%% 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}
\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 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.
-\paragraph{SIMGRID Simulator parameters}
+\paragraph{Simgrid Simulator parameters}
\begin{itemize}
\item hostfile: Hosts description file.
\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 determine the computational power (here 19GFlops) of the simulator host machine.
+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.
%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%
\label{sec:expe}
-\subsection{Setup study and Methodology}
+\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.
\textbf{Step 2} : Collect the software materials needed for the
experimentation. In our case, we have two variants algorithms for the
-resolution of three 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 (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. \\
\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
and in the other hand the execution time and the number of iterations of
the application before obtaining the convergence. \\
-\textbf{Step 4 }: Setup up the different grid testbeds environment
+\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 8E-6
-microseconds (resp. 5E-5) for the intra-clusters links (resp.
+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). \\
\textbf{Step 5}: Conduct an extensive and comprehensive testings
within these configurations in varying the key parameters, especially
the CPU power capacity, the network parameters and also the size of the
-input matrix. Note that some parameters should be fixed to be invariant to allow the
-comparison like some program input arguments. \\
+input matrix. Note that some parameters like some program input arguments should be fixed to be invariant to allow the comparison. \\
\textbf{Step 6} : Collect and analyze the output results.
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 in transit between the clusters and nodes during the code
+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
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
-\textbf{\textit{synchronous mode}}. Better algorithm performance
+\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
\textit{3.a Executing the algorithms on various computational grid
-architecture scaling up the input matrix size}
+architecture and scaling up the input matrix size}
\\
% environment
%\RCE{J'ai voulu mettre les tableaux des données mais je pense que c'est inutile et ça va surcharger}
-The results in figure 3 show the non-variation of the number of
-iterations of classical GMRES for a given input matrix size; it is not
+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}
\end{figure}
%\end{wrapfigure}
-Unless the 8x8 cluster, the time
-execution difference between the two algorithms is important when
+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 to higher input
-matrix size.
+(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 various computational grid architecture\\}
+\textit{\\3.b Running on two different speed cluster inter-networks\\}
% environment
\begin{footnotesize}
%\end{wrapfigure}
The experiments compare the behavior of the algorithms running first on
-a speed inter- cluster network (N1) and a less performant network (N2).
+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, the difference between the execution
+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\\}
-The results of increasing the network bandwidth depict the improvement
-of the performance by reducing the execution time for both of the two
-algorithms (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.
+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\\}
%\label{overflow}}
\end{figure}
-Using the SIMGRID simulator flexibility, we have tried to determine the
+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. Note that the execution time axis in the
-figure is in logarithmic scale.
+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. In the same line, the study compares
-the performance of the two proposed methods in \textbf{synchronous mode
-}. In this section, with the same previous methodology, the goal is to
-demonstrate the efficiency of the multisplitting method in \textbf{
-asynchronous mode} compare with the classical GMRES staying in the
-synchronous mode.
+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
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
+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 "\,relative gain" in comparison with the
-classical GMRES time.
+\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 : \\
\begin{tabular}{r c }
\hline
Grid & 2x50 totaling 100 processors\\ %\hline
- Processors & 1 GFlops to 1.5 GFlops\\
+ 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
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 I below has recorded the best grid configurations
+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.
|*{#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"}
+% \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
\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}
