[rce2015.git] / paper.tex

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\begin{document} \RCE{Titre a confirmer.} \title{Comparative performance
analysis of simulated grid-enabled numerical iterative algorithms}
%\itshape{\journalnamelc}\footnotemark[2]}

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

\address{
        \centering
    Femto-ST Institute - DISC Department\\
    Université de Franche-Comté\\
    Belfort\\
    Email: \email{{raphael.couturier,arnaud.giersch,david.laiymani,charles.ramamonjisoa}@univ-fcomte.fr}
}

%% Lilia Ziane Khodja: Department of Aerospace \& Mechanical Engineering\\ Non Linear Computational Mechanics\\ University of Liege\\ Liege, Belgium. 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 developpers to better tune their applications for a given multi-core
architecture.

In particular we focus our attention on two parallel iterative algorithms based
on the  Multisplitting algorithm  and we  compare them  to the  GMRES algorithm.
These algorithms  are used to  solve linear  systems. Two different  variants of
the Multisplitting are studied: one  using synchronoous  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 obtain simulated results confirm the real results
previously obtained on different real multi-core architectures and also confirm
the efficiency of the asynchronous multisplitting algorithm compared to the
synchronous GMRES method.

\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~\ldots. 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 communication 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 can lead to very different numbers
of iterations to reach the converge 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 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 application for a given multi-core architecture. To show the validity
of this approach we first compare the simulated execution of the multisplitting
algorithm  with  the  GMRES   (Generalized   Minimal  Residual)
solver~\cite{saad86} in synchronous mode. The obtained results on different
simulated multi-core architectures confirm the real results previously obtained
on non simulated architectures.  We also confirm  the efficiency  of the
asynchronous  multisplitting algorithm  compared to the synchronous  GMRES. 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 theoritecally 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 iterations  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 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
architecture are often very important. So, prior to delpoyment 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 produce
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}

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

\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 ({\it Generalized Minimal RESidual})~\cite{saad86} as an inner iteration to approximate the solutions of the different splittings arising from the block Jacobi multisplitting of matrix $A$. The algorithm in Figure~\ref{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}[t]
%\begin{algorithm}[t]
%\caption{Block Jacobi two-stage multisplitting method}
\begin{algorithmic}[1]
  \Input $A_\ell$ (sparse matrix), $b_\ell$ (right-hand side)
  \Output $x_\ell$ (solution vector)\vspace{0.2cm}
  \State Set the initial guess $x^0$
  \For {$k=1,2,3,\ldots$ until convergence}
    \State $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m^{k-1}$
    \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}[t]
%\begin{algorithm}[t]
%\caption{Krylov two-stage method using block Jacobi multisplitting}
\begin{algorithmic}[1]
  \Input $A_\ell$ (sparse matrix), $b_\ell$ (right-hand side)
  \Output $x_\ell$ (solution vector)\vspace{0.2cm}
  \State Set the initial guess $x^0$
  \For {$k=1,2,3,\ldots$ until convergence}
    \State $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m^{k-1}$
    \State $x^k_\ell=Solve_{gmres}(A_{\ell\ell},c_\ell,x^{k-1}_\ell,\MIG,\TOLG)$
    \State $S_{\ell,k\mod s}=x_\ell^k$
    \If{$k\mod s = 0$}
       \State $\alpha = Solve_{cgls}(AS,b,\MIC,\TOLC)$\label{cgls}
       \State $\tilde{x_\ell}=S_\ell\alpha$
       \State Send $\tilde{x_\ell}$ to neighboring clusters
       \Else
         \State Send $x_\ell^k$ to neighboring clusters
    \EndIf
    \State Receive $\{x_m^k\}_{m\neq\ell}$ from neighboring clusters
  \EndFor
\end{algorithmic}
\caption{Krylov two-stage method using block Jacobi multisplitting}
\label{alg:02}
%\end{algorithm}
\end{figure}

\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  (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.

%\RC{On vire cette  phrase ?} \RCE {Si c'est la phrase d'avant sur les threads, je pense qu'on peut la retenir car c'est l'explication du pourquoi Simgrid n'aime pas les variables globales. Si c'est pas bien dit, on peut la reformuler. Si c'est la phrase ci-apres, effectivement, on peut la virer si elle preterais a discussion}The
%last modification on the  MPI program pointed out for some  cases, the review of
%the sequence of  the MPI\_Isend, MPI\_Irecv and  MPI\_Waitall instructions which
%might cause an infinite loop.


