10-12-2014 v01

[Krylov_multi.git] / krylov_multi.tex

diff --git a/krylov_multi.tex b/krylov_multi.tex
index 1010f0775f13dc0ca215b7fe3e5c99b3dcae10ea..1ab94c6f15f34cce6269b9ba1d8248f0ba8505cd 100644 (file)
--- a/krylov_multi.tex
+++ b/krylov_multi.tex
@@ -5,14 +5,36 @@
 \usepackage{graphicx}
 \usepackage{algorithm}
 \usepackage{algpseudocode}
 \usepackage{graphicx}
 \usepackage{algorithm}
 \usepackage{algpseudocode}

-
-
-\title{A scalable multisplitting algorithm for solving large sparse linear systems} 
-
-
-
+\usepackage{multirow}
+\usepackage{authblk}
+
+\algnewcommand\algorithmicinput{\textbf{Input:}}
+\algnewcommand\Input{\item[\algorithmicinput]}
+
+\algnewcommand\algorithmicoutput{\textbf{Output:}}
+\algnewcommand\Output{\item[\algorithmicoutput]}
+
+\newcommand{\Time}[1]{\mathit{Time}_\mathit{#1}}
+\newcommand{\Prec}{\mathit{prec}}
+\newcommand{\Ratio}{\mathit{Ratio}}
+
+\def\changemargin#1#2{\list{}{\rightmargin#2\leftmargin#1}\item[]}
+\let\endchangemargin=\endlist
+
+\title{A scalable multisplitting algorithm to solve large sparse linear systems} 
+\date{}
+
+\author[1]{Raphaël Couturier}
+\author[2]{ Lilia Ziane Khodja}
+\affil[1]{ Femto-ST Institute\\
+    University of Franche Comte\\
+    France\\
+    email: raphael.couturier@univ-fcomte.fr}
+\affil[2]{Inria Bordeaux Sud-Ouest\\
+    France\\
+    email: lilia.ziane@inria.fr}

 \begin{document}
 \begin{document}

-\author{Raphaël Couturier \and Lilia Ziane Khodja}
+

 
 \maketitle
 
 
 \maketitle
 


@@ -20,140 +42,136 @@
 %%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%
 
 %%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%
 

-

 \begin{abstract}
 \begin{abstract}

-In  this  paper we  revist  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  sequential  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
 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
 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.
+problem  are  presented  with  up   to  8,192  cores.   They  show  the  obtained
+improvements compared to a classical GMRES both in terms of number of iterations
+and in terms of execution times.

 \end{abstract}
 
 \end{abstract}
 

-

 %%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%
 
 %%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%
 

-

 \section{Introduction}
 \section{Introduction}

-

 Iterative methods are used to solve  large sparse linear systems of equations of
 the form  $Ax=b$ because they are  easier to parallelize than  direct ones. Many
 Iterative methods are used to solve  large sparse linear systems of equations of
 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
+iterative  methods have  been proposed  and  adapted by  different researchers.   For

 example, the GMRES method and the  Conjugate Gradient method are very well known
 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~\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.
 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.
 
 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
+%traditional  multisplitting  method  that  suffer  from  slow  convergence,  as
+%proposed  in~\cite{huang1993krylov},  the  use  of a  minimization  process  can
+%drastically improve the convergence.
+
+Traditional parallel iterative solvers are based on fine-grain computations that
+frequently  require  data exchanges  between  computing  nodes  and have  global
+synchronizations  that penalize  the  scalability. Particularly,  they are  more
+penalized on large  scale architectures or on distributed  platforms composed of
+distant  clusters interconnected  by  a high-latency  network.  It is  therefore
+imperative to develop coarse-grain based algorithms to reduce the communications
+in the  parallel iterative  solvers. Two possible  solutions consists  either in
+using  asynchronous  iterative  methods~\cite{ref18}  or in  using  multisplitting
+algorithmss.  In this  paper,  we will  reconsider  the use  of a  multisplitting
+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.
+
+The present paper is  organized as follows. First, Section~\ref{sec:02} presents
+some  related  works and  the  principle  of  multisplitting methods.  Then,  in
+Section~\ref{sec:03}  the algorithm  of our  Krylov multisplitting
+method, based  on inner-outer  iterations, is presented. Finally, in  Section~\ref{sec:04}, the
+parallel experiments on Hector architecture  show the performances of the Krylov
+multisplitting algorithm compared to the classical GMRES algorithm to solve a 3D
+Poisson problem.

