%\keywords{Algorithm; distributed; iterative; asynchronous; simulation; simgrid;
%performance}
-\keywords{Multisplitting algorithms, Synchronous and asynchronous iterations, SimGrid, Simulation, Performance evaluation}
+\keywords{ Performance evaluation, Simulation, SimGrid, Synchronous and asynchronous iterations, Multisplitting algorithms}
\maketitle
\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
+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 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 maximum number of the gmres's restarts in the Arnorldi process;
+ \item maximum number of iterations qnd the tolerance threshold in classical GMRES;
+ \item tolerance threshold for outer and inner-iterations;
+ \item matrix size (N$_{x}$, N$_{y}$ and N$_{z}$) respectively on x, y, z axis;
+ \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 matrix off-diagonal value;
+ \item execution mode: synchronous or asynchronous;
+ \RCE {C'est ok la liste des arguments du programme mais si Lilia ou toi pouvez preciser pour les arguments pour CGLS ci dessous} \RC{Vu que tu n'as pas fait varier ce paramètre, on peut ne pas en parler}
+ \item Size of matrix S;
+ \item Maximum number of iterations and tolerance threshold for CGLS.
\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{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{3D Poisson}
+
+
+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}
-To conduct our study, we have put in place the following methodology
-which can be reused for any grid-enabled applications.
+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
+\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
+\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 (Algo-1); (2) and the multisplitting method (Algo-2). In addition, Simgrid simulator has been chosen to simulate the behaviors of the
-distributed applications. Simgrid is running on the Mesocentre datacenter in Franche-Comte University but also in a virtual machine on a laptop. \\
+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 simple laptop. \\
-\textbf{Step 3} : Fix the criteria which will be used for the future
+\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
-in one hand the algorithm execution mode (synchronous and asynchronous)
-and in the other hand the execution time and the number of iterations of
-the application before obtaining the convergence. \\
-
-\textbf{Step 4 }: Set up the different grid testbed environments
-which will be simulated in the simulator tool to run the program. The
-following architecture has been configured in Simgrid : 2x16 - that is a
-grid containing 2 clusters with 16 hosts (processors/cores) each -, 4x8,
-4x16, 8x8 and 2x50. The network has been designed to operate with a
-bandwidth equals to 10Gbits (resp. 1Gbits/s) and a latency of 8.10$^{-6}$
-microseconds (resp. 5.10$^{-5}$) for the intra-clusters links (resp.
-inter-clusters backbone links). \\
+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 in varying the key parameters, especially
+within these configurations by varying the key parameters, especially
the CPU power capacity, the network parameters and also the size of the
-input matrix. Note that some parameters like some program input arguments should be fixed to be invariant to allow the comparison. \\
+input data. \\
\textbf{Step 6} : Collect and analyze the output results.
\subsection{Factors impacting distributed applications performance in
a grid environment}
-From our previous experience on running distributed application in a
-computational grid, many factors are identified to have an impact on the
-program behavior and performance on this specific environment. Mainly,
-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 : (i) 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 pass
-from one point to another in a unit of time. (ii) 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. 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 in one hand, the
-"\,intra-network" which refers to the links between nodes within a
-cluster and in the other hand, the "\,inter-network" which is the
-backbone link between clusters. By design, these two networks perform
-with different speed. 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 thru 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
+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 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 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.
+In the scope of this paper, our first objective is to analyze when the Krylov
+Multisplitting method has better performances than the classical GMRES
+method. With an iterative method, better performances 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.\\
-\textit{3.a Executing the algorithms on various computational grid
+\subsubsection{Execution of the the algorithms on various computational grid
architecture and scaling up the input matrix size}
-\\
-
+\ \\
% environment
\begin{footnotesize}
\begin{tabular}{r c }
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
-input matrix size. But the interesting results here direct on (i) 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
\includegraphics[width=100mm]{cpu_power_impact_on_execution_time.pdf}
\caption{CPU Power impact on execution time}
%\label{overflow}}
-\end{figure}
+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
