[rce2015.git] / paper.tex

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\begin{document}
\title{Grid-enabled simulation of large-scale linear iterative solvers}
%\itshape{\journalnamelc}\footnotemark[2]}

\author{Charles Emile Ramamonjisoa\affil{1},
  Lilia Ziane Khodja\affil{2},
  David Laiymani\affil{1},
  Raphaël Couturier\affil{1} and
  Arnaud Giersch\affil{1}
}

\address{
  \affilnum{1}%
  Femto-ST Institute, DISC Department,
  University of Franche-Comté,
  Belfort, France.
  Email:~\email{{charles.ramamonjisoa,david.laiymani,raphael.couturier,arnaud.giersch}@univ-fcomte.fr}\break
  \affilnum{2}
  Department of Aerospace \& Mechanical Engineering,
  Non Linear Computational Mechanics,
  University of Liege, Liege, Belgium.
  Email:~\email{l.zianekhodja@ulg.ac.be}
}

\begin{abstract} %% The behavior of multi-core applications is always a challenge
%% to predict, especially with a new architecture for which no experiment has been
%% performed. With some applications, it is difficult, if not impossible, to build
%% accurate performance models. That is why another solution is to use a simulation
%% tool which allows us to change many parameters of the architecture (network
%% bandwidth, latency, number of processors) and to simulate the execution of such
%% applications. The main contribution of this paper is to show that the use of a
%% simulation tool (here we have decided to use the SimGrid toolkit) can really
%% help developers to better tune their applications for a given multi-core
%% architecture.

%% In this paper we focus our attention on the simulation of iterative algorithms to solve sparse linear systems on large clusters. We study the behavior of the widely used GMRES algorithm and two different variants of the Multisplitting algorithms: one using synchronous iterations and another one with asynchronous iterations.
%% For each algorithm we have simulated
%% different architecture parameters to evaluate their influence on the overall
%% execution time.
%% The simulations confirm the real results previously obtained on different real multi-core architectures and also confirm the efficiency of the asynchronous Multisplitting algorithm on distant clusters compared to the synchronous GMRES algorithm.

The behavior of multi-core applications always proves quite challenging to predict, especially with a new architecture for which no experiment has yet been performed. With some applications, it is difficult, if not impossible, to build accurate performance models. That is why another solution is to use a simulation tool which allows us to change many parameters of the architecture (network bandwidth, latency, number of processors) and to simulate the execution of such applications.

In this paper we focus on the simulation of iterative algorithms to solve sparse linear systems. We study the behavior of the GMRES algorithm and two different variants of the multisplitting algorithms: using synchronous or asynchronous iterations. For each algorithm we have simulated different architecture parameters to evaluate their influence on the overall execution time. The simulations confirm the real results previously obtained on different real multi-core architectures and also confirm the efficiency of the asynchronous multisplitting algorithm on distant clusters compared to the GMRES algorithm.

\end{abstract}

%\keywords{Algorithm; distributed; iterative; asynchronous; simulation; simgrid;
%performance}
\keywords{ Performance evaluation, Simulation, SimGrid,  Synchronous and asynchronous iterations, Multisplitting algorithms}

\maketitle

\section{Introduction}  The use of multi-core architectures to solve large
scientific problems seems to  become imperative  in  many situations.
Whatever the scale of these architectures (distributed clusters, computational
grids, embedded multi-core,~\ldots) they  are generally  well adapted to execute
complex parallel applications operating on a large amount of data.
Unfortunately,  users (industrials or scientists),  who need such computational
resources, may not have an easy access to such efficient architectures. The cost
of using the platform and/or the cost of  testing and deploying an application
are often very important. So, in this context, it is difficult to optimize a
given application for a given  architecture. In this way and in order to reduce
the access cost to these computing resources it seems very interesting to use a
simulation environment.  The advantages are numerous: life cycle development,
code debugging, ability to obtain results quickly\dots{} In return, the simulation results need to be consistent with the real ones.

In this paper we focus on a class of highly efficient parallel algorithms called
\emph{iterative algorithms}. The parallel scheme of iterative methods is quite
simple. It generally involves the division of the problem into  several
\emph{blocks}  that  will  be  solved  in  parallel  on  multiple processing
units.  Each processing unit has to compute an iteration to send/receive some
data dependencies to/from its neighbors and to iterate this process until the
convergence of the method. Several well-known studies demonstrate the
convergence of these algorithms~\cite{BT89,bahi07}. In this processing mode a
task cannot begin a new iteration while it has not received data dependencies
from its neighbors. The iteration computation is said to follow a
\textit{synchronous} scheme. In the asynchronous scheme a task can compute a new
iteration without having to wait for the data dependencies coming from its
neighbors. Both communications and computations are \textit{asynchronous}
inducing that there is no more idle time, due to synchronizations, between two
iterations~\cite{bcvc06:ij}. This model presents some advantages and drawbacks
that we detail in Section~\ref{sec:asynchro}. Even if the number of
iterations required to converge is generally  greater  than for the synchronous
case, it appears that the asynchronous  iterative scheme  can significantly
reduce  overall execution times by  suppressing idle  times due to
synchronizations~(see~\cite{bahi07} for more details).

Nevertheless,  in both  cases  (synchronous  or asynchronous)  it  is extremely  time
consuming to find optimal configurations  and deployment requirements for a given
application  on   a  given   multi-core  architecture.  Finding   good  resource
allocations policies under  varying CPU power, network speeds and  loads is very
challenging and  labor intensive~\cite{Calheiros:2011:CTM:1951445.1951450}. This
problematic is  even more difficult  for the  asynchronous scheme where  a small
parameter variation of the execution platform and of the application data can
lead to very different numbers of iterations to reach the convergence and consequently to
very different execution times. In this challenging context we think that the
use of a simulation tool can greatly leverage the possibility of testing various
platform scenarios.