 
 

-A completer...
-On ne peut pas parler de tout...\\

 
 

+%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%
+
+\section{Related works and presentation of the multisplitting method}
+\label{sec:02}
+A general framework  to study parallel multisplitting methods has  been presented in~\cite{o1985multi}
+by O'Leary and White. Convergence conditions are given for the
+most general cases.  Many authors have improved multisplitting algorithms by proposing,
+for  example,  an  asynchronous  version~\cite{bru1995parallel} or  convergence
+conditions~\cite{bai1999block,bahi2000asynchronous}     or  other
+two-stage algorithms~\cite{frommer1992h,bru1995parallel}.
+
+In~\cite{huang1993krylov},  the  authors  have 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 solutions to  the first task which  is in
+charge of  combining 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 the
+convergence of the global system. 
+
+In~\cite{couturier2008gremlins}, the  authors have developed practical implementations
+of multisplitting algorithms to solve  large scale linear systems. Inner solvers
+could be  based on sequential direct method  with the LU method  or sequential iterative
+one with GMRES.

 
 

+In~\cite{prace-multi},  the  authors have  designed a  parallel  multisplitting
+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
+an asynchronous  multisplitting algorithm  could be more  efficient  than traditional
+solvers on an exascale architecture with hundreds of thousands of cores.

 
 

+So, compared to these works, we propose in this paper a practical multisplitting method based on parallel iterative blocks which gives better results than classical GMRES method for the 3D Poisson problem we considered.
+\\

 
 

-%%%%%%%%%%%%%%%%%%%%%%%
-%% BEGIN
-%%%%%%%%%%%%%%%%%%%%%%%
-The key idea  of the multisplitting method for  solving a large system
-of linear equations $Ax=b$ consists  in partitioning the matrix $A$ in
-$L$ several ways
+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}
 \begin{equation}

-A = M_l - N_l,~l\in\{1,\ldots,L\},
+A = M_\ell - N_\ell,

 \label{eq01}
 \end{equation}
 \label{eq01}
 \end{equation}

-where $M_l$ are nonsingular matrices. Then the linear system is solved
-by iteration based on the multisplittings as follows
+where for all $\ell\in\{1,\ldots,L\}$ $M_\ell$ are non-singular matrices. Then the linear system is solved by an iteration based on the obtained splittings as follows

 \begin{equation}
 \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}
 \label{eq02}
 \end{equation}

-where $E_l$ are non-negative and diagonal weighting matrices such that
-$\sum^L_{l=1}E_l=I$ ($I$ is an identity matrix).  Thus 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}
 \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}
 \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
+The advantage of the multisplitting method is that at each iteration $k$ there are $L$ different linear sub-systems

 \begin{equation}
 \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}
 \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
-calculations  of $v_l^k$  may be  performed in  parallel by  a  set of
-processors.   A multisplitting  method using  an iterative  method for
-solving the $L$ linear  sub-systems is called an inner-outer iterative
-method or a  two-stage method.  The results $v_l^k$  obtained from the
-different splittings~(\ref{eq04}) are combined to compute the solution
-$x^k$ 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}
 \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}    
 \label{eq05}
 \end{equation}    

-In the case where the diagonal weighting matrices $E_l$ have only zero
-and   one   factors  (i.e.   $v_l^k$   are   disjoint  vectors),   the
-multisplitting method is non-overlapping  and corresponds to the block
-Jacobi method.
-%%%%%%%%%%%%%%%%%%%%%%%
-%% END
-%%%%%%%%%%%%%%%%%%%%%%%
-
-\section{Related works}
-
-
-A general framework  for studying parallel multisplitting has  been presented in
-\cite{o1985multi} by O'Leary and White. Convergence conditions are given for the
-most general case.  Many authors improved multisplitting algorithms by proposing
-for  example  an  asynchronous  version  \cite{bru1995parallel}  and  convergence
-conditions  \cite{bai1999block,bahi2000asynchronous}   in  this  case   or  other
-two-stage algorithms~\cite{frommer1992h,bru1995parallel}.
-
-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
-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 update 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
-solve large scale linear systems. Inner  solvers could be based on scalar direct
-method with the LU method or scalar iterative one with GMRES.
-
-In~\cite{prace-multi},  the  authors  have  proposed a  parallel  multisplitting
-algorithm in which large block are solved using a GMRES solver. The authors have
-performed large scale experimentations upto  32.768 cores and they conclude that
-asynchronous  multisplitting algorithm  could more  efficient  than traditionnal
-solvers on exascale architecture with hunders of thousands of cores.
-
+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.

