11-12-2014 v00

[Krylov_multi.git] / krylov_multi_reviewed.tex

diff --git a/krylov_multi_reviewed.tex b/krylov_multi_reviewed.tex
index 98b443336a9663165dbd49f94758f9dbedb3ff52..4cad7d73ec9fa029376ed483855a918820bf61c7 100644 (file)
--- a/krylov_multi_reviewed.tex
+++ b/krylov_multi_reviewed.tex
@@ -8,7 +8,8 @@
 \usepackage{multirow}
 \usepackage{authblk}
 
 \usepackage{multirow}
 \usepackage{authblk}
 

-\algnewcommand\algorithmicinput{\textbf{Input:}}
+
+\algnewcommand\algorithmicinput{\textbf{I1nput:}}

 \algnewcommand\Input{\item[\algorithmicinput]}
 
 \algnewcommand\algorithmicoutput{\textbf{Output:}}
 \algnewcommand\Input{\item[\algorithmicinput]}
 
 \algnewcommand\algorithmicoutput{\textbf{Output:}}


@@ -94,7 +95,12 @@ method. In opposition to traditional multisplitting method that suffer from slow
 convergence, as  proposed in~\cite{huang1993krylov},  the use of  a minimization
 process can drastically improve the convergence.
 
 convergence, as  proposed in~\cite{huang1993krylov},  the use of  a minimization
 process can drastically improve the convergence.
 

+
+%%% AJOUTE************************
+%%%*******************************

 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 others 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.
 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 others 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 present paper is  organized as follows. First, Section~\ref{sec:02} presents
 some  related  works and  the  principle  of  multisplitting methods.  Then,  in
 
 The present paper is  organized as follows. First, Section~\ref{sec:02} presents
 some  related  works and  the  principle  of  multisplitting methods.  Then,  in


@@ -173,7 +179,14 @@ In the case where the diagonal weighting matrices $E_\ell$ have only zero and on
 
 \section{A two-stage method with a minimization}
 \label{sec:03}
 
 \section{A two-stage method with a minimization}
 \label{sec:03}

-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 multisplittig method to solve the linear system on a large computing platform in order to reduce the communications. Let the computing platform be composed of $L$ clusters of processors physically adjacent or geographically distant. In this work we apply the block Jacobi multisplitting to the linear system as follows
+
+%%% 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 $L$ clusters of processors physically adjacent or geographically distant. In this work we apply the block Jacobi multisplitting to the linear system as follows
+%%%*********************************
+%%%*********************************
+
+

 \begin{equation}
 \left\{
 \begin{array}{lll}
 \begin{equation}
 \left\{
 \begin{array}{lll}


@@ -184,7 +197,13 @@ b & = & [B_{1}, \ldots, B_{L}]
 \right.
 \label{sec03:eq01}
 \end{equation}  
 \right.
 \label{sec03:eq01}
 \end{equation}  

-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 $\sum_\ell n_\ell=n$. 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 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 $\sum_\ell n_\ell=n$. 
+%%% 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 cluster. 
+%%%********************************
+%%%********************************
+So, the multisplitting format of the linear system is defined as follows

 \begin{equation}
 \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}
 \begin{equation}
 \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}


@@ -201,7 +220,13 @@ Y_\ell = B_\ell - \displaystyle\sum_{\substack{m=1\\m\neq \ell}}^{L}A_{\ell m}X_
 \right.
 \label{sec03:eq03}
 \end{equation}
 \right.
 \label{sec03:eq03}
 \end{equation}

-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 by minimizing the residuals over an orthonormal basis of a Krylov subspace. %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}). 
+%%% MODIFIE ***********************
+%%%********************************
+GMRES is one of the most used Krylov iterative methods to solve sparse linear systems by minimizing the residuals over an orthonormal basis of a Krylov subspace. 
+%%%********************************
+%%%********************************
+%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
 
 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


@@ -214,12 +239,22 @@ number of  splitting is, the larger the  spectral radius is.  In  this paper, ou
 work is based  on   the  work   presented  in~\cite{huang1993krylov}  to   increase  the
 convergence and improve the scalability of the multisplitting methods.
 
 work is based  on   the  work   presented  in~\cite{huang1993krylov}  to   increase  the
 convergence and improve the scalability of the multisplitting methods.
 

-Krylov subspace methods are well-known for their good convergence compared to other iterative methods. In order to accelerate the convergence, we implemented the outer iteration of our 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})
+%%% AJOUTE ************************
+%%%********************************
+Krylov subspace methods are well-known for their good convergence compared to other iterative methods. 
+%%%********************************
+%%%********************************
+In order to accelerate the convergence, we implemented the outer iteration of our 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}
 \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 neither need an orthonormal basis nor any synchronization between clusters is necessary to orthogonalize the generated basis. This avoids to perform other synchronizations between the blocks of processors.
+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 clusters is necessary 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 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}
 
 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}


@@ -360,6 +395,9 @@ $743^3$ & 8,192 (4x2,048)        & 704.4     & 87,822    & 4.80e-07 &  110.3   &
 \end{table}
 
 
 \end{table}
 
 

+
+
+

 From these  experiments, it can be  observed that the  multisplitting version is
 always  faster   than  the  GMRES   version.   The  acceleration  gain   of  the
 multisplitting version ranges between 4 and 6.  It can be noticed that the number of
 From these  experiments, it can be  observed that the  multisplitting version is
 always  faster   than  the  GMRES   version.   The  acceleration  gain   of  the
 multisplitting version ranges between 4 and 6.  It can be noticed that the number of


@@ -369,6 +407,40 @@ better performance than simply using 2 clusters. In fact, we can notice that the
 precision with 2 clusters is slightly  better but in both cases the precision is
 under the specified threshold.
 
 precision with 2 clusters is slightly  better but in both cases the precision is
 under the specified threshold.
 

+
+%%% AJOUTE************************
+%%%*******************************
+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
+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 clusters  is approximately
+equals to 115. So it is not different from GMRES.
+
+
+\begin{figure}[htbp]
+\centering
+  \includegraphics[width=0.7\textwidth]{nb_iter_sec}
+\caption{Number of iterations per second  with the same parameters as in Table~\ref{tab1} (weak scaling) with only 2 clusters}
+\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  method. 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.
+%%%*******************************
+%%%*******************************
+

 \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
 \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