The  {\bf main  contribution  of  this paper}  is  to show  that  the  use of  a
simulation tool (i.e. the SimGrid toolkit~\cite{SimGrid}) in the context of real
parallel applications (i.e. large linear  system solvers) can help developers to
better tune their  applications for a given multi-core architecture.  To show the
validity of this approach we first compare the simulated execution of the Krylov
multisplitting  algorithm   with  the   GMRES  (Generalized   Minimal  RESidual)
solver~\cite{saad86} in  synchronous mode.  The simulation  results allow  us to
determine  which method  to choose  for a given multi-core  architecture.
Moreover, the  obtained results  on different simulated  multi-core architectures
confirm the  real results  previously obtained  on real physical architectures.
More precisely the simulated results are in accordance (i.e. with the same order
of magnitude)  with the works  presented in~\cite{couturier15}, which  show that
the synchronous  Krylov multisplitting method  is more efficient  than GMRES  for large
scale  clusters.   Simulated   results  also  confirm  the   efficiency  of  the
asynchronous  multisplitting   algorithm  compared  to  the   synchronous  GMRES
especially in case of geographically distant clusters.

Thus, with a simple computing architecture (a laptop) SimGrid allows us
to run a test campaign  of  real parallel iterative  applications on
different simulated multi-core architectures.  To our knowledge, there is no
related work on the large-scale multi-core simulation of a real synchronous and
asynchronous iterative application.

This paper is organized as follows. Section~\ref{sec:asynchro} presents the
iteration model we use and more particularly the asynchronous scheme.  In
Section~\ref{sec:simgrid} the SimGrid simulation toolkit is presented.
Section~\ref{sec:04} details the different solvers that we use.  Finally our
experimental results are presented in Section~\ref{sec:expe} followed by some
concluding remarks and perspectives.


\section{The asynchronous iteration model and the motivations of our work}
\label{sec:asynchro}

Asynchronous iterative methods have been  studied for many years both theoretically and
practically. Many methods have been considered and convergence results have been
proved. These  methods can  be used  to solve, in  parallel, fixed  point problems
(i.e. problems  for which  the solution is  $x^\star =f(x^\star)$).  In practice,
asynchronous iteration  methods can be used  to solve, for example,  linear and
non-linear systems of equations or optimization problems. Interested readers are
invited to read~\cite{BT89,bahi07}.

Before  using  an  asynchronous  iterative   method,  the  convergence  must  be
studied. Otherwise, there is no guarantee that the  application will reach  the convergence. An
algorithm that supports both the synchronous or the asynchronous iteration model
requires very few modifications  to be able to be executed  in both variants. In
practice, only  the communications management and  the convergence detection are  different. In
the synchronous  mode, iterations are  synchronized, whereas, in  the asynchronous
one, they are not.  It should be noticed that non-blocking communications can be
used in both  modes. Concerning the convergence  detection, synchronous variants
can use  a global convergence procedure  which acts as a  global synchronization
point. In the  asynchronous model, the convergence detection is  more tricky as
it   must  not   synchronize  all   the  processors.   Interested  readers   can
consult~\cite{myBCCV05c,bahi07,ccl09:ij}.

The number of iterations required to reach the convergence is generally greater
for the asynchronous scheme (this number depends on  the delay of the
messages). Note that, it is not the case in the synchronous mode where the
number of iterations is the same as in the sequential mode. Thus, the
set of the parameters  of the  platform (number  of nodes,  power of nodes,
inter and  intra clusters  bandwidth  and  latency,~\ldots) and  of  the
application can drastically change the number of iterations required to get the
convergence. It follows that asynchronous iterative algorithms are difficult to
optimize since the financial and deployment costs on large scale multi-core
architectures are often very important. So, prior to deployment and tests it
seems very promising to be able to simulate the behavior of asynchronous
iterative algorithms. The problematic is then to show that the results produced
by simulation are in accordance with reality (i.e. of the same order of
magnitude). To our knowledge, there is no study on this problematic.

\section{SimGrid}
\label{sec:simgrid}
<<<<<<< HEAD
In the scope of this paper, we have chosen the SimGrid toolkit~\cite{SimGrid,casanova+giersch+legrand+al.2014.versatile} to simulate the behavior of parallel iterative linear solvers on different computational grid configurations. Contrary to most simulators which remain very application-oriented, the SimGrid framework is designed to study the behavior of many large-scale distributed computing platforms as Grids, Peer-to-Peer systems, Clouds or High Performance Computation systems. It is still actively developed by the scientific community and distributed as an open source software.

SimGrid provides four user interfaces which can be convenient for different distributed applications~\cite{casanova+legrand+quinson.2008.simgrid}. In this paper we are interested in the SMPI user interface (Simulator MPI) which implements about \np[\%]{80} of the MPI 2.0 standard and allows minor modifications of the initial code~\cite{bedaride+degomme+genaud+al.2013.toward} (see Section~\ref{sec:04.02}). SMPI enables the direct simulation of the execution, as in real life, of an unmodified MPI distributed application, and gets accurate results with the detailed resources consumption.

