From: couturie <couturie@carcariass.(none)>
Date: Mon, 25 Feb 2013 10:23:12 +0000 (+0100)
Subject: ch13
X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/book_gpu.git/commitdiff_plain/18cb85fd3967a4c46c9582e8515cef9af3448269?ds=inline

ch13
---

diff --git a/BookGPU/BookGPU.tex b/BookGPU/BookGPU.tex
index 76dc8f6..f4f8323 100755
--- a/BookGPU/BookGPU.tex
+++ b/BookGPU/BookGPU.tex
@@ -148,6 +148,7 @@
 \include{Chapters/chapter9/ch9}
 \include{Chapters/chapter11/ch11}
 \include{Chapters/chapter12/ch12}
+\include{Chapters/chapter13/ch13}
 \include{Chapters/chapter14/ch14}
 \include{Chapters/chapter15/ch15}
 \include{Chapters/chapter16/ch16}
diff --git a/BookGPU/Chapters/chapter13/biblio13.bib b/BookGPU/Chapters/chapter13/biblio13.bib
new file mode 100644
index 0000000..b5ca7f0
--- /dev/null
+++ b/BookGPU/Chapters/chapter13/biblio13.bib
@@ -0,0 +1,155 @@
+@article{ref1,
+title = {Asynchronous grid computation for {A}merican options derivatives},
+author = {Chau, Ming and Couturier, Rapha\"el and Bahi, Jacques M. and Spiteri, Pierre},
+journal = {Advances in Engineering Software},
+volume = {},
+number = {0},
+pages = {},
+note = {},
+year = {2012},
+}
+
+@article{ref2,
+title = {Matrix iterative analysis},
+author = {Varga, Richard S.},
+journal = {Prentice Hall},
+volume = {},
+number = {},
+pages = {},
+note = {},
+year = {},
+}
+
+@article{ref3,
+ author = {Baudet, G\'erard M.},
+ title = {Asynchronous iterative methods for multiprocessors},
+ journal = {Journal Assoc. Comput. Mach.},
+ volume = {25},
+ number = {},
+ pages = {226--244},
+ year = {1978},
+}
+
+@article{ref4,
+ author = {Bertsekas, Dimitri P. and Tsitsiklis, John N.},
+ title = {Parallel and distributed computation, numerical methods},
+ journal = {Prentice Hall Englewood Cliffs N. J. (1989)},
+ volume = {},
+ number = {},
+ pages = {},
+ year = {1989},
+}
+
+@article{ref5,
+ author = {Bahi, Jacques M. and Contassot-Vivier, Sylvain and Couturier, Rapha\"el},
+ title = {Parallel iterative algorithms: from sequential to grid computing},
+ journal = {Chapman \& Hall/CRC, Numerical Analysis \& Scientific Computating 1 (2007)},
+ volume = {},
+ number = {},
+ pages = {},
+ year = {2007},
+}
+
+@article{ref6,
+ author = {Miellou, Jean-Claude and Spiteri, Pierre},
+ title = {Two criteria for the convergence of asynchronous iterations},
+ journal = {in Computers and computing, P. Chenin et al. ed., Wiley Masson},
+ volume = {},
+ number = {},
+ pages = {91--95},
+ year = {1985},
+}
+
+@article{ref7,
+ author = {Miellou, Jean-Claude},
+ title = {Algorithmes de relaxation chaotique \`a retards},
+ journal = {RAIRO Analyse num\'erique},
+ volume = {R1},
+ number = {},
+ pages = {55--82},
+ year = {1975},
+}
+
+@article{ref8,
+title = {{CUDA} Toolkit 4.2 {CUBLAS} Library},
+author = {NVIDIA Corporation},
+journal = {},
+volume = {},
+number = {},
+pages = {},
+note = {\url{http://developer.download.nvidia.com/compute/DevZone/docs/html/CUDALibraries/doc/CUBLAS_Library.pdf}},
+year = {2012},
+}
+
+@article{ref9,
+ author = {Micikevicius, Paulius},
+ title = {{3D} finite difference computation on {GPUs} using {CUDA}},
+ journal = {Proceedings of 2nd Workshop on General Purpose Processing on Graphics Processing Units},
+ volume = {},
+ number = {},
+ pages = {79--84},
+ year = {2009},
+}
+
+@article{ref10,
+ author = {Leist, Aron and Playne, Daniel P. and Hawick, Ken A.},
+ title = {Exploiting graphical processing units for data-parallel scientific applications},
+ journal = {Concurrency and Computation: Practice and Experience},
+ volume = {21},
+ number = {18},
+ pages = {2400--2437},
+ year = {2009},
+}
+
+@article{ref11,
+ author = {Chau, Ming and Couturier, Rapha\"el and Bahi, Jacques M. and Spiteri, Pierre},
+ title = {Parallel solution of the obstacle problem in grid environments},
+ journal = {International Journal of High Performance Computing Applications},
+ volume = {25},
+ number = {4},
+ pages = {488--495},
+ year = {2011},
+}
+
+@article{ref12,
+ author = {Nvidia},
+ title = {{NVIDIA} {CUDA} {C} {P}rogramming {G}uide},
+ journal = {Version 4.2 (2012)},
+ volume = {},
+ number = {},
+ pages = {},
+ year = {2012},
+}
+
+@article{ref13,
+ author = {Evans, David J.},
+ title = {Parallel {S}.{O}.{R}. iterative methods},
+ journal = {Parallel Computing},
+ volume = {1},
+ number = {1},
+ pages = {3--18},
+ year = {1984},
+}
+
+@article{ref14,
+ author = {Iwashita, Takeshi and Shimasaki, Masaaki},
+ title = {Block red-black ordering method for parallel processing of {ICCG} solver},
+ journal = {High Performance Computing},
+ volume = {2327},
+ number = {},
+ pages = {},
+ year = {2006},
+}
+
+@article{ref15,
+title = {Iterative methods for sparse linear systems},
+author = {Saad, Yousef},
+journal = {Society for Industrial and Applied Mathematics, 2nd edition},
+volume = {},
+number = {},
+pages = {},
+year = {2003},
+}
+
+
+
diff --git a/BookGPU/Chapters/chapter13/bibunits.sty b/BookGPU/Chapters/chapter13/bibunits.sty
new file mode 100644
index 0000000..e5e7bb9
--- /dev/null
+++ b/BookGPU/Chapters/chapter13/bibunits.sty
@@ -0,0 +1,314 @@
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diff --git a/BookGPU/Chapters/chapter13/ch12.tex~ b/BookGPU/Chapters/chapter13/ch12.tex~
new file mode 100755
index 0000000..370c6c7
--- /dev/null
+++ b/BookGPU/Chapters/chapter13/ch12.tex~
@@ -0,0 +1,1080 @@
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%                          %%
+%%       CHAPTER 12         %%
+%%                          %%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ 
+\chapterauthor{}{}
+\chapter{Solving sparse linear systems with GMRES and CG methods on GPU clusters}
+
+%%--------------------------%%
+%%       SECTION 1          %%
+%%--------------------------%%
+\section{Introduction}
+\label{sec:01}
+The sparse linear systems are used to model many scientific and industrial problems, such as the environmental simulations or
+the industrial processing of the complex or non-Newtonian fluids. Moreover, the resolution of these problems often involves the
+solving of such linear systems which is considered as the most expensive process in terms of time execution and memory space.
+Therefore, solving sparse linear systems must be as efficient as possible in order to deal with problems of ever increasing size.
+
+There are, in the jargon of numerical analysis, different methods of solving sparse linear systems that we can classify in two
+classes: the direct and iterative methods. However, the iterative methods are often more suitable than their counterpart, direct
+methods, for solving large sparse linear systems. Indeed, they are less memory consuming and easier to parallelize on parallel
+computers than direct methods. Different computing platforms, sequential and parallel computers, are used for solving sparse
+linear systems with iterative solutions. Nowadays, graphics processing units (GPUs) have become attractive for solving these
+linear systems, due to their computing power and their ability to compute faster than traditional CPUs.
+
+In Section~\ref{sec:02}, we describe the general principle of two well-known iterative methods: the conjugate gradient method and
+the generalized minimal residual method. In Section~\ref{sec:03}, we give the main key points of the parallel implementation of both
+methods on a cluster of GPUs. Then, in Section~\ref{sec:04}, we present the experimental results obtained on a CPU cluster and on
+a GPU cluster, for solving sparse linear systems associated to matrices of different structures. Finally, in Section~\ref{sec:05},
+we apply the hypergraph partitioning technique to reduce the total communication volume between the computing nodes and, thus, to
+improve the execution times of the parallel algorithms of both iterative methods.   
+
+
+%%--------------------------%%
+%%       SECTION 2          %%
+%%--------------------------%%
+\section{Krylov iterative methods}
+\label{sec:02}
+Let us consider the following system of $n$ linear equations in $\mathbb{R}$: 
+\begin{equation}
+Ax=b,
+\label{eq:01}
+\end{equation}
+where $A\in\mathbb{R}^{n\times n}$ is a sparse nonsingular square matrix, $x\in\mathbb{R}^{n}$ is the solution vector,
+$b\in\mathbb{R}^{n}$ is the right-hand side and $n\in\mathbb{N}$ is a large integer number. 
+
+The iterative methods for solving the large sparse linear system~(\ref{eq:01}) proceed by successive iterations of a same
+block of elementary operations, during which an infinite number of approximate solutions $\{x_k\}_{k\geq 0}$ are computed.
+Indeed, from an initial guess $x_0$, an iterative method determines at each iteration $k>0$ an approximate solution $x_k$
+which, gradually, converges to the exact solution $x^{*}$ as follows:
+\begin{equation}
+x^{*}=\lim\limits_{k\to\infty}x_{k}=A^{-1}b.
+\label{eq:02}
+\end{equation}
+The number of iterations necessary to reach the exact solution $x^{*}$ is not known beforehand and can be infinite. In
+practice, an iterative method often finds an approximate solution $\tilde{x}$ after a fixed number of iterations and/or
+when a given convergence criterion is satisfied as follows:   
+\begin{equation}
+\|b-A\tilde{x}\| < \varepsilon,
+\label{eq:03}
+\end{equation}
+where $\varepsilon<1$ is the required convergence tolerance threshold. 
+
+Some of the most iterative methods that have proven their efficiency for solving large sparse linear systems are those
+called \textit{Krylov sub-space methods}~\cite{ref1}. In the present chapter, we describe two Krylov methods which are
+widely used: the conjugate gradient method (CG) and the generalized minimal residual method (GMRES). In practice, the
+Krylov sub-space methods are usually used with preconditioners that allow to improve their convergence. So, in what
+follows, the CG and GMRES methods are used for solving the left-preconditioned sparse linear system:
+\begin{equation}
+M^{-1}Ax=M^{-1}b,
+\label{eq:11}
+\end{equation}
+where $M$ is the preconditioning matrix.
+
+%%****************%%
+%%****************%%
+\subsection{CG method}
+\label{sec:02.01}
+The conjugate gradient method is initially developed by Hestenes and Stiefel in 1952~\cite{ref2}. It is one of the well
+known iterative method for solving large sparse linear systems. In addition, it can be adapted for solving nonlinear 
+equations and optimization problems. However, it can only be applied to problems with positive definite symmetric matrices.
+
+The main idea of the CG method is the computation of a sequence of approximate solutions $\{x_k\}_{k\geq 0}$ in a Krylov
+sub-space of order $k$ as follows: 
+\begin{equation}
+x_k \in x_0 + \mathcal{K}_k(A,r_0),
+\label{eq:04}
+\end{equation}
+such that the Galerkin condition must be satisfied:
+\begin{equation}
+r_k \bot \mathcal{K}_k(A,r_0),
+\label{eq:05}
+\end{equation}
+where $x_0$ is the initial guess, $r_k=b-Ax_k$ is the residual of the computed solution $x_k$ and $\mathcal{K}_k$ the Krylov
+sub-space of order $k$: \[\mathcal{K}_k(A,r_0) \equiv\text{span}\{r_0, Ar_0, A^2r_0,\ldots, A^{k-1}r_0\}.\]
+In fact, CG is based on the construction of a sequence $\{p_k\}_{k\in\mathbb{N}}$ of direction vectors in $\mathcal{K}_k$
+which are pairwise $A$-conjugate ($A$-orthogonal):
+\begin{equation}
+\begin{array}{ll}
+p_i^T A p_j = 0, & i\neq j. 
+\end{array} 
+\label{eq:06}
+\end{equation}
+At each iteration $k$, an approximate solution $x_k$ is computed by recurrence as follows:  
+\begin{equation}
+\begin{array}{ll}
+x_k = x_{k-1} + \alpha_k p_k, & \alpha_k\in\mathbb{R}.
+\end{array} 
+\label{eq:07}
+\end{equation}
+Consequently, the residuals $r_k$ are computed in the same way:
+\begin{equation}
+r_k = r_{k-1} - \alpha_k A p_k. 
+\label{eq:08}
+\end{equation}
+In the case where all residuals are nonzero, the direction vectors $p_k$ can be determined so that the following recurrence 
+holds:
+\begin{equation}
+\begin{array}{lll}
+p_0=r_0, & p_k=r_k+\beta_k p_{k-1}, & \beta_k\in\mathbb{R}.
+\end{array} 
+\label{eq:09}
+\end{equation}
+Moreover, the scalars $\{\alpha_k\}_{k>0}$ are chosen so as to minimize the $A$-norm error $\|x^{*}-x_k\|_A$ over the Krylov
+sub-space $\mathcal{K}_{k}$ and the scalars $\{\beta_k\}_{k>0}$ are chosen so as to ensure that the direction vectors are
+pairwise $A$-conjugate. So, the assumption that matrix $A$ is symmetric and the recurrences~(\ref{eq:08}) and~(\ref{eq:09})
+allow to deduce that:
+\begin{equation}
+\begin{array}{ll}
+\alpha_{k}=\frac{r^{T}_{k-1}r_{k-1}}{p_{k}^{T}Ap_{k}}, & \beta_{k}=\frac{r_{k}^{T}r_{k}}{r_{k-1}^{T}r_{k-1}}.
+\end{array}
+\label{eq:10}
+\end{equation}
+
+Algorithm~\ref{alg:01} shows the main key points of the preconditioned CG method. It allows to solve the left-preconditioned
+sparse linear system~(\ref{eq:11}). In this algorithm, $\varepsilon$ is the convergence tolerance threshold, $maxiter$ is the maximum
+number of iterations and $(\cdot,\cdot)$ defines the dot product between two vectors in $\mathbb{R}^{n}$. At every iteration, a direction
+vector $p_k$ is determined, so that it is orthogonal to the preconditioned residual $z_k$ and to the direction vectors $\{p_i\}_{i<k}$
+previously determined (from line~$8$ to line~$13$). Then, at lines~$16$ and~$17$ , the iterate $x_k$ and the residual $r_k$ are computed
+using formulas~(\ref{eq:07}) and~(\ref{eq:08}), respectively. The CG method converges after, at most, $n$ iterations. In practice, the CG
+algorithm stops when the tolerance threshold $\varepsilon$ and/or the maximum number of iterations $maxiter$ are reached.
+
+\begin{algorithm}
+  \SetLine
+  \linesnumbered
+  Choose an initial guess $x_0$\;
+  $r_{0} = b - A x_{0}$\;
+  $convergence$ = false\;
+  $k = 1$\;
+  \Repeat{convergence}{
+    $z_{k} = M^{-1} r_{k-1}$\;
+    $\rho_{k} = (r_{k-1},z_{k})$\;
+    \eIf{$k = 1$}{
+      $p_{k} = z_{k}$\;
+    }{
+      $\beta_{k} = \rho_{k} / \rho_{k-1}$\;
+      $p_{k} = z_{k} + \beta_{k} \times p_{k-1}$\;
+    }
+    $q_{k} = A \times p_{k}$\;
+    $\alpha_{k} = \rho_{k} / (p_{k},q_{k})$\;
+    $x_{k} = x_{k-1} + \alpha_{k} \times p_{k}$\;
+    $r_{k} = r_{k-1} - \alpha_{k} \times q_{k}$\;
+    \eIf{$(\rho_{k} < \varepsilon)$ {\bf or} $(k \geq maxiter)$}{
+      $convergence$ = true\;
+    }{
+      $k = k + 1$\;
+    }
+  }
+\caption{Left-preconditioned CG method}
+\label{alg:01}
+\end{algorithm}
+
+%%****************%%
+%%****************%%
+\subsection{GMRES method} 
+\label{sec:02.02}
+The iterative method GMRES is developed by Saad and Schultz in 1986~\cite{ref3} as a generalization of the minimum residual method
+MINRES~\cite{ref4}. Indeed, GMRES can be applied for solving symmetric or asymmetric linear systems. 
+
+The main principle of the GMRES method is to find an approximation minimizing at best the residual norm. In fact, GMRES
+computes a sequence of approximate solutions $\{x_k\}_{k>0}$ in a Krylov sub-space $\mathcal{K}_k$ as follows:
+\begin{equation}
+\begin{array}{ll}
+x_k \in x_0 + \mathcal{K}_k(A, v_1),& v_1=\frac{r_0}{\|r_0\|_2},
+\end{array}
+\label{eq:12}
+\end{equation} 
+so that the Petrov-Galerkin condition is satisfied:
+\begin{equation}
+\begin{array}{ll}
+r_k \bot A \mathcal{K}_k(A, v_1).
+\end{array}
+\label{eq:13}
+\end{equation}
+GMRES uses the Arnoldi process~\cite{ref5} to construct an orthonormal basis $V_k$ for the Krylov sub-space $\mathcal{K}_k$
+and an upper Hessenberg matrix $\bar{H}_k$ of order $(k+1)\times k$:
+\begin{equation}
+\begin{array}{ll}
+V_k = \{v_1, v_2,\ldots,v_k\}, & \forall k>1, v_k=A^{k-1}v_1,
+\end{array}
+\label{eq:14}
+\end{equation}
+and
+\begin{equation}
+V_k A = V_{k+1} \bar{H}_k.
+\label{eq:15}
+\end{equation}
+
+Then, at each iteration $k$, an approximate solution $x_k$ is computed in the Krylov sub-space $\mathcal{K}_k$ spanned by $V_k$
+as follows:
+\begin{equation}
+\begin{array}{ll}
+x_k = x_0 + V_k y, & y\in\mathbb{R}^{k}.
+\end{array}
+\label{eq:16}
+\end{equation}
+From both formulas~(\ref{eq:15}) and~(\ref{eq:16}) and $r_k=b-Ax_k$, we can deduce that:
+\begin{equation}
+\begin{array}{lll}
+  r_{k} & = & b - A (x_{0} + V_{k}y) \\
+        & = & r_{0} - AV_{k}y \\
+        & = & \beta v_{1} - V_{k+1}\bar{H}_{k}y \\
+        & = & V_{k+1}(\beta e_{1} - \bar{H}_{k}y),
+\end{array}
+\label{eq:17}
+\end{equation}
+such that $\beta=\|r_0\|_2$ and $e_1=(1,0,\cdots,0)$ is the first vector of the canonical basis of $\mathbb{R}^k$. So,
+the vector $y$ is chosen in $\mathbb{R}^k$ so as to minimize at best the Euclidean norm of the residual $r_k$. Consequently,
+a linear least-squares problem of size $k$ is solved:
+\begin{equation}
+\underset{y\in\mathbb{R}^{k}}{min}\|r_{k}\|_{2}=\underset{y\in\mathbb{R}^{k}}{min}\|\beta e_{1}-\bar{H}_{k}y\|_{2}.
