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

\address{
  \affilnum{1}%
  Femto-ST Institute, DISC Department,
  University of Franche-Comté,
  Belfort, France.
  Email:~\email{{charles.ramamonjisoa,david.laiymani,arnaud.giersch,raphael.couturier}@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 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. 

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: development life cycle,
code debugging, ability to obtain results quickly\dots{} In counterpart, 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. We say that the iteration computation follows 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} but 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 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~\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 so 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 non  simulated 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.

In this way and with a simple computing architecture (a laptop) SimGrid allows us
to run a test campaign  of  a  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 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, the  application is not ensure to 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 and  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 than in the sequential mode. In this way, 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}
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 their both versions (synchronous and asynchronous)~\cite{Frommer92,Szyld92,Bru95}. These iterative methods are based on multisplitting methods~\cite{O'leary85,White86,Alefeld97} and use two nested iterations: the outer iteration and the inner iteration. Let us consider the following sparse linear system of $n$ equations in $\mathbb{R}$:
\begin{equation}
Ax=b,
\label{eq:01}
\end{equation}
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 such 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 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. In line~\ref{solve}, the linear sub-system~(\ref{eq:03}) is solved in parallel using GMRES method where $\MIG$ and $\TOLG$ are the maximum number of inner iterations and the tolerance threshold for GMRES respectively. The convergence of the two-stage multisplitting methods, based on synchronous or asynchronous iterations, 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 model 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 CGLS method~\cite{Hestenes52} such that $\MIC$ is the maximum number of iterations and $\TOLC$ is the tolerance threshold for this method (line~\ref{cgls} in Figure~\ref{alg:02}).

\begin{figure}[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 the same  conditions.  According to  our experience,
very  few modifications  are required  to adapt  a MPI  program for  the SimGrid
simulator using SMPI (Simulator 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{Parameters of the simulation in 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 the  one hand,  the
 "intra-network" which refers  to the links between nodes within  a cluster and
 on  the other  hand, the  "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, 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$\times$10$^{-6}$. 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$\times$10$^{-6}$ \\
                                        & $N2$: $bw$=1Gbs, $lat$=5$\times$10$^{-5}$ \\ 
\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 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 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}[t]
\begin{center}
\includegraphics[width=100mm]{cluster_x_nodes_nx_150_and_nx_170.pdf}
\end{center}
\caption{Various grid configurations with the matrix sizes 150$^3$ and 170$^3$}
\label{fig:01}
\end{figure}

\subsubsection{Simulations for two different inter-clusters network speeds\\}

In this section, the experiments  compare the  behavior of  the algorithms  running on a
speeder inter-cluster  network (N2) and  also on  a less performant  network (N1) respectively defined in the test conditions Table~\ref{tab:02}.
%\RC{Il faut définir cela avant...}
Figure~\ref{fig:02} shows that end users will reduce the execution time
for  both  algorithms when using  a  grid  architecture  like  4 $\times$ 16 or  8 $\times$ 8: the reduction factor is around $2$. The results depict  also that when
the  network speed  drops down (variation of 12.5\%), the  difference between  the two Multisplitting algorithms execution times can reach more than 25\%.

\begin{figure}[t]
\centering
\includegraphics[width=100mm]{cluster_x_nodes_n1_x_n2.pdf}
\caption{Various grid configurations with networks $N1$ vs. $N2$}
\label{fig:02}
\end{figure}






















\subsubsection{Network latency impacts on performance\\}

\begin{table} [ht!]
\centering
\begin{tabular}{r c }
 \hline
 Grid Architecture & 2 $\times$ 16\\ %\hline
 \multirow{2}{*}{Inter Network N1} & $bw$=1Gbs, \\ %\hline
                          & $lat$= From 8$\times$10$^{-6}$ to  $6.10^{-5}$ second \\
 Input matrix size & $N_{x} \times N_{y} \times N_{z} = 150 \times 150 \times 150$\\ \hline
 \end{tabular}
\caption{Test conditions: network latency impacts}
\label{tab:03}
\end{table}

\begin{figure} [htbp]
\centering
\includegraphics[width=100mm]{network_latency_impact_on_execution_time.pdf}
\caption{Network latency impacts on execution time}
%\AG{\np{E-6}}}
\label{fig:03}
\end{figure}

In Table~\ref{tab:03}, parameters  for the influence of the  network latency are
reported.  According to the results of Figure~\ref{fig:03}, a degradation of the
network  latency  from  $8.10^{-6}$  to $6.10^{-5}$  implies  an  absolute  time
increase of more than $75\%$ (resp.   $82\%$) of the execution for the classical
GMRES  (resp.   Krylov  multisplitting)  algorithm. The  execution  time  factor
between the two algorithms  varies from 2.2 to 1.5 times  with a network latency
decreasing from $8.10^{-6}$ to $6.10^{-5}$ second.