The SimGrid simulator uses at least three XML input files describing the computational grid resources: the number of clusters in the grid, the number of processors/cores in each cluster, the detailed description of the intra and inter networks and the list of the hosts in each cluster (see the details in Section~\ref{sec:expe}). SimGrid uses a fluid model to simulate the program execution. It allows several simulation modes which produce accurate results~\cite{bedaride+degomme+genaud+al.2013.toward,velho+schnorr+casanova+al.2013.validity}. For instance, the "in vivo" mode really executes the computation but "intercepts" the communications (the execution time is then evaluated according to the parameters of the simulated platform). It is also possible for SimGrid/SMPI to only keep the duration of large computations by skipping them. Moreover the application can be run in an "in vitro" mode by sharing some structures between the simulated processes and thus allowing the use of very large-scale data.

The choice of SimGrid/SMPI as a simulator tool in this study has been emphasized by the results obtained by several studies to validate, in real environments, the behavior of different network models simulated in SimGrid~\cite{velho+schnorr+casanova+al.2013.validity}. Other studies underline the comparison between the real MPI application executions and the SimGrid/SMPI ones~\cite{guermouche+renard.2010.first,clauss+stillwell+genaud+al.2011.single,bedaride+degomme+genaud+al.2013.toward}. These works show the accuracy of SimGrid simulations compared to the executions on real physical architectures.











=======
>>>>>>> e2082ae796c90fe550e1ad8e44f43f9d5bfedf95

In the scope of this paper, we have chosen the SimGrid
toolkit~\cite{SimGrid,casanova+legrand+quinson.2008.simgrid,casanova+giersch+legrand+al.2014.versatile}
to simulate the behavior of parallel iterative linear solvers on different
computational grid configurations. In opposite to most of the simulators which
are stayed very application-oriented, the SimGrid framework is designed to study
the behavior of many large-scale distributed computing platforms as Grids,
Peer-to-Peer systems, Clouds or High Performance Computation systems. It is
still actively developed by the scientific community and distributed as an open
source software.

SimGrid provides four user interfaces which can be convenient for different
distributed applications.  In this paper we are interested on the SMPI
(Simulated MPI) user interface which implements about \np[\%]{80} of the MPI 2.0
standard~\cite{bedaride+degomme+genaud+al.2013.toward}, and allows minor
modifications of the initial code (see Section~\ref{sec:04.02}). SMPI enables
the direct simulation of the execution, as in the real life, of an unmodified
MPI distributed application, and gets accurate results with the detailed
resources consumption.

SimGrid simulator uses an XML input file describing the computational grid
resources: the number of clusters in the grid, the number of processors/cores in
each cluster, the detailed description of the intra and inter networks and the
list of the hosts in each cluster (see the details in
Section~\ref{sec:expe}). SimGrid employs a fluid model to simulate the use of
these resources along the program execution.  This model produces accurate
results while still running relatively
fast~\cite{bedaride+degomme+genaud+al.2013.toward,velho+schnorr+casanova+al.2013.validity}.
During the simulation, the computation is really executed, but the communications
are intercepted and their execution time evaluated according to the parameters
of the simulated platform. It is also possible for SimGrid/SMPI to only keep the
duration of large computations by skipping them.  Moreover, when applicable, the
application can be run by sharing some in-memory structures between the
simulated processes and thus allowing the use of very large-scale data.

The choice of SimGrid/SMPI as a simulator tool in this study has been emphasized
by the results obtained by several studies to validate, in the real
environments, the behavior of different network models simulated in
SimGrid~\cite{velho+schnorr+casanova+al.2013.validity}. Other studies underline
the comparison between the real MPI application executions and the SimGrid/SMPI
ones~\cite{guermouche+renard.2010.first,clauss+stillwell+genaud+al.2011.single,bedaride+degomme+genaud+al.2013.toward}. These
works show the accuracy of SimGrid simulations compared to the executions on
real physical architectures.

%% In the scope of this paper, the SimGrid toolkit~\cite{SimGrid,casanova+legrand+quinson.2008.simgrid,casanova+giersch+legrand+al.2014.versatile},
%% an open source framework actively developed by its scientific community, has been chosen to simulate the behavior of iterative linear solvers in different computational grid configurations. SimGrid pretends to be non-specialized in opposite to some other simulators which stayed to be very specific oriented-application. One of the well-known SimGrid advantage is its SMPI (Simulated MPI) user interface. SMPI purpose is to execute by simulation in a similar way as in real life, an MPI distributed application and to get accurate results with the detailed resources
%% consumption.Several studies have demonstrated the accuracy of the simulation
%% compared with execution on real physical architectures. In addition of SMPI,
%% Simgrid provides other API which can be convienent for different distrbuted
%% applications: computational grid applications, High Performance Computing (HPC),
%% P2P but also clouds applications. In this paper we use the SMPI API. It
%% implements about \np[\%]{80} of the MPI 2.0 standard and allows minor
%% modifications of the initial code~\cite{bedaride+degomme+genaud+al.2013.toward}
%% (see Section~\ref{sec:04.02}).


%%  Provided as an input to the simulator, at least $3$ XML files describe the
%%  computational grid resources: number of clusters in the grid, number of
%%  processors/cores in each cluster, detailed description of the intra and inter
%%  networks and the list of the hosts in each cluster (see the details in Section~\ref{sec:expe}). Simgrid uses a fluid model to simulate the program execution.
%%  This gives several simulation modes which produce accurate
%%  results~\cite{bedaride+degomme+genaud+al.2013.toward,
%%  velho+schnorr+casanova+al.2013.validity}. For instance, the "in vivo" mode
%%  really executes the computation but "intercepts" the communications (running
%%  time is then evaluated according to the parameters of the simulated platform).
%%  It is also possible for SimGrid/SMPI to only keep duration of large
%%  computations by skipping them. Moreover the application can be run "in vitro"
%%  by sharing some in-memory structures between the simulated processes and
%%  thus allowing the use of very large data scale.


