University of Franche-Comté\\
IUT de Belfort-Montbéliard, 19 avenue du Maréchal Juin, BP 527, 90016 Belfort cedex, France\\
Fax : +33~3~84~58~77~32\\
- Email: \{jean-claude.charr, ahmed.fanfakh\_badri\_muslim, raphael.couturier, arnaud.giersch\}@univ-fcomte.fr
+ Email: \{jean-claude.charr,raphael.couturier,ahmed.fanfakh\_badri\_muslim,arnaud.giersch\}@univ-fcomte.fr
}
}
\maketitle
\AG{Is the fax number correct? Shall we add a telephone number?}
-\JC{Use Capital letters for only the first letter in the title of a section, table, figure, ...}
\begin{abstract}
Dynamic Voltage Frequency Scaling (DVFS) can be applied to modern CPUs.
This technique is usually used to reduce the energy consumed by a CPU while
the power consumed by the CPU. However, it can also significantly affect the
performance of the executed program if it is compute bound and a low CPU
frequency is selected. The performance degradation ratio can even be higher than
-the saved energy ratio.Therefore, the chosen scaling factor must give the best possible tradeoff
+the saved energy ratio. Therefore, the chosen scaling factor must give the best possible trade-off
between energy reduction and performance.
In this paper we present an algorithm
using the simulator SimGrid/SMPI v3.10~\cite{Casanova:2008:SGF:1397760.1398183}
over an homogeneous distributed memory architecture. Furthermore, we compare the
proposed algorithm with Rauber and Rünger methods~\cite{3}.
-The comparison's results show that our
-algorithm gives better energy-time tradeoff.
+The comparison's results show that our algorithm gives better energy-time trade-off.
This paper is organized as follows: Section~\ref{sec.relwork} presents related works
from other authors. Section~\ref{sec.exe} shows the execution of parallel
tasks and sources of idle times. It resumes the energy
model of homogeneous platform. Section~\ref{sec.mpip} evaluates the performance
-of MPI program. Section~\ref{sec.compet} presents the energy-performance tradeoffs
-objective function. Section~\ref{sec.optim} demonstrates the proposed
-energy-performance algorithm. Section~\ref{sec.expe} verifies the performance prediction
+of MPI program. Section~\ref{sec.compet} presents the energy-performance trade-offs
+objective function. Section~\ref{sec.optim} demonstrates the proposed energy-performance algorithm. Section~\ref{sec.expe} verifies the performance prediction
model and presents the results of the proposed algorithm. Also, It shows the comparison results. Finally,
we conclude in Section~\ref{sec.concl}.
-\section{Related Works}
+\section{Related works}
\label{sec.relwork}
-\AG{Consider introducing the models sec.~\ref{sec.exe} maybe before related works}
+\AG{Consider introducing the models (sec.~\ref{sec.exe}) before related works}
In this section, some heuristics to compute the scaling factor are
presented and classified into two categories: offline and online methods.
The online scaling factor selection methods are executed during the runtime of the program. They are usually integrated into iterative programs where the same block of instructions is executed many times. During the first few iterations, many informations are measured such as the execution time, the energy consumed using a multimeter, the slack times, ... Then a method will exploit these measurements to compute the scaling factor values for each processor. This operation, measurements and computing new scaling factor, can be repeated as much as needed if the iterations are not regular. Kimura, Peraza, Yu-Liang et al. ~\cite{11,2,31} used learning methods to select the appropriate scaling factor values to eliminate the slack times during runtime. However, as seen in ~\cite{39,19}, machine learning methods can take a lot of time to converge when the number of available gears is big. To reduce the impact of slack times, in~\cite{1}, Lim et al. developed an algorithm that detects the
communication sections and changes the frequency during these sections
only. This approach might change the frequency of each processor many times per iteration if an iteration
-contains more than one communication section. In ~\cite{3}, Rauber et al. used an analytical model that after measuring the energy consumed and the execution time with the highest frequency gear, it can predict the energy consumed and the execution time for every frequency gear . These predictions may be used to choose the optimal gear for each processor executing the parallel program to reduce energy consumption.
+contains more than one communication section. In ~\cite{3}, Rauber and Rünger used an analytical model that after measuring the energy consumed and the execution time with the highest frequency gear, it can predict the energy consumed and the execution time for every frequency gear . These predictions may be used to choose the optimal gear for each processor executing the parallel program to reduce energy consumption.
