%% 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 multicore applications is always a challenge
+\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
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 variantsof
+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
\maketitle
-\section{Introduction} The use of multi-core architectures for solving large
-scientific problems seems to become imperative in a lot of cases.
+\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.
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 at the condition that
-the simulation results are in education with the real ones.
+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
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 times, due to synchronizations, between two iterations~\cite{bcvc06:ij}.
+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
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 variations
-of the parameters of the execution platform can lead to very different number of
-iterations required to converge and so to very different execution times. In
-this challenging context we think that the use of a simulation tool can greatly
+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
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 comparing to the synchronous GMRES. In
+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
\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}
\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 $\mathbb{R}$
+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). The two-stage multisplitting methods solve the linear system~(\ref{eq:01}) iteratively as follows
+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{algorithm}
\end{figure}
-In this paper, we propose two algorithms of two-stage multisplitting methods. The first algorithm is based on 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
+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.
+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}
\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 our experience, very few modifications are required to adapt a MPI program to run in Simgrid simulator using SMPI (Simulator MPI).The first modification is to include SMPI libraries and related header files (smpi.h). The second and important modification is to eliminate all global variables in moving them to local subroutine 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.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.
+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 plarform: File describing the platform architecture : clusters (CPU power,
+ \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
+ \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;
- \item Execution Mode: synchronous or asynchronous.
+ \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 vérifie, je dis ca de tête}
+ \item execution mode: synchronous or asynchronous.
\end{itemize}
-At last, note that the two solver algorithms have been executed with the Simgrid selector -cfg=smpi/running\_power which determines the computational power (here 19GFlops) of the simulator host machine.
+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*{Acknowledgment}
-
-The authors would like to thank\dots{}
+This work is partially funded by the Labex ACTION program (contract ANR-11-LABX-01-01).
\bibliographystyle{wileyj}
