method. Indeed, the GMRES method has difficulties to scale with many cores while the Krylov multisplitting method allows to hide latency and reduce the inter-cluster communications.
In future works, we plan to conduct experiments on larger number of cores and test the scalability of our Krylov multisplitting method. It would be interesting to validate its performances for solving other linear/nonlinear and symmetric/nonsymmetric problems. Moreover, we intend to develop multisplitting methods based on asynchronous iteration in which communications are overlapped by computations. These methods would be interesting for platforms composed of distant clusters interconnected by a high-latency network. In addition, we intend to investigate the convergence improvements of our method by using preconditioning techniques for Krylov iterative methods and multisplitting methods with overlapping blocks.
