Desription du problème à résoudre: 3D Poisson equation

[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. We have decided to use SimGrid as it enables to benchmark MPI
applications.

In this paper, 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 libear  systems. Two different  variants of
the Multisplitting are studied: one  using synchronoous  iterations and  another
one  with asynchronous iterations. For each algorithm we have  tested different
parameters to see their influence.  We strongly  recommend people  interested
by investing  into a  new expensive  hardware  architecture  to   benchmark
their  applications  using  a simulation tool before.




\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 methods 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 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 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}
\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}.

\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{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, is 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 the 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 two-stage methods using SimGrid framework}
\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  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, generated by
the Simgrid  to simulate the  grid environment.  \RC{On vire cette  phrase ?}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 and outer iterations;
        \item inner and outer precisions;
        \item matrix size (N$_{x}$, N$_{y}$ and N$_{z}$);
        \item matrix diagonal value = 6.0 (for synchronous Krylov multisplitting experiments and 6.2 for asynchronous block Jacobi experiments); \RC{CE tu vérifies, je dis ca de tête}
        \item execution mode: synchronous or asynchronous.
\end{itemize}

It should also be noticed that both solvers have been executed with the Simgrid selector -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 problem sued in our experiments is described.

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+h,y,z) \\
                  & +\phi(x,y-h,z)+\phi(x,y+h,z) \\
                  & +\phi(x,y,z-h)+\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 sub-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 on the Mesocentre datacenter in the University of  Franche-Comte and also in a virtual machine on a 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 performances.  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 performances  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  the data from a source and the  final time the destination
  have finished to receive it.
\end{enumerate}
Upon  the   network  characteristics,  another  impacting   factor  is  the
application dependent volume of data exchanged  between the nodes in the cluster
and  between distant  clusters.  Large volume  of data  can  be transferred  and
transit between the clusters and nodes during the code execution.

 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   practse;  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{Comparing GMRES and Multisplitting algorithms in
synchronous mode}

In the scope of this paper, our first objective is to demonstrate the
Algo-2 (Multisplitting method) shows a better performance in grid
architecture compared with Algo-1 (Classical GMRES) both running in
\textit{synchronous mode}. Better algorithm performance
should means a less number of iterations output and a less 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.\\


\textit{3.a Executing the algorithms on various computational grid
architecture and scaling up the input matrix size}
\\

% environment
\begin{footnotesize}
\begin{tabular}{r c }
 \hline
 Grid & 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}
Table 1 : Clusters x Nodes with N$_{x}$=150 or N$_{x}$=170 \\

\end{footnotesize}



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


In this section, we compare the algorithms performance running on various grid configuration (2x16, 4x8, 4x16 and 8x8). First, the results in figure 3 show for all grid configuration 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.

%\begin{wrapfigure}{l}{100mm}
\begin{figure} [ht!]
\centering
\includegraphics[width=100mm]{cluster_x_nodes_nx_150_and_nx_170.pdf}
\caption{Cluster x Nodes N$_{x}$=150 and N$_{x}$=170}
%\label{overflow}}
\end{figure}
%\end{wrapfigure}

The execution time difference between the two algorithms is important when
comparing between different grid architectures, even with the same number of
processors (like 2x16 and 4x8 = 32 processors for example). The
experiment concludes the low sensitivity of the 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\%) less when running from 2x16=32 to 8x8=64 processors.

\textit{\\3.b Running on two different speed cluster inter-networks\\}

% environment
\begin{footnotesize}
\begin{tabular}{r c }
 \hline
 Grid & 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}
Table 2 : Clusters x Nodes - Networks N1 x N2 \\

 \end{footnotesize}



%\begin{wrapfigure}{l}{100mm}
\begin{figure} [ht!]
\centering
\includegraphics[width=100mm]{cluster_x_nodes_n1_x_n2.pdf}
\caption{Cluster x Nodes N1 x N2}
%\label{overflow}}
\end{figure}
%\end{wrapfigure}

The 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 4 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 in a factor of 2. The results depict also that
when the network speed drops down (12.5\%), the difference between the execution
times can reach more than 25\%.

\textit{\\3.c Network latency impacts on performance\\}

% environment
\begin{footnotesize}
\begin{tabular}{r c }
 \hline
 Grid & 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}
Table 3 : Network latency impact \\

\end{footnotesize}



\begin{figure} [ht!]
\centering
\includegraphics[width=100mm]{network_latency_impact_on_execution_time.pdf}
\caption{Network latency impact on execution time}
%\label{overflow}}
\end{figure}


According the results in figure 5, degradation of the network
latency from 8.10$^{-6}$ to 6.10$^{-5}$ implies an absolute time
increase more than 75\% (resp. 82\%) of the execution for the classical
GMRES (resp. multisplitting) algorithm. In addition, it appears that the
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 of the time for
the multisplitting, even though, the performance was on the same order
of magnitude with a latency of 8.10$^{-6}$.