To maintain the performance of the parallel program , they
-set the processor with the biggest load to the highest gear and then compute the scaling factor values for the rest pf the processors. Although this model was built for parallel architectures, it can be adapted to distributed architectures by taking into account the communications.
+set the processor with the biggest load to the highest gear and then compute the scaling factor values for the rest of the processors. Although this model was built for parallel architectures, it can be adapted to distributed architectures by taking into account the communications.
The primary contribution of this paper is presenting a new online scaling factor selection method which has the following characteristics :
\begin{enumerate}
\item Based on Rauber's analytical model to predict the energy consumption and the execution time of the application with different frequency gears.
\end{enumerate}
-\section{Execution and Energy of Parallel Tasks on Homogeneous Platform}
+\section{Execution and energy of parallel tasks on homogeneous platform}
\label{sec.exe}
-\JC{The whole subsection ``Parallel Tasks Execution on Homogeneous Platform'', can be deleted if we need space, we can just say we are interested in this paper in homogeneous clusters}
-\subsection{Parallel Tasks Execution on Homogeneous Platform}
+%\JC{The whole subsection ``Parallel Tasks Execution on Homogeneous Platform'', can be deleted if we need space, we can just say we are interested in this paper in homogeneous clusters}
+\subsection{Parallel tasks execution on homogeneous platform}
A homogeneous cluster consists of identical nodes in terms of hardware and software.
Each node has its own memory and at least one processor which can
be a multi-core. The nodes are connected via a high bandwidth network. Tasks
platform. These tasks can exchange the data via synchronous message passing.
\begin{figure*}[t]
\centering
- \subfloat[Sync. Imbalanced Communications]{\includegraphics[scale=0.67]{commtasks}\label{fig:h1}}
- \subfloat[Sync. Imbalanced Computations]{\includegraphics[scale=0.67]{compt}\label{fig:h2}}
- \caption{Parallel Tasks on Homogeneous Platform}
+ \subfloat[Sync. imbalanced communications]{\includegraphics[scale=0.67]{fig/commtasks}\label{fig:h1}}
+ \subfloat[Sync. imbalanced computations]{\includegraphics[scale=0.67]{fig/compt}\label{fig:h2}}
+ \caption{Parallel tasks on homogeneous platform}
\label{fig:homo}
\end{figure*}
Therefore, the execution time of a task consists of the computation time and the
This happens when processing different amounts of data on each processor (see figure~(\ref{fig:h2})).
In this case the fastest tasks have to wait at the synchronization barrier for the
slowest ones to begin the next task. In both cases the overall execution time
-of the program is the execution time of the slowest task as in equation \ref{eq:T1}.
+of the program is the execution time of the slowest task as in EQ~(\ref{eq:T1}).
\begin{equation}
\label{eq:T1}
\textit{Program Time} = \max_{i=1,2,\dots,N} T_i
\end{equation}
where $T_i$ is the execution time of task $i$ and all the tasks are executed concurrently on different processors.
-\subsection{Energy Model for Homogeneous Platform}
+\subsection{Energy model for homogeneous platform}
Many researchers~\cite{9,3,15,26} divide the power consumed by a processor to two power metrics: the
static and the dynamic power. While the first one is consumed as long as the computing unit is on, the latter is only consumed during computation times. The dynamic power
$P_{dyn}$ is related to the switching activity $\alpha$, load capacitance $C_L$,
-the supply voltage $V$ and operational frequency $f$, as shown in equation ~\ref{eq:pd}.
+the supply voltage $V$ and operational frequency $f$, as shown in EQ~(\ref{eq:pd}).
\begin{equation}
\label{eq:pd}
P_\textit{dyn} = \alpha \cdot C_L \cdot V^2 \cdot f
quadratically the dynamic power which may cause degradation in performance and thus, the increase of the static energy because the execution time is increased~\cite{36}. If the tasks are sorted according to their execution times before scaling in a descending order, the total energy consumption model for a parallel
homogeneous platform, as presented by Rauber et al.~\cite{3}, can be written as a function of the scaling factor \emph S, as in EQ~(\ref{eq:energy}).
-\JC{Are you sure of the following equation}
\begin{equation}
\label{eq:energy}
E = P_\textit{dyn} \cdot S_1^{-2} \cdot
\end{equation}
where \emph N is the number of parallel nodes, $T_i \ and \ S_i \ for \ i=1,...,N$ are the execution times and scaling factors of the sorted tasks. Therefore, $T1$ is the time of the slowest task, and $S_1$ its scaling factor which should be the highest because they are proportional to
the time values $T_i$. The scaling factors are computed as in EQ~(\ref{eq:si}).
