Desription du problème à résoudre: 3D Poisson equation

[rce2015.git] / paper.tex

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+++ b/paper.tex
@@ -115,7 +115,7 @@ their  applications  using  a simulation tool before.
 
 %\keywords{Algorithm; distributed; iterative; asynchronous; simulation; simgrid;
 %performance} 
 
 %\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
 
 
 \maketitle
 


@@ -358,79 +358,99 @@ It should also be noticed that both solvers have been executed with the Simgrid
 
 In this section, experiments for both Multisplitting algorithms are reported. First the problem sued in our experiments is described.
 
 
 In this section, experiments for both Multisplitting algorithms are reported. First the problem sued in our experiments is described.
 

-\RC{Lilia a toi de jouer}
+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.\\
 
 
 \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
+\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. \\
 
 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
 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
+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. \\
 
 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
+\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
 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
 
 \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
 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}
 
 
 \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.
+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}
 
 \subsection{Comparing GMRES and Multisplitting algorithms in
 synchronous mode}


@@ -611,7 +631,7 @@ 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
 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
 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


@@ -641,7 +661,7 @@ Table 6 : CPU Power impact \\
 \includegraphics[width=100mm]{cpu_power_impact_on_execution_time.pdf}
 \caption{CPU Power impact on execution time}
 %\label{overflow}}
 \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
 
 Using the Simgrid simulator flexibility, we have tried to determine the
 impact on the algorithms performance in varying the CPU power of the