+\label{eq:18}
+\end{equation}
+The QR factorization of matrix $\bar{H}_k$ is used to compute the solution of this problem by using Givens rotations~\cite{ref1,ref3},
+such that:
+\begin{equation}
+\begin{array}{lll}
+\bar{H}_{k}=Q_{k}R_{k}, & Q_{k}\in\mathbb{R}^{(k+1)\times (k+1)}, & R_{k}\in\mathbb{R}^{(k+1)\times k},
+\end{array}
+\label{eq:19}
+\end{equation}
+where $Q_kQ_k^T=I_k$ and $R_k$ is an upper triangular matrix.
+
+The GMRES method computes an approximate solution with a sufficient precision after, at most, $n$ iterations ($n$ is the size of the 
+sparse linear system to be solved). However, the GMRES algorithm must construct and store in the memory an orthonormal basis $V_k$ whose
+size is proportional to the number of iterations required to achieve the convergence. Then, to avoid a huge memory storage, the GMRES
+method must be restarted at each $m$ iterations, such that $m$ is very small ($m\ll n$), and with $x_m$ as the initial guess to the
+next iteration. This allows to limit the size of the basis $V$ to $m$ orthogonal vectors.
+
+Algorithm~\ref{alg:02} shows the main key points of the GMRES method with restarts. It solves the left-preconditioned sparse linear
+system~(\ref{eq:11}), such that $M$ is the preconditioning matrix. At each iteration $k$, GMRES uses the Arnoldi process (defined
+from line~$7$ to line~$17$) to construct a basis $V_m$ of $m$ orthogonal vectors and an upper Hessenberg matrix $\bar{H}_m$ of size
+$(m+1)\times m$. Then, it solves the linear least-squares problem of size $m$ to find the vector $y\in\mathbb{R}^{m}$ which minimizes
+at best the residual norm (line~$18$). Finally, it computes an approximate solution $x_m$ in the Krylov sub-space spanned by $V_m$ 
+(line~$19$). The GMRES algorithm is stopped when the residual norm is sufficiently small ($\|r_m\|_2<\varepsilon$) and/or the maximum
+number of iterations ($maxiter$) is reached.
+
+\begin{algorithm}
+  \SetLine
+  \linesnumbered
+  Choose an initial guess $x_0$\;
+  $convergence$ = false\;
+  $k = 1$\;
+  $r_{0} = M^{-1}(b-Ax_{0})$\;
+  $\beta = \|r_{0}\|_{2}$\;
+  \While{$\neg convergence$}{
+    $v_{1} = r_{0}/\beta$\;
+    \For{$j=1$ \KwTo $m$}{ 
+      $w_{j} = M^{-1}Av_{j}$\;
+      \For{$i=1$ \KwTo $j$}{
+        $h_{i,j} = (w_{j},v_{i})$\;
+        $w_{j} = w_{j}-h_{i,j}v_{i}$\;
+      }
+      $h_{j+1,j} = \|w_{j}\|_{2}$\;
+      $v_{j+1} = w_{j}/h_{j+1,j}$\;
+    }
+    Set $V_{m}=\{v_{j}\}_{1\leq j \leq m}$ and $\bar{H}_{m}=(h_{i,j})$ a $(m+1)\times m$ upper Hessenberg matrix\;
+    Solve a least-squares problem of size $m$: $min_{y\in\mathrm{I\!R}^{m}}\|\beta e_{1}-\bar{H}_{m}y\|_{2}$\;
+    $x_{m} = x_{0}+V_{m}y_{m}$\;
+    $r_{m} = M^{-1}(b-Ax_{m})$\;
+    $\beta = \|r_{m}\|_{2}$\;   
+    \eIf{ $(\beta<\varepsilon)$ {\bf or} $(k\geq maxiter)$}{
+      $convergence$ = true\;
+    }{
+      $x_{0} = x_{m}$\;
+      $r_{0} = r_{m}$\;
+      $k = k + 1$\;
+    }
+  }
+\caption{Left-preconditioned GMRES method with restarts}
+\label{alg:02}
+\end{algorithm}
+
+
+%%--------------------------%%
+%%       SECTION 3          %%
+%%--------------------------%%
+\section{Parallel implementation on a GPU cluster}
+\label{sec:03}
+In this section, we present the parallel algorithms of both iterative CG and GMRES methods for GPU clusters.
+The implementation is performed on a GPU cluster composed of different computing nodes, such that each node
+is a CPU core managed by a MPI process and equipped with a GPU card. The parallelization of these algorithms
+is carried out by using the MPI communication routines between the GPU computing nodes and the CUDA programming
+environment inside each node. In what follows, the algorithms of the iterative methods are called iterative
+solvers.
+
+%%****************%%
+%%****************%%
+\subsection{Data partitioning}
+\label{sec:03.01}
+The parallel solving of the large sparse linear system~(\ref{eq:11}) requires a data partitioning between the computing
+nodes of the GPU cluster. Let $p$ denotes the number of the computing nodes on the GPU cluster. The partitioning operation
+consists in the decomposition of the vectors and matrices, involved in the iterative solver, in $p$ portions. Indeed, this
+operation allows to assign to each computing node $i$:
+\begin{itemize*}
+\item a portion of size $\frac{n}{p}$ elements of each vector,
+\item a sparse rectangular sub-matrix $A_i$ of size $(n,\frac{n}{p})$ and,
+\item a square preconditioning sub-matrix $M_i$ of size $(\frac{n}{p},\frac{n}{p})$, 
+\end{itemize*} 
+where $n$ is the size of the sparse linear system to be solved. In the first instance, we perform a naive row-wise partitioning
+(decomposition row-by-row) on the data of the sparse linear systems to be solved. Figure~\ref{fig:01} shows an example of a row-wise
+data partitioning between four computing nodes of a sparse linear system (sparse matrix $A$, solution vector $x$ and right-hand
+side $b$) of size $16$ unknown values. 
+
+\begin{figure}
+\centerline{\includegraphics[scale=0.35]{Chapters/chapter12/figures/partition}}
+\caption{A data partitioning of the sparse matrix $A$, the solution vector $x$ and the right-hand side $b$ into four portions.}
+\label{fig:01}
+\end{figure}
+
+%%****************%%
+%%****************%%
+\subsection{GPU computing}
+\label{sec:03.02}
+After the partitioning operation, all the data involved from this operation must be transferred from the CPU memories to the GPU
+memories, in order to be processed by GPUs. We use two functions of the CUBLAS library (CUDA Basic Linear Algebra Subroutines),
+developed by Nvidia~\cite{ref6}: \verb+cublasAlloc()+ for the memory allocations on GPUs and \verb+cublasSetVector()+ for the
+memory copies from the CPUs to the GPUs.
+
+An efficient implementation of CG and GMRES solvers on a GPU cluster requires to determine all parts of their codes that can be
+executed in parallel and, thus, take advantage of the GPU acceleration. As many Krylov sub-space methods, the CG and GMRES methods
+are mainly based on arithmetic operations dealing with vectors or matrices: sparse matrix-vector multiplications, scalar-vector
+multiplications, dot products, Euclidean norms, AXPY operations ($y\leftarrow ax+y$ where $x$ and $y$ are vectors and $a$ is a
+scalar) and so on. These vector operations are often easy to parallelize and they are more efficient on parallel computers when
+they work on large vectors. Therefore, all the vector operations used in CG and GMRES solvers must be executed by the GPUs as kernels.
+
+We use the kernels of the CUBLAS library to compute some vector operations of CG and GMRES solvers. The following kernels of CUBLAS
+(dealing with double floating point) are used: \verb+cublasDdot()+ for the dot products, \verb+cublasDnrm2()+ for the Euclidean
+norms and \verb+cublasDaxpy()+ for the AXPY operations. For the rest of the data-parallel operations, we code their kernels in CUDA.
+In the CG solver, we develop a kernel for the XPAY operation ($y\leftarrow x+ay$) used at line~$12$ in Algorithm~\ref{alg:01}. In the
+GMRES solver, we program a kernel for the scalar-vector multiplication (lines~$7$ and~$15$ in Algorithm~\ref{alg:02}), a kernel for
+solving the least-squares problem and a kernel for the elements updates of the solution vector $x$.
+
+The least-squares problem in the GMRES method is solved by performing a QR factorization on the Hessenberg matrix $\bar{H}_m$ with
+plane rotations and, then, solving the triangular system by backward substitutions to compute $y$. Consequently, solving the least-squares
+problem on the GPU is not interesting. Indeed, the triangular solves are not easy to parallelize and inefficient on GPUs. However,
+the least-squares problem to solve in the GMRES method with restarts has, generally, a very small size $m$. Therefore, we develop
+an inexpensive kernel which must be executed in sequential by a single CUDA thread. 
+
+The most important operation in CG and GMRES methods is the sparse matrix-vector multiplication (SpMV), because it is often an
+expensive operation in terms of execution time and memory space. Moreover, it requires to take care of the storage format of the
+sparse matrix in the memory. Indeed, the naive storage, row-by-row or column-by-column, of a sparse matrix can cause a significant
+waste of memory space and execution time. In addition, the sparsity nature of the matrix often leads to irregular memory accesses
+to read the matrix nonzero values. So, the computation of the SpMV multiplication on GPUs can involve non coalesced accesses to
+the global memory, which slows down even more its performances. One of the most efficient compressed storage formats of sparse
+matrices on GPUs is HYB format~\cite{ref7}. It is a combination of ELLpack (ELL) and Coordinate (COO) formats. Indeed, it stores
+a typical number of nonzero values per row in ELL format and remaining entries of exceptional rows in COO format. It combines
+the efficiency of ELL due to the regularity of its memory accesses and the flexibility of COO which is insensitive to the matrix
+structure. Consequently, we use the HYB kernel~\cite{ref8} developed by Nvidia to implement the SpMV multiplication of CG and
+GMRES methods on GPUs. Moreover, to avoid the non coalesced accesses to the high-latency global memory, we fill the elements of
+the iterate vector $x$ in the cached texture memory.
+
+%%****************%%
+%%****************%%
+\subsection{Data communications}
+\label{sec:03.03}
+All the computing nodes of the GPU cluster execute in parallel the same iterative solver (Algorithm~\ref{alg:01} or Algorithm~\ref{alg:02})
+adapted to GPUs, but on their own portions of the sparse linear system: $M^{-1}_iA_ix_i=M^{-1}_ib_i$, $0\leq i<p$. However in order to solve
+the complete sparse linear system~(\ref{eq:11}), synchronizations must be performed between the local computations of the computing nodes
+over the cluster. In what follows, two computing nodes sharing data are called neighboring nodes.
+
+As already mentioned, the most important operation of CG and GMRES methods is the SpMV multiplication. In the parallel implementation of
+the iterative methods, each computing node $i$ performs the SpMV multiplication on its own sparse rectangular sub-matrix $A_i$. Locally, it
+has only sub-vectors of size $\frac{n}{p}$ corresponding to rows of its sub-matrix $A_i$. However, it also requires the vector elements
+of its neighbors, corresponding to the column indices on which its sub-matrix has nonzero values (see Figure~\ref{fig:01}). So, in addition
+to the local vectors, each node must also manage vector elements shared with neighbors and required to compute the SpMV multiplication.
+Therefore, the iterate vector $x$ managed by each computing node is composed of a local sub-vector $x^{local}$ of size $\frac{n}{p}$ and a
+sub-vector of shared elements $x^{shared}$. In the same way, the vector used to construct the orthonormal basis of the Krylov sub-space 
+(vectors $p$ and $v$ in CG and GMRES methods, respectively) is composed of a local sub-vector and a shared sub-vector. 
+
+Therefore, before computing the SpMV multiplication, the neighboring nodes over the GPU cluster must exchange between them the shared 
+vector elements necessary to compute this multiplication. First, each computing node determines, in its local sub-vector, the vector
+elements needed by other nodes. Then, the neighboring nodes exchange between them these shared vector elements. The data exchanges 
+are implemented by using the MPI point-to-point communication routines: blocking sends with \verb+MPI_Send()+ and nonblocking receives
+with \verb+MPI_Irecv()+. Figure~\ref{fig:02} shows an example of data exchanges between \textit{Node 1} and its neighbors \textit{Node 0},
+\textit{Node 2} and \textit{Node 3}. In this example, the iterate matrix $A$ split between these four computing nodes is that presented
+in Figure~\ref{fig:01}.
+
+\begin{figure}
+\centerline{\includegraphics[scale=0.30]{Chapters/chapter12/figures/compress}}
+\caption{Data exchanges between \textit{Node 1} and its neighbors \textit{Node 0}, \textit{Node 2} and \textit{Node 3}.}
+\label{fig:02}
+\end{figure}
+
+After the synchronization operation, the computing nodes receive, from their respective neighbors, the shared elements in a sub-vector
+stored in a compressed format. However, in order to compute the SpMV multiplication, the computing nodes operate on sparse global vectors
+(see Figure~\ref{fig:02}). In this case, the received vector elements must be copied to the corresponding indices in the global vector.
+So as not to need to perform this at each iteration, we propose to reorder the columns of each sub-matrix $\{A_i\}_{0\leq i<p}$, so that
+the shared sub-vectors could be used in their compressed storage formats. Figure~\ref{fig:03} shows a reordering of a sparse sub-matrix
+(sub-matrix of \textit{Node 1}). Furthermore, we use the texture memory to cache the global vector. This allows to avoid the non coalesced
+accesses to the GPU global memory. 
+
+\begin{figure}
+\centerline{\includegraphics[scale=0.30]{Chapters/chapter12/figures/reorder}}
+\caption{Columns reordering of a sparse sub-matrix.}
+\label{fig:03}
+\end{figure}
+
+A GPU cluster is a parallel platform with a distributed memory. So, the synchronizations and communication data between GPU nodes are
+carried out by passing messages. However, GPUs can not communicate between them in direct way. Then, CPUs via MPI processes are in charge
+of the synchronizations within the GPU cluster. Consequently, the vector elements to be exchanged must be copied from the GPU memory
+to the CPU memory and vice-versa before and after the synchronization operation between CPUs. We have used the CBLAS communication subroutines
+to perform the data transfers between a CPU core and its GPU: \verb+cublasGetVector()+ and \verb+cublasSetVector()+. Finally, in addition
+to the data exchanges, GPU nodes perform reduction operations to compute in parallel the dot products and Euclidean norms. This is implemented
+by using the MPI global communication \verb+MPI_Allreduce()+.
+
+
+%%--------------------------%%
+%%       SECTION 4          %%
+%%--------------------------%%
+\section{Experimental results}
+\label{sec:04}
+In this section, we present the performances of the parallel CG and GMRES linear solvers obtained on a cluster of $12$ GPUs. Indeed, this GPU
+cluster of tests is composed of six machines connected by $20$Gbps InfiniBand network. Each machine is a Quad-Core Xeon E5530 CPU running at
+$2.4$GHz and providing $12$GB of RAM with a memory bandwidth of $25.6$GB/s. In addition, two Tesla C1060 GPUs are connected to each machine via
+a PCI-Express 16x Gen 2.0 interface with a throughput of $8$GB/s. A Tesla C1060 GPU contains $240$ cores running at $1.3$GHz and providing a
+global memory of $4$GB with a memory bandwidth of $102$GB/s. Figure~\ref{fig:04} shows the general scheme of the GPU cluster that we used in
+the experimental tests.
+
+\begin{figure}
+\centerline{\includegraphics[scale=0.25]{Chapters/chapter12/figures/cluster}}
+\caption{General scheme of the GPU cluster of tests composed of six machines, each with two GPUs.}
+\label{fig:04}
+\end{figure}
+
+Linux cluster version 2.6.39 OS is installed on CPUs. C programming language is used for coding the parallel algorithms of both methods on the
+GPU cluster. CUDA version 4.0~\cite{ref9} is used for programming GPUs, using CUBLAS library~\cite{ref6} to deal with vector operations in GPUs
+and, finally, MPI routines of OpenMPI 1.3.3 are used to carry out the communications between CPU cores. Indeed, the experiments are done on a
+cluster of $12$ computing nodes, where each node is managed by a MPI process and it is composed of one CPU core and one GPU card.
+
+All tests are made on double-precision floating point operations. The parameters of both linear solvers are initialized as follows: the residual
+tolerance threshold $\varepsilon=10^{-12}$, the maximum number of iterations $maxiter=500$, the right-hand side $b$ is filled with $1.0$ and the
+initial guess $x_0$ is filled with $0.0$. In addition, we limited the Arnoldi process used in the GMRES method to $16$ iterations ($m=16$). For
+the sake of simplicity, we have chosen the preconditioner $M$ as the main diagonal of the sparse matrix $A$. Indeed, it allows to easily compute
+the required inverse matrix $M^{-1}$ and it provides a relatively good preconditioning for not too ill-conditioned matrices. In the GPU computing,
+the size of thread blocks is fixed to $512$ threads. Finally, the performance results, presented hereafter, are obtained from the mean value over
+$10$ executions of the same parallel linear solver and for the same input data.
+
+To get more realistic results, we tested the CG and GMRES algorithms on sparse matrices of the Davis's collection~\cite{ref10}, that arise in a wide
+spectrum of real-world applications. We chose six symmetric sparse matrices and six nonsymmetric ones from this collection. In Figure~\ref{fig:05},
+we show structures of these matrices and in Table~\ref{tab:01} we present their main characteristics which are the number of rows, the total number
+of nonzero values (nnz) and the maximal bandwidth. In the present chapter, the bandwidth of a sparse matrix is defined as the number of matrix columns
+separating the first and the last nonzero value on a matrix row.