-\JC{This equation does not make sense either, what's S? there is no F}
\begin{equation}
\label{eq:si}
S_i = S \cdot \frac{T_1}{T_i}
= \frac{F_\textit{max}}{F_\textit{new}} \cdot \frac{T_1}{T_i}
\end{equation}
-\JC{The Rauber model was used for a parallel machine not a homogeneous platform}
-where $F$ is the number of available frequencies. In this paper we depend on
+In this paper we depend on
Rauber and Rünger energy model EQ~(\ref{eq:energy}) for two reasons: (1) this
-model is used for homogeneous platform that we work on in this paper, and (2) we
+model is used for any number of concurrent tasks, and (2) we
compare our algorithm with Rauber and Rünger scaling factor selection method which is based on
EQ~(\ref{eq:energy}). The optimal scaling factor is computed by minimizing the derivation for this equation which produces EQ~(\ref{eq:sopt}).
-\JC{what's the small n in the equation}
\begin{equation}
\label{eq:sopt}
- S_\textit{opt} = \sqrt[3]{\frac{2}{n} \cdot \frac{P_\textit{dyn}}{P_\textit{static}} \cdot
+ S_\textit{opt} = \sqrt[3]{\frac{2}{N} \cdot \frac{P_\textit{dyn}}{P_\textit{static}} \cdot
\left( 1 + \sum_{i=2}^{N} \frac{T_i^3}{T_1^3} \right) }
\end{equation}
\JC{The following 2 sections can be merged easily}
-\section{Performance Evaluation of MPI Programs}
+\section{Performance evaluation of MPI programs}
\label{sec.mpip}
The performance (execution time) of parallel synchronous MPI applications depend on
\end{equation}
In this paper, this prediction method is used to select the best scaling factor for each processor as presented in the next section.
-
-\section{Performance to Energy Competition}
+\section{Performance to energy competition}
\label{sec.compet}
This section demonstrates our approach for choosing the optimal scaling
\begin{multline}
\label{eq:enorm}
E_\textit{Norm} = \frac{ E_\textit{Reduced}}{E_\textit{Original}} \\
- {} = \frac{P_\textit{dyn} \cdot S_i^{-2} \cdot
+ {} = \frac{P_\textit{dyn} \cdot S_1^{-2} \cdot
\left( T_1 + \sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) +
- P_\textit{static} \cdot T_1 \cdot S_i \cdot N }{
+ P_\textit{static} \cdot T_1 \cdot S_1 \cdot N }{
P_\textit{dyn} \cdot \left(T_1+\sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) +
P_\textit{static} \cdot T_1 \cdot N }
\end{multline}
\end{equation}
\begin{figure*}
\centering
- \subfloat[Converted Relation.]{%
- \includegraphics[width=.4\textwidth]{file.eps}\label{fig:r1}}%
+ \subfloat[Converted relation.]{%
+ \includegraphics[width=.4\textwidth]{fig/file}\label{fig:r1}}%
\qquad%
- \subfloat[Real Relation.]{%
- \includegraphics[width=.4\textwidth]{file3.eps}\label{fig:r2}}
+ \subfloat[Real relation.]{%
+ \includegraphics[width=.4\textwidth]{fig/file3}\label{fig:r2}}
\label{fig:rel}
- \caption{The Energy and Performance Relation}
+ \caption{The energy and performance relation}
\end{figure*}
Then, we can modelize our objective function as finding the maximum distance
between the energy curve EQ~(\ref{eq:enorm}) and the inverse of performance
following form:
\begin{equation}
\label{eq:max}
- \textit{MaxDist} = \max (\overbrace{P^{-1}_\textit{Norm}}^{\text{Maximize}} -
- \overbrace{E_\textit{Norm}}^{\text{Minimize}} )
+ Max Dist = \max_{j=1,2,\dots,F} (\overbrace{P^{-1}_\textit{Norm}(S_j)}^{\text{Maximize}} -
+ \overbrace{E_\textit{Norm}(S_j)}^{\text{Minimize}} )
\end{equation}
-Then we can select the optimal scaling factor that satisfy
+where F is the number of available frequencies. Then we can select the optimal scaling factor that satisfy
EQ~(\ref{eq:max}). Our objective function can work with any energy model or
static power values stored in a data file. Moreover, this function works in
optimal way when the energy curve has a convex form over the available frequency scaling
factors as shown in~\cite{15,3,19}.