%% The choice of Simgrid/SMPI as a simulator tool in this study has been emphasized
%% by the results obtained by several studies to validate, in real environments,
%% the behavior of different network models simulated in
%% Simgrid~\cite{velho+schnorr+casanova+al.2013.validity}. Other studies underline
%% the comparison between real MPI executions  and SimGrid/SMPI
%% ones\cite{guermouche+renard.2010.first, clauss+stillwell+genaud+al.2011.single,
%% bedaride+degomme+genaud+al.2013.toward}. These works show the accuracy of
%% SimGrid simulations.






% SimGrid~\cite{SimGrid,casanova+legrand+quinson.2008.simgrid,casanova+giersch+legrand+al.2014.versatile} is a discrete event simulation framework to study the behavior of large-scale distributed computing platforms as Grids, Peer-to-Peer systems, Clouds and High Performance Computation systems. It is widely used to simulate and evaluate heuristics, prototype applications or even assess legacy MPI applications. It is still actively developed by the scientific community and distributed as an open source software.
%
% %%%%%%%%%%%%%%%%%%%%%%%%%
% % SimGrid~\cite{SimGrid,casanova+legrand+quinson.2008.simgrid,casanova+giersch+legrand+al.2014.versatile}
% % is a simulation framework to study the behavior of large-scale distributed
% % systems.  As its name suggests, it emanates from the grid computing community,
% % but is nowadays used to study grids, clouds, HPC or peer-to-peer systems.  The
% % early versions of SimGrid date back from 1999, but it is still actively
% % developed and distributed as an open source software.  Today, it is one of the
% % major generic tools in the field of simulation for large-scale distributed
% % systems.
%
% SimGrid provides several programming interfaces: MSG to simulate Concurrent
% Sequential Processes, SimDAG to simulate DAGs of (parallel) tasks, and SMPI to
% run real applications written in MPI~\cite{MPI}.  Apart from the native C
% interface, SimGrid provides bindings for the C++, Java, Lua and Ruby programming
% languages.  SMPI is the interface that has been used for the work described in
% this paper.  The SMPI interface implements about \np[\%]{80} of the MPI 2.0
% standard~\cite{bedaride+degomme+genaud+al.2013.toward}, and supports
% applications written in C or Fortran, with little or no modifications (cf Section IV - paragraph B).
%
% Within SimGrid, the execution of a distributed application is simulated by a
% single process.  The application code is really executed, but some operations,
% like communications, are intercepted, and their running time is computed
% according to the characteristics of the simulated execution platform.  The
% description of this target platform is given as an input for the execution, by
% means of an XML file.  It describes the properties of the platform, such as
% the computing nodes with their computing power, the interconnection links with
% their bandwidth and latency, and the routing strategy.  The scheduling of the
% simulated processes, as well as the simulated running time of the application
% are computed according to these properties.
%
% To compute the durations of the operations in the simulated world, and to take
% into account resource sharing (e.g. bandwidth sharing between competing
% communications), SimGrid uses a fluid model.  This allows users to run relatively fast
% simulations, while still keeping accurate
% results~\cite{bedaride+degomme+genaud+al.2013.toward,
%   velho+schnorr+casanova+al.2013.validity}.  Moreover, depending on the
% simulated application, SimGrid/SMPI allows to skip long lasting computations and
% to only take their duration into account.  When the real computations cannot be
% skipped, but the results are unimportant for the simulation results, it is
% also possible to share dynamically allocated data structures between
% several simulated processes, and thus to reduce the whole memory consumption.
% These two techniques can help to run simulations on a very large scale.
%
% The validity of simulations with SimGrid has been asserted by several studies.
% See, for example, \cite{velho+schnorr+casanova+al.2013.validity} and articles
% referenced therein for the validity of the network models.  Comparisons between
% real execution of MPI applications on the one hand, and their simulation with
% SMPI on the other hand, are presented in~\cite{guermouche+renard.2010.first,
%   clauss+stillwell+genaud+al.2011.single,
%   bedaride+degomme+genaud+al.2013.toward}.  All these works conclude that
% SimGrid is able to simulate pretty accurately the real behavior of the
% applications.
%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Two-stage multisplitting methods}
\label{sec:04}
\subsection{Synchronous and asynchronous two-stage methods for sparse linear systems}
\label{sec:04.01}
In this paper we focus on two-stage multisplitting methods in both their 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}
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). Two-stage multisplitting methods solve the linear system~(\ref{eq:01}) iteratively as follows:
\begin{equation}
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}
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 so that each processor or cluster of processors is responsible for solving one splitting as a linear sub-system:
\begin{equation}
A_{\ell\ell} x_\ell = c_\ell,\mbox{~for~}\ell=1,\ldots,L,
\label{eq:03}
\end{equation}
where the right-hand sides $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m$ are computed using the shared vectors $x_m$. In this paper, we use the well-known iterative method GMRES~\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. Line~\ref{solve}, the linear sub-system~(\ref{eq:03}) is solved in parallel using the 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, has been studied by many authors for example~\cite{Bru95,bahi07}.

\begin{figure}[htpb]
%\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 $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m^{k-1}$
    \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}
\caption{Block Jacobi two-stage multisplitting method}
\label{alg:01}
%\end{algorithm}
\end{figure}

In this paper, we propose two algorithms of two-stage multisplitting methods. The first algorithm is based on the asynchronous scheme which allows 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,
\label{eq:04}
\end{equation}
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 the CGLS method~\cite{Hestenes52} sosuch 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}[htbp]
%\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}

\subsection{Simulation of the two-stage methods using SimGrid toolkit}
\label{sec:04.02}

One of our objectives when simulating the  application in SimGrid is, as in real
life, to  get accurate results  (solutions of the  problem) but also to ensure the
test reproducibility  under similar  conditions.  According to  our experience,
very  few modifications  are required  to adapt  a MPI  program for  the SimGrid
simulator using SMPI (Simulated MPI). The  first modification is to include SMPI
libraries  and related  header files  (\verb+smpi.h+).  The  second modification  is to
suppress all global variables by replacing  them with local variables or using a
SimGrid selector       called      "runtime       automatic      switching"
(smpi/privatize\_global\_variables). Indeed, global  variables can generate side
effects on runtime between the threads running in the same process and generated by
SimGrid  to simulate the  grid environment.