+
+\begin{figure}
+\centerline{\includegraphics[scale=0.30]{Chapters/chapter12/figures/matrices}}
+\caption{Sketches of sparse matrices chosen from the Davis's collection.}
+\label{fig:05}
+\end{figure}
+
+\begin{table}[!h]
+\centering
+\begin{tabular}{|c|c|c|c|c|}
+\hline
+{\bf Matrix type}             & {\bf Matrix name} & {\bf \# rows} & {\bf \# nnz} & {\bf Bandwidth} \\ \hline \hline
+
+\multirow{6}{*}{Symmetric}    & 2cubes\_sphere    & $101,492$     & $1,647,264$  & $100,464$ \\
+
+                              & ecology2          & $999,999$     & $4,995,991$  & $2,001$   \\ 
+
+                              & finan512          & $74,752$      & $596,992$    & $74,725$  \\ 
+
+                              & G3\_circuit       & $1,585,478$   & $7,660,826$  & $1,219,059$ \\
+            
+                              & shallow\_water2   & $81,920$      & $327,680$    & $58,710$ \\
+
+                              & thermal2          & $1,228,045$   & $8,580,313$  & $1,226,629$ \\ \hline \hline
+            
+\multirow{6}{*}{Nonsymmetric} & cage13            & $445,315$     & $7,479,343$  & $318,788$\\
+
+                              & crashbasis        & $160,000$     & $1,750,416$  & $120,202$ \\
+
+                              & FEM\_3D\_thermal2 & $147,900$     & $3,489.300$  & $117,827$ \\
+
+                              & language          & $399,130$     & $1,216,334$  & $398,622$\\
+ 
+                              & poli\_large       & $15,575$      & $33,074$     & $15,575$ \\
+
+                              & torso3            & $259,156$     & $4,429,042$  & $216,854$  \\ \hline
+\end{tabular}
+\vspace{0.5cm}
+\caption{Main characteristics of sparse matrices chosen from the Davis's collection.}
+\label{tab:01}
+\end{table}
+
+\begin{table}
+\begin{center}
+\begin{tabular}{|c|c|c|c|c|c|c|} 
+\hline
+{\bf Matrix}     & $\mathbf{Time_{cpu}}$ & $\mathbf{Time_{gpu}}$ & $\mathbf{\tau}$  & $\mathbf{\# iter.}$ & $\mathbf{prec.}$     & $\mathbf{\Delta}$   \\ \hline \hline
+
+2cubes\_sphere    & $0.132s$           & $0.069s$            & $1.93$        & $12$           & $1.14e$-$09$     & $3.47e$-$18$ \\
+
+ecology2          & $0.026s$           & $0.017s$            & $1.52$        & $13$           & $5.06e$-$09$     & $8.33e$-$17$ \\
+
+finan512          & $0.053s$           & $0.036s$            & $1.49$        & $12$           & $3.52e$-$09$     & $1.66e$-$16$ \\
+
+G3\_circuit       & $0.704s$           & $0.466s$            & $1.51$        & $16$           & $4.16e$-$10$     & $4.44e$-$16$ \\
+
+shallow\_water2   & $0.017s$           & $0.010s$            & $1.68$        & $5$            & $2.24e$-$14$     & $3.88e$-$26$ \\
+
+thermal2          & $1.172s$           & $0.622s$            & $1.88$        & $15$           & $5.11e$-$09$     & $3.33e$-$16$ \\ \hline   
+\end{tabular}
+\caption{Performances of the parallel CG method on a cluster of 24 CPU cores vs. on a cluster of 12 GPUs.}
+\label{tab:02}
+\end{center}
+\end{table}
+
+\begin{table}[!h]
+\begin{center}
+\begin{tabular}{|c|c|c|c|c|c|c|} 
+\hline
+{\bf Matrix}     & $\mathbf{Time_{cpu}}$ & $\mathbf{Time_{gpu}}$ & $\mathbf{\tau}$  & $\mathbf{\# iter.}$ & $\mathbf{prec.}$     & $\mathbf{\Delta}$   \\ \hline \hline
+
+2cubes\_sphere    & $0.234s$           & $0.124s$            & $1.88$        & $21$           & $2.10e$-$14$     & $3.47e$-$18$ \\
+
+ecology2          & $0.076s$           & $0.035s$            & $2.15$        & $21$           & $4.30e$-$13$     & $4.38e$-$15$ \\
+
+finan512          & $0.073s$           & $0.052s$            & $1.40$        & $17$           & $3.21e$-$12$     & $5.00e$-$16$ \\
+
+G3\_circuit       & $1.016s$           & $0.649s$            & $1.56$        & $22$           & $1.04e$-$12$     & $2.00e$-$15$ \\
+
+shallow\_water2   & $0.061s$           & $0.044s$            & $1.38$        & $17$           & $5.42e$-$22$     & $2.71e$-$25$ \\
+
+thermal2          & $1.666s$           & $0.880s$            & $1.89$        & $21$           & $6.58e$-$12$     & $2.77e$-$16$ \\ \hline \hline
+
+cage13            & $0.721s$           & $0.338s$            & $2.13$        & $26$           & $3.37e$-$11$     & $2.66e$-$15$ \\
+
+crashbasis        & $1.349s$           & $0.830s$            & $1.62$        & $121$          & $9.10e$-$12$     & $6.90e$-$12$ \\
+
+FEM\_3D\_thermal2 & $0.797s$           & $0.419s$            & $1.90$        & $64$           & $3.87e$-$09$     & $9.09e$-$13$ \\
+
+language          & $2.252s$           & $1.204s$            & $1.87$        & $90$           & $1.18e$-$10$     & $8.00e$-$11$ \\
+
+poli\_large       & $0.097s$           & $0.095s$            & $1.02$        & $69$           & $4.98e$-$11$     & $1.14e$-$12$ \\
+
+torso3            & $4.242s$           & $2.030s$            & $2.09$        & $175$          & $2.69e$-$10$     & $1.78e$-$14$ \\ \hline
+\end{tabular}
+\caption{Performances of the parallel GMRES method on a cluster 24 CPU cores vs. on cluster of 12 GPUs.}
+\label{tab:03}
+\end{center}
+\end{table}
+
+Tables~\ref{tab:02} and~\ref{tab:03} shows the performances of the parallel CG and GMRES solvers, respectively, for solving linear systems associated to
+the sparse matrices presented in Tables~\ref{tab:01}. They allow to compare the performances obtained on a cluster of $24$ CPU cores and on a cluster
+of $12$ GPUs. However, Table~\ref{tab:02} shows only the solving performances of symmetric sparse linear systems, due to the inability of the CG method
+to solve the nonsymmetric systems. In both tables, the second and third columns give, respectively, the execution times in seconds obtained on $24$ CPU
+cores($Time_{gpu}$) and that obtained on $12$ GPUs ($Time_{gpu}$). Moreover, we take into account the relative gains $\tau$ of a solver implemented on the
+GPU cluster compared to the same solver implemented on the CPU cluster. The relative gains, presented in the fourth column, are computed as a ratio of
+the CPU execution time over the GPU execution time:
+\begin{equation}
+\tau = \frac{Time_{cpu}}{Time_{gpu}}.
+\label{eq:20}
+\end{equation}
+In addition, Tables~\ref{tab:02} and~\ref{tab:03} give the number of iterations ($iter$), the precision $prec$ of the solution computed on the GPU cluster
+and the difference $\Delta$ between the solution computed on the CPU cluster and that computed on the GPU cluster. Both parameters $prec$ and $\Delta$
+allow to validate and verify the accuracy of the solution computed on the GPU cluster. We have computed them as follows:
+\begin{eqnarray}
+\Delta = max|x^{cpu}-x^{gpu}|,\\
+prec = max|M^{-1}r^{gpu}|,
+\end{eqnarray}
+where $\Delta$ is the maximum vector element, in absolute value, of the difference between the two solutions $x^{cpu}$ and $x^{gpu}$ computed, respectively,
+on CPU and GPU cluster and $prec$ is the maximum element, in absolute value, of the residual vector $r^{gpu}\in\mathbb{R}^{n}$ of the solution $x^{gpu}$.
+Thus, we can see that the solutions obtained on the GPU cluster were computed with a sufficient accuracy (about $10^{-10}$) and they are, more or less,
+equivalent to those computed on the CPU cluster with a small difference ranging from $10^{-10}$ and $10^{-26}$. However, we can notice from the relative
+gains $\tau$ that is not interesting to use multiple GPUs for solving small sparse linear systems. in fact, a small sparse matrix does not allow to maximize
+utilization of GPU cores. In addition, the communications required to synchronize the computations over the cluster increase the idle times of GPUs and
+slow down further the parallel computations.
+
+Consequently, in order to test the performances of the parallel solvers, we developed in C programming language a generator of large sparse matrices. 
+This generator takes a matrix from the Davis's collection~\cite{ref10} as an initial matrix to construct large sparse matrices exceeding ten million
+of rows. It must be executed in parallel by the MPI processes of the computing nodes, so that each process could construct its sparse sub-matrix. In
+first experimental tests, we are focused on sparse matrices having a banded structure, because they are those arise in the most of numerical problems.
+So to generate the global sparse matrix, each MPI process constructs its sub-matrix by performing several copies of an initial sparse matrix chosen
+from the Davis's collection. Then, it puts all these copies on the main diagonal of the global matrix (see Figure~\ref{fig:06}). Moreover, the empty
+spaces between two successive copies in the main diagonal are filled with sub-copies (left-copy and right-copy in Figure~\ref{fig:06}) of the same
+initial matrix.
+
+\begin{figure}
+\centerline{\includegraphics[scale=0.30]{Chapters/chapter12/figures/generation}}
+\caption{Parallel generation of a large sparse matrix by four computing nodes.}
+\label{fig:06}
+\end{figure}
+
+We have used the parallel CG and GMRES algorithms for solving sparse linear systems of $25$ million of unknown values. The sparse matrices associated 
+to these linear systems are generated from those presented in Table~\ref{tab:01}. Their main characteristics are given in Table~\ref{tab:04}. Tables~\ref{tab:05}
+and~\ref{tab:06} shows the performances of the parallel CG and GMRES solvers, respectively, obtained on a cluster of $24$ CPU cores and on a cluster 
+of $12$ GPUs. Obviously, we can notice from these tables that solving large sparse linear systems on a GPU cluster is more efficient than on a CPU 
+cluster (see relative gains $\tau$). We can also notice that the execution times of the CG method, whether in a CPU cluster or on a GPU cluster, are
+better than those the GMRES method for solving large symmetric linear systems. In fact, the CG method is characterized by a better convergence rate 
+and a shorter execution time of an iteration than those of the GMRES method. Moreover, an iteration of the parallel GMRES method requires more data
+exchanges between computing nodes compared to the parallel CG method.
+ 
+\begin{table}
+\centering
+\begin{tabular}{|c|c|c|c|}
+\hline
+{\bf Matrix type}             & {\bf Matrix name} & {\bf \# nnz} & {\bf Bandwidth} \\ \hline \hline
+
+\multirow{6}{*}{Symmetric}    & 2cubes\_sphere    & $413,703,602$ & $198,836$     \\
+
+                              & ecology2          & $124,948,019$ & $2,002$          \\ 
+
+                              & finan512          & $278,175,945$ & $123,900$        \\ 
+
+                              & G3\_circuit       & $125,262,292$ & $1,891,887$      \\
+            
+                              & shallow\_water2   & $100,235,292$ & $62,806$      \\
+
+                              & thermal2          & $175,300,284$ & $2,421,285$ \\ \hline \hline
+            
+\multirow{6}{*}{Nonsymmetric} & cage13            & $435,770,480$ & $352,566$        \\
+
+                              & crashbasis        & $409,291,236$ & $200,203$        \\
+
+                              & FEM\_3D\_thermal2 & $595,266,787$ & $206,029$ \\
+
+                              & language          & $76,912,824$  & $398,626$ \\
+
+                              & poli\_large       & $53,322,580$  & $15,576$         \\
+
+                              & torso3            & $433,795,264$ & $328,757$        \\ \hline
+\end{tabular}
+\vspace{0.5cm}
+\caption{Main characteristics of sparse banded matrices generated from those of the Davis's collection.}
+\label{tab:04}
+\end{table}
+ 
+\begin{table}
+\begin{center}
+\begin{tabular}{|c|c|c|c|c|c|c|} 
+\hline
+{\bf Matrix}    & $\mathbf{Time_{cpu}}$ & $\mathbf{Time_{gpu}}$ & $\mathbf{\tau}$ & $\mathbf{\# iter.}$ & $\mathbf{prec.}$ & $\mathbf{\Delta}$   \\ \hline \hline
+
+2cubes\_sphere  & $1.625s$             & $0.401s$              & $4.05$          & $14$                & $5.73e$-$11$     & $5.20e$-$18$ \\
+
+ecology2        & $0.856s$             & $0.103s$              & $8.27$          & $15$                & $3.75e$-$10$     & $1.11e$-$16$ \\
+
+finan512        & $1.210s$             & $0.354s$              & $3.42$          & $14$                & $1.04e$-$10$     & $2.77e$-$16$ \\
+
+G3\_circuit     & $1.346s$             & $0.263s$              & $5.12$          & $17$                & $1.10e$-$10$     & $5.55e$-$16$ \\
+
+shallow\_water2 & $0.397s$             & $0.055s$              & $7.23$          & $7$                 & $3.43e$-$15$     & $5.17e$-$26$ \\
+
+thermal2        & $1.411s$             & $0.244s$              & $5.78$          & $16$                & $1.67e$-$09$     & $3.88e$-$16$ \\ \hline  
+\end{tabular}
+\caption{Performances of the parallel CG method for solving linear systems associated to sparse banded matrices on a cluster of 24 CPU cores vs. 
+on a cluster of 12 GPUs.}
+\label{tab:05}
+\end{center}
+\end{table}
+
+\begin{table}
+\begin{center}
+\begin{tabular}{|c|c|c|c|c|c|c|} 
+\hline
+{\bf Matrix}      & $\mathbf{Time_{cpu}}$ & $\mathbf{Time_{gpu}}$ & $\mathbf{\tau}$ & $\mathbf{\# iter.}$ & $\mathbf{prec.}$ & $\mathbf{\Delta}$   \\ \hline \hline
+
+2cubes\_sphere    & $3.597s$             & $0.514s$              & $6.99$          & $21$                & $2.11e$-$14$     & $8.67e$-$18$ \\
+
+ecology2          & $2.549s$             & $0.288s$              & $8.83$          & $21$                & $4.88e$-$13$     & $2.08e$-$14$ \\
+
+finan512          & $2.660s$             & $0.377s$              & $7.05$          & $17$                & $3.22e$-$12$     & $8.82e$-$14$ \\
+
+G3\_circuit       & $3.139s$             & $0.480s$              & $6.53$          & $22$                & $1.04e$-$12$     & $5.00e$-$15$ \\
+
+shallow\_water2   & $2.195s$             & $0.253s$              & $8.68$          & $17$                & $5.54e$-$21$     & $7.92e$-$24$ \\
+
+thermal2          & $3.206s$             & $0.463s$              & $6.93$          & $21$                & $8.89e$-$12$     & $3.33e$-$16$ \\ \hline \hline
+
+cage13            & $5.560s$             & $0.663s$              & $8.39$          & $26$                & $3.29e$-$11$     & $1.59e$-$14$ \\
+
+crashbasis        & $25.802s$            & $3.511s$              & $7.35$          & $135$               & $6.81e$-$11$     & $4.61e$-$15$ \\
+
+FEM\_3D\_thermal2 & $13.281s$            & $1.572s$              & $8.45$          & $64$                & $3.88e$-$09$     & $1.82e$-$12$ \\
+
+language          & $12.553s$            & $1.760s$              & $7.13$          & $89$                & $2.11e$-$10$     & $1.60e$-$10$ \\
+
+poli\_large       & $8.515s$             & $1.053s$              & $8.09$          & $69$                & $5.05e$-$11$     & $6.59e$-$12$ \\
+
+torso3            & $31.463s$            & $3.681s$              & $8.55$          & $175$               & $2.69e$-$10$     & $2.66e$-$14$ \\ \hline
+\end{tabular}
+\caption{Performances of the parallel GMRES method for solving linear systems associated to sparse banded matrices on a cluster of 24 CPU cores vs. 
+on a cluster of 12 GPUs.}
+\label{tab:06}
+\end{center}
+\end{table}
+
+
+%%--------------------------%%
+%%       SECTION 5          %%
+%%--------------------------%%
+\section{Hypergraph partitioning}
+\label{sec:05}
+In this section, we present the performances of both parallel CG and GMRES solvers for solving linear systems associated to sparse matrices having
+large bandwidths. Indeed, we are interested on sparse matrices having the nonzero values distributed along their bandwidths. 
+
+We have developed in C programming language a generator of large sparse matrices having five bands distributed along their bandwidths (see Figure~\ref{fig:07}).
+The principle of this generator is equivalent to that in Section~\ref{sec:04}. However, the copies performed on the initial matrix (chosen from the
+Davis's collection) are placed on the main diagonal and on four off-diagonals, two on the right and two on the left of the main diagonal. Figure~\ref{fig:07}
+shows an example of a generation of a sparse five-bands matrix by four computing nodes. Table~\ref{tab:07} shows the main characteristics of sparse
+five-bands matrices generated from those presented in Table~\ref{tab:01} and associated to linear systems of $25$ million of unknown values.   
+
+\begin{figure}[!h]
+\centerline{\includegraphics[scale=0.23]{Chapters/chapter12/figures/generation_1}}
+\caption{Parallel generation of a large sparse five-bands matrix by four computing nodes.}
+\label{fig:07}
+\end{figure}
+
+\begin{table}
+\begin{center}
+\begin{tabular}{|c|c|c|c|} 
+\hline
+{\bf Matrix type}             & {\bf Matrix name} & {\bf \# nnz}  & {\bf Bandwidth} \\ \hline \hline
+
+\multirow{6}{*}{Symmetric}    & 2cubes\_sphere    & $829,082,728$ & $24,999,999$     \\
+
+                              & ecology2          & $254,892,056$ & $25,000,000$     \\ 
+
+                              & finan512          & $556,982,339$ & $24,999,973$     \\ 
+
+                              & G3\_circuit       & $257,982,646$ & $25,000,000$     \\
+            
+                              & shallow\_water2   & $200,798,268$ & $25,000,000$     \\
+
+                              & thermal2          & $359,340,179$ & $24,999,998$     \\ \hline \hline
+            
+\multirow{6}{*}{Nonsymmetric} & cage13            & $879,063,379$ & $24,999,998$     \\
+
+                              & crashbasis        & $820,373,286$ & $24,999,803$     \\
+
+                              & FEM\_3D\_thermal2 & $1,194,012,703$ & $24,999,998$     \\
+
+                              & language          & $155,261,826$ & $24,999,492$     \\
+
+                              & poli\_large       & $106,680,819$ & $25,000,000$    \\
+
+                              & torso3            & $872,029,998$ & $25,000,000$\\ \hline
+\end{tabular}
+\caption{Main characteristics of sparse five-bands matrices generated from those of the Davis's collection.}
+\label{tab:07}
+\end{center}
+\end{table}
+
+
+\begin{table}
+\begin{center}
+\begin{tabular}{|c|c|c|c|c|c|c|} 
+\hline
+{\bf Matrix}      & $\mathbf{Time_{cpu}}$ & $\mathbf{Time_{gpu}}$ & $\mathbf{\tau}$ & $\mathbf{\# iter.}$ & $\mathbf{prec.}$ & $\mathbf{\Delta}$   \\ \hline \hline
+
+2cubes\_sphere    & $6.041s$     & $3.338s$      & $1.81$ & $30$ & $6.77e$-$11$ & $3.25e$-$19$ \\
+
+ecology2          & $1.404s$     & $1.301s$      & $1.08$ & $13$     & $5.22e$-$11$ & $2.17e$-$18$ \\
+
+finan512          & $1.822s$     & $1.299s$      & $1.40$ & $12$     & $3.52e$-$11$ & $3.47e$-$18$ \\
+
+G3\_circuit       & $2.331s$     & $2.129s$      & $1.09$ & $15$     & $1.36e$-$11$ & $5.20e$-$18$ \\
+
+shallow\_water2   & $0.541s$     & $0.504s$      & $1.07$ & $6$      & $2.12e$-$16$ & $5.05e$-$28$ \\
+
+thermal2          & $2.549s$     & $1.705s$      & $1.49$ & $14$     & $2.36e$-$10$ & $5.20e$-$18$ \\ \hline  
+\end{tabular}
+\caption{Performances of parallel CG solver for solving linear systems associated to sparse five-bands matrices
+on a cluster of 24 CPU cores vs. on a cluster of 12 GPUs}
+\label{tab:08}
+\end{center}
+\end{table}
+
+\begin{table}
+\begin{center}
+\begin{tabular}{|c|c|c|c|c|c|c|} 
+\hline
+{\bf Matrix}      & $\mathbf{Time_{cpu}}$ & $\mathbf{Time_{gpu}}$ & $\mathbf{\tau}$ & $\mathbf{\# iter.}$ & $\mathbf{prec.}$ & $\mathbf{\Delta}$   \\ \hline \hline
+
+2cubes\_sphere    & $15.963s$    & $7.250s$      & $2.20$  & $58$     & $6.23e$-$16$ & $3.25e$-$19$ \\
+
+ecology2          & $3.549s$     & $2.176s$      & $1.63$  & $21$     & $4.78e$-$15$ & $1.06e$-$15$ \\
+
+finan512          & $3.862s$     & $1.934s$      & $1.99$  & $17$     & $3.21e$-$14$ & $8.43e$-$17$ \\
+
+G3\_circuit       & $4.636s$     & $2.811s$      & $1.65$  & $22$     & $1.08e$-$14$ & $1.77e$-$16$ \\
+
+shallow\_water2   & $2.738s$     & $1.539s$      & $1.78$  & $17$     & $5.54e$-$23$ & $3.82e$-$26$ \\
+
+thermal2          & $5.017s$     & $2.587s$      & $1.94$  & $21$     & $8.25e$-$14$ & $4.34e$-$18$ \\ \hline \hline
+
+cage13            & $9.315s$     & $3.227s$      & $2.89$  & $26$     & $3.38e$-$13$ & $2.08e$-$16$ \\
+
+crashbasis        & $35.980s$    & $14.770s$     & $2.43$  & $127$    & $1.17e$-$12$ & $1.56e$-$17$ \\
+
+FEM\_3D\_thermal2 & $24.611s$    & $7.749s$      & $3.17$  & $64$     & $3.87e$-$11$ & $2.84e$-$14$ \\
+
+language          & $16.859s$    & $9.697s$      & $1.74$  & $89$     & $2.17e$-$12$ & $1.70e$-$12$ \\
+
+poli\_large       & $10.200s$    & $6.534s$      & $1.56$  & $69$     & $5.14e$-$13$ & $1.63e$-$13$ \\
+
+torso3            & $49.074s$    & $19.397s$     & $2.53$  & $175$    & $2.69e$-$12$ & $2.77e$-$16$ \\ \hline
+\end{tabular}
+\caption{Performances of parallel GMRES solver for solving linear systems associated to sparse five-bands matrices
+on a cluster of 24 CPU cores vs. on a cluster of 12 GPUs}
+\label{tab:09}
+\end{center}
+\end{table}
+
+Tables~\ref{tab:08} and~\ref{tab:09} shows the performaces of the parallel CG and GMRES solvers, respectively, obtained on
+a cluster of $24$ CPU cores and on a cluster of $12$ GPUs. The linear systems solved in these tables are associated to the
+sparse five-bands matrices presented on Table~\ref{tab:07}. We can notice from both Tables~\ref{tab:08} and~\ref{tab:09} that 
+using a GPU cluster is not efficient for solving these kind of sparse linear systems. We can see that the execution times obtained
+on the GPU cluster are almost equivalent to those obtained on the CPU cluster (see the relative gains presented in column~$4$
+of each table). This is due to the large number of communications necessary to synchronize the computations over the cluster.