 
 %%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%
 
 
 %%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%
 

-

 \section{A two-stage method with a minimization}
 \section{A two-stage method with a minimization}

-Let $Ax=b$ be a given sparse  and large linear system of $n$ equations
-to  solve  in  parallel   on  $L$  clusters,  physically  adjacent  or
-geographically distant, where $A\in\mathbb{R}^{n\times n}$ is a square
-and  nonsingular 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:
+\label{sec:03}
+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}
 \begin{equation}
 \left\{
 \begin{array}{lll}


@@ -164,78 +182,49 @@ b & = & [B_{1}, \ldots, B_{L}]
 \right.
 \label{sec03:eq01}
 \end{equation}  
 \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 $n_l$, such
-that  $\sum_ln_l=n$.  In this  case,  we  use  a row-by-row  splitting
-without overlapping in  such a way that successive  rows of the 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}
 \begin{equation}

-\forall l\in\{1,\ldots,L\} \mbox{,~} \displaystyle\sum_{i=1}^{l-1}A_{li}X_i + A_{ll}X_l + \displaystyle\sum_{i=l+1}^{L}A_{li}X_i = 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} 
 \label{sec03:eq02}
 \end{equation} 

-where $A_{li}$ is  a block of size $n_l\times  n_i$ of the rectangular
-matrix  $A_l$, $X_i\neq  X_l$ is  a sub-vector  of size  $n_i$  of the
-solution vector  $x$ and $\sum_{i<l}n_i+\sum_{i>l}n_i+n_l=n$,  for all
-$i\in\{1,\ldots,l-1,l+1,\ldots,L\}$.
+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 $m\in\{1,\ldots,L\}$.

 
 

-The multisplitting method proceeds by iteration for solving the linear
-system in such a way each sub-system
+Our multisplitting method proceeds by iteration to solve the linear system in such a way that each sub-system

 \begin{equation}
 \left\{
 \begin{array}{l}
 \begin{equation}
 \left\{
 \begin{array}{l}

-A_{ll}X_l = Y_l \mbox{,~such that}\\
-Y_l = B_l - \displaystyle\sum_{i=1,i\neq l}^{L}A_{li}X_i,
+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}
 \end{array}
 \right.
 \label{sec03:eq03}
 \end{equation}

-is solved  independently by a cluster of  processors and communication
-are required  to update the  right-hand side vectors $Y_l$,  such that
-the  vectors  $X_i$  represent   the  data  dependencies  between  the
-clusters. In this work,  we use the parallel GMRES method~\cite{ref34}
-as     an     inner    iteration     method     for    solving     the
-sub-systems~(\ref{sec03:eq03}).  It  is a well-known  iterative method
-which  gives good performances  for solving  sparse linear  systems in
-parallel on a cluster of processors.
-
-It should be noted that  the convergence of the inner iterative solver
-for  the  different  linear  sub-systems~(\ref{sec03:eq03})  does  not
-necessarily involve  the convergence of the  multisplitting method. It
-strongly depends on  the properties of the 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  implement  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  space of  the method that  we used  is spanned by  a basis
-composed  of   successive  solutions  issued  from   solving  the  $L$
-splittings~(\ref{sec03:eq03})
+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~\cite{o1985multi,ref18}. Furthermore, the  splitting of the linear system
+among  several clusters  of  processors  increases the  spectral  radius of  the
+iteration  matrix, thereby  slowing the  convergence.  In  fact, the  larger the
+number of  splitting is, the larger the  spectral radius is.  In  this paper, our
+work is 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 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 need  neither an orthogonal basis  nor synchronizations between  the different 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  the vectors  of  $S$.  So,  $\alpha$  is  defined as  the
-solution of the large overdetermined linear system
+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 orthogonal basis nor any synchronization 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

 \begin{equation}
 R\alpha=b,
 \label{sec03:eq05}
 \end{equation}
 \begin{equation}
 R\alpha=b,
 \label{sec03:eq05}
 \end{equation}