\paragraph{Simulation parameters for SimGrid}
\  \\ \noindent  Before running  a SimGrid  benchmark, many  parameters for  the
computation platform must be defined. For our experiments, we consider platforms
in which  several clusters are  geographically distant,  so there are  intra and
inter-cluster communications. In the following, these parameters are described:

\begin{itemize}
        \item hostfile: hosts description file,
        \item platform: file describing the platform architecture: clusters (CPU power,
\dots{}), intra cluster network description, inter cluster network (bandwidth $bw$,
latency $lat$, \dots{}),
        \item archi   : grid computational description (number of clusters, number of
nodes/processors in each cluster).
\end{itemize}
\noindent
In addition, the following arguments are given to the programs at runtime:

\begin{itemize}
        \item maximum number of inner iterations $\MIG$ and outer iterations $\MIM$,
        \item inner precision $\TOLG$ and outer precision $\TOLM$,
        \item matrix sizes of the problem: N$_{x}$, N$_{y}$ and N$_{z}$ on axis $x$, $y$ and $z$ respectively (in our experiments, we solve 3D problem, see Section~\ref{3dpoisson}),
        \item matrix diagonal value is fixed to $6.0$ for synchronous experiments and $6.2$ for asynchronous ones,
        \item matrix off-diagonal value is fixed to $-1.0$,
        \item number of vectors in matrix $S$ (i.e. value of $s$),
        \item maximum number of iterations $\MIC$ and precision $\TOLC$ for CGLS method,
        \item maximum number of iterations and precision for the classical GMRES method,
        \item maximum number of restarts for the Arnorldi process in GMRES method,
        \item execution mode: synchronous or asynchronous.
\end{itemize}

It should also be noticed that both solvers have been executed with the SimGrid selector \texttt{-cfg=smpi/running\_power} which determines the computational power (here 19GFlops) of the simulator host machine.

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

\section{Experimental results}
\label{sec:expe}

In this section, experiments for both multisplitting algorithms are reported. First the 3D Poisson problem used in our experiments is described.

\subsection{The 3D Poisson problem}
\label{3dpoisson}
We use our two-stage algorithms to solve the well-known Poisson problem $\nabla^2\phi=f$~\cite{Polyanin01}. In three-dimensional Cartesian coordinates in $\mathbb{R}^3$, the problem takes the following form:
\begin{equation}
\frac{\partial^2}{\partial x^2}\phi(x,y,z)+\frac{\partial^2}{\partial y^2}\phi(x,y,z)+\frac{\partial^2}{\partial z^2}\phi(x,y,z)=f(x,y,z)\mbox{~in the domain~}\Omega
\label{eq:07}
\end{equation}
such that:
\begin{equation*}
\phi(x,y,z)=0\mbox{~on the boundary~}\partial\Omega
\end{equation*}
where the real-valued function $\phi(x,y,z)$ is the solution sought, $f(x,y,z)$ is a known function and $\Omega=[0,1]^3$. The 3D discretization of the Laplace operator $\nabla^2$ with the finite difference scheme includes 7 points stencil on the computational grid. The numerical approximation of the Poisson problem on three-dimensional grid is repeatedly computed as $\phi=\phi^\star$ such that:
\begin{equation}
\begin{array}{ll}
\phi^\star(x,y,z)=&\frac{1}{6}(\phi(x-h,y,z)+\phi(x,y-h,z)+\phi(x,y,z-h)\\&+\phi(x+h,y,z)+\phi(x,y+h,z)+\phi(x,y,z+h)\\&-h^2f(x,y,z))
\end{array}
\label{eq:08}
\end{equation}
until convergence where $h$ is the grid spacing between two adjacent elements in the 3D computational grid.

In the parallel context, the 3D Poisson problem is partitioned into $L\times p$
sub-problems such that $L$ is the number of clusters and $p$ is the number of
processors in each cluster. We apply the three-dimensional partitioning instead
of the row-by-row one in order to reduce the size of the data shared at the
sub-problems boundaries. In this case, each processor is in charge of
parallelepipedic block of the problem and has at most six neighbors in the same
cluster or in distant clusters with which it shares data at boundaries.