+Indeed, the naive partitioning, row-by-row or column-by-column, of sparse matrices having large bandwidths can link a computing
+node to many neighbors and then generate a large number of data dependencies between these computing nodes in the cluster. 
+
+Therefore, we have chosen to use a hypergraph partitioning method, which is well-suited to numerous kinds of sparse matrices~\cite{ref11}.
+Indeed, it can well model the communications between the computing nodes, particularly in the case of nonsymmetric and irregular
+matrices, and it gives good reduction of the total communication volume. In contrast, it is an expensive operation in terms of 
+execution time and memory space. 
+
+The sparse matrix $A$ of the linear system to be solved is modeled as a hypergraph $\mathcal{H}=(\mathcal{V},\mathcal{E})$ as
+follows:
+\begin{itemize*}
+\item each matrix row $\{i\}_{0\leq i<n}$ corresponds to a vertex $v_i\in\mathcal{V}$ and,
+\item each matrix column $\{j\}_{0\leq j<n}$ corresponds to a hyperedge $e_j\in\mathcal{E}$, where:
+\begin{equation}
+\forall a_{ij} \neq 0 \mbox{~is a nonzero value of matrix~} A \mbox{~:~} v_i \in pins[e_j],
+\end{equation} 
+\item $w_i$ is the weight of vertex $v_i$ and,
+\item $c_j$ is the cost of hyperedge $e_j$.
+\end{itemize*}
+A $K$-way partitioning of a hypergraph $\mathcal{H}=(\mathcal{V},\mathcal{E})$ is defined as $\mathcal{P}=\{\mathcal{V}_1,\ldots,\mathcal{V}_K\}$
+a set of pairwise disjoint non-empty subsets (or parts) of the vertex set $\mathcal{V}$, so that each subset is attributed to a computing node.
+Figure~\ref{fig:08} shows an example of the hypergraph model of a  $(9\times 9)$ sparse matrix in three parts. The circles and squares correspond,
+respectively, to the vertices and hyperedges of the hypergraph. The solid squares define the cut hyperedges connecting at least two different parts. 
+The connectivity $\lambda_j$ of a cut hyperedge $e_j$ denotes the number of different parts spanned by $e_j$.
+
+\begin{figure}
+\centerline{\includegraphics[scale=0.5]{Chapters/chapter12/figures/hypergraph}}
+\caption{An example of the hypergraph partitioning of a sparse matrix decomposed between three computing nodes.}
+\label{fig:08}
+\end{figure}
+
+The cut hyperedges model the total communication volume between the different computing nodes in the cluster, necessary to perform the parallel SpMV
+multiplication. Indeed, each hyperedge $e_j$ defines a set of atomic computations $b_i\leftarrow b_i+a_{ij}x_j$, $0\leq i,j<n$, of the SpMV multiplication
+$Ax=b$ that need the $j^{th}$ unknown value of solution vector $x$. Therefore, pins of hyperedge $e_j$, $pins[e_j]$, are the set of matrix rows sharing
+and requiring the same unknown value $x_j$. For example in Figure~\ref{fig:08}, hyperedge $e_9$ whose pins are: $pins[e_9]=\{v_2,v_5,v_9\}$ represents the
+dependency of matrix rows $2$, $5$ and $9$ to unknown $x_9$ needed to perform in parallel the atomic operations: $b_2\leftarrow b_2+a_{29}x_9$,
+$b_5\leftarrow b_5+a_{59}x_9$ and $b_9\leftarrow b_9+a_{99}x_9$. However, unknown $x_9$ is the third entry of the sub-solution vector $x$ of part (or node) $3$.
+So the computing node $3$ must exchange this value with nodes $1$ and $2$, which leads to perform two communications.
+
+The hypergraph partitioning allows to reduce the total communication volume required to perform the parallel SpMV multiplication, while maintaining the 
+load balancing between the computing nodes. In fact, it allows to minimize at best the following amount:
+\begin{equation}
+\mathcal{X}(\mathcal{P})=\sum_{e_{j}\in\mathcal{E}_{C}}c_{j}(\lambda_{j}-1),
+\end{equation}
+where $\mathcal{E}_{C}$ denotes the set of the cut hyperedges coming from the hypergraph partitioning $\mathcal{P}$ and $c_j$ and $\lambda_j$ are, respectively,
+the cost and the connectivity of cut hyperedge $e_j$. Moreover, it also ensures the load balancing between the $K$ parts as follows: 
+\begin{equation}
+  W_{k}\leq (1+\epsilon)W_{avg}, \hspace{0.2cm} (1\leq k\leq K) \hspace{0.2cm} \text{and} \hspace{0.2cm} (0<\epsilon<1),
+\end{equation} 
+where $W_{k}$ is the sum of all vertex weights ($w_{i}$) in part $\mathcal{V}_{k}$, $W_{avg}$ is the average weight of all $K$ parts and $\epsilon$ is the 
+maximum allowed imbalanced ratio.
+
+The hypergraph partitioning is a NP-complete problem but software tools using heuristics are developed, for example: hMETIS~\cite{ref12}, PaToH~\cite{ref13}
+and Zoltan~\cite{ref14}. Since our objective is solving large sparse linear systems, we use the parallel hypergraph partitioning which must be performed by
+at least two MPI processes. It allows us to accelerate the data partitioning of large sparse matrices. For this, the hypergraph $\mathcal{H}$ must be partitioned
+in $p$ (number of MPI processes) sub-hypergraphs $\mathcal{H}_k=(\mathcal{V}_k,\mathcal{E}_k)$, $0\leq k<p$, and then we performed the parallel hypergraph
+partitioning method using some functions of the MPI library between the $p$ processes.
+
+Tables~\ref{tab:10} and~\ref{tab:11} shows the performances of the parallel CG and GMRES solvers, respectively, using the hypergraph partitioning for solving
+large linear systems associated to the sparse five-bands matrices presented in Table~\ref{tab:07}. For these experimental tests, we have applied the parallel
+hypergraph partitioning~\cite{ref15} developed in Zoltan tool~\ref{ref14}. We have initialized the parameters of the partitioning as follows:
+\begin{itemize*}
+\item the weight $w_{i}$ of each vertex $v_{j}\in\mathcal{V}$ is set to the number of nonzero values on matrix row $i$,
+\item for the sake of simplicity, the cost $c_{j}$ of each hyperedge $e_{j}\in\mathcal{E}$ is fixed to $1$,
+\item the maximum imbalanced load ratio $\epsilon$ is limited to $10\%$.\\
+\end{itemize*}  
+
+\begin{table}[!h]
+\begin{center}
+\begin{tabular}{|c|c|c|c|c|} 
+\hline
+{\bf Matrix}    & $\mathbf{Time_{cpu}}$ & $\mathbf{Time_{gpu}}$ & $\mathbf{\tau}$ & $\mathbf{Gains \%}$ \\ \hline \hline
+
+2cubes\_sphere  & $5.935s$             & $1.213s$              & $4.89$          & $63.66\%$ \\
+
+ecology2        & $1.093s$             & $0.136s$              & $8.00$          & $89.55\%$ \\
+
+finan512        & $1.762s$             & $0.475s$              & $3.71$          & $63.43\%$ \\
+
+G3\_circuit     & $2.095s$             & $0.558s$              & $3.76$          & $73.79\%$ \\
+
+shallow\_water2 & $0.498s$             & $0.068s$              & $7.31$          & $86.51\%$ \\
+
+thermal2        & $1.889s$             & $0.348s$              & $5.43$          & $79.59\%$ \\ \hline  
+\end{tabular}
+\caption{Performances of the parallel CG solver using hypergraph partitioning for solving linear systems associated to
+sparse five-bands matrices on a cluster of 24 CPU cores vs. on a cluster of 12 GPU.}
+\label{tab:10}
+\end{center}
+\end{table}
+
+\begin{table}[!h]
+\begin{center}
+\begin{tabular}{|c|c|c|c|c|} 
+\hline
+{\bf Matrix}      & $\mathbf{Time_{cpu}}$ & $\mathbf{Time_{gpu}}$ & $\mathbf{\tau}$ & $\mathbf{Gains \%}$ \\ \hline \hline
+
+2cubes\_sphere    & $16.430s$            & $2.840s$              & $5.78$          & $60.83\%$ \\
+
+ecology2          & $3.152s$             & $0.367s$              & $8.59$          & $83.13\%$ \\
+
+finan512          & $3.672s$             & $0.723s$              & $5.08$          & $62.62\%$ \\
+
+G3\_circuit       & $4.468s$             & $0.971s$              & $4.60$          & $65.46\%$ \\
+
+shallow\_water2   & $2.647s$             & $0.312s$              & $8.48$          & $79.73\%$ \\
+
+thermal2          & $4.190s$             & $0.666s$              & $6.29$          & $74.25\%$ \\ \hline \hline
+
+cage13            & $8.077s$             & $1.584s$              & $5.10$          & $50.91\%$ \\
+
+crashbasis        & $35.173s$            & $5.546s$              & $6.34$          & $62.43\%$ \\
+
+FEM\_3D\_thermal2 & $24.825s$            & $3.113s$              & $7.97$          & $59.83\%$ \\
+
+language          & $16.706s$            & $2.522s$              & $6.62$          & $73.99\%$ \\
+
+poli\_large       & $12.715s$            & $3.989s$              & $3.19$          & $38.95\%$ \\
+
+torso3            & $48.459s$            & $6.234s$              & $7.77$          & $67.86\%$ \\ \hline
+\end{tabular}
+\caption{Performances of the parallel GMRES solver using hypergraph partitioning for solving linear systems associated to
+sparse five-bands matrices on a cluster of 24 CPU cores vs. on a cluster of 12 GPU.}
+\label{tab:11}
+\end{center}
+\end{table}
+
+We can notice from both Tables~\ref{tab:10} and~\ref{tab:11} that the hypergraph partitioning has improved the performances of both
+parallel CG and GMRES algorithms for solving large linear systems associated to matrices having large bandwidths. The execution times
+on the GPU cluster of both parallel solvers are significantly improved compared to those obtained by using the partitioning row-by-row.
+For these examples of sparse matrices, the execution times of CG and GMRES solvers are reduced on average about $76\%$ and $65\%$ respectively
+(see column~$5$ of each table) compared to those obtained in Tables~\ref{tab:08} and~\ref{tab:09}.
+
+In fact, the hypergraph partitioning applied to sparse matrices having large bandwidths allows to reduce the total communication volume
+necessary to synchronize the computations between the computing nodes in the GPU cluster. Table~\ref{tab:12} presents, for each sparse
+matrix, the total communication volume between $12$ GPU computing nodes obtained by using the partitioning row-by-row (column~$2$), the
+total communication volume obtained by using the hypergraph partitioning (column~$3$) and the execution times in minutes of the hypergraph
+partitioning operation performed by $12$ MPI processes (column~$4$). The total communication volume defines the total number of the vector
+elements exchanged by the computing nodes. Then, Table~\ref{tab:12} shows that the hypergraph partitioning method can split the sparse 
+matrix so as to minimize the data dependencies between the computing nodes and thus to reduce the total communication volume.
+
+
+\begin{table}[!h]
+\begin{center}
+\begin{tabular}{|c|c|c|c|} 
+\hline
+\multirow{4}{*}{\bf Matrix}  & {\bf Total comms.}      & {\bf Total comms.}      & {\bf Execution} \\
+                             & {\bf volume without}    & {\bf volume with}       & {\bf trime}  \\
+                             & {\bf hypergraph}        & {\bf hypergraph }       & {\bf of the parti.}  \\  
+                             & {\bf parti.}            & {\bf parti.}            & {\bf in minutes}\\ \hline \hline
+
+2cubes\_sphere               & $25,360,543$            & $240,679$               & $68.98$         \\
+
+ecology2                     & $26,044,002$            & $73,021$                & $4.92$          \\
+
+finan512                     & $26,087,431$            & $900,729$               & $33.72$         \\
+
+G3\_circuit                  & $31,912,003$            & $5,366,774$             & $11.63$         \\ 
+
+shallow\_water2              & $25,105,108$            & $60,899$                & $5.06$          \\ 
+
+thermal2                     & $30,012,846$            & $1,077,921$             & $17.88$         \\ \hline \hline
+
+cage13                       & $28,254,282$            & $3,845,440$             & $196.45$        \\
+
+crashbasis                   & $29,020,060$            & $2,401,876$             & $33.39$         \\
+
+FEM\_3D\_thermal2            & $25,263,767$            & $250,105$               & $49.89$         \\
+
+language                     & $27,291,486$            & $1,537,835$             & $9.07$          \\
+
+poli\_large                  & $25,053,554$            & $7,388,883$             & $5.92$          \\
+
+torso3                       & $25,682,514$            & $613,250$               & $61.51$         \\ \hline       
+\end{tabular}
+\caption{The total communication volume between 12 GPU computing nodes without and with the hypergraph partitioning method.}
+\label{tab:12}
+\end{center}
+\end{table}
+
+Nevertheless, as we can see from the fourth column of Table~\ref{tab:12}, the hypergraph partitioning takes longer compared
+to the execution times of the resolutions. As previously mentioned, the hypergraph partitioning method is less efficient in
+terms of memory consumption and partitioning time than its graph counterpart, but the hypergraph well models the nonsymmetric
+and irregular problems. So for the applications which often use the same sparse matrices, we can perform the hypergraph partitioning
+on these matrices only once for each and then, we save the traces of these partitionings in files to be reused several times.
+Therefore, this allows us to avoid the partitioning of the sparse matrices at each resolution of the linear systems.
+
+However, the most important performance parameter is the scalability of the parallel CG and GMRES solvers on a GPU cluster.
+Particularly, we have taken into account the weak-scaling of both parallel algorithms on a cluster of one to 12 GPU computing
+nodes. We have performed a set of experimental tests on both matrix structures: band matrices and five-bands matrices. The
+sparse matrices of tests are generated from the symmetric sparse matrix {\it thermal2} chosen from the Davis's collection.
+Figures~\ref{fig:09}-$(a)$ and~\ref{fig:09}-$(b)$ show the execution times of both parallel methods for solving large linear
+systems associated to band matrices and those associated to five-bands matrices, respectively. The size of a sparse sub-matrix
+per computing node, for each matrix structure, is fixed as follows:
+\begin{itemize*}
+\item band matrix: $15$ million of rows and $105,166,557$ of nonzero values,
+\item five-bands matrix: $5$ million of rows and $78,714,492$ of nonzero values. 
+\end{itemize*}
+We can see from these figures that both parallel solvers are quite scalable on a GPU cluster. Indeed, the execution times remains
+almost constant while the size of the size of the sparse linear systems to be solved increases proportionally with the number of 
+the GPU computing nodes. This means that the communication cost is relatively constant regardless of the number the computing nodes
+in the GPU cluster.
+
+\begin{figure}
+\centering
+\begin{tabular}{c}
+\includegraphics[scale=0.7]{Chapters/chapter12/figures/scale_band} \\
+\small{(a) Sparse band matrices} \\
+\\
+\includegraphics[scale=0.7]{Chapters/chapter12/figures/scale_5band} \\
+\small{(b) Sparse five-bands matrices}
+\end{tabular}
+\caption{Weak-scaling of the parallel CG and GMRES solvers on a GPU cluster for solving large sparse linear systems.}
+\label{fig:09}
+\end{figure}
+
+%%--------------------------%%
+%%       SECTION 6          %%
+%%--------------------------%%
+\section{Conclusion}
+\label{sec:06}
+In this chapter, we have aimed at harnessing the computing power of a cluster of GPUs for solving large sparse linear systems.
+For this, we have used two Krylov sub-space iterative methods: the CG and GMRES methods. The first method is well-known to its
+efficiency for solving symmetric linear systems and the second one is used, particularly, for solving nonsymmetric linear systems. 
+
+We have presentend the parallel implementation of both iterative methods on a GPU cluster. Particularly, the operations dealing with
+the vectors and/or matrices, of these methods, are parallelized between the different GPU computing nodes of the cluster. Indeed,
+the data-parallel vector operations are accelerated by GPUs and the communications required to synchronize the parallel computations
+are carried out by CPU cores. For this, we have used a heterogeneous CUDA/MPI programming to implement the parallel iterative
+algorithms.
+
+In the experimental tests, we have shown that using a GPU cluster is efficient for solving linear systems associated to very large
+sparse matrices. The experimental results, obtained in the present chapter, showed that a cluster of $12$ GPUs is about $7$
+times faster than a cluster of $24$ CPU cores for solving large sparse linear systems of $25$ million unknown values. This is due
+to the GPU ability to compute the data-parallel operations faster than the CPUs. However, we have shown that solving linear systems
+associated to matrices having large bandwidths uses many communications to synchronize the computations of GPUs, which slow down 
+even more the resolution. Moreover, there two kinds of communications: between a CPU and its GPU and between CPUs of the computing
+nodes, such that the first ones are the slowest communications on a GPU cluster. So, we have proposed to use the hypergraph partitioning
+instead of the row-by-row partitioning. This allows to minimize the data dependencies between the GPU computing nodes and thus to
+reduce the total communication volume. The experimental results showed that using the hypergraph partitioning technique improve the
+execution times on average of $76\%$ to the CG method and of $65\%$ to the GMRES method on a cluster of $12$ GPUs. 