-\section{Optimal Scaling Factor for Performance and Energy}
+\section{Optimal scaling factor for performance and energy}
\label{sec.optim}
Algorithm~\ref{EPSA} compute the optimal scaling factor according to the objective function described above.
\begin{algorithm}[tp]
\State Set $P_{states}$ to the number of available frequencies.
\State Set the variable $F_{new}$ to max. frequency, $F_{new} = F_{max} $
\State Set the variable $F_{diff}$ to the difference between two successive frequencies.
- \For {$i=1$ to $P_{states} $}
+ \For {$j:=1$ to $P_{states} $}
\State - $F_{new}=F_{new} - F_{diff} $
\State - $S = \frac{F_\textit{max}}{F_\textit{new}}$
- \State - $S_i = S \cdot \frac{T_1}{T_i}= \frac{F_\textit{max}}{F_\textit{new}} \cdot \frac{T_1}{T_i} \ for \ i=1,...,N$
- \State - $E_\textit{Norm} = \frac{P_\textit{dyn} \cdot S_i^{-2} \cdot
+ \State - $S_i = S \cdot \frac{T_1}{T_i}= \frac{F_\textit{max}}{F_\textit{new}} \cdot \frac{T_1}{T_i} \
+ for \ i=1,...,N$
+ \State - $E_\textit{Norm} = \frac{P_\textit{dyn} \cdot S_1^{-2} \cdot
\left( T_1 + \sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) +
- P_\textit{static} \cdot T_1 \cdot S_i \cdot N }{
+ P_\textit{static} \cdot T_1 \cdot S_1 \cdot N }{
P_\textit{dyn} \cdot \left(T_1+\sum_{i=2}^{N}\frac{T_i^3}{T_1^2}\right) +
P_\textit{static} \cdot T_1 \cdot N }$
\State - $P_{NormInv}=T_{old}/T_{new}$
\If{ $(P_{NormInv}-E_{Norm} > Dist$) }
- \State $S_{optimal} = S$
+ \State $S_{opt} = S$
\State $Dist = P_{NormInv} - E_{Norm}$
\EndIf
\EndFor
- \State Return $S_{optimal}$
+ \State Return $S_{opt}$
\end{algorithmic}
\end{algorithm}
\caption{DVFS}
\label{dvfs}
\begin{algorithmic}[1]
- \For {$J:=1$ to $Some-Iterations \; $}
+ \For {$k:=1$ to $Some-Iterations \; $}
\State -Computations Section.
\State -Communications Section.
- \If {$(J=1)$}
+ \If {$(k=1)$}
\State -Gather all times of computation and\par\hspace{13 pt} communication from each node.
\State -Call algorithm~\ref{EPSA} with these times.
- \State -Compute the new frequency from the returned optimal scaling factor.
+ \State -Compute the new frequency from the \par\hspace{13 pt} returned optimal scaling factor.
\State -Set the new frequency to the CPU.
\EndIf
\EndFor
they have balanced workloads, otherwise, they take different frequencies when
having imbalanced workloads. Thus, EQ~(\ref{eq:fi}) adapts the frequency of the CPU to the nodes' workloads to maintain performance.
-\section{Experimental Results}
+\section{Experimental results}
\label{sec.expe}
Our experiments are executed on the simulator SimGrid/SMPI
-v3.10. We configure the simulator to use a a homogeneous cluster with one core per
+v3.10. We configure the simulator to use a homogeneous cluster with one core per
node. The
detailed characteristics of our platform file are shown in the
table~(\ref{table:platform}).
frequencies. The simulated network link is 1 GB Ethernet (TCP/IP).
The backbone of the cluster simulates a high performance switch.
\begin{table}[htb]
- \caption{Platform File Parameters}
+ \caption{Platform file parameters}
% title of Table
\centering
\begin{tabular}{|*{7}{l|}}
\end{tabular}
\label{table:platform}
\end{table}
-\subsection{Performance Prediction Verification}
+\subsection{Performance prediction verification}
In this section we evaluate the precision of our performance prediction method based on EQ~(\ref{eq:tnew}) by applying it the NAS benchmarks. The NAS programs are executed with the class B option for comparing the
real execution time with the predicted execution time. Each program runs offline
frequency by the new one see EQ~(\ref{eq:s}).