\textit{\\3.d Network bandwidth impacts on performance\\}

% environment
\begin{footnotesize}
\begin{tabular}{r c }
 \hline
 Grid & 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}
Table 4 : Network bandwidth impact \\

\end{footnotesize}


\begin{figure} [ht!]
\centering
\includegraphics[width=100mm]{network_bandwith_impact_on_execution_time.pdf}
\caption{Network bandwith impact on execution time}
%\label{overflow}
\end{figure}



The results of increasing the network bandwidth show the improvement
of the performance for both of the two algorithms by reducing the execution time (Figure 6). However, and again in this case, the multisplitting method presents a better performance in the considered bandwidth interval with a gain of 40\% which is only around 24\% for classical GMRES.

\textit{\\3.e Input matrix size impacts on performance\\}

% environment
\begin{footnotesize}
\begin{tabular}{r c }
 \hline
 Grid & 4x8\\ %\hline
 Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\hline
 Input matrix size & N$_{x}$ = From 40 to 200\\ \hline \\
 \end{tabular}
Table 5 : Input matrix size impact\\

\end{footnotesize}


\begin{figure} [ht!]
\centering
\includegraphics[width=100mm]{pb_size_impact_on_execution_time.pdf}
\caption{Pb size impact on execution time}
%\label{overflow}}
\end{figure}

In this experimentation, 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 the figure 7,
the execution time for the two algorithms convergence increases with the
iinput matrix size. But the interesting results here direct on (i) the
drastic increase (300 times) of the number of iterations needed before
the convergence for the classical GMRES algorithm when the matrix size
go beyond N$_{x}$=150; (ii) the classical GMRES execution time also almost
the double from N$_{x}$=140 compared with the convergence time of the
multisplitting method. 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. Note that the
same test has been done with the grid 2x16 getting the same conclusion.

\textit{\\3.f CPU Power impact on performance\\}

% environment
\begin{footnotesize}
\begin{tabular}{r c }
 \hline
 Grid & 2x16\\ %\hline
 Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\hline
 Input matrix size & N$_{x}$ = 150 x 150 x 150\\ \hline
 \end{tabular}
Table 6 : CPU Power impact \\

\end{footnotesize}


\begin{figure} [ht!]
\centering
\includegraphics[width=100mm]{cpu_power_impact_on_execution_time.pdf}
\caption{CPU Power impact on execution time}
%\label{overflow}}
s\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 the figure 6
confirm the performance gain, around 95\% for both of the two methods,
after adding more powerful CPU. 

\subsection{Comparing GMRES in native synchronous mode and
Multisplitting algorithms 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.
We have focused the study on analyzing the performance in varying the
key factors impacting the results. The study compares
the performance of the two proposed algorithms both in \textit{synchronous mode
}. In this section, following the same previous methodology, the goal is to
demonstrate the efficiency of the multisplitting method in \textit{
asynchronous mode} compared with the classical GMRES staying in
\textit{synchronous mode}.

Note that the interest of using the asynchronous mode for data exchange
is mainly, in opposite of the synchronous mode, the non-wait aspects of
the current computation after a communication operation like sending
some data between nodes. Each processor can continue their local
calculation without waiting for the end of the communication. Thus, the
asynchronous may theoretically reduce the overall execution time and can
improve the algorithm performance.

As stated supra, Simgrid simulator tool has been used to prove 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 below : \\

% environment
\begin{footnotesize}
\begin{tabular}{r c }
 \hline
 Grid & 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}
\end{footnotesize}

Again, comprehensive and extensive tests have been conducted varying the
CPU power and the network parameters (bandwidth and latency) in the
simulator tool with different problem size. The relative gains greater
than 1 between the two algorithms have been captured after each step of
the test. Table 7 below has recorded the best grid configurations
allowing the multisplitting method execution time more performant 2.5 times than
the classical GMRES execution and convergence time. The experimentation has demonstrated the relative multisplitting algorithm tolerance when using a low speed network that we encounter usually with distant clusters thru 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
%  \caption{Relative gain of the multisplitting algorithm compared with the classical GMRES}
%  \label{"Table 7"}
Table 7. Relative gain of the multisplitting algorithm compared with
the classical GMRES \\

  \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}

\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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