+
+In the recent GPU hardware and software architectures, the GPU-Direct system with CUDA version 5.0 is used so that two GPUs located on
+the same node or on distant nodes can communicate between them directly without CPUs. This allows to improve the data transfers between
+GPUs.          
+
+\putbib[Chapters/chapter12/biblio12]
+
diff --git a/BookGPU/Chapters/chapter13/ch13.aux b/BookGPU/Chapters/chapter13/ch13.aux
new file mode 100644
index 0000000..140e1c4
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diff --git a/BookGPU/Chapters/chapter13/ch13.tex b/BookGPU/Chapters/chapter13/ch13.tex
new file mode 100755
index 0000000..635181d
--- /dev/null
+++ b/BookGPU/Chapters/chapter13/ch13.tex
@@ -0,0 +1,710 @@
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%                          %%
+%%       CHAPTER 13         %%
+%%                          %%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ 
+\chapterauthor{}{}
+\newcommand{\scalprod}[2]%
+{\ensuremath{\langle #1 \, , #2 \rangle}}
+\chapter{Solving sparse nonlinear systems of obstacle problems on GPU clusters}
+
+%%--------------------------%%
+%%       SECTION 1          %%
+%%--------------------------%%
+\section{Introduction}
+\label{sec:01}
+The obstacle problem is one kind of free boundary problems. It allows to model, for example, an elastic membrane covering a solid obstacle.
+In this case, the objective is to find an equilibrium position of this membrane constrained to be above the obstacle and which tends to minimize
+its surface and/or its energy. The study of such problems occurs in many applications, for example: fluid mechanics, bio-mathematics (tumour growth
+process) or financial mathematics (American or European option pricing).
+
+In this chapter, we focus on solutions of large obstacle problems defined in a three-dimensional domain. Particularly, the present study consists
+in solving large nonlinear systems derived from the spatial discretization of these problems. Owing to the great size of such systems, in order to
+reduce computation times, we proceed by solving them by parallel synchronous or asynchronous iterative algorithms. Moreover, we aim at harnessing
+the computing power of GPUs to accelerate computations of these parallel algorithms. For this, we use an iterative method involving a projection
+on a convex set, which is: the projected Richardson method. We choose this method among other iterative methods because it is easy to implement on
+parallel computers and easy to adapt to GPU architectures.
+
+In Section~\ref{sec:02}, we present the mathematical model of obstacle problems then, in Section~\ref{sec:03}, we describe the general principle of 
+the parallel projected Richardson method. Next, in Section~\ref{sec:04}, we give the main key points of the parallel implementation of both synchronous
+and asynchronous algorithms of the projected Richardson method on a GPU cluster. In Section~\ref{sec:05}, we present the performances of both parallel
+algorithms obtained from simulations carried out on a CPU and GPU clusters. Finally, in Section~\ref{sec:06}, we use the read-black ordering technique
+to improve the convergence and, thus, the execution times of the parallel projected Richardson algorithms on the GPU cluster. 
+
+
+%%--------------------------%%
+%%       SECTION 2          %%
+%%--------------------------%%
+\section{Obstacle problems}
+\label{sec:02}
+In this section, we present the mathematical model of obstacle problems defined in a three-dimensional domain.
+This model is based on that presented in~\cite{ref1}.
+
+%%*******************
+%%*******************
+\subsection{Mathematical model}
+\label{sec:02.01}
+An obstacle problem, arising for example in mechanics or financial derivatives, consists in solving a time dependent
+nonlinear equation:
+\begin{equation}
+\left\{
+\begin{array}{l}
+\frac{\partial u}{\partial t}+b^t.\nabla u-\eta.\Delta u+c.u-f\geq 0\mbox{,~}u\geq\phi\mbox{,~a.e.w. in~}\lbrack 0,T\rbrack\times\Omega\mbox{,~}\eta>0,\\
+(\frac{\partial u}{\partial t}+b^t.\nabla u-\eta.\Delta u+c.u-f)(u-\phi)=0\mbox{,~a.e.w. in~}\lbrack 0,T\rbrack\times\Omega,\\
+u(0,x,y,z)=u_0(x,y,z),\\
+\mbox{B.C. on~}u(t,x,y,z)\mbox{~defined on~}\partial\Omega,
+\end{array}
+\right.
+\label{eq:01}
+\end{equation}
+where $u_0$ is the initial condition, $c\geq 0$, $b$ and $\eta$ are physical parameters, $T$ is the final time, $u=u(t,x,y,z)$
+is an element of the solution vector $U$ to compute, $f$ is the right-hand right that could represent, for example, the external
+forces, B.C. describes the boundary conditions on the boundary $\partial\Omega$ of the domain $\Omega$, $\phi$ models a constraint 
+imposed to $u$, $\Delta$ is the Laplacian operator, $\nabla$ is the gradient operator, a.e.w. means almost every where and ``.''
+defines the products between two scalars, a scalar and a vector or a matrix and a vector. In practice the boundary condition,
+generally considered, is the Dirichlet condition (where $u$ is fixed on $\partial\Omega$) or the Neumann condition (where the
+normal derivative of $u$ is fixed on $\partial\Omega$). 
+
+The time dependent equation~(\ref{eq:01}) is numerically solved by considering an implicit or a semi-implicit time marching,
+where at each time step $k$ a stationary nonlinear problem is solved:
+\begin{equation}
+\left\{
+\begin{array}{l}
+b^t.\nabla u-\eta.\Delta u+(c+\delta).u-g\geq 0\mbox{,~}u\geq\phi\mbox{,~a.e.w. in~}\lbrack 0,T\rbrack\times\Omega\mbox{,~}\eta>0, \\
+(b^t.\nabla u-\eta.\Delta u+(c+\delta).u- g)(u-\phi)=0\mbox{,~a.e.w. in~}\lbrack 0,T\rbrack\times\Omega, \\
+\mbox{B.C. on~}u(t,x,y,z)\mbox{~defined on~}\partial\Omega,
+\end{array}
+\right.
+\label{eq:02}
+\end{equation}
+where $\delta=\frac{1}{k}$ is the inverse of the time step $k$, $g=f+\delta u^{prev}$ and $u^{prev}$ is the solution computed at the
+previous time step. 
+
+
+%%*******************
+%%*******************
+\subsection{Discretization}
+\label{sec:02.02}
+First, we note that the spatial discretization of the previous stationary problem~(\ref{eq:02}) does not provide a symmetric matrix,
+because the convection-diffusion operator is not self-adjoint. Moreover, the fact that the operator is self-adjoint or not plays an
+important role in the choice of the appropriate algorithm for solving nonlinear systems derived from the discretization of the obstacle
+problem. Nevertheless, since the convection coefficients arising in the operator~(\ref{eq:02}) are constant, we can formulate the same 
+problem by an self-adjoint operator by performing a classical change of variables. Then, we can replace the stationary convection-diffusion
+problem:
+\begin{equation}
+b^{t}.\nabla v-\eta.\Delta v+(c+\delta).v=g\mbox{,~a.e.w. in~}\lbrack 0,T\rbrack\times\Omega\mbox{,~}c\geq 0\mbox{,~}\delta\geq~0,
+\label{eq:03}
+\end{equation}
+by the following stationary diffusion operator:
+\begin{equation}
+-\eta.\Delta u+(\frac{\|b\|^{2}_{2}}{4\eta}+c+\delta).u=e^{-a}g=f,
+\label{eq:04}
+\end{equation}
+where $b=\{b_{1},b_{2},b_{3}\}$, $\|b\|_{2}$ denotes the Euclidean norm of $b$ and $v=e^{-a}.u$ represents the general change of variables
+such that $a=\frac{b^{t}(x,y,z)}{2\eta}$. Consequently, the numerical resolution of the diffusion problem (the self-adjoint operator~(\ref{eq:04}))
+is done by optimization algorithms, in contrast, that of the convection-diffusion problem (non self-adjoint operator~(\ref{eq:03})) is
+done by relaxation algorithms. In the case of our studied algorithm, the convergence is ensured by M-matrix property then, the performance
+is linked to the magnitude of the spectral radius of the iteration matrix, which is independent of the condition number.
+
+Next, the three-dimensional domain $\Omega\subset\mathbb{R}^{3}$ is set to $\Omega=\lbrack 0,1\rbrack^{3}$ and discretized with an uniform 
+Cartesian mesh constituted by $M=m^3$ discretization points, where $m$ related to the spatial discretization step by $h=\frac{1}{m+1}$. This
+is carried out by using a classical order 2 finite difference approximation of the Laplacian. So, the complete discretization of both stationary
+boundary value problems~(\ref{eq:03}) and~(\ref{eq:04}) leads to the solution of a large discrete complementary problem of the following
+form, when both Dirichlet or Neumann boundary conditions are used:
+\begin{equation}
+\left\{
+\begin{array}{l}
+\mbox{Find~}U^{*}\in\mathbb{R}^{M}\mbox{~such~that} \\
+(A+\delta I)U^{*}-G\geq 0\mbox{,~}U^{*}\geq\bar{\Phi},\\
+((A+\delta I)U^{*}-G)^{T}(U^{*}-\bar{\Phi})=0,\\
+\end{array}
+\right.
+\label{eq:05}
+\end{equation}
+where $A$ is a matrix obtained after the spatial discretization by a finite difference method, $G$ is derived from the Euler first order implicit time
+marching scheme and from the discretized right-hand side of the obstacle problem, $\delta$ is the inverse of the time step $k$ and $I$ is the identity
+matrix. The matrix $A$ is symmetric when the self-adjoint operator is considered and nonsymmetric otherwise.
+
+According to the chosen discretization scheme of the Laplacian, $A$ is an M-matrix (irreducibly diagonal dominant, see~\cite{ref2}) and, consequently,
+the matrix $(A+\delta I)$ is also an M-matrix. This property is important to the convergence of iterative methods.
+
+
+%%--------------------------%%
+%%       SECTION 3          %%
+%%--------------------------%%
+\section{Parallel iterative method}
+\label{sec:03}
+Owing to the large size of the previous discrete complementary problem~(\ref{eq:05}), we will solve it by parallel synchronous or asynchronous iterative
+algorithms (see~\cite{ref3,ref4,ref5}). In this chapter, we aim at harnessing the computing power of GPU clusters for solving these large nonlinear systems.
+Then, we choose to use the projected Richardson iterative method for solving the diffusion problem~(\ref{eq:04}). Indeed, this method is based on the iterations
+of the Jacobi method which are easy to parallelize on parallel computers and easy to adapt to GPU architectures. Then, according to the boundary value problem
+formulation with a self-adjoint operator~(\ref{eq:04}), we can consider here the equivalent optimization problem and the fixed point mapping associated to
+its solution.
+
+Assume that $E=\mathbb{R}^{M}$ is a Hilbert space, in which $\scalprod{.}{.}$ is the scalar product and $\|.\|$ its associated norm. So, the general fixed
+point problem to be solved is defined as follows:
+\begin{equation}
+\left\{
+\begin{array}{l}
+\mbox{Find~} U^{*} \in E \mbox{~such that} \\
+U^{*} = F(U^{*}), \\
+\end{array}
+\right.
+\label{eq:06}
+\end{equation}
+where $U\mapsto F(U)$ is an application from $E$ to $E$.
+
+Let $K$ be a closed convex set defined by:
+\begin{equation}
+K = \{U | U \geq \Phi \mbox{~everywhere in~} E\},
+\label{eq:07}
+\end{equation}
+where $\Phi$ is the discrete obstacle function. In fact, the obstacle problem~(\ref{eq:05}) is formulated as the following constrained optimization problem:
+\begin{equation}
+\left\{
+\begin{array}{l}
+\mbox{Find~} U^{*} \in K \mbox{~such that} \\
+\forall V \in K, J(U^{*}) \leq J(V), \\
+\end{array}
+\right.
+\label{eq:08}
+\end{equation}
+where the cost function is given by:
+\begin{equation}
+J(U) = \frac{1}{2}\scalprod{\mathcal{A}.U}{U} - \scalprod{G}{U},
+\label{eq:09}
+\end{equation}
+in which $\scalprod{.}{.}$ denotes the scalar product in $E$, $\mathcal{A}=A+\delta I$ is a symmetric positive definite, $A$ is the discretization matrix
+associated with the self-adjoint operator~(\ref{eq:04}) after change of variables.
+
+For any $U\in E$, let $P_K(U)$ be the projection of $U$ on $K$. For any $\gamma\in\mathbb{R}$, $\gamma>0$, the fixed point mapping $F_{\gamma}$ of the projected
+Richardson method is defined as follows:
+\begin{equation}
+U^{*} = F_{\gamma}(U^{*}) = P_K(U^{*} - \gamma(\mathcal{A}.U^{*} - G)).
+\label{eq:10}
+\end{equation}
+In order to reduce the computation time, the large optimization problem is solved in a numerical way by using a parallel asynchronous algorithm of the projected
+Richardson method on the convex set $K$. Particularly, we will consider an asynchronous parallel adaptation of the projected Richardson method~\cite{ref6}.
+
+Let $\alpha\in\mathbb{N}$ be a positive integer. We consider that the space $E=\displaystyle\prod_{i=1}^{\alpha} E_i$ is a product of $\alpha$ subspaces $E_i$
+where $i\in\{1,\ldots,\alpha\}$. Note that $E_i=\mathbb{R}^{m_i}$, where $\displaystyle\sum_{i=1}^{\alpha} m_{i}=M$, is also a Hilbert space in which $\scalprod{.}{.}_i$
+denotes the scalar product and $|.|_i$ the associated norm, for all $i\in\{1,\ldots,\alpha\}$. Then, for all $u,v\in E$, $\scalprod{u}{v}=\displaystyle\sum_{i=1}^{\alpha}\scalprod{u_i}{v_i}_i$
+is the scalar product on $E$.
+
+Let $U\in E$, we consider the following decomposition of $U$ and the corresponding decomposition of $F_\gamma$ into $\alpha$ blocks:
+\begin{equation}
+\begin{array}{rcl}
+U    & = & (U_1,\ldots,U_{\alpha}), \\
+F_{\gamma}(U) & = & (F_{1,\gamma}(U),\ldots,F_{\alpha,\gamma}(U)). \\
+\end{array}
+\label{eq:11}
+\end{equation}
+Assume that the convex set $K=\displaystyle\prod_{i=1}^{\alpha}K_{i}$, such that $\forall i\in\{1,\ldots,\alpha\},K_i\subset E_i$ and $K_i$ is a closed convex set.
+Let also $G=(G_1,\ldots,G_{\alpha})\in E$ and, for any $U\in E$, $P_K(U)=(P_{K_1}(U_1),\ldots,P_{K_{\alpha}}(U_{\alpha}))$ is the projection of $U$ on $K$ where $\forall i\in\{1,\ldots,\alpha\},P_{K_i}$
+is the projector from $E_i$ onto $K_i$. So, the fixed point mapping of the projected Richardson method~(\ref{eq:10}) can be written in the following way:
+\begin{equation}
+\forall U\in E\mbox{,~}\forall i\in\{1,\ldots,\alpha\}\mbox{,~}F_{i,\gamma}(U) = P_{K_i}(U_i - \gamma(\mathcal{A}_i.U - G_i)).
+\label{eq:12}
+\end{equation}
+Note that $\displaystyle\mathcal{A}_i.U= \sum_{j=1}^{\alpha}\mathcal{A}_{i,j}.U_j$, where $\mathcal{A}_{i,j}$ denote block matrices of $\mathcal{A}$.
+
+The parallel asynchronous iterations of the projected Richardson method for solving the obstacle problem~(\ref{eq:08}) are defined as follows: let $U^0\in E,U^0=(U^0_1,\ldots,U^0_\alpha)$ be
+the initial solution, then for all $p\in\mathbb{N}$, the iterate $U^{p+1}=(U^{p+1}_1,\ldots,U^{p+1}_{\alpha})$ is recursively defined by:
+\begin{equation}
+U_i^{p+1} = 
+\left\{
+\begin{array}{l}
+F_{i,\gamma}(U_1^{\rho_1(p)}, \ldots, U_{\alpha}^{\rho_{\alpha}(p)}) \mbox{~if~} i\in s(p), \\
+U_i^p \mbox{~otherwise}, \\
+\end{array}
+\right.
+\label{eq:13}
+\end{equation}
+where
+\begin{equation}
+\left\{
+\begin{array}{l}
+\forall p\in\mathbb{N}, s(p)\subset\{1,\ldots,\alpha\}\mbox{~and~} s(p)\ne\emptyset, \\
+\forall i\in\{1,\ldots,\alpha\},\{p \ | \ i \in s(p)\}\mbox{~is denombrable},
+\end{array}
+\right.
+\label{eq:14}
+\end{equation}
+and $\forall j\in\{1,\ldots,\alpha\}$,
+\begin{equation}
+\left\{
+\begin{array}{l}
+\forall p\in\mathbb{N}, \rho_j(p)\in\mathbb{N}, 0\leq\rho_j(p)\leq p\mbox{~and~}\rho_j(p)=p\mbox{~if~} j\in s(p),\\
+\displaystyle\lim_{p\to\infty}\rho_j(p) = +\infty.\\
+\end{array}
+\right.
+\label{eq:15}
+\end{equation}
+
+The previous asynchronous scheme of the projected Richardson method models computations that are carried out in parallel
+without order nor synchronization (according to the behavior of the parallel iterative method) and describes a subdomain
+method without overlapping. It is a general model that takes into account all possible situations of parallel computations
+and non-blocking message passing. So, the synchronous iterative scheme is defined by:
+\begin{equation}
+\forall j\in\{1,\ldots,\alpha\} \mbox{,~} \forall p\in\mathbb{N} \mbox{,~} \rho_j(p)=p.
+\label{eq:16}
+\end{equation}
+The values of $s(p)$ and $\rho_j(p)$ are defined dynamically and not explicitly by the parallel asynchronous or synchronous
+execution of the algorithm. Particularly, it enables one to consider distributed computations whereby processors compute at
+their own pace according to their intrinsic characteristics and computational load. The parallelism between the processors is
+well described by the set $s(p)$ which contains at each step $p$ the index of the components relaxed by each processor on a
+parallel way while the use of delayed components in~(\ref{eq:13}) permits one to model nondeterministic behavior and does not
+imply inefficiency of the considered distributed scheme of computation. Note that, according to~\cite{ref7}, theoretically,
+each component of the vector must be relaxed an infinity of time. The choice of the relaxed components to be used in the
+computational process may be guided by any criterion and, in particular, a natural criterion is to pick-up the most recently
+available values of the components computed by the processors. Furthermore, the asynchronous iterations are implemented by
+means of non-blocking MPI communication subroutines (asynchronous communications).
+
+The important property ensuring the convergence of the parallel projected Richardson method, both  synchronous and asynchronous
+algorithms, is the fact that $\mathcal{A}$ is an M-matrix. Moreover, the convergence proceeds from a result of~\cite{ref6}.
+Indeed, there exists a value $\gamma_0>0$, such that $\forall\gamma\in ]0,\gamma_0[$, the parallel iterations~(\ref{eq:13}), 
+(\ref{eq:14}) and~(\ref{eq:15}), associated to the fixed point mapping $F_\gamma$~(\ref{eq:12}), converge to the unique solution
+$U^{*}$ of the discretized problem. 