\begin{figure*}[t]
\centering
- \includegraphics[width=.4\textwidth]{cg_per.eps}\qquad%
- \includegraphics[width=.4\textwidth]{mg_pre.eps}
- \includegraphics[width=.4\textwidth]{bt_pre.eps}\qquad%
- \includegraphics[width=.4\textwidth]{lu_pre.eps}
+ \includegraphics[width=.328\textwidth]{fig/cg_per}\hfill%
+ \includegraphics[width=.328\textwidth]{fig/mg_pre}\hfill%
+ % \includegraphics[width=.4\textwidth]{fig/bt_pre}\qquad%
+ \includegraphics[width=.328\textwidth]{fig/lu_pre}\hfill%
\caption{Comparing predicted to real execution time}
\label{fig:pred}
\end{figure*}
In our cluster there are 18 available frequency states for each processor.
This lead to 18 run states for each program. We use seven MPI programs of the
NAS parallel benchmarks: CG, MG, EP, FT, BT, LU
-and SP. Figure~(\ref{fig:pred}) presents plots of the real execution times and the simulated ones. The average normalized errors between the predicted execution time and
-the real time (SimGrid time) for all programs is between 0.0032 to 0.0133.
-\JC{why compute the average error not the max}
-\subsection{The experimental results}
+and SP. Figure~(\ref{fig:pred}) presents plots of the real execution times and the simulated ones. The maximum normalized error between the predicted execution time and the real time (SimGrid time) for all programs is between 0.0073 to 0.031. The better case is for CG and the worse case is for LU.
+\subsection{The experimental results for the scaling algorithm }
The proposed algorithm was applied to seven MPI programs of the NAS
benchmarks (EP, CG, MG, FT, BT, LU and SP) which were run with three classes (A, B and
C). For each instance the benchmarks were executed on a number of processors
same time from all available scaling factors.
\begin{figure*}[t]
\centering
- \includegraphics[width=.33\textwidth]{ep.eps}\hfill%
- \includegraphics[width=.33\textwidth]{cg.eps}\hfill%
- \includegraphics[width=.33\textwidth]{sp.eps}
- \includegraphics[width=.33\textwidth]{lu.eps}\hfill%
- \includegraphics[width=.33\textwidth]{bt.eps}\hfill%
- \includegraphics[width=.33\textwidth]{ft.eps}
- \caption{Optimal scaling factors for The parallel NAS benchmarks}
+ \includegraphics[width=.328\textwidth]{fig/ep}\hfill%
+ \includegraphics[width=.328\textwidth]{fig/cg}\hfill%
+ \includegraphics[width=.328\textwidth]{fig/sp}
+ \includegraphics[width=.328\textwidth]{fig/lu}\hfill%
+ \includegraphics[width=.328\textwidth]{fig/bt}\hfill%
+ \includegraphics[width=.328\textwidth]{fig/ft}
+ \caption{Optimal scaling factors for the predicted energy and performance of NAS benchmarks}
\label{fig:nas}
\end{figure*}
\begin{table}[htb]
- \caption{The Scaling Factors Results}
+ \caption{The scaling factors results}
% title of Table
\centering
\begin{tabular}{|l|*{4}{r|}}
example BT and EP. Our algorithm selects big scaling factor value when the
communication and the other slacks times are big and smaller ones in opposite
cases. In EP there are no communications inside the iterations. This make our
-EPSA to selects smaller scaling factor values (inducing smaller energy savings).
+algorithm to selects smaller scaling factor values (inducing smaller energy savings).
-\subsection{Comparing Results}
+\subsection{Results comparison}
In this section, we compare our scaling factor selection method with Rauber and Rünger
methods~\cite{3}. They had two scenarios, the first is to reduce energy to the
refer to this scenario as $R_{E}$. The second scenario is similar to the first
except setting the slower task to the maximum frequency (when the scale $S=1$)
to keep the performance from degradation as mush as possible. We refer to this
-scenario as $R_{E-P}$. The comparison is made in tables~(\ref{table:compare
- Class A},\ref{table:compare Class B},\ref{table:compare Class C}). These
+scenario as $R_{E-P}$. While we refer to our algorithm as EPSA. The comparison
+is made in tables \ref{table:compareA}, \ref{table:compareB},
+and~\ref{table:compareC}. These
tables show the results of our method and Rauber and Rünger scenarios for all the
NAS benchmarks programs for classes A,B and C.