\subsection{Study setup and simulation methodology}

First, to conduct our study, we propose the following methodology
which can be reused for any grid-enabled applications.\\

\textbf{Step 1}: Choose with the end users the class of algorithms or
the application to be tested. Numerical parallel iterative algorithms
have been chosen for the study in this paper. \\

\textbf{Step 2}: Collect the software materials needed for the experimentation.
In our case, we have two variants algorithms for the resolution of the
3D-Poisson problem: (1) using the classical GMRES; (2) and the multisplitting
method. In addition, the SimGrid simulator has been chosen to simulate the
behaviors of the distributed applications. SimGrid is running in a virtual
machine on a simple laptop. \\

\textbf{Step 3}: Fix the criteria which will be used for the future
results comparison and analysis. In the scope of this study, we retain
on the  one hand the algorithm execution mode (synchronous and asynchronous)
and on the other hand the execution time and the number of iterations to reach the convergence. \\

\textbf{Step 4}: Set up the  different grid testbed environments  that will be
simulated in the  simulator tool to run the program.  The following architectures
have been configured in SimGrid : 2$\times$16, 4$\times$8, 4$\times$16, 8$\times$8 and 2$\times$50. The first number
represents the number  of clusters in the grid and  the second number represents
the number  of hosts (processors/cores)  in each  cluster. \\

\textbf{Step 5}: Conduct an extensive and comprehensive testings
within these configurations by varying the key parameters, especially
the CPU power capacity, the network parameters and also the size of the
input data.  \\

\textbf{Step 6} : Collect and analyze the output results.

\subsection{Factors impacting distributed applications performance in a grid environment}

When running a distributed application in a computational grid, many factors may
have a strong impact on the performance.  First of all, the architecture of the
grid itself can obviously influence the  performance results of the program. The
performance gain  might be important  theoretically when the number  of clusters
and/or  the  number  of  nodes (processors/cores)  in  each  individual  cluster
increase.

Another important factor  impacting the overall performance  of the application
is the network configuration. Two main network parameters can modify drastically
the program output results:
\begin{enumerate}
\item  the network  bandwidth  ($bw$ in bits/s) also  known  as "the  data-carrying
    capacity" of the network is defined as  the maximum of data that can transit
    from one point to another in a unit of time.
\item the  network latency  ($lat$ in microseconds) defined as  the delay  from the
  start time to send  a simple data from a source to a destination.
\end{enumerate}
Upon  the   network  characteristics,  another  impacting   factor  is  the volume of data exchanged  between the nodes in the cluster
and  between distant  clusters.  This parameter is application dependent.

 In  a grid  environment, it  is common  to distinguish,  on one hand,  the
 \textit{intra-network} which refers  to the links between nodes within  a
 cluster and on  the other  hand, the  \textit{inter-network} which  is the
 backbone link  between clusters.  In   practice,  these  two   networks  have
 different   speeds. The intra-network  generally works  like a  high speed
 local network  with a high bandwidth and very low latency. In opposite, the
 inter-network connects clusters sometime via  heterogeneous networks components
 through internet with a lower speed.  The network  between distant  clusters
 might  be a  bottleneck for  the global performance of the application.


\subsection{Comparison between GMRES and two-stage multisplitting algorithms in
synchronous mode}
In the scope of this paper, our first objective is to analyze
when the synchronous Krylov two-stage method has better performance than the
classical GMRES method. With a synchronous iterative method, better performance
means a smaller number of iterations and execution time before reaching the
convergence.

Table~\ref{tab:01} summarizes the parameters used in the different simulations:
the grid architectures (i.e. the number of clusters and the number of nodes per
cluster), the network of inter-clusters backbone links and the matrix sizes of
the 3D Poisson problem. However, for all simulations we fix the network
parameters of the intra-clusters links: the bandwidth $bw$=10Gbs and the latency
$lat=8\mu$s. In what follows, we will present the test conditions, the output
results and our comments.

\begin{table} [ht!]
\begin{center}
\begin{tabular}{ll}
\hline
Grid architecture                       & 2$\times$16, 4$\times$8, 4$\times$16 and 8$\times$8\\
\multirow{2}{*}{Network inter-clusters} & $N1$: $bw$=10Gbs, $lat=8\mu$s \\
                                        & $N2$: $bw$=1Gbs, $lat=50\mu$s \\
\multirow{2}{*}{Matrix size}            & $Mat1$: N$_{x}\times$N$_{y}\times$N$_{z}$=150$\times$150$\times$150\\
                                        & $Mat2$: N$_{x}\times$N$_{y}\times$N$_{z}$=170$\times$170$\times$170 \\ \hline
\end{tabular}
\caption{Parameters for the different simulations}
\label{tab:01}
\end{center}
\end{table}

\subsubsection{Simulations for various grid architectures and scaling-up matrix sizes\\}

In  this  section,  we  analyze   the  simulations  conducted  on  various  grid
configurations and for different sizes of the 3D Poisson problem. The parameters
of    the    network    between    clusters    is    fixed    to    $N2$    (see
Table~\ref{tab:01}). Figure~\ref{fig:01} shows, for all grid configurations and
a given matrix size of 170$^3$ elements, a  non-variation in the number of
iterations for the classical GMRES algorithm, which is not the case of the
Krylov two-stage algorithm. In fact, with multisplitting  algorithms, the number
of splitting (in our case, it is equal to the number of clusters) influences on the
convergence speed. The higher the number  of splitting is, the slower the
convergence of the algorithm is (see the output results obtained from
configurations 2$\times$16 vs. 4$\times$8 and configurations 4$\times$16 vs.
8$\times$8).