\paragraph{Simgrid Simulator parameters}
\  \\ \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 for 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 3D Poisson problem: N$_{x}$, N$_{y}$ and N$_{z}$ on axis $x$, $y$ and $z$ respectively,
        \item matrix diagonal value is fixed to $6.0$ for synchronous Krylov multisplitting experiments and $6.2$ for asynchronous block Jacobi experiments,
        \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}
\LZK{CE pourrais tu vérifier et confirmer les valeurs des éléments diag et off-diag de la matrice?}
\RCE{oui, les valeurs de diag et off-diag donnees sont ok}

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}


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 architecture
has been configured in Simgrid : 2x16, 4x8, 4x16, 8x8 and 2x50. 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. The network  has been
designed to  operate with a bandwidth  equals to 10Gbits (resp.  1Gbits/s) and a
latency of 8.10$^{-6}$ seconds (resp.  5.10$^{-5}$) for the intra-clusters links
(resp.  inter-clusters backbone links). \\

\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=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 :  microsecond) 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 bandwith and very low latency. In opposite, the inter-network connects
 clusters sometime via  heterogeneous networks components  throuth internet with
 a lower speed.  The network  between distant  clusters might  be a  bottleneck
 for  the global performance of the application.

\subsection{Comparison of GMRES and Krylov Multisplitting algorithms in synchronous mode}

In the scope  of this paper, our  first objective is to analyze  when the Krylov
Multisplitting  method   has  better  performance  than   the  classical  GMRES
method. With a synchronous  iterative method, better performance mean a
smaller number of iterations and execution time before reaching the convergence.
For a systematic study,  the experiments  should figure  out  that, for  various
grid  parameters values, the simulator will confirm  the targeted outcomes,
particularly for poor and slow  networks, focusing on the  impact on the
communication  performance on the chosen class of algorithm.

The following paragraphs present the test conditions, the output results
and our comments.\\


\subsubsection{Execution of the algorithms on various computational grid
architectures and scaling up the input matrix size}
\ \\
% environment

\begin{table} [ht!]
\begin{center}
\begin{tabular}{r c }
 \hline
 Grid Architecture & 2x16, 4x8, 4x16 and 8x8\\ %\hline
 Network & N2 : bw=1Gbits/s - lat=5.10$^{-5}$ \\ %\hline
 Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ %\hline
 - &  N$_{x}$ x N$_{y}$ x N$_{z}$  =170 x 170 x 170    \\ \hline
 \end{tabular}
\caption{Test conditions: Various grid configurations with the input matix size N$_{x}$=150 or N$_{x}$=170 \RC{je ne comprends pas la légende... Ca ne serait pas plutot Characteristics of cluster (mais il faudrait lui donner un nom)}}
\label{tab:01}
\end{center}
\end{table}




%\RCE{J'ai voulu mettre les tableaux des données mais je pense que c'est inutile et ça va surcharger}


In this  section, we analyze the  performance of algorithms running  on various
grid configurations  (2x16, 4x8, 4x16  and 8x8). First,  the results in  Figure~\ref{fig:01}
show for all grid configurations the non-variation of the number of iterations of
classical  GMRES for  a given  input matrix  size; it is not  the case  for the
multisplitting method.

\RC{CE attention tu n'as pas mis de label dans tes figures, donc c'est le bordel, j'en mets mais vérifie...}
\RC{Les légendes ne sont pas explicites...}


\begin{figure} [ht!]
  \begin{center}
    \includegraphics[width=100mm]{cluster_x_nodes_nx_150_and_nx_170.pdf}
  \end{center}
  \caption{Various grid configurations with the input matrix size N$_{x}$=150 and N$_{x}$=170}
  \label{fig:01}
\end{figure}


The execution  times between  the two algorithms  is significant  with different
grid architectures, even  with the same number of processors  (for example, 2x16
and  4x8). We  can  observ  the low  sensitivity  of  the Krylov multisplitting  method
(compared with the classical GMRES) when scaling up the number of the processors
in the  grid: in  average, the GMRES  (resp. Multisplitting)  algorithm performs
$40\%$ better (resp. $48\%$) when running from 2x16=32 to 8x8=64 processors.