\begin{table}[p]
- \caption{Comparing Results for The NAS Class A}
+ \caption{Comparing results for the NAS class A}
% title of Table
\centering
\begin{tabular}{|l|l|*{4}{r|}}
% is used to refer this table in the text
\end{table}
\begin{table}[p]
- \caption{Comparing Results for The NAS Class B}
+ \caption{Comparing results for the NAS class B}
% title of Table
\centering
\begin{tabular}{|l|l|*{4}{r|}}
\end{table}
\begin{table}[p]
- \caption{Comparing Results for The NAS Class C}
+ \caption{Comparing results for the NAS class C}
% title of Table
\centering
\begin{tabular}{|l|l|*{4}{r|}}
Figure~(\ref{fig:compare}) shows the maximum distance between the energy saving
percent and the performance degradation percent.
-Negative values mean that one of the two objectives (energy or performance) have been degraded more than the other. The positive tradeoffs with the highest values lead to maximum energy savings
+Negative values mean that one of the two objectives (energy or performance) have been degraded more than the other. The positive trade-offs with the highest values lead to maximum energy savings
while keeping the performance degradation as low as possible. Our algorithm always
-gives the highest positive energy to performance tradeoffs while Rauber and Rünger method
-($R_{E-P}$) gives in some time negative tradeoffs such as in BT and
+gives the highest positive energy to performance trade-offs while Rauber and Rünger method
+($R_{E-P}$) gives in some time negative trade-offs such as in BT and
EP.
-\begin{figure}[t]
+\begin{figure*}[t]
\centering
- \includegraphics[width=.33\textwidth]{compare_class_A.pdf}
- \includegraphics[width=.33\textwidth]{compare_class_B.pdf}
- \includegraphics[width=.33\textwidth]{compare_class_c.pdf}
- \caption{Comparing our method to Rauber and Rünger Methods}
+ \includegraphics[width=.328\textwidth]{fig/compare_class_A}
+ \includegraphics[width=.328\textwidth]{fig/compare_class_B}
+ \includegraphics[width=.328\textwidth]{fig/compare_class_C}
+ \caption{Comparing our method to Rauber and Rünger methods}
\label{fig:compare}
-\end{figure}
+\end{figure*}
\section{Conclusion}
\label{sec.concl}
-In this paper, we have presented a new online scaling factor selection method that optimizes simultaneously the energy and performance of a distributed application running on an homogeneous cluster. It uses the computation and communication times measured at the first iteration to predict energy consumption and the performance of the parallel application at every available frequency. Then, it selects the scaling factor that gives the best tradeoff between energy reduction and performance which is the maximum distance between the energy and the inverted performance curves. To evaluate this method, we have applied it to the NAS benchmarks and it was compared to Rauber's methods while being executed on the simulator SimGrid. The results showed that our method, outperforms Rauber's methods in terms of energy-performance ratio.
+In this paper, we have presented a new online scaling factor selection method that optimizes simultaneously the energy and performance of a distributed application running on an homogeneous cluster. It uses the computation and communication times measured at the first iteration to predict energy consumption and the performance of the parallel application at every available frequency. Then, it selects the scaling factor that gives the best trade-off between energy reduction and performance which is the maximum distance between the energy and the inverted performance curves. To evaluate this method, we have applied it to the NAS benchmarks and it was compared to Rauber and Rünger methods while being executed on the simulator SimGrid. The results showed that our method, outperforms Rauber and Rünger methods in terms of energy-performance ratio.
In the near future, we would like to adapt this scaling factor selection method to heterogeneous platforms where each node has different characteristics. In particular, each CPU has different available frequencies, energy consumption and performance. It would be also interesting to develop a new energy model for asynchronous parallel iterative methods where the number of iterations is not known in advance and depends on the global convergence of the iterative system.
\section*{Acknowledgment}
+\AG{Jean-Claude, why did you remove the Mésocentre here?}
As a PhD student, M. Ahmed Fanfakh, would like to thank the University of
Babylon (Iraq) for supporting his work.
-\JC{delete the online paths for each reference}
% trigger a \newpage just before the given reference
% number - used to balance the columns on the last page
% adjust value as needed - may need to be readjusted if
%%% End:
% LocalWords: Fanfakh Charr FIXME Tianhe DVFS HPC NAS NPB SMPI Rauber's Rauber
-% LocalWords: CMOS EQ EPSA Franche Comté Tflop Rünger
+% LocalWords: CMOS EQ EPSA Franche Comté Tflop Rünger IUT Maréchal Juin cedex