+
+
+%%--------------------------%%
+%%       SECTION 4          %%
+%%--------------------------%%
+\section{Parallel implementation on a GPU cluster}
+\label{sec:04}
+In this section, we give the main key points of the parallel implementation of the projected Richardson method, both synchronous
+and asynchronous versions, on a GPU cluster, for solving the nonlinear systems derived from the discretization of large obstacle
+problems. More precisely, each nonlinear system is solved iteratively using the whole cluster. We use a heteregeneous CUDA/MPI
+programming. Indeed, the communication of data, at each iteration between the GPU computing nodes, can be either synchronous
+or asynchronous using the MPI communication subroutines, whereas inside each GPU node, a CUDA parallelization is performed.
+
+\begin{figure}[!h]
+\centerline{\includegraphics[scale=0.30]{Chapters/chapter13/figures/splitCPU}}
+\caption{Data partitioning of a problem to be solved among $S=3\times 4$ computing nodes.}
+\label{fig:01}
+\end{figure}
+
+Let $S$ denote the number of computing nodes on the GPU cluster, where a computing node is composed of CPU core holding one MPI
+process and a GPU card. So, before starting computations, the obstacle problem of size $(NX\times NY\times NZ)$ is split into $S$
+parallelepipedic sub-problems, each for a node (MPI process, GPU), as is shown in Figure~\ref{fig:01}. Indeed, the $NY$ and $NZ$
+dimensions (according to the $y$ and $z$ axises) of the three-dimensional problem are, respectively, split into $Sy$ and $Sz$ parts,
+such that $S=Sy\times Sz$. In this case, each computing node has at most four neighboring nodes. This kind of the data partitioning
+reduces the data exchanges at subdomain boundaries compared to a naive $z$-axis-wise partitioning.
+
+\begin{algorithm}[!t]
+%\SetLine
+%\linesnumbered
+Initialization of the parameters of the sub-problem\;
+Allocate and fill the data in the global memory GPU\;
+\For{$i=1$ {\bf to} $NbSteps$}{
+   $G = \frac{1}{k}.U^0 + F$\;
+   Solve($A$, $U^0$, $G$, $U$, $\varepsilon$, $MaxRelax$)\;
+   $U^0 = U$\;
+}
+Copy the solution $U$ back from GPU memory\;
+\caption{Parallel solving of the obstacle problem on a GPU cluster}
+\label{alg:01}
+\end{algorithm}
+
+All the computing nodes of the GPU cluster execute in parallel the same Algorithm~\ref{alg:01} but on different three-dimensional
+sub-problems of size $(NX\times ny\times nz)$. This algorithm gives the main key points for solving an obstacle problem defined in
+a three-dimensional domain, where $A$ is the discretization matrix, $G$ is the right-hand side and $U$ is the solution vector. After
+the initialization step, all the data generated from the partitioning operation are copied from the CPU memories to the GPU global
+memories, to be processed on the GPUs. Next, the algorithm uses $NbSteps$ time steps to solve the global obstacle problem. In fact,
+it uses a parallel algorithm adapted to GPUs of the projected Richardson iterative method for solving the nonlinear systems of the
+obstacle problem. This function is defined by {\it Solve()} in Algorithm~\ref{alg:01}. At every time step, the initial guess $U^0$
+for the iterative algorithm is set to the solution found at the previous time step. Moreover, the right-hand side $G$ is computed
+as follows: \[G = \frac{1}{k}.U^{prev} + F\] where $k$ is the time step, $U^{prev}$ is the solution computed in the previous time
+step and each element $f(x, y, z)$ of the vector $F$ is computed as follows:
+\begin{equation}
+f(x,y,z)=\cos(2\pi x)\cdot\cos(4\pi y)\cdot\cos(6\pi z).
+\label{eq:18}
+\end{equation}
+Finally, the solution $U$ of the obstacle problem is copied back from the GPU global memories to the CPU memories. We use the
+communication subroutines of the CUBLAS library~\cite{ref8} (CUDA Basic Linear Algebra Subroutines) for the memory allocations in
+the GPU (\verb+cublasAlloc()+) and the data transfers between the CPU and its GPU: \verb+cublasSetVector()+ and \verb+cublasGetVector()+. 
+
+\begin{algorithm}[!t]
+%  \SetLine
+%  \linesnumbered
+  $p = 0$\;
+  $conv = false$\;
+  $U = U^{0}$\;
+  \Repeat{$(conv=true)$}{
+    Determine\_Bordering\_Vector\_Elements($U$)\;
+    Compute\_New\_Vector\_Elements($A$, $G$, $U$)\;
+    $tmp = U^{0} - U$\;
+    $error = \|tmp\|_{2}$\;
+    $U^{0} = U$\;
+    $p = p + 1$\;
+    $conv$ = Convergence($error$, $p$, $\varepsilon$, $MaxRelax$)\;
+  }
+\caption{Parallel iterative solving of the nonlinear systems on a GPU cluster ($Solve()$ function)}
+\label{alg:02}
+\end{algorithm}
+
+As many other iterative methods, the algorithm of the projected Richardson method is based on algebraic functions operating on vectors
+and/or matrices, which are more efficient on parallel computers when they work on large vectors. Its parallel implementation on the GPU
+cluster is carried out so that the GPUs execute the vector operations as kernels and the CPUs execute the serial codes, supervise the
+kernel executions and the data exchanges with the neighboring nodes and supply the GPUs with data. Algorithm~\ref{alg:02} shows the
+main key points of the parallel iterative algorithm (function $Solve()$ in Algorithm~\ref{alg:01}). All the vector operations inside
+the main loop ({\bf repeat} ... {\bf until}) are executed by the GPU. We use the following functions of the CUBLAS library:
+\begin{itemize}
+\item \verb+cublasDaxpy()+ to compute the difference between the solution vectors $U^{p}$ and $U^{p+1}$ computed in two successive relaxations
+$p$ and $p+1$ (line~$7$ in Algorithm~\ref{alg:02}),
+\item \verb+cublasDnrm2()+ to perform the Euclidean norm (line~$8$) and,
+\item \verb+cublasDcpy()+ for the data copy of a vector to another one in the GPU memory (lines~$3$ and~$9$).
+\end{itemize}
+
+The dimensions of the grid and blocks of threads that execute a given kernel depend on the resources of the GPU multiprocessor and the
+resource requirements of the kernel. So, if $block$ defines the size of a thread block, which must not exceed the maximum size of a thread
+block, then the number of thread blocks in the grid, denoted by $grid$, can be computed according to the size of the local sub-problem
+as follows: \[grid = \frac{(NX\times ny\times nz)+block-1}{block}.\] However, when solving very large problems, the size of the thread
+grid can exceed the maximum number of thread blocks that can be executed on the GPUs (up-to $65.535$ thread blocks) and, thus, the kernel
+will fail to launch. Therefore, for each kernel, we decompose the three-dimensional sub-problem into $nz$ two-dimensional slices of size
+($NX\times ny$), as is shown in Figure~\ref{fig:02}. All slices of the same kernel are executed using {\bf for} loop by $NX\times ny$ parallel
+threads organized in a two-dimensional grid of two-dimensional thread blocks, as is shown in Listing~\ref{list:01}. Each thread is in charge
+of $nz$ discretization points (one from each slice), accessed in the GPU memory with a constant stride $(NX\times ny)$.
+
+\begin{figure}
+\centerline{\includegraphics[scale=0.30]{Chapters/chapter13/figures/splitGPU}}
+\caption{Decomposition of a sub-problem in a GPU into $nz$ slices.}
+\label{fig:02}
+\end{figure}
+
+\begin{center}
+\lstinputlisting[label=list:01,caption=Skeleton codes of a GPU kernel and a CPU function]{Chapters/chapter13/ex1.cu}
+\end{center}
+The function $Determine\_Bordering\_Vector\_Elements()$ (line~$5$ in Algorithm~\ref{alg:02}) determines the values of the vector
+elements shared at the boundaries with neighboring computing nodes. Its main operations are defined as follows:
+\begin{enumerate}
+\item define the values associated to the bordering points needed by the neighbors,
+\item copy the values associated to the bordering points from the GPU to the CPU,
+\item exchange the values associated to the bordering points between the neighboring CPUs,
+\item copy the received values associated to the bordering points from the CPU to the GPU,
+\end{enumerate}
+The first operation of this function is implemented as kernels to be performed by the GPU:
+\begin{itemize}
+\item a kernel executed by $NX\times nz$ threads to define the values associated to the bordering vector elements along $y$-axis and,
+\item a kernel executed by $NX\times ny$ threads to define the values associated to the bordering vector elements along $z$-axis.  
+\end{itemize}
+As mentioned before, we develop the \emph{synchronous} and \emph{asynchronous} algorithms of the projected Richardson method. Obviously,
+in this scope, the synchronous or asynchronous communications refer to the communications between the CPU cores (MPI processes) on the
+GPU cluster, in order to exchange the vector elements associated to subdomain boundaries. For the memory copies between a CPU core and
+its GPU, we use the synchronous communication routines of the CUBLAS library: \verb+cublasSetVector()+ and \verb+cublasGetVector()+
+in the synchronous algorithm and the asynchronous ones: \verb+cublasSetVectorAsync()+ and \verb+cublasGetVectorAsync()+ in the
+asynchronous algorithm. Moreover, we use the communication routines of the MPI library to carry out the data exchanges between the neighboring
+nodes. We use the following communication routines: \verb+MPI_Isend()+ and \verb+MPI_Irecv()+ to perform non-blocking sends and receptions,
+respectively. For the synchronous algorithm, we use the MPI routine \verb+MPI_Waitall()+ which puts the MPI process of a computing node
+in blocking status until all data exchanges with neighboring nodes (sends and receptions) are completed. In contrast, for the asynchronous
+algorithms, we use the MPI routine \verb+MPI_Test()+ which tests the completion of a data exchange (send or reception) without putting the
+MPI process in blocking status.   
+
+The function $Compute\_New\_Vector\_Elements()$ (line~$6$ in Algorithm~\ref{alg:02}) computes, at each iteration, the new elements
+of the iterate vector $U$. Its general code is presented in Listing~\ref{list:01} (CPU function). The iterations of the projected
+Richardson method, based on those of the Jacobi method, are defined as follows: 
+\begin{equation}
+\begin{array}{ll}
+u^{p+1}(x,y,z) =& \frac{1}{Center}(g(x,y,z) - (Center\cdot u^{p}(x,y,z) + \\
+& West\cdot u^{p}(x-h,y,z) + East\cdot u^{p}(x+h,y,z) + \\
+& South\cdot u^{p}(x,y-h,z) + North\cdot u^{p}(x,y+h,z) + \\
+& Rear\cdot u^{p}(x,y,z-h) + Front\cdot u^{p}(x,y,z+h))),  
+\end{array}
+\label{eq:17}
+\end{equation}
+where $u^{p}(x,y,z)$ is an element of the iterate vector $U$ computed at the iteration $p$ and $g(x,y,z)$ is a vector element of the
+right-hand side $G$. The scalars $Center$, $West$, $East$, $South$, $North$, $Rear$ and $Front$ define constant coefficients of the
+block matrix $A$. Figure~\ref{fig:03} shows the positions of these coefficients in a three-dimensional domain.  
+
+\begin{figure}
+\centerline{\includegraphics[scale=0.35]{Chapters/chapter13/figures/matrix}}
+\caption{Matrix constant coefficients in a three-dimensional domain.}
+\label{fig:03}
+\end{figure}
+
+The kernel implementations of the projected Richardson method on GPUs uses a perfect fine-grain multithreading parallelism. Since the 
+projected Richardson algorithm is implemented as a fixed point method, each kernel is executed by a large number of GPU threads such
+that each thread is in charge of the computation of one element of the iterate vector $U$. Moreover, this method uses the vector elements
+updates of the Jacobi method, which means that each thread $i$ computes the new value of its element $u_{i}^{p+1}$ independently of the
+new values $u_{j}^{p+1}$, where $j\neq i$, of those computed in parallel by other threads at the same iteration $p+1$. Listing~\ref{list:02}
+shows the GPU implementations of the main kernels of the projected Richardson method, which are: the matrix-vector multiplication
+(\verb+MV_Multiplication()+) and the vector elements updates (\verb+Vector_Updates()+). The codes of these kernels are based on
+that presented in Listing~\ref{list:01}.
+
+\lstinputlisting[label=list:02,caption=GPU kernels of the projected Richardson method]{Chapters/chapter13/ex2.cu}
+
+\begin{figure}
+\centerline{\includegraphics[scale=0.3]{Chapters/chapter13/figures/points3D}}
+\caption{Computation of a vector element with the projected Richardson method.}
+\label{fig:04}
+\end{figure}
+
+Each kernel is executed by $NX\times ny$ GPU threads so that $nz$ slices of $(NX\times ny)$ vector elements are computed in
+a {\bf for} loop. In this case, each thread is in charge of one vector element from each slice (in total $nz$ vector elements
+along $z$-axis). We can notice from the formula~(\ref{eq:17}) that the computation of a vector element $u^{p+1}(x,y,z)$, by
+a thread at iteration $p+1$, requires seven vector elements computed at the previous iteration $p$: two vector elements in
+each dimension plus the vector element at the intersection of the three axises $x$, $y$ and $z$ (see Figure~\ref{fig:04}). 
+So, to reduce the memory accesses to the high-latency global memory, the vector elements of the current slice can be stored
+in the low-latency shared memories of thread blocks, as is described in~\cite{ref9}. Nevertheless, the fact that the computation
+of a vector element requires only two elements in each dimension does not allow to maximize the data reuse from the shared memories.
+The computation of a slice involves in total $(bx+2)\times(by+2)$ accesses to the global memory per thread block, to fill the
+required vector elements in the shared memory where $bx$ and $by$ are the dimensions of a thread block. Then, in order to optimize
+the memory accesses on GPUs, the elements of the iterate vector $U$ are filled in the cache texture memory (see~\cite{ref10}).
+In new GPU generations as Fermi or Kepler, the global memory accesses are always cached in L1 and L2 caches. For example, for
+a given kernel, we can favour the use of the L1 cache to that of the shared memory by using the function \verb+cudaFuncSetCacheConfig(Kernel,cudaFuncCachePreferL1)+.
+So, the initial access to the global memory loads the vector elements required by the threads of a block into the cache memory
+(texture or L1/L2 caches). Then, all the following memory accesses read from this cache memory. In Listing~\ref{list:02}, the
+function \verb+fetch_double(v,i)+ is used to read from the texture memory the $i^{th}$ element of the double-precision vector
+\verb+v+ (see Listing~\ref{list:03}). Moreover, the seven constant coefficients of matrix $A$ can be stored in the constant memory
+but, since they are reused $nz$ times by each thread, it is more interesting to fill them on the low-latency registers of each thread.    
+
+\lstinputlisting[label=list:03,caption=Memory access to the cache texture memory]{Chapters/chapter13/ex3.cu}
+
+The function $Convergence()$ (line~$11$ in Algorithm~\ref{alg:02}) allows to detect the convergence of the parallel iterative algorithm
+and is based on the tolerance threshold $\varepsilon$ and the maximum number of relaxations $MaxRelax$. We take into account the number 
+of relaxations since that of iterations cannot be computed in the asynchronous case. Indeed, a relaxation is the update~(\ref{eq:13}) of
+a local iterate vector $U_i$ according to $F_i$. Then, counting the number of relaxations is possible in both synchronous and asynchronous
+cases. On the other hand, an iteration is the update of at least all vector components with $F_i$.
+
+In the synchronous algorithm, the global convergence is detected when the maximal value of the absolute error, $error$, is sufficiently small
+and/or the maximum number of relaxations, $MaxRelax$, is reached, as follows:
+$$
+\begin{array}{l}
+error=\|U^{p}-U^{p+1}\|_{2}; \\
+AllReduce(error,\hspace{0.1cm}maxerror,\hspace{0.1cm}MAX); \\
+\text{if}((maxerror<\varepsilon)\hspace{0.2cm}\text{or}\hspace{0.2cm}(p\geq MaxRelax)) \\
+conv \leftarrow true;
+\end{array}
+$$
+where the function $AllReduce()$ uses the MPI reduction subroutine \verb+MPI_Allreduce()+ to compute the maximal value, $maxerror$, among the
+local absolute errors, $error$, of all computing nodes and $p$ (in Algorithm~\ref{alg:02}) is used as a counter of the local relaxations carried
+out by a computing node. In the asynchronous algorithms, the global convergence is detected when all computing nodes locally converge. For this,
+we use a token ring architecture around which a boolean token travels, in one direction, from a computing node to another. Starting from node $0$,
+the boolean token is set to $true$ by node $i$ if the local convergence is reached or to $false$ otherwise and, then, it is sent to node $i+1$.
+Finally, the global convergence is detected when node $0$ receives from its neighbor node $S-1$, in the ring architecture, a token set to $true$.
+In this case, node $0$ sends a stop message (end of parallel solving) to all computing nodes in the cluster.
+
+
+%%--------------------------%%
+%%       SECTION 5          %%
+%%--------------------------%%
+\section{Experimental tests on a GPU cluster}
+\label{sec:05}
+The GPU cluster of tests, that we used in this chapter, is an $20Gbps$ Infiniband network of six machines. Each machine is a Quad-Core Xeon
+E5530 CPU running at $2.4$GHz. It provides a RAM memory of $12$GB with a memory bandwidth of $25.6$GB/s and it is equipped with two Nvidia
+Tesla C1060 GPUs. A Tesla GPU contains in total $240$ cores running at $1.3$GHz. It provides $4$GB of global memory with a memory bandwidth
+of $102$GB/s, accessible by all its cores and also by the CPU through the PCI-Express 16x Gen 2.0 interface with a throughput of $8$GB/s.
+Hence, the memory copy operations between the GPU and the CPU are about $12$ times slower than those of the Tesla GPU memory. We have performed
+our simulations on a cluster of $24$ CPU cores and on a cluster of $12$ GPUs. Figure~\ref{fig:05} describes the components of the GPU cluster
+of tests.
+
+\begin{figure}
+\centerline{\includegraphics[scale=0.25]{Chapters/chapter13/figures/cluster}}
+\caption{GPU cluster of tests composed of 12 computing nodes (six machines, each with two GPUs.}
+\label{fig:05}
+\end{figure}
+
+Linux cluster version 2.6.39 OS is installed on CPUs. C programming language is used for coding the parallel algorithms of the methods on both
+GPU cluster and CPU cluster. CUDA version 4.0~\cite{ref12} is used for programming GPUs, using CUBLAS library~\cite{ref8} to deal with vector
+operations in GPUs and, finally, MPI functions of OpenMPI 1.3.3 are used to carry out the synchronous and asynchronous communications between
+CPU cores. Indeed, in our experiments, a computing node is managed by a MPI process and it is composed of one CPU core and one GPU card.
+ 
+All experimental results of the parallel projected Richardson algorithms are obtained from simulations made in double precision data. The obstacle
+problems to be solved are defined in constant three-dimensional domain $\Omega\subset\mathbb{R}^{3}$. The numerical values of the parameters of the
+obstacle problems are: $\eta=0.2$, $c=1.1$, $f$ is computed by formula~(\ref{eq:18}) and final time $T=0.02$. Moreover, three time steps ($NbSteps=3$)
+are computed with $k=0.0066$. As the discretization matrix is constant along the time steps, the convergence properties of the iterative algorithms
+do not change. Thus, the performance characteristics obtained with three time steps will still be valid for more time steps. The initial function
+$u(0,x,y,z)$ of the obstacle problem~(\ref{eq:01}) is set to $0$, with a constraint $u\geq\phi=0$. The relaxation parameter $\gamma$ used by the
+projected Richardson method is computed automatically thanks to the diagonal entries of the discretization matrix. The formula and its proof can be
+found in~\cite{ref11}, Section~2.3. The convergence tolerance threshold $\varepsilon$ is set to $1e$-$04$ and the maximum number of relaxations is
+limited to $10^{6}$ relaxations. Finally, the number of threads per block is set to $256$ threads, which gives, in general, good performances for
+most GPU applications. We have performed some tests for the execution configurations and we have noticed that the best configuration of the $256$
+threads per block is an organization into two dimensions of sizes $(64,4)$. 
+
+\begin{table}[!h]
+\centering
+\begin{tabular}{|c|c|c|c|c|c|}
+\hline
+\multirow{2}{*}{\bf Pb. size} & \multicolumn{2}{c|}{\bf Synchronous} & \multicolumn{2}{c|}{\bf Asynchronous} & \multirow{2}{*}{\bf Gain\%} \\ \cline{2-5}
+
+                              & $\mathbf{T_{cpu}}$ & {\bf \#relax.}  & $\mathbf{T_{cpu}}$ & {\bf \#relax.}  &  \\ \hline \hline
+
+$256^{3}$                     & $575.22$           & $198,288$        & $539.25$          & $198,613$        & $6.25$ \\ \hline \hline
+
+$512^{3}$                     & $19,250.25$        & $750,912$        & $18,237.14$       & $769,611$        & $5.26$ \\ \hline \hline 
+
+$768^{3}$                     & $206,159.44$       & $1,635,264$      & $183,582.60$      & $1,577,004$      & $10.95$ \\ \hline \hline
+
+$800^{3}$                     & $222,108.09$       & $1,769,232$      & $188,790.04$      & $1,701,735$      & $15.00$ \\ \hline
+\end{tabular}
+\vspace{0.5cm}
+\caption{Execution times in seconds of the parallel projected Richardson method implemented on a cluster of 24 CPU cores.}
+\label{tab:01}
+\end{table}
+
+\begin{table}[!h]
+\centering
+\begin{tabular}{|c|c|c|c|c|c|c|c|}
+\hline
+\multirow{2}{*}{\bf Pb. size} & \multicolumn{3}{c|}{\bf Synchronous}                   & \multicolumn{3}{c|}{\bf Asynchronous}                   & \multirow{2}{*}{\bf Gain\%}  \\ \cline{2-7}
+
+                              & $\mathbf{T_{gpu}}$ & {\bf \#relax.}  & $\mathbf{\tau}$ & $\mathbf{T_{gpu}}$ & {\bf \#relax.}  & $\mathbf{\tau}$ &           \\  \hline \hline
+
+$256^{3}$                     & $29.67$            & $100,692$        &  $19.39$        & $18.00$           & $94,215$         & $29.96$         & $39.33$  \\ \hline \hline
+
+$512^{3}$                     & $521.83$           & $381,300$        & $36.89$         & $425.15$          & $347,279$        & $42.89$         & $18.53$ \\ \hline \hline 
+
+$768^{3}$                     & $4,112.68$         & $831,144$        & $50.13$         & $3,313.87$        & $750,232$        & $55.40$         & $19.42$ \\ \hline \hline 
+
+$800^{3}$                     & $3,950.87$         & $899,088$        & $56.22$         & $3,636.57$        & $834,900$        & $51.91$         & $7.95$ \\ \hline 
+\end{tabular}
+\vspace{0.5cm}
+\caption{Execution times in seconds of the parallel projected Richardson method implemented on a cluster of 12 GPUs.}
+\label{tab:02}
+\end{table}
+
+The performance measures that we took into account are the execution times and the number of relaxations performed by the parallel iterative algorithms,
+both synchronous and asynchronous versions, on the GPU and CPU clusters. These algorithms are used for solving nonlinear systems derived from the discretization
+of obstacle problems of sizes $256^{3}$, $512^{3}$, $768^{3}$ and $800^{3}$. In Table~\ref{tab:01} and Table~\ref{tab:02}, we show the performances
+of the parallel synchronous and asynchronous algorithms of the projected Richardson method implemented, respectively, on a cluster of $24$ CPU cores
+and on a cluster of $12$ GPUs. In these tables, the execution time defines the time spent by the slowest computing node and the number of relaxations
+is computed as the summation of those carried out by all computing nodes.
+
+In the sixth column of Table~\ref{tab:01} and in the eighth column of Table~\ref{tab:02}, we give the gains in $\%$ obtained by using an
+asynchronous algorithm compared to a synchronous one. We can notice that the asynchronous version on CPU and GPU clusters is slightly faster
+than the synchronous one for both methods. Indeed, the cluster of tests is composed of local and homogeneous nodes communicating via low-latency
+connections. So, in the case of distant and/or heterogeneous nodes (or even with geographically distant clusters) the asynchronous version
+would be faster than the synchronous one. However, the gains obtained on the GPU cluster are better than those obtained on the CPU cluster.
+In fact, the computation times are reduced by accelerating the computations on GPUs while the communication times still unchanged.  
+
+The fourth and seventh columns of Table~\ref{tab:02} show the relative gains obtained by executing the parallel algorithms on the cluster
+of $12$ GPUs instead on the cluster of $24$ CPU cores. We compute the relative gain $\tau$ as a ratio between the execution time $T_{cpu}$
+spent on the CPU cluster over that $T_{gpu}$ spent on the GPU cluster: \[\tau=\frac{T_{cpu}}{T_{gpu}}.\] We can see from these ratios that
+solving large obstacle problems is faster on the GPU cluster than on the CPU cluster. Indeed, the GPUs are more efficient than their
+counterpart CPUs to execute large data-parallel operations. In addition, the projected Richardson method is implemented as a fixed point-based
+iteration and uses the Jacobi vector updates that allow a well thread-parallelization on GPUs, such that each GPU thread is in charge
+of one vector component at a time without being dependent on other vector components computed by other threads. Then, this allow to exploit
+at best the high performance computing of the GPUs by using all the GPU resources and avoiding the idle cores.
+
+Finally, the number of relaxations performed by the parallel synchronous algorithm is different in the CPU and GPU versions, because the number
+of computing nodes involved in the GPU cluster and in the CPU cluster is different. In the CPU case, $24$ computing nodes ($24$ CPU cores) are
+considered, whereas in the GPU case, $12$ computing nodes ($12$ GPUs) are considered. As the number of relaxations depends on the domain decomposition,
+consequently it also depends on the number of computing nodes.
+
+
+%%--------------------------%%
+%%       SECTION 6          %%
+%%--------------------------%%
+\section{Red-Black ordering technique}
+\label{sec:06}
+As is well-known, the Jacobi method is characterized by a slow convergence rate compared to some iterative methods (for example Gauss-Seidel method).
+So, in this section, we present some solutions to reduce the execution time and the number of relaxations and, more specifically, to speed up the
+convergence of the parallel projected Richardson method on the GPU cluster. We propose to use the point red-black ordering technique to accelerate
+the convergence. This technique is often used to increase the parallelism of iterative methods for solving linear systems~\cite{ref13,ref14,ref15}.
+We apply it to the projected Richardson method as a compromise between the Jacobi and Gauss-Seidel iterative methods. 
+
+The general principle of the red-black technique is as follows. Let $t$ be the summation of the integer $x$-, $y$- and $z$-coordinates of a vector
+element $u(x,y,z)$ on a three-dimensional domain: $t=x+y+z$. As is shown in Figure~\ref{fig:06.01}, the red-black ordering technique consists in the
+parallel computing of the red vector elements having even value $t$ by using the values of the black ones then, the parallel computing of the black
+vector elements having odd values $t$ by using the new values of the red ones.
+
+\begin{figure}
+\centering
+  \mbox{\subfigure[Red-black ordering on x, y and z axises]{\includegraphics[width=2.3in]{Chapters/chapter13/figures/rouge-noir}\label{fig:06.01}}\quad
+        \subfigure[Red-black ordering on y axis]{\includegraphics[width=2.3in]{Chapters/chapter13/figures/rouge-noir-y}\label{fig:06.02}}}
+\caption{Red-black ordering for computing the iterate vector elements in a three-dimensional space.}
+\end{figure}
+
+This technique can be implemented on the GPU in two different manners:
+\begin{itemize}
+\item among all launched threads ($NX\times ny$ threads), only one thread out of two computes its red or black vector element at a time or,
+\item all launched threads (on average half of $NX\times ny$ threads) compute the red vector elements first and, then, the black ones.
+\end{itemize}
+However, in both solutions, for each memory transaction, only half of the memory segment addressed by a half-warp is used. So, the computation of the
+red and black vector elements leads to use twice the initial number of memory transactions. Then, we apply the point red-black ordering accordingly to
+the $y$-coordinate, as is shown in Figure~\ref{fig:06.02}. In this case, the vector elements having even $y$-coordinate are computed in parallel using
+the values of those having odd $y$-coordinate and then vice-versa. Moreover, in the GPU implementation of the parallel projected Richardson method (Section~\ref{sec:04}),
+we have shown that a sub-problem of size $(NX\times ny\times nz)$ is decomposed into $nz$ grids of size $(NX\times ny)$. Then, each kernel is executed
+in parallel by $NX\times ny$ GPU threads, so that each thread is in charge of $nz$ vector elements along $z$-axis (one vector element in each grid of
+the sub-problem). So, we propose to use the new values of the vector elements computed in grid $i$ to compute those of the vector elements in grid $i+1$.
+Listing~\ref{list:04} describes the kernel of the matrix-vector multiplication and the kernel of the vector elements updates of the parallel projected
+Richardson method using the red-black ordering technique.
+
+\lstinputlisting[label=list:04,caption=GPU kernels of the projected Richardson method using the red-black technique]{Chapters/chapter13/ex4.cu}
+
+Finally, we exploit the concurrent executions between the host functions and the GPU kernels provided by the GPU hardware and software. In fact, the kernel
+launches are asynchronous (when this environment variable is not disabled on the GPUs), such that the control is returned to the host (MPI process) before
+the GPU has completed the requested task (kernel)~\cite{ref12}. Therefore, all the kernels necessary to update the local vector elements, $u(x,y,z)$ where
+$0<y<(ny-1)$ and $0<z<(nz-1)$, are executed first. Then, the values associated to the bordering vector elements are exchanged between the neighbors. Finally,
+the values of the vector elements associated to the bordering vector elements are updated. In this case, the computation of the local vector elements is
+performed concurrently with the data exchanges between neighboring CPUs and this in both synchronous and asynchronous cases.
+
+\begin{table}[!h]
+\centering
+\begin{tabular}{|c|c|c|c|c|c|}
+\hline
+\multirow{2}{*}{\bf Pb. size} & \multicolumn{2}{c|}{\bf Synchronous} & \multicolumn{2}{c|}{\bf Asynchronous} & \multirow{2}{*}{\bf Gain\%}  \\ \cline{2-5}
+
+                              & $\mathbf{T_{gpu}}$ & {\bf \#relax.}   & $\mathbf{T_{gpu}}$ & {\bf \#relax.}   &           \\  \hline \hline
+
+$256^{3}$                     & $18.37$            & $71,988$         & $12.58$           & $67,638$         & $31.52$  \\ \hline \hline
+
+$512^{3}$                     & $349.23$           & $271,188$        & $289.41$          & $246,036$        & $17.13$ \\ \hline \hline 
+
+$768^{3}$                     & $2,773.65$         & $590,652$        & $2,222.22$        & $532,806$        & $19.88$ \\ \hline \hline 
+
+$800^{3}$                     & $2,748.23$         & $638,916$        & $2,502.61$        & $592,525$        & $8.92$ \\ \hline 
+\end{tabular}
+\vspace{0.5cm}
+\caption{Execution times in seconds of the parallel projected Richardson method using read-black ordering technique implemented on a cluster of 12 GPUs.}
+\label{tab:03}
+\end{table}
+
+In Table~\ref{tab:03}, we report the execution times and the number of relaxations performed on a cluster of $12$ GPUs by the parallel projected Richardson
+algorithms; it can be noted that the performances of the projected Richardson are improved by using the point read-black ordering. We compare the performances
+of the parallel projected Richardson method with and without this later ordering (Tables~\ref{tab:02} and~\ref{tab:03}). We can notice that both parallel synchronous
+and asynchronous algorithms are faster when they use the red-black ordering. Indeed, we can see in Table~\ref{tab:03} that the execution times of these algorithms
+are reduced, on average, by $32\%$ compared to those shown in Table~\ref{tab:02}.
+
+\begin{figure}
+\centerline{\includegraphics[scale=0.9]{Chapters/chapter13/figures/scale}}
+\caption{Weak scaling of both synchronous and asynchronous algorithms of the projected Richardson method using red-black ordering technique.}
+\label{fig:07}
+\end{figure}
+
+In Figure~\ref{fig:07}, we study the ratio between the computation time and the communication time of the parallel projected Richardson algorithms on a GPU cluster.
+The experimental tests are carried out on a cluster composed of one to ten Tesla GPUs. We have focused on the weak scaling of both parallel, synchronous and asynchronous,
+algorithms using the red-black ordering technique. For this, we have fixed the size of a sub-problem to $256^{3}$ per computing node (a CPU core and a GPU). Then,
+Figure~\ref{fig:07} shows the number of relaxations performed, on average, per second by a computing node. We can see from this figure that the efficiency of the
+asynchronous algorithm is almost stable, while that of the synchronous algorithm decreases (down to $81\%$ in this example) with the increasing of the number of
+computing nodes on the cluster. This is due to the fact that the ratio between the time of the computation over that of the communication is reduced when the computations
+are performed on GPUs. Indeed, GPUs compute faster than CPUs and communications are more time consuming. In this context, asynchronous algorithms are more scalable
+than synchronous ones. So, with large scale GPU clusters, synchronous algorithms might be more penalized by communications, as can be deduced from Figure~\ref{fig:07}.
+That is why we think that asynchronous iterative algorithms are all the more interesting in this case.
+
+
+%%--------------------------%%
+%%       SECTION 7          %%
+%%--------------------------%%
+\section{Conclusion}
+\label{sec:07}
+Our main contribution, in this chapter, is the parallel implementation of an asynchronous iterative method on GPU clusters for solving large scale nonlinear
+systems derived from the spatial discretization of three-dimensional obstacle problems. For this, we have implemented both synchronous and asynchronous algorithms of the
+Richardson iterative method using a projection on a convex set. Indeed, this method uses point-based iterations of the Jacobi method that are very easy to parallelize on
+parallel computers. We have shown that its adapted parallel algorithms to GPU architectures allows to exploit at best the computing power of the GPUs and to accelerate the
+resolution of large nonlinear systems. Consequently, the experimental results have shown that solving nonlinear systems of large obstacle problems with this method is about
+fifty times faster on a cluster of $12$ GPUs than on a cluster of $24$ CPU cores. Moreover, we have applied to this projected Richardson method the red-black ordering technique
+which allows it to improve its convergence rate. Thus, the execution times of both parallel algorithms performed on the cluster of $12$ GPUs are reduced on average of $32\%$.
+
+Afterwards, the experiments have shown that the asynchronous version is slightly more efficient than the synchronous one. In fact, the computations are accelerated by using GPUs
+while the communication times still unchanged. In addition, we have studied the weak-scaling in the synchronous and asynchronous cases, which has confirmed that the ratio between
+the computations and the communications are reduced when using a cluster of GPUs. We highlight that asynchronous iterative algorithms are more scalable than synchronous ones.
+Therefore, we can conclude that asynchronous iterations are well suited to tackle scalability issues on GPU clusters.
+
+In future works, we plan to perform experiments on large scale GPU clusters and on geographically distant GPU clusters, because we expect that asynchronous versions would
+be faster and more scalable on such architectures. Furthermore, we want to study the performance behavior and the scalability of other numerical algorithms which support,
+if possible, the model of asynchronous iterations.
+
+\putbib[Chapters/chapter13/biblio13]
+
diff --git a/BookGPU/Chapters/chapter13/ch13.tex~ b/BookGPU/Chapters/chapter13/ch13.tex~
new file mode 100755
index 0000000..65a9980
--- /dev/null
+++ b/BookGPU/Chapters/chapter13/ch13.tex~
@@ -0,0 +1,710 @@
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%                          %%
+%%       CHAPTER 13         %%
+%%                          %%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ 
+\chapterauthor{}{}
+\newcommand{\scalprod}[2]%
+{\ensuremath{\langle #1 \, , #2 \rangle}}
+\chapter{Solving sparse nonlinear systems of obstacle problems on GPU clusters}
+
+%%--------------------------%%
+%%       SECTION 1          %%
+%%--------------------------%%
+\section{Introduction}
+\label{sec:01}
+The obstacle problem is one kind of free boundary problems. It allows to model, for example, an elastic membrane covering a solid obstacle.
+In this case, the objective is to find an equilibrium position of this membrane constrained to be above the obstacle and which tends to minimize
+its surface and/or its energy. The study of such problems occurs in many applications, for example: fluid mechanics, bio-mathematics (tumour growth
+process) or financial mathematics (American or European option pricing).
+
+In this chapter, we focus on solutions of large obstacle problems defined in a three-dimensional domain. Particularly, the present study consists
+in solving large nonlinear systems derived from the spatial discretization of these problems. Owing to the great size of such systems, in order to
+reduce computation times, we proceed by solving them by parallel synchronous or asynchronous iterative algorithms. Moreover, we aim at harnessing
+the computing power of GPUs to accelerate computations of these parallel algorithms. For this, we use an iterative method involving a projection
+on a convex set, which is: the projected Richardson method. We choose this method among other iterative methods because it is easy to implement on
+parallel computers and easy to adapt to GPU architectures.
+
+In Section~\ref{sec:02}, we present the mathematical model of obstacle problems then, in Section~\ref{sec:03}, we describe the general principle of 
+the parallel projected Richardson method. Next, in Section~\ref{sec:04}, we give the main key points of the parallel implementation of both synchronous
+and asynchronous algorithms of the projected Richardson method on a GPU cluster. In Section~\ref{sec:05}, we present the performances of both parallel
+algorithms obtained from simulations carried out on a CPU and GPU clusters. Finally, in Section~\ref{sec:06}, we use the read-black ordering technique
+to improve the convergence and, thus, the execution times of the parallel projected Richardson algorithms on the GPU cluster. 
+
+
+%%--------------------------%%
+%%       SECTION 2          %%
+%%--------------------------%%
+\section{Obstacle problems}
+\label{sec:02}
+In this section, we present the mathematical model of obstacle problems defined in a three-dimensional domain.
+This model is based on that presented in~\cite{ref1}.
+
+%%*******************
+%%*******************
+\subsection{Mathematical model}
+\label{sec:02.01}
+An obstacle problem, arising for example in mechanics or financial derivatives, consists in solving a time dependent
+nonlinear equation:
+\begin{equation}
+\left\{
+\begin{array}{l}
+\frac{\partial u}{\partial t}+b^t.\nabla u-\eta.\Delta u+c.u-f\geq 0\mbox{,~}u\geq\phi\mbox{,~a.e.w. in~}\lbrack 0,T\rbrack\times\Omega\mbox{,~}\eta>0,\\
+(\frac{\partial u}{\partial t}+b^t.\nabla u-\eta.\Delta u+c.u-f)(u-\phi)=0\mbox{,~a.e.w. in~}\lbrack 0,T\rbrack\times\Omega,\\
+u(0,x,y,z)=u_0(x,y,z),\\
+\mbox{B.C. on~}u(t,x,y,z)\mbox{~defined on~}\partial\Omega,
+\end{array}
+\right.
+\label{eq:01}
+\end{equation}
+where $u_0$ is the initial condition, $c\geq 0$, $b$ and $\eta$ are physical parameters, $T$ is the final time, $u=u(t,x,y,z)$
+is an element of the solution vector $U$ to compute, $f$ is the right-hand right that could represent, for example, the external
+forces, B.C. describes the boundary conditions on the boundary $\partial\Omega$ of the domain $\Omega$, $\phi$ models a constraint 
+imposed to $u$, $\Delta$ is the Laplacian operator, $\nabla$ is the gradient operator, a.e.w. means almost every where and ``.''
+defines the products between two scalars, a scalar and a vector or a matrix and a vector. In practice the boundary condition,
+generally considered, is the Dirichlet condition (where $u$ is fixed on $\partial\Omega$) or the Neumann condition (where the
+normal derivative of $u$ is fixed on $\partial\Omega$). 
+
+The time dependent equation~(\ref{eq:01}) is numerically solved by considering an implicit or a semi-implicit time marching,
+where at each time step $k$ a stationary nonlinear problem is solved:
+\begin{equation}
+\left\{
+\begin{array}{l}
+b^t.\nabla u-\eta.\Delta u+(c+\delta).u-g\geq 0\mbox{,~}u\geq\phi\mbox{,~a.e.w. in~}\lbrack 0,T\rbrack\times\Omega\mbox{,~}\eta>0, \\
+(b^t.\nabla u-\eta.\Delta u+(c+\delta).u- g)(u-\phi)=0\mbox{,~a.e.w. in~}\lbrack 0,T\rbrack\times\Omega, \\
+\mbox{B.C. on~}u(t,x,y,z)\mbox{~defined on~}\partial\Omega,
+\end{array}
+\right.
+\label{eq:02}
+\end{equation}
+where $\delta=\frac{1}{k}$ is the inverse of the time step $k$, $g=f+\delta u^{prev}$ and $u^{prev}$ is the solution computed at the
+previous time step. 
+
+
+%%*******************
+%%*******************
+\subsection{Discretization}
+\label{sec:02.02}
+First, we note that the spatial discretization of the previous stationary problem~(\ref{eq:02}) does not provide a symmetric matrix,
+because the convection-diffusion operator is not self-adjoint. Moreover, the fact that the operator is self-adjoint or not plays an
+important role in the choice of the appropriate algorithm for solving nonlinear systems derived from the discretization of the obstacle
+problem. Nevertheless, since the convection coefficients arising in the operator~(\ref{eq:02}) are constant, we can formulate the same 
+problem by an self-adjoint operator by performing a classical change of variables. Then, we can replace the stationary convection-diffusion
+problem:
+\begin{equation}
+b^{t}.\nabla v-\eta.\Delta v+(c+\delta).v=g\mbox{,~a.e.w. in~}\lbrack 0,T\rbrack\times\Omega\mbox{,~}c\geq 0\mbox{,~}\delta\geq~0,
+\label{eq:03}
+\end{equation}
+by the following stationary diffusion operator:
+\begin{equation}
+-\eta.\Delta u+(\frac{\|b\|^{2}_{2}}{4\eta}+c+\delta).u=e^{-a}g=f,
+\label{eq:04}
+\end{equation}
+where $b=\{b_{1},b_{2},b_{3}\}$, $\|b\|_{2}$ denotes the Euclidean norm of $b$ and $v=e^{-a}.u$ represents the general change of variables
+such that $a=\frac{b^{t}(x,y,z)}{2\eta}$. Consequently, the numerical resolution of the diffusion problem (the self-adjoint operator~(\ref{eq:04}))
+is done by optimization algorithms, in contrast, that of the convection-diffusion problem (non self-adjoint operator~(\ref{eq:03})) is
+done by relaxation algorithms. In the case of our studied algorithm, the convergence is ensured by M-matrix property then, the performance
+is linked to the magnitude of the spectral radius of the iteration matrix, which is independent of the condition number.
+
+Next, the three-dimensional domain $\Omega\subset\mathbb{R}^{3}$ is set to $\Omega=\lbrack 0,1\rbrack^{3}$ and discretized with an uniform 
+Cartesian mesh constituted by $M=m^3$ discretization points, where $m$ related to the spatial discretization step by $h=\frac{1}{m+1}$. This
+is carried out by using a classical order 2 finite difference approximation of the Laplacian. So, the complete discretization of both stationary
+boundary value problems~(\ref{eq:03}) and~(\ref{eq:04}) leads to the solution of a large discrete complementary problem of the following
+form, when both Dirichlet or Neumann boundary conditions are used:
+\begin{equation}
+\left\{
+\begin{array}{l}
+\mbox{Find~}U^{*}\in\mathbb{R}^{M}\mbox{~such~that} \\
+(A+\delta I)U^{*}-G\geq 0\mbox{,~}U^{*}\geq\bar{\Phi},\\
+((A+\delta I)U^{*}-G)^{T}(U^{*}-\bar{\Phi})=0,\\
+\end{array}
+\right.
+\label{eq:05}
+\end{equation}
+where $A$ is a matrix obtained after the spatial discretization by a finite difference method, $G$ is derived from the Euler first order implicit time
+marching scheme and from the discretized right-hand side of the obstacle problem, $\delta$ is the inverse of the time step $k$ and $I$ is the identity
+matrix. The matrix $A$ is symmetric when the self-adjoint operator is considered and nonsymmetric otherwise.
+
+According to the chosen discretization scheme of the Laplacian, $A$ is an M-matrix (irreducibly diagonal dominant, see~\cite{ref2}) and, consequently,
+the matrix $(A+\delta I)$ is also an M-matrix. This property is important to the convergence of iterative methods.
+
+
+%%--------------------------%%
+%%       SECTION 3          %%
+%%--------------------------%%
+\section{Parallel iterative method}
+\label{sec:03}
+Owing to the large size of the previous discrete complementary problem~(\ref{eq:05}), we will solve it by parallel synchronous or asynchronous iterative
+algorithms (see~\cite{ref3,ref4,ref5}). In this chapter, we aim at harnessing the computing power of GPU clusters for solving these large nonlinear systems.
+Then, we choose to use the projected Richardson iterative method for solving the diffusion problem~(\ref{eq:04}). Indeed, this method is based on the iterations
+of the Jacobi method which are easy to parallelize on parallel computers and easy to adapt to GPU architectures. Then, according to the boundary value problem
+formulation with a self-adjoint operator~(\ref{eq:04}), we can consider here the equivalent optimization problem and the fixed point mapping associated to
+its solution.
+
+Assume that $E=\mathbb{R}^{M}$ is a Hilbert space, in which $\scalprod{.}{.}$ is the scalar product and $\|.\|$ its associated norm. So, the general fixed
+point problem to be solved is defined as follows:
+\begin{equation}
+\left\{
+\begin{array}{l}
+\mbox{Find~} U^{*} \in E \mbox{~such that} \\
+U^{*} = F(U^{*}), \\
+\end{array}
+\right.
+\label{eq:06}
+\end{equation}
+where $U\mapsto F(U)$ is an application from $E$ to $E$.
+
+Let $K$ be a closed convex set defined by:
+\begin{equation}
+K = \{U | U \geq \Phi \mbox{~everywhere in~} E\},
+\label{eq:07}
+\end{equation}
+where $\Phi$ is the discrete obstacle function. In fact, the obstacle problem~(\ref{eq:05}) is formulated as the following constrained optimization problem:
+\begin{equation}
+\left\{
+\begin{array}{l}
+\mbox{Find~} U^{*} \in K \mbox{~such that} \\
+\forall V \in K, J(U^{*}) \leq J(V), \\
+\end{array}
+\right.
+\label{eq:08}
+\end{equation}
+where the cost function is given by:
+\begin{equation}
+J(U) = \frac{1}{2}\scalprod{\mathcal{A}.U}{U} - \scalprod{G}{U},
+\label{eq:09}
+\end{equation}
+in which $\scalprod{.}{.}$ denotes the scalar product in $E$, $\mathcal{A}=A+\delta I$ is a symmetric positive definite, $A$ is the discretization matrix
+associated with the self-adjoint operator~(\ref{eq:04}) after change of variables.
+
+For any $U\in E$, let $P_K(U)$ be the projection of $U$ on $K$. For any $\gamma\in\mathbb{R}$, $\gamma>0$, the fixed point mapping $F_{\gamma}$ of the projected
+Richardson method is defined as follows:
+\begin{equation}
+U^{*} = F_{\gamma}(U^{*}) = P_K(U^{*} - \gamma(\mathcal{A}.U^{*} - G)).
+\label{eq:10}
+\end{equation}
+In order to reduce the computation time, the large optimization problem is solved in a numerical way by using a parallel asynchronous algorithm of the projected
+Richardson method on the convex set $K$. Particularly, we will consider an asynchronous parallel adaptation of the projected Richardson method~\cite{ref6}.
+
+Let $\alpha\in\mathbb{N}$ be a positive integer. We consider that the space $E=\displaystyle\prod_{i=1}^{\alpha} E_i$ is a product of $\alpha$ subspaces $E_i$
+where $i\in\{1,\ldots,\alpha\}$. Note that $E_i=\mathbb{R}^{m_i}$, where $\displaystyle\sum_{i=1}^{\alpha} m_{i}=M$, is also a Hilbert space in which $\scalprod{.}{.}_i$
+denotes the scalar product and $|.|_i$ the associated norm, for all $i\in\{1,\ldots,\alpha\}$. Then, for all $u,v\in E$, $\scalprod{u}{v}=\displaystyle\sum_{i=1}^{\alpha}\scalprod{u_i}{v_i}_i$
+is the scalar product on $E$.
+
+Let $U\in E$, we consider the following decomposition of $U$ and the corresponding decomposition of $F_\gamma$ into $\alpha$ blocks:
+\begin{equation}
+\begin{array}{rcl}
+U    & = & (U_1,\ldots,U_{\alpha}), \\
+F_{\gamma}(U) & = & (F_{1,\gamma}(U),\ldots,F_{\alpha,\gamma}(U)). \\
+\end{array}
+\label{eq:11}
+\end{equation}
+Assume that the convex set $K=\displaystyle\prod_{i=1}^{\alpha}K_{i}$, such that $\forall i\in\{1,\ldots,\alpha\},K_i\subset E_i$ and $K_i$ is a closed convex set.
+Let also $G=(G_1,\ldots,G_{\alpha})\in E$ and, for any $U\in E$, $P_K(U)=(P_{K_1}(U_1),\ldots,P_{K_{\alpha}}(U_{\alpha}))$ is the projection of $U$ on $K$ where $\forall i\in\{1,\ldots,\alpha\},P_{K_i}$
+is the projector from $E_i$ onto $K_i$. So, the fixed point mapping of the projected Richardson method~(\ref{eq:10}) can be written in the following way:
+\begin{equation}
+\forall U\in E\mbox{,~}\forall i\in\{1,\ldots,\alpha\}\mbox{,~}F_{i,\gamma}(U) = P_{K_i}(U_i - \gamma(\mathcal{A}_i.U - G_i)).
+\label{eq:12}
+\end{equation}
+Note that $\displaystyle\mathcal{A}_i.U= \sum_{j=1}^{\alpha}\mathcal{A}_{i,j}.U_j$, where $\mathcal{A}_{i,j}$ denote block matrices of $\mathcal{A}$.
+
+The parallel asynchronous iterations of the projected Richardson method for solving the obstacle problem~(\ref{eq:08}) are defined as follows: let $U^0\in E,U^0=(U^0_1,\ldots,U^0_\alpha)$ be
+the initial solution, then for all $p\in\mathbb{N}$, the iterate $U^{p+1}=(U^{p+1}_1,\ldots,U^{p+1}_{\alpha})$ is recursively defined by:
+\begin{equation}
+U_i^{p+1} = 
+\left\{
+\begin{array}{l}
+F_{i,\gamma}(U_1^{\rho_1(p)}, \ldots, U_{\alpha}^{\rho_{\alpha}(p)}) \mbox{~if~} i\in s(p), \\
+U_i^p \mbox{~otherwise}, \\
+\end{array}
+\right.
+\label{eq:13}
+\end{equation}
+where
+\begin{equation}
+\left\{
+\begin{array}{l}
+\forall p\in\mathbb{N}, s(p)\subset\{1,\ldots,\alpha\}\mbox{~and~} s(p)\ne\emptyset, \\
+\forall i\in\{1,\ldots,\alpha\},\{p \ | \ i \in s(p)\}\mbox{~is denombrable},
+\end{array}
+\right.
+\label{eq:14}
+\end{equation}
+and $\forall j\in\{1,\ldots,\alpha\}$,
+\begin{equation}
+\left\{
+\begin{array}{l}
+\forall p\in\mathbb{N}, \rho_j(p)\in\mathbb{N}, 0\leq\rho_j(p)\leq p\mbox{~and~}\rho_j(p)=p\mbox{~if~} j\in s(p),\\
+\displaystyle\lim_{p\to\infty}\rho_j(p) = +\infty.\\
+\end{array}
+\right.
+\label{eq:15}
+\end{equation}
+
+The previous asynchronous scheme of the projected Richardson method models computations that are carried out in parallel
+without order nor synchronization (according to the behavior of the parallel iterative method) and describes a subdomain
+method without overlapping. It is a general model that takes into account all possible situations of parallel computations
+and non-blocking message passing. So, the synchronous iterative scheme is defined by:
+\begin{equation}
+\forall j\in\{1,\ldots,\alpha\} \mbox{,~} \forall p\in\mathbb{N} \mbox{,~} \rho_j(p)=p.
+\label{eq:16}
+\end{equation}
+The values of $s(p)$ and $\rho_j(p)$ are defined dynamically and not explicitly by the parallel asynchronous or synchronous
+execution of the algorithm. Particularly, it enables one to consider distributed computations whereby processors compute at
+their own pace according to their intrinsic characteristics and computational load. The parallelism between the processors is
+well described by the set $s(p)$ which contains at each step $p$ the index of the components relaxed by each processor on a
+parallel way while the use of delayed components in~(\ref{eq:13}) permits one to model nondeterministic behavior and does not
+imply inefficiency of the considered distributed scheme of computation. Note that, according to~\cite{ref7}, theoretically,
+each component of the vector must be relaxed an infinity of time. The choice of the relaxed components to be used in the
+computational process may be guided by any criterion and, in particular, a natural criterion is to pick-up the most recently
+available values of the components computed by the processors. Furthermore, the asynchronous iterations are implemented by
+means of non-blocking MPI communication subroutines (asynchronous communications).
+
+The important property ensuring the convergence of the parallel projected Richardson method, both  synchronous and asynchronous
+algorithms, is the fact that $\mathcal{A}$ is an M-matrix. Moreover, the convergence proceeds from a result of~\cite{ref6}.
+Indeed, there exists a value $\gamma_0>0$, such that $\forall\gamma\in ]0,\gamma_0[$, the parallel iterations~(\ref{eq:13}), 
+(\ref{eq:14}) and~(\ref{eq:15}), associated to the fixed point mapping $F_\gamma$~(\ref{eq:12}), converge to the unique solution
+$U^{*}$ of the discretized problem. 
+
+
+%%--------------------------%%
+%%       SECTION 4          %%
+%%--------------------------%%
+\section{Parallel implementation on a GPU cluster}
+\label{sec:04}
+In this section, we give the main key points of the parallel implementation of the projected Richardson method, both synchronous
+and asynchronous versions, on a GPU cluster, for solving the nonlinear systems derived from the discretization of large obstacle
+problems. More precisely, each nonlinear system is solved iteratively using the whole cluster. We use a heteregeneous CUDA/MPI
+programming. Indeed, the communication of data, at each iteration between the GPU computing nodes, can be either synchronous
+or asynchronous using the MPI communication subroutines, whereas inside each GPU node, a CUDA parallelization is performed.
+
+\begin{figure}[!h]
+\centerline{\includegraphics[scale=0.30]{Chapters/chapter13/figures/splitCPU}}
+\caption{Data partitioning of a problem to be solved among $S=3\times 4$ computing nodes.}
+\label{fig:01}
+\end{figure}
+
+Let $S$ denote the number of computing nodes on the GPU cluster, where a computing node is composed of CPU core holding one MPI
+process and a GPU card. So, before starting computations, the obstacle problem of size $(NX\times NY\times NZ)$ is split into $S$
+parallelepipedic sub-problems, each for a node (MPI process, GPU), as is shown in Figure~\ref{fig:01}. Indeed, the $NY$ and $NZ$
+dimensions (according to the $y$ and $z$ axises) of the three-dimensional problem are, respectively, split into $Sy$ and $Sz$ parts,
+such that $S=Sy\times Sz$. In this case, each computing node has at most four neighboring nodes. This kind of the data partitioning
+reduces the data exchanges at subdomain boundaries compared to a naive $z$-axis-wise partitioning.
+
+\begin{algorithm}[!t]
+\SetLine
+\linesnumbered
+Initialization of the parameters of the sub-problem\;
+Allocate and fill the data in the global memory GPU\;
+\For{$i=1$ {\bf to} $NbSteps$}{
+   $G = \frac{1}{k}.U^0 + F$\;
+   Solve($A$, $U^0$, $G$, $U$, $\varepsilon$, $MaxRelax$)\;
+   $U^0 = U$\;
+}
+Copy the solution $U$ back from GPU memory\;
+\caption{Parallel solving of the obstacle problem on a GPU cluster}
+\label{alg:01}
+\end{algorithm}
+
+All the computing nodes of the GPU cluster execute in parallel the same Algorithm~\ref{alg:01} but on different three-dimensional
+sub-problems of size $(NX\times ny\times nz)$. This algorithm gives the main key points for solving an obstacle problem defined in
+a three-dimensional domain, where $A$ is the discretization matrix, $G$ is the right-hand side and $U$ is the solution vector. After
+the initialization step, all the data generated from the partitioning operation are copied from the CPU memories to the GPU global
+memories, to be processed on the GPUs. Next, the algorithm uses $NbSteps$ time steps to solve the global obstacle problem. In fact,
+it uses a parallel algorithm adapted to GPUs of the projected Richardson iterative method for solving the nonlinear systems of the
+obstacle problem. This function is defined by {\it Solve()} in Algorithm~\ref{alg:01}. At every time step, the initial guess $U^0$
+for the iterative algorithm is set to the solution found at the previous time step. Moreover, the right-hand side $G$ is computed
+as follows: \[G = \frac{1}{k}.U^{prev} + F\] where $k$ is the time step, $U^{prev}$ is the solution computed in the previous time
+step and each element $f(x, y, z)$ of the vector $F$ is computed as follows:
+\begin{equation}
+f(x,y,z)=\cos(2\pi x)\cdot\cos(4\pi y)\cdot\cos(6\pi z).
+\label{eq:18}
+\end{equation}
+Finally, the solution $U$ of the obstacle problem is copied back from the GPU global memories to the CPU memories. We use the
+communication subroutines of the CUBLAS library~\cite{ref8} (CUDA Basic Linear Algebra Subroutines) for the memory allocations in
+the GPU (\verb+cublasAlloc()+) and the data transfers between the CPU and its GPU: \verb+cublasSetVector()+ and \verb+cublasGetVector()+. 
+
+\begin{algorithm}[!t]
+  \SetLine
+  \linesnumbered
+  $p = 0$\;
+  $conv = false$\;
+  $U = U^{0}$\;
+  \Repeat{$(conv=true)$}{
+    Determine\_Bordering\_Vector\_Elements($U$)\;
+    Compute\_New\_Vector\_Elements($A$, $G$, $U$)\;
+    $tmp = U^{0} - U$\;
+    $error = \|tmp\|_{2}$\;
+    $U^{0} = U$\;
+    $p = p + 1$\;
+    $conv$ = Convergence($error$, $p$, $\varepsilon$, $MaxRelax$)\;
+  }
+\caption{Parallel iterative solving of