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-Iterative methods are become more attractive than direct ones to solve large sparse linear systems. They are more effective in a parallel context and require less memory and arithmetic operations than direct methods. A number of iterative methods are proposed and adapted by many researchers and the increased need for solving very large sparse linear systems triggered the development of efficient iterative techniques suitable for the parallel processing.
-
-The most successful iterative methods currently available are those based on Krylov subspaces which consist in forming a basis of a sequence of successive matrix powers times an initial vector for example the residual. These methods are based on orthogonality of vectors of the Krylov subspace basis to solve linear systems. The most well-known iterative Krylov subspace methods are Conjugate Gradient method and GMRES method (generalized minimal residual).
-
-However, the iterative methods suffer from scalability problems on parallel computing platforms with many processors due to their need for reduction operations and collective communications to perform matrix-vector multiplications. The communications on large clusters with thousands of cores and large sizes of messages can significantly affect the performances of iterative methods. In practice, Krylov subspace iteration methods are often used with preconditioners in order to increase their convergence and accelerate their performances. However, most of the good preconditioners are not scalable on large clusters.
-
-In this paper we propose a two-stage algorithm based on two nested iterations called inner-outer iterations. The algorithm consists in solving the sparse linear system iteratively with a small number of inner iterations and restarts the outer step with a new solution minimizing some error function over a Krylov subspace. The algorithm is iterative and easy to parallelize on large clusters and the minimization technique improves its convergence and performances.
-
-The present paper is organized as follows. In Section~\ref{sec:02} some related works are presented. Section~\ref{sec:03} presents our two-stage algorithm based on Krylov subspace iteration methods. Section~\ref{sec:04} shows some experimental results obtained on large clusters of our algorithm using routines of PETSc toolkit.
+Iterative methods are become more attractive than direct ones to solve very
+large sparse linear systems. They are more effective in a parallel context and
+require less memory and arithmetic operations than direct methods. A number of
+iterative methods are proposed and adapted by many researchers and the increased
+need for solving very large sparse linear systems triggered the development of
+efficient iterative techniques suitable for the parallel processing.
+
+Most of the successful iterative methods currently available are based on Krylov
+subspaces which consist in forming a basis of a sequence of successive matrix
+powers times an initial vector for example the residual. These methods are based
+on orthogonality of vectors of the Krylov subspace basis to solve linear
+systems. The most well-known iterative Krylov subspace methods are Conjugate
+Gradient method and GMRES method (generalized minimal residual).
+
+However, iterative methods suffer from scalability problems on parallel
+computing platforms with many processors due to their need for reduction
+operations and collective communications to perform matrix-vector
+multiplications. The communications on large clusters with thousands of cores
+and large sizes of messages can significantly affect the performances of
+iterative methods. In practice, Krylov subspace iteration methods are often used
+with preconditioners in order to increase their convergence and accelerate their
+performances. However, most of the good preconditioners are not scalable on
+large clusters.
+
+In this paper we propose a two-stage algorithm based on two nested iterations
+called inner-outer iterations. This algorithm consists in solving the sparse
+linear system iteratively with a small number of inner iterations and restarts
+the outer step with a new solution minimizing some error functions over a Krylov
+subspace. This algorithm is iterative and easy to parallelize on large clusters
+and the minimization technique improves its convergence and performances.
+
+The present paper is organized as follows. In Section~\ref{sec:02} some related
+works are presented. Section~\ref{sec:03} presents our two-stage algorithm based
+on Krylov subspace iteration methods. Section~\ref{sec:04} shows some
+experimental results obtained on large clusters of our algorithm using routines
+of PETSc toolkit.
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\section{A Krylov two-stage algorithm}
\label{sec:03}
-We propose a two-stage algorithm to solve large sparse linear systems of the form $Ax=b$, where $A\in\mathbb{R}^{n\times n}$ is a sparse and square nonsingular matrix, $x\in\mathbb{R}^n$ is the solution vector and $b\in\mathbb{R}^n$ is the right-hand side. The algorithm is implemented as an inner-outer iteration solver based on iterative Krylov methods. The main key points of our solver are given in Algorithm~\ref{algo:01}.