\subsubsection{Network bandwidth impacts on performance\\}

\begin{table} [ht!]
\centering
\begin{tabular}{r c }
 \hline
 Grid Architecture & 2 $\times$ 16\\ %\hline
\multirow{2}{*}{Inter Network N1} & $bw$=From 1Gbs to 10 Gbs \\ %\hline
                          & $lat$= 5.10$^{-5}$ second \\
 Input matrix size & $N_{x} \times N_{y} \times N_{z} =150 \times 150 \times 150$\\ \hline \\
 \end{tabular}
\caption{Test conditions: Network bandwidth impacts}
%  \RC{Qu'est ce qui varie ici? Il n'y a pas de variation dans le tableau}
%\RCE{C est le bw}
\label{tab:04}
\end{table}


\begin{figure} [htbp]
\centering
\includegraphics[width=100mm]{network_bandwith_impact_on_execution_time.pdf}
\caption{Network bandwith impacts on execution time}
%\AG{``Execution time'' avec un 't' minuscule}. Idem autres figures.}
%\RCE{Corrige}
\label{fig:04}
\end{figure}

The results  of increasing  the network  bandwidth show  the improvement  of the
performance  for   both  algorithms   by  reducing   the  execution   time  (see
Figure~\ref{fig:04}). However,  in this  case, the Krylov  multisplitting method
presents a better  performance in the considered bandwidth interval  with a gain
of $40\%$ which is only around $24\%$ for the classical GMRES.

\subsubsection{Input matrix size impacts on performance\\}

\begin{table} [ht!]
\centering
\begin{tabular}{r c }
 \hline
 Grid Architecture & 4 $\times$ 8\\ %\hline
 Inter Network & $bw$=1Gbs - $lat$=5.10$^{-5}$ \\
 Input matrix size & $N_{x} \times N_{y} \times N_{z}$ = From 50$^{3}$ to 190$^{3}$\\ \hline
 \end{tabular}
\caption{Test conditions: Input matrix size impacts}
\label{tab:05}
\end{table}


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

In  these  experiments, the  input  matrix  size has  been  set  from $50^3$  to
$190^3$. Obviously, as shown in Figure~\ref{fig:05}, the execution time for both
algorithms increases when the input matrix size also increases.  For all problem
sizes, GMRES is always slower than the Krylov multisplitting. Moreover, for this
benchmark, it seems that  the greater the problem size is,  the bigger the ratio
between both  algorithm execution  times is.  We can also  observ that  for some
problem   sizes,  the   Krylov   multisplitting  convergence   varies  quite   a
lot. Consequently the execution times in that cases also varies.


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.  It  should be noticed that the same test has  been done with the
grid 4 $\times$ 8 leading to the same conclusion.

\subsubsection{CPU Power impacts on performance\\}


\begin{table} [htbp]
\centering
\begin{tabular}{r c }
 \hline
 Grid architecture & 2 $\times$ 16\\ %\hline
 Inter Network & N2 : $bw$=1Gbs - $lat$=5.10$^{-5}$ \\ %\hline
 Input matrix size & $N_{x} = 150 \times 150 \times 150$\\ 
 CPU Power & From 3 to 19 GFlops \\ \hline
 \end{tabular}
\caption{Test conditions: CPU Power impacts}
\label{tab:06}
\end{table}

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

Using the Simgrid  simulator flexibility, we have tried to  determine the impact
on the  algorithms performance in  varying the CPU  power of the  clusters nodes
from $1$ to $19$ GFlops.  The outputs  depicted in Figure~\ref{fig:06}  confirm the
performance gain,  around $95\%$ for  both of the  two methods, after  adding more
powerful CPU.
\ \\
%\DL{il faut une conclusion sur ces tests : ils confirment les résultats déjà
%obtenus en grandeur réelle. Donc c'est une aide précieuse pour les dev. Pas
%besoin de déployer sur une archi réelle}

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. Moreover the behavior of
both GMRES and  Krylov multisplitting methods is in accordance  with larger real
executions on large scale supercomputer~\cite{couturier15}.


\subsection{Comparing GMRES in native synchronous mode and the multisplitting algorithm in asynchronous mode}

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  iteration  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
multisplitting in  asynchronous mode  with GMRES  in synchronous  mode.  Several
benchmarks have  been performed with  various combination of the  grid resources
(CPU, Network, input  matrix size, \ldots ). The test  conditions are summarized
in  Table~\ref{tab:07}. In  order to  compare  the execution  times, this  table
reports the  relative gain between both  algorithms. It is defined  by the ratio
between  the   execution  time  of   GMRES  and   the  execution  time   of  the
multisplitting.  The  ration  is  greater  than  one  because  the  asynchronous
multisplitting version is faster than GMRES.



\begin{table} [htbp]
\centering
\begin{tabular}{r c }
 \hline
 Grid Architecture & 2 $\times$ 50 totaling 100 processors\\ %\hline
 Processors Power & 1 GFlops to 1.5 GFlops\\
   Intra-Network & bw=1.25 Gbits - lat=5.10$^{-5}$ \\ %\hline
   Inter-Network & bw=5 Mbits - lat=2.10$^{-2}$\\
 Input matrix size & $N_{x}$ = From 62 to 150\\ %\hline
 Residual error precision & 10$^{-5}$ to 10$^{-9}$\\ \hline \\
 \end{tabular}
\caption{Test conditions: GMRES in synchronous mode vs Krylov Multisplitting in asynchronous mode}
\label{tab:07}
\end{table}

Again,  comprehensive and  extensive tests  have been  conducted with  different
parameters as  the CPU power, the  network parameters (bandwidth and  latency)
and with different problem size. The  relative gains greater than $1$  between the
two algorithms have  been captured after  each step  of the test.   In
Table~\ref{tab:08}  are  reported the  best  grid  configurations allowing
the  multisplitting method 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.

% 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 gain of the multisplitting algorithm compared with the classical GMRES}
 \label{tab:08}
\end{table}


\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 show 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
or core,  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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