-where $R=AS$  is a  dense rectangular matrix  of size $n\times  s$ and
-$s\ll n$. This leads us to solve the system of normal equations
+where $R=AS$ is a dense rectangular matrix of size $n\times s$ and $s\ll n$. This leads us to solve a system of normal equations

 \begin{equation}
 R^TR\alpha=R^Tb,
 \label{sec03:eq06}
 \begin{equation}
 R^TR\alpha=R^Tb,
 \label{sec03:eq06}


@@ -245,72 +234,180 @@ which is associated with the least squares problem
 \text{minimize}~\|b-R\alpha\|_2,
 \label{sec03:eq07}
 \end{equation}  
 \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 for  solving 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]
 
 \begin{algorithm}[!t]
 \caption{A two-stage linear solver with inner iteration GMRES method}
 \begin{algorithmic}[1]

-\State Load $A_l$, $B_l$, initial guess $x^0$
-\State Initialize the minimizer $\tilde{x}^0=x^0$
+\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}
 \For {$k=1,2,3,\ldots$ until the global convergence}

-\State Restart with $x^0=\tilde{x}^{k-1}$: \textbf{for} $j=1,2,\ldots,s$ \textbf{do}
-\State\hspace{0.5cm} Inner iteration solver: \Call{InnerSolver}{$x^0$, $j$}
-\State\hspace{0.5cm} Construct the basis $S$: add the column vector $X_l^j$ to the matrix $S_l^k$
-\State\hspace{0.5cm} Exchange the local solution vector $X_l^j$ with the neighboring clusters
-\State\hspace{0.5cm} Compute the dense matrix $R$: $R_l^{k,j}=\sum^L_{i=1}A_{li}X_i^j$ 
-\State\textbf{end for} 
-\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 Restart with $x^0=\tilde{x}^{k-1}$:
+\For {$j=1,2,\ldots,s$}
+\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 \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}{$x^0$, $j$}
 \EndFor
 
 \Statex
 
 \Function {InnerSolver}{$x^0$, $j$}

-\State Compute the local right-hand side: $Y_l = B_l - \sum^L_{i=1,i\neq l}A_{li}X_i^0$
-\State Solving the local splitting $A_{ll}X_l^j=Y_l$ using the 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
 
 \EndFunction
 
 \Statex
 

-\Function {UpdateMinimizer}{$S_l$, $R$, $b$, $k$}
-\State Solving the normal equations $(R^k)^TR^k\alpha^k=(R^k)^Tb$ in parallel by $L$ clusters using the parallel CGNR method
-\State Compute the local minimizer: $\tilde{X}_l^k=S_l^k\alpha^k$
-\State \Return $\tilde{X}_l^k$
+\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 $\tilde{X}_\ell^k=S_\ell^k\alpha^k$
+\State \Return $\tilde{X}_\ell^k$

 \EndFunction
 \end{algorithmic}
 \label{algo:01}
 \end{algorithm}
 
 \EndFunction
 \end{algorithmic}
 \label{algo:01}
 \end{algorithm}
 

-The main  key points  of the multisplitting  method for  solving large
-sparse  linear  systems are  given  in Algorithm~\ref{algo:01}.   This
-algorithm is based on a two-stage method with a minimization using the
-GMRES iterative method as an  inner solver. It is executed in parallel
-by  each cluster  of processors.   The matrices  and vectors  with the
-subscript  $l$ represent  the local  data for  the cluster  $l$, where
-$l\in\{1,\ldots,L\}$. The two-stage solver uses two different parallel
-iterative algorithms: the GMRES method for solving each splitting on a
-cluster of processors, and the CGNR method executed in parallel by all
-clusters for  minimizing the function  error over the  Krylov subspace
-spanned by  $S$.  The  algorithm requires two  global synchronizations
-between the $L$  clusters. The first one is  performed at line~$12$ in
-Algorithm~\ref{algo:01}  to exchange  the local  values of  the 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 choose  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
-matrix-matrix    multiplication     $R=AS$.     We    implement    all
-synchronizations   by   using   the   MPI   collective   communication
-subroutines.
+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: the GMRES method to solve each splitting~(\ref{sec03:eq03}) on a cluster of processors, and the 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  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.
+
+%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%