The execution times between both algorithms is significant with different grid
architectures. The synchronous Krylov two-stage algorithm presents better
performances than the GMRES algorithm, even for a high number of clusters (about
$32\%$ more efficient on a grid of 8$\times$8 than GMRES). In addition, we can
observe a better sensitivity of the Krylov two-stage algorithm (compared to the
GMRES one) when scaling up the number of the processors in the computational
grid: the Krylov two-stage algorithm is about $48\%$ and the GMRES algorithm is
about $40\%$ better on $64$ processors (grid of 8$\times$8) than $32$ processors
(grid of 2$\times$16).

\begin{figure}[ht]
\begin{center}
\includegraphics[width=100mm]{cluster_x_nodes_nx_150_and_nx_170.pdf}
\end{center}
\caption{Various grid configurations with two matrix sizes: $150^3$ and $170^3$}
\label{fig:01}
\end{figure}

\subsubsection{Simulations for two different inter-clusters network speeds\\}
In  Figure~\ref{fig:02} we  present the  execution times  of both  algorithms to
solve a  3D Poisson problem of  size $150^3$ on two  different simulated network
$N1$ and $N2$ (see Table~\ref{tab:01}). As previously mentioned, we can see from
this figure  that the Krylov two-stage  algorithm is sensitive to  the number of
clusters (i.e. it is better to have a small number of clusters). However, we can
notice an  interesting behavior of  the Krylov  two-stage algorithm. It  is less
sensitive to bad network bandwidth and latency for the inter-clusters links than
the  GMRES algorithms.  This  means  that the  multisplitting  methods are  more
efficient for distributed systems with high latency networks.

\begin{figure}[ht]
\centering
\includegraphics[width=100mm]{cluster_x_nodes_n1_x_n2.pdf}
\caption{Various grid configurations with two networks parameters: $N1$ vs. $N2$}
%\LZK{CE, remplacer les ``,'' des décimales par un ``.''}
%\RCE{ok}
\label{fig:02}
\end{figure}

\subsubsection{Network latency impacts on performances\\}
Figure~\ref{fig:03} shows the impact of the network latency on the performances of both algorithms. The simulation is conducted on a computational grid of 2 clusters of 16 processors each (i.e. configuration 2$\times$16) interconnected by a network of bandwidth $bw$=1Gbs to solve a 3D Poisson problem of size $150^3$. According to the results, a degradation of the network latency from $8\mu$s to $60\mu$s implies an absolute execution time increase for both algorithms, but not with the same rate of degradation. The GMRES algorithm is more sensitive to the latency degradation than the Krylov two-stage algorithm.

\begin{figure}[ht]
\centering
\includegraphics[width=100mm]{network_latency_impact_on_execution_time.pdf}
\caption{Network latency impacts on performances}
\label{fig:03}
\end{figure}

\subsubsection{Network bandwidth impacts on performances\\}

Figure~\ref{fig:04} reports the results obtained for the simulation of a grid of
$2\times16$ processors interconnected by a network of latency $lat=50\mu$s to
solve a 3D Poisson problem of size $150^3$. The results of increasing the
network bandwidth from $1$Gbs to $10$Gbs show the performances improvement for
both algorithms by reducing the execution times. However, the Krylov two-stage
algorithm presents a better performance gain in the considered bandwidth
interval with a gain of $40\%$ compared to only about $24\%$ for the classical
GMRES algorithm.

\begin{figure}[ht]
\centering
\includegraphics[width=100mm]{network_bandwith_impact_on_execution_time.pdf}
\caption{Network bandwith impacts on performances}
\label{fig:04}
\end{figure}

\subsubsection{Matrix size impacts on performances\\}

In these experiments, the matrix size of the 3D Poisson problem is varied from
$50^3$ to $190^3$ elements. The simulated computational grid is composed of $4$
clusters of $8$ processors each interconnected by the network $N2$ (see
Table~\ref{tab:01}). As shown in Figure~\ref{fig:05}, the execution
times for both algorithms increase with increased matrix sizes.  For all problem
sizes, the GMRES algorithm is always slower than the Krylov two-stage algorithm.
Moreover, for this benchmark, it seems that the greater the problem size is, the
bigger the ratio between execution times of both algorithms is. We can also
observe that for some problem sizes, the convergence (and thus the execution
time) of the Krylov two-stage algorithm varies quite a lot.
%This is due to the 3D partitioning of the 3D matrix of the Poisson problem.
These findings may help a lot end users to setup the best and the optimal targeted environment for the application deployment when focusing on the problem size scale up.

\begin{figure}[ht]
\centering
\includegraphics[width=100mm]{pb_size_impact_on_execution_time.pdf}
\caption{Problem size impacts on performances}
\label{fig:05}
\end{figure}

\subsubsection{CPU power impacts on performances\\}

Using the SimGrid simulator flexibility, we have tried to determine the impact
of the CPU power of the processors in the different clusters on performances of
both algorithms. We have varied the CPU power from $1$GFlops to $19$GFlops. The
simulation is conducted on a grid of $2\times16$ processors interconnected by
the network $N2$ (see Table~\ref{tab:01}) to solve a 3D Poisson problem of size
$150^3$. The results depicted in Figure~\ref{fig:06} confirm the performance
gain, about $95\%$ for both algorithms, after improving the CPU power of
processors.

\begin{figure}[ht]
\centering
\includegraphics[width=100mm]{cpu_power_impact_on_execution_time.pdf}
\caption{CPU Power impacts on performances}
\label{fig:06}
\end{figure}
\ \\

To conclude these series of experiments, with  SimGrid we have been able to make
many simulations  with many parameters  variations. Doing all  these experiments
with a real platform is most of the time not possible or very costly. Moreover
the behavior of both GMRES and  Krylov two-stage algorithms is in accordance
with larger real executions on large scale supercomputers~\cite{couturier15}.


\subsection{Comparison between synchronous GMRES and asynchronous two-stage multisplitting algorithms}

The previous paragraphs  put in evidence the interests to  simulate the behavior
of  the application  before  any  deployment in  a  real  environment.  In  this
section, following  the same previous  methodology, our  goal is to  compare the
efficiency of the multisplitting method  in \textit{ asynchronous mode} compared with the
classical GMRES in \textit{synchronous mode}.