-
-In order to accelerate the convergence, the outer iteration is implemented as an iterative Krylov method which minimizes some error function over a Krylov subspace~\cite{saad96}. At every iteration, the sparse linear system $Ax=b$ is solved iteratively with an iterative method for example GMRES method~\cite{saad86} and the Krylov subspace that we used is spanned by a basis $S$ composed of successive solutions issued from the inner iteration
+A two-stage algorithm is proposed to solve large sparse linear systems of the
+form $Ax=b$, where $A\in\mathbb{R}^{n\times n}$ is a sparse and square
+nonsingular matrix, $x\in\mathbb{R}^n$ is the solution vector and
+$b\in\mathbb{R}^n$ is the right-hand side. The algorithm is implemented as an
+inner-outer iteration solver based on iterative Krylov methods. The main key
+points of our solver are given in Algorithm~\ref{algo:01}.
+
+In order to accelerate the convergence, the outer iteration is implemented as an
+iterative Krylov method which minimizes some error functions over a Krylov
+subspace~\cite{saad96}. At each iteration, the sparse linear system $Ax=b$ is
+solved iteratively with an iterative method, for example GMRES
+method~\cite{saad86} or some of its variants, and the Krylov subspace that we
+used is spanned by a basis $S$ composed of successive solutions issued from the
+inner iteration
\begin{equation}
S = \{x^1, x^2, \ldots, x^s\} \text{,~} s\leq n.
\end{equation}
-The advantage of such a Krylov subspace is that we neither need an orthogonal basis nor any synchronization between processors to generate this basis. The algorithm is periodically restarted every $s$ iterations with a new initial guess $x=S\alpha$ which minimizes the residual norm $\|b-Ax\|_2$ over the Krylov subspace spanned by vectors of $S$, where $\alpha$ is a solution of the normal equations
+The advantage of such a Krylov subspace is that we neither need an orthogonal
+basis nor any synchronization between processors to generate this basis. The
+algorithm is periodically restarted every $s$ iterations with a new initial
+guess $x=S\alpha$ which minimizes the residual norm $\|b-Ax\|_2$ over the Krylov
+subspace spanned by vectors of $S$, where $\alpha$ is a solution of the normal
+equations
\begin{equation}
R^TR\alpha = R^Tb,
\end{equation}
\underset{\alpha\in\mathbb{R}^{s}}{min}\|b-R\alpha\|_2
\label{eq:01}
\end{equation}
-such that $R=AS$ is a dense rectangular matrix in $\mathbb{R}^{n\times s}$, $s\ll n$, and $R^T$ denotes the transpose of matrix $R$. We use an iterative method to solve the least-squares problem~(\ref{eq:01}) as CGLS~\cite{hestenes52} or LSQR~\cite{paige82} methods which is more appropriate than a direct method in the parallel context.
+such that $R=AS$ is a dense rectangular matrix in $\mathbb{R}^{n\times s}$,
+$s\ll n$, and $R^T$ denotes the transpose of matrix $R$. We use an iterative
+method to solve the least-squares problem~(\ref{eq:01}) such as CGLS
+~\cite{hestenes52} or LSQR~\cite{paige82} which are more appropriate than a
+direct method in the parallel context.
\begin{algorithm}[t]
\caption{A Krylov two-stage algorithm}
\State Set the initial guess $x^0$
\For {$k=1,2,3,\ldots$ until convergence}
\State Solve iteratively $Ax^k=b$
- \State Add vector $x^k$ to Krylov subspace basis $S$
+ \State $S_{k~mod~s}=x^k$
\If {$k$ mod $s=0$ {\bf and} not convergence}
\State Compute dense matrix $R=AS$
\State Solve least-squares problem $\underset{\alpha\in\mathbb{R}^{s}}{min}\|b-R\alpha\|_2$
\State Compute minimizer $x^k=S\alpha$
- \State Reinitialize Krylov subspace basis $S$
\EndIf
\EndFor
\end{algorithmic}
\label{algo:01}
\end{algorithm}
+
+Operation $S_{k~ mod~ s}=x^k$ consists in copying the residual $x_k$ into the
+column $k~ mod~ s$ of the matrix $S$. After the minimization, the matrix $S$ is
+reused with the new values of the residuals.
+
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