 
 

+\section{Experiments}
+\label{sec:04}
+In order to illustrate  the interest  of our algorithm, we have  compared our
+algorithm  with  the  GMRES  method  which  is a commonly  used  method  in  many
+situations.  We have chosen to focus on only one problem which is very simple to
+implement: a 3 dimension Poisson problem.
+
+\begin{equation}
+\left\{
+                \begin{array}{ll}
+                  \nabla u&=f \mbox{~in~} \omega\\
+                  u &=0 \mbox{~on~}  \Gamma=\partial \omega
+                \end{array}
+              \right.
+\end{equation}

 
 

+After discretization, with a finite  difference scheme, a seven point stencil is
+used. It  is well-known that the  spectral radius of  matrices representing such
+problems are very close to 1.  Moreover, the larger the number of discretization
+points is,  the closer to 1  the spectral radius  is.  Hence, to solve  a matrix
+obtained for  a 3D Poisson  problem, the number  of iterations is high.  Using a
+preconditioner  it  is   possible  to  reduce  the  number   of  iterations  but
+preconditioners are not scalable when using many cores.
+
+%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
+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.
+
+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
+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
+brackets. For  the GMRES code (alone  and in the  multisplitting version) the
+restart parameter is fixed to 16. The precision of the GMRES version is fixed to
+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  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.
+
+\begin{table}[htbp]
+\begin{center}
+\begin{changemargin}{-1.8cm}{0cm}
+\begin{small}
+\begin{tabular}{|c|c||c|c|c||c|c|c||c|} 
+\hline
+\multirow{2}{*}{Pb size}&\multirow{2}{*}{Nb. cores} &  \multicolumn{3}{c||}{GMRES} &  \multicolumn{3}{c||}{Krylov Multisplitting} & \multirow{2}{*}{Ratio}\\
+ \cline{3-8}
+           &                   &  Time (s) & nb Iter. & $\Delta$  &   Time (s)& nb Iter. & $\Delta$ & \\
+\hline
+$468^3$ & 2,048 (2x1,024)        &  299.7    & 41,028    & 5.02e-8  &  48.4    & 691(6,146) & 8.24e-08  & 6.19   \\
+\hline
+$590^3$ & 4,096 (2x2,048)        &  433.1    & 55,494    & 4.92e-7  &  74.1    & 1,101(8,211) & 6.62e-08  & 5.84   \\
+\hline
+$743^3$ & 8,192 (2x4,096)        & 704.4     & 87,822    & 4.80e-07 &  151.2   & 3,061(14,914) & 5.87e-08 & 4.65    \\
+\hline
+$743^3$ & 8,192 (4x2,048)        & 704.4     & 87,822    & 4.80e-07 &  110.3   & 1,531(12,721) & 1.47e-07& 6.39  \\
+\hline
+
+\end{tabular}
+\caption{Results}
+\label{tab1}
+\end{small}
+\end{changemargin}
+\end{center}
+\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
+iterations is drastically reduced with the multisplitting version even it is not
+negligible. Moreover, with 8,192 cores, we  can see that using 4 clusters gives a
+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.
+
+\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  multisaplitting which  uses GMRES
+iterative  method as  an inner  iteration.  Our  contribution in  this  paper is
+twofold. First we provide a multi cluster decomposition that allows us to choose
+the  appropriate size  of  the clusters  according  to the  architecures of  the
+supercomputer.  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 tested  our multisplitting  method to  solve the  sparse  linear system
+issued from  the discretization of  a 3D Poisson  problem. We have  compared its
+performances to the  classical GMRES method on a  supercomputer composed of 2,048
+to 8,192 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 the  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 numbers  of cores and
+test  the  scalability  of  our   Krylov  multisplitting  method.  It  would  be
+interesting  to validate its  performances to  solve other  linear/nonlinear and
+symmetric/nonsymmetric problems.  Moreover, we intend  to develop multisplitting
+methods based  on asynchronous iterations 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.
+
+\section{Acknowledgement}
+The authors would like to thank Mark Bull of the EPCC his fruitful remarks and the facilities of HECToR.
+
+%Other applications (=> other matrices)\\
+%Larger experiments\\
+%Async\\
+%Overlapping\\
+%preconditioning

 
 
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