The  interest of  using  an asynchronous  algorithm  is that  there  is no  more
synchronization. With  geographically distant  clusters, this may  be essential.
In  this case,  each  processor can  compute its  iterations  freely without  any
synchronization  with   the  other   processors.  Thus,  the   asynchronous  may
theoretically reduce  the overall execution  time and can improve  the algorithm
performance.

In this section,  the SimGrid simulator is  used to compare the  behavior of the
two-stage  algorithm  in  asynchronous  mode with  GMRES  in  synchronous  mode.
Several benchmarks  have been  performed with various  combinations of  the grid
resources  (CPU,  Network,  matrix  size,   \ldots).  The  test  conditions  are
summarized in Table~\ref{tab:02}.



%\LZK{Quelle table repporte les gains relatifs?? Sûrement pas Table II !!}
%\RCE{Table III avec la nouvelle numerotation}


\begin{table}[htbp]
\centering
\begin{tabular}{ll}
 \hline
 Grid architecture                       & 2$\times$50 totaling 100 processors\\
 Processors Power                        & 1 GFlops to 1.5 GFlops \\
 \multirow{2}{*}{Network inter-clusters} & $bw$: 5 Mbits to 50 Mbits\\
                                         & $lat$: 20 ms\\
 Matrix size                             & from $62^3$ to $150^3$\\
 Residual error precision                & $10^{-5}$ to $10^{-11}$\\ \hline \\
 \end{tabular}
\caption{Test conditions: GMRES in synchronous mode vs. two-stage multisplitting in asynchronous mode}
\label{tab:02}
\end{table}


% use the same column width for the following three tables
\newlength{\mytablew}\settowidth{\mytablew}{\footnotesize\np{E-11}}
\newenvironment{mytable}[1]{% #1: number of columns for data
  \renewcommand{\arraystretch}{1.3}%
  \begin{tabular}{|>{\bfseries}r%
                  |*{#1}{>{\centering\arraybackslash}p{\mytablew}|}}}{%
    \end{tabular}}


\begin{table}[!t]
\centering
%\begin{table}
%  \caption{Relative gain of the multisplitting algorithm compared with the classical GMRES}
%  \label{"Table 7"}
 \begin{mytable}{11}
    \hline
    bandwidth (Mbit/s)
    & 5     & 5     & 5         & 5         & 5  & 50        & 50        & 50        & 50        & 50 \\
    \hline
    latency (ms)
    & 20      & 20      & 20      & 20      & 20 & 20      & 20      & 20      & 20      & 20 \\
    \hline
    power (GFlops)
    & 1    & 1    & 1    & 1.5       & 1.5  & 1.5         & 1.5         & 1         & 1.5       & 1.5 \\
    \hline
    size ($N^3$)
    & 62  & 62   & 62        & 100       & 100 & 110       & 120       & 130       & 140       & 150 \\
    \hline
    Precision
    & \np{E-5}  & \np{E-8}  & \np{E-9}  & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11}\\
    \hline
    Relative gain
    & 2.52     & 2.55     & 2.52     & 2.57     & 2.54 & 2.53     & 2.51     & 2.58     & 2.55     & 2.54 \\
    \hline
  \end{mytable}
%\end{table}
 \caption{Relative gains of the asynchronous two-stage multisplitting algorithm compared to the classical synchronous GMRES algorithm}
 \label{tab:03}
\end{table}


Table~\ref{tab:03} reports  the relative gains  between both algorithms.   It is
defined by the ratio between the execution  time of GMRES and the execution time
of the  multisplitting. The ratio is  greater than one because  the asynchronous
multisplitting  version  is  faster  than   GMRES.  In  average,  the  two-stage
multisplitting algorithm to  be more than $2.5$ times faster  than the classical
GMRES.  These experiments also show the relative tolerance of the multisplitting
algorithm when using a low speed network as usually observed with geographically
distant clusters through the internet.


\section{Conclusion}
In this paper we have presented the simulation of the execution of three
different parallel solvers on some multi-core architectures. We have shown that
the SimGrid toolkit is an interesting simulation tool that has allowed us to
determine  which method  to choose  given a  specified multi-core  architecture.
Moreover the simulated results are in accordance (i.e. with the same order of
magnitude)  with the works  presented in~\cite{couturier15}. Simulated   results
also  confirm  the   efficiency  of  the asynchronous  multisplitting
algorithm  compared  to  the   synchronous  GMRES especially in case of
geographically distant clusters.

These results are important since it is very  time consuming to find optimal
configuration  and deployment requirements for a given application  on   a given
multi-core  architecture. Finding   good  resource allocations policies under
varying CPU power, network speeds and  loads is very challenging and  labor
intensive. This problematic is  even more difficult  for the  asynchronous
scheme where  a small parameter variation of the execution platform and of the
application data can lead to very different numbers of iterations to reach the
converge and so to very different execution times.


In future works, we  plan to investigate how to simulate  the behavior of really
large scale  applications. For  example, if  we are  interested to  simulate the
execution of the solvers of this paper with thousand or even dozens of thousands
of cores,  it is not possible  to do that with  SimGrid. In fact, this  tool will
make the real computation. So we plan to focus our research on that problematic.



%\section*{Acknowledgment}
\ack
This work is partially funded by the Labex ACTION program (contract ANR-11-LABX-01-01).

\bibliographystyle{wileyj}
\bibliography{biblio}


\end{document}

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%  LocalWords:  Ramamonjisoa Ziane Khodja Laiymani Raphaël Arnaud Giersch Femto
%  LocalWords:  Franche Comté Belfort GMRES multisplitting SimGrid Krylov SMPI
%  LocalWords:  MPI