\subsubsection{Running on two different inter-clusters network speeds \\}

\begin{table} [ht!]
\begin{center}
\begin{tabular}{r c }
 \hline
 Grid Architecture & 2x16, 4x8\\ %\hline
 Network & N1 : bw=10Gbs-lat=8.10$^{-6}$ \\ %\hline
 - & N2 : bw=1Gbs-lat=5.10$^{-5}$ \\
 Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ \hline
 \end{tabular}
\caption{Test conditions: Grid 2x16 and 4x8 - Networks N1 vs N2}
\label{tab:02}
\end{center}
\end{table}

These experiments  compare the  behavior of  the algorithms  running first  on a
speed inter-cluster  network (N1) and  also on  a less performant  network (N2).
Figure~\ref{fig:02} shows that end users will  gain to reduce the execution time
for  both  algorithms  in using  a  grid  architecture  like  4x16 or  8x8:  the
performance was increased  by a factor of  $2$. 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\%.
%\RC{c'est pas clair : la différence entre quoi et quoi?}
%\DL{pas clair}
%\RCE{Modifie}


%\begin{wrapfigure}{l}{100mm}
\begin{figure} [ht!]
\centering
\includegraphics[width=100mm]{cluster_x_nodes_n1_x_n2.pdf}
\caption{Grid 2x16 and 4x8 - Networks N1 vs N2}
\label{fig:02}
\end{figure}
%\end{wrapfigure}


\subsubsection{Network latency impacts on performance}
\ \\
\begin{table} [ht!]
\centering
\begin{tabular}{r c }
 \hline
 Grid Architecture & 2x16\\ %\hline
 Network & N1 : bw=1Gbs \\ %\hline
 Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ \hline
 \end{tabular}
\caption{Test conditions: Network latency impacts}
\label{tab:03}
\end{table}



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


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.   In   addition,   it  appears   that  the   Krylov
multisplitting method tolerates  more the network latency variation  with a less
rate  increase  of  the  execution   time.   Consequently,  in  the  worst  case
($lat=6.10^{-5 }$), the  execution time for GMRES is almost  the double than the
time of the Krylov multisplitting, even  though, the performance was on the same
order of magnitude with a latency of $8.10^{-6}$.

\subsubsection{Network bandwidth impacts on performance}
\ \\
\begin{table} [ht!]
\centering
\begin{tabular}{r c }
 \hline
 Grid Architecture & 2x16\\ %\hline
 Network & N1 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\hline
 Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ \hline \\
 \end{tabular}
\caption{Test conditions: Network bandwidth impacts}
\label{tab:04}
\end{table}


\begin{figure} [ht!]
\centering
\includegraphics[width=100mm]{network_bandwith_impact_on_execution_time.pdf}
\caption{Network bandwith impacts on execution time}
\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 & 4x8\\ %\hline
 Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\
 Input matrix size & N$_{x}$ = From 40 to 200\\ \hline
 \end{tabular}
\caption{Test conditions: Input matrix size impacts}
\label{tab:05}
\end{table}


\begin{figure} [ht!]
\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 $N_{x} = N_{y}
= N_{z} = 40$ to $200$ side elements  that is from $40^{3} = 64.000$ to $200^{3}
= 8,000,000$  points. Obviously, as  shown in Figure~\ref{fig:05},  the execution
time for  both algorithms increases when  the input matrix size  also increases.
But the interesting results are:
\begin{enumerate}
  \item the drastic increase ($10$ times) \RC{Je ne vois pas cela sur la figure}
\RCE{Corrige} of the  number of  iterations needed  to reach the  convergence for  the classical
GMRES algorithm when  the matrix size go beyond $N_{x}=150$;
\item the  classical GMRES execution time  is almost the double  for $N_{x}=140$
  compared with the Krylov multisplitting method.
\end{enumerate}

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 2x16 leading to the same conclusion.

\subsubsection{CPU Power impacts on performance}

\begin{table} [ht!]
\centering
\begin{tabular}{r c }
 \hline
 Grid architecture & 2x16\\ %\hline
 Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\hline
 Input matrix size & N$_{x}$ = 150 x 150 x 150\\ \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}


\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.

\RC{la phrase suivante est bizarre, je ne comprends pas pourquoi elle vient ici}
In this section, Simgrid simulator tool has been successfully used to show
the efficiency of  the multisplitting in asynchronous mode and  to find the best
combination of the grid resources (CPU,  Network, input matrix size, \ldots ) to
get    the   highest    \textit{"relative    gain"}   (exec\_time$_{GMRES}$    /
exec\_time$_{multisplitting}$) in comparison with the classical GMRES time.


The test conditions are summarized in the table~\ref{tab:07}: \\

\begin{table} [ht!]
\centering
\begin{tabular}{r c }
 \hline
 Grid Architecture & 2x50 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
Figure~\ref{fig:07}  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{figure}[!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)
    & 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{fig:07}
\end{figure}


\section{Conclusion}
CONCLUSION


\section*{Acknowledgment}

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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