Krylov two-stage multisplitting method

[rce2015.git] / paper.tex

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+++ b/paper.tex
@@ -56,6 +56,8 @@
 \newcommand{\MIG}{\mathit{maxit_{gmres}}}
 \newcommand{\TOLM}{\mathit{tol_{multi}}}
 \newcommand{\MIM}{\mathit{maxit_{multi}}}
 \newcommand{\MIG}{\mathit{maxit_{gmres}}}
 \newcommand{\TOLM}{\mathit{tol_{multi}}}
 \newcommand{\MIM}{\mathit{maxit_{multi}}}

+\newcommand{\TOLC}{\mathit{tol_{cgls}}}
+\newcommand{\MIC}{\mathit{maxit_{cgls}}}

 
 \usepackage{array}
 \usepackage{color, colortbl}
 
 \usepackage{array}
 \usepackage{color, colortbl}


@@ -106,77 +108,93 @@ ABSTRACT
 %%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%
 
 %%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%
 

-\section{Two-stage splitting methods}
+\section{Two-stage multisplitting methods}

 \label{sec:04}
 \label{sec:04}

-\subsection{Multisplitting methods for sparse linear systems}
+\subsection{Synchronous and asynchronous two-stage methods for sparse linear systems}

 \label{sec:04.01}
 \label{sec:04.01}

-Let us consider the following sparse linear system of $n$ equations in $\mathbb{R}$
+In this paper we focus on two-stage multisplitting methods in their both versions synchronous and asynchronous~\cite{Frommer92,Szyld92,Bru95}. These iterative methods are based on multisplitting methods~\cite{O'leary85,White86,Alefeld97} and use two nested iterations: the outer iteration and the inner iteration. Let us consider the following sparse linear system of $n$ equations in $\mathbb{R}$

 \begin{equation}
 Ax=b,
 \label{eq:01}
 \end{equation}
 \begin{equation}
 Ax=b,
 \label{eq:01}
 \end{equation}

-where $A$ is a sparse square and nonsingular matrix, $b$ is the right-hand side and $x$ is the solution of the system. The multisplitting methods solve the linear system~(\ref{eq:01}) iteratively as follows
+where $A$ is a sparse square and nonsingular matrix, $b$ is the right-hand side and $x$ is the solution of the system. Our work in this paper is restricted to the block Jacobi splitting method. This approach of multisplitting consists in partitioning the matrix $A$ into $L$ horizontal band matrices of order $\frac{n}{L}\times n$ without overlapping (i.e. sub-vectors $\{x_\ell\}_{1\leq\ell\leq L}$ are disjoint). The two-stage multisplitting methods solve the linear system~(\ref{eq:01}) iteratively as follows

 \begin{equation}
 \begin{equation}

-x^{k+1}=\displaystyle\sum^L_{\ell=1} E_\ell M^{-1}_\ell (N_\ell x^k + b),~k=1,2,3,\ldots
+x_\ell^{k+1} = A_{\ell\ell}^{-1}(b_\ell - \displaystyle\sum^{L}_{\substack{m=1\\m\neq\ell}}{A_{\ell m}x^k_m}),\mbox{~for~}\ell=1,\ldots,L\mbox{~and~}k=1,2,3,\ldots

 \label{eq:02}
 \end{equation}
 \label{eq:02}
 \end{equation}

-where a collection of $L$ triplets $(M_\ell, N_\ell, E_\ell)$ defines the multisplitting of matrix $A$~\cite{O'leary85,White86}, such that: the different splittings are defined as $A=M_\ell-N_\ell$ where $M_\ell$ are nonsingular matrices, and $\sum_\ell{E_\ell=I}$ are diagonal nonnegative weighting matrices and $I$ is the identity matrix. The iterations of the multisplitting methods can naturally be computed in parallel such that each processor or cluster of processors is responsible for solving one splitting as a linear sub-system   
+where $x_\ell$ are sub-vectors of the solution $x$, $b_\ell$ are the sub-vectors of the right-hand side $b$, and $A_{\ell\ell}$ and $A_{\ell m}$ are diagonal and off-diagonal blocks of matrix $A$ respectively. The iterations of these methods can naturally be computed in parallel such that each processor or cluster of processors is responsible for solving one splitting as a linear sub-system 

 \begin{equation}
 \begin{equation}

-M_\ell y_\ell = c_\ell^k,\mbox{~such that~} c_\ell^k = N_\ell x^k + b,
+A_{\ell\ell} x_\ell = c_\ell,\mbox{~for~}\ell=1,\ldots,L,

 \label{eq:03}
 \end{equation}
 \label{eq:03}
 \end{equation}

-then the weighting matrices $E_\ell$ are used to compute the solution of the global system~(\ref{eq:01})
-\begin{equation}
-x^{k+1}=\displaystyle\sum^L_{\ell=1} E_\ell y_\ell.
-\label{eq:04}
-\end{equation}
-The convergence of the multisplitting methods, based on synchronous or asynchronous iterations, is studied by many authors for example~\cite{O'leary85,bahi97,Bai99,bahi07}. %It is dependent on the condition  
-%\begin{equation}
-%\rho(\displaystyle\sum_{\ell=1}^L E_\ell M^{-1}_\ell N_\ell) < 1,
-%\label{eq:05}
-%\end{equation}
-%where $\rho$ is the spectral radius of the square matrix. 
-The multisplitting methods are convergent: 
-\begin{itemize}
-\item if $A^{-1}>0$ and the splittings of matrix $A$ are weak regular (i.e. $M^{-1}\geq 0$ and $M^{-1}N\geq 0$) when the iterations are synchronous, or
-\item if $A$ is M-matrix and its splittings are regular (i.e. $M^{-1}\geq 0$ and $N\geq 0$) when the iterations are asynchronous.
-\end{itemize}
-The solutions of the different linear sub-systems~(\ref{eq:03}) arising from the multisplitting of matrix $A$ can be either computed exactly with a direct method or approximated with an iterative method. In the latter case, the multisplitting methods are called {\it inner-outer iterative methods} or {\it two-stage multisplitting methods}. This kind of methods uses two nested iterations: the outer iteration and the inner iteration (that of the iterative method).
-
-In this paper we are focused on two-stage multisplitting methods, in their both versions synchronous and asynchronous, where the well-known iterative method GMRES ({\it Generalized Minimal RESidual})~\cite{saad86} is used as an inner iteration. Furthermore, our work in this paper is restricted to the block Jacobi splitting method. This approach of multisplitting consists in partitioning the matrix $A$ into $L$ horizontal band matrices of order $\frac{n}{L}\times n$ without overlapping (i.e. weighting matrices $E_\ell$ have only zero and one factors). In this case, the iteration of the multisplitting method presented by (\ref{eq:03}) and~(\ref{eq:04}) can be rewritten in the following form 
-\begin{equation}
-A_{\ell\ell} x_\ell^{k+1} = b_\ell - \displaystyle\sum^{L}_{\substack{m=1\\m\neq\ell}}{A_{\ell m}x^k_m},\mbox{~for~}\ell=1,\ldots,L\mbox{~and~}k=1,2,3,\ldots
-\label{eq:05}
-\end{equation}
-where $x_\ell$ are sub-vectors of the solution $x$, $b_\ell$ are the sub-vectors of the right-hand side $b$, and $A_{\ell\ell}$ and $A_{\ell m}$ are diagonal and off-diagonal blocks of matrix $A$ respectively. In each outer iteration $k$ until the convergence, each sub-system arising from the block Jacobi multisplitting
-\begin{equation}
-A_{\ell\ell} x_\ell = c_\ell,
-\label{eq:06}
-\end{equation}
-is solved iteratively using GMRES method and independently from other sub-systems by a cluster of processors. The right-hand sides $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m$ are computed using the shared vectors $x_m$. Algorithm~\ref{alg:01} shows the main key points of our block Jacobi two-stage method executed by a cluster of processors. In line~\ref{solve}, the linear sub-system~(\ref{eq:06}) is solved in parallel using GMRES method where $\MIG$ and $\TOLG$ are the maximum number of inner iterations and the tolerance threshold of GMRES respectively.  
+where right-hand sides $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m$ are computed using the shared vectors $x_m$. In this paper, we use the well-known iterative method GMRES ({\it Generalized Minimal RESidual})~\cite{saad86} as an inner iteration to approximate the solutions of the different splittings arising from the block Jacobi multisplitting of matrix $A$. The algorithm in Figure~\ref{alg:01} shows the main key points of our block Jacobi two-stage method executed by a cluster of processors. In line~\ref{solve}, the linear sub-system~(\ref{eq:03}) is solved in parallel using GMRES method where $\MIG$ and $\TOLG$ are the maximum number of inner iterations and the tolerance threshold for GMRES respectively. The convergence of the two-stage multisplitting methods, based on synchronous or asynchronous iterations, is studied by many authors for example~\cite{Bru95,bahi07}. 

 
 

-\begin{algorithm}[t]
-\caption{Block Jacobi two-stage multisplitting method}
+\begin{figure}[t]
+%\begin{algorithm}[t]
+%\caption{Block Jacobi two-stage multisplitting method}

 \begin{algorithmic}[1]
   \Input $A_\ell$ (sparse matrix), $b_\ell$ (right-hand side)
   \Output $x_\ell$ (solution vector)\vspace{0.2cm}
   \State Set the initial guess $x^0$
   \For {$k=1,2,3,\ldots$ until convergence}
     \State $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m^{k-1}$
 \begin{algorithmic}[1]
   \Input $A_\ell$ (sparse matrix), $b_\ell$ (right-hand side)
   \Output $x_\ell$ (solution vector)\vspace{0.2cm}
   \State Set the initial guess $x^0$
   \For {$k=1,2,3,\ldots$ until convergence}
     \State $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m^{k-1}$

-    \State $x^k_\ell=Solve(A_{\ell\ell},c_\ell,x^{k-1}_\ell,\MIG,\TOLG)$\label{solve}
+    \State $x^k_\ell=Solve_{gmres}(A_{\ell\ell},c_\ell,x^{k-1}_\ell,\MIG,\TOLG)$\label{solve}

     \State Send $x_\ell^k$ to neighboring clusters\label{send}
     \State Receive $\{x_m^k\}_{m\neq\ell}$ from neighboring clusters\label{recv}
   \EndFor
 \end{algorithmic}
     \State Send $x_\ell^k$ to neighboring clusters\label{send}
     \State Receive $\{x_m^k\}_{m\neq\ell}$ from neighboring clusters\label{recv}
   \EndFor
 \end{algorithmic}

+\caption{Block Jacobi two-stage multisplitting method}

 \label{alg:01}
 \label{alg:01}

-\end{algorithm}
+%\end{algorithm} 
+\end{figure}

 
 

-Multisplitting methods are more advantageous for large distributed computing platforms composed of hundreds or even thousands of processors interconnected by high latency networks. In this context, the parallel asynchronous model is preferred to the synchronous one to reduce overall execution times of the algorithms, even if it generally requires more iterations to converge. The asynchronous model allows the communications to be overlapped by computations which suppresses the idle times resulting from the synchronizations. So in asynchronous mode, our two-stage algorithm uses asynchronous outer iterations and asynchronous communications between clusters. The communications (i.e. lines~\ref{send} and~\ref{recv} in Algorithm~\ref{alg:01}) are performed by message passing using MPI non-blocking communication routines. The convergence of the asynchronous iterations is detected when all clusters have locally converged
+In this paper, we propose two algorithms of two-stage multisplitting methods. The first algorithm is based on asynchronous model which allows the communications to be overlapped by computations and reduces the idle times resulting from the synchronizations. So in the asynchronous mode, our two-stage algorithm uses asynchronous outer iterations and asynchronous communications between clusters. The communications (i.e. lines~\ref{send} and~\ref{recv} in Figure~\ref{alg:01}) are performed by message passing using MPI non-blocking communication routines. The convergence of the asynchronous iterations is detected when all clusters have locally converged

 \begin{equation}
 k\geq\MIM\mbox{~or~}\|x_\ell^{k+1}-x_\ell^k\|_{\infty }\leq\TOLM,
 \begin{equation}
 k\geq\MIM\mbox{~or~}\|x_\ell^{k+1}-x_\ell^k\|_{\infty }\leq\TOLM,

-\label{eq:07}
+\label{eq:04}

 \end{equation}    
 \end{equation}    

-where $\MIM$ is the maximum number of outer iterations and $\TOLM$ is the tolerance threshold of the two-stage algorithm. The procedure of the convergence detection is implemented as follows. All clusters are interconnected by a virtual unidirectional ring network around which a Boolean token circulates from a cluster to another.  
+where $\MIM$ is the maximum number of outer iterations and $\TOLM$ is the tolerance threshold for the two-stage algorithm. 
+
+The second two-stage algorithm is based on synchronous outer iterations. We propose to use the Krylov iteration based on residual minimization to improve the slow convergence of the multisplitting methods. In this case, a $n\times s$ matrix $S$ is set using solutions issued from the inner iteration 
+\begin{equation}
+S=[x^1,x^2,\ldots,x^s],~s\ll n.
+\label{eq:05}
+\end{equation}   
+At each $s$ outer iterations, the algorithm computes a new approximation $\tilde{x}=S\alpha$ which minimizes the residual
+\begin{equation}
+\min_{\alpha\in\mathbb{R}^s}{\|b-AS\alpha\|_2}.
+\label{eq:06}
+\end{equation} 
+The algorithm in Figure~\ref{alg:02} includes the procedure of the residual minimization and the outer iteration is restarted with a new approximation $\tilde{x}$ at every $s$ iterations. The least-squares problem~(\ref{eq:06}) is solved in parallel by all clusters using CGLS method~\cite{Hestenes52} such that $\MIC$ is the maximum number of iterations and $\TOLC$ is the tolerance threshold for this method (line~\ref{cgls} in Figure~\ref{alg:02}).
+ 
+\begin{figure}[t]
+%\begin{algorithm}[t]
+%\caption{Krylov two-stage method using block Jacobi multisplitting}
+\begin{algorithmic}[1]
+  \Input $A_\ell$ (sparse matrix), $b_\ell$ (right-hand side)
+  \Output $x_\ell$ (solution vector)\vspace{0.2cm}
+  \State Set the initial guess $x^0$
+  \For {$k=1,2,3,\ldots$ until convergence}
+    \State $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m^{k-1}$
+    \State $x^k_\ell=Solve_{gmres}(A_{\ell\ell},c_\ell,x^{k-1}_\ell,\MIG,\TOLG)$
+    \State $S_{\ell,k\mod s}=x_\ell^k$
+    \If{$k\mod s = 0$}
+       \State $\alpha = Solve_{cgls}(AS,b,\MIC,\TOLC)$\label{cgls}
+       \State $\tilde{x_\ell}=S_\ell\alpha$
+       \State Send $\tilde{x_\ell}$ to neighboring clusters
+       \Else 
+         \State Send $x_\ell^k$ to neighboring clusters
+    \EndIf
+    \State Receive $\{x_m^k\}_{m\neq\ell}$ from neighboring clusters
+  \EndFor
+\end{algorithmic}
+\caption{Krylov two-stage method using block Jacobi multisplitting}
+\label{alg:02}
+%\end{algorithm} 
+\end{figure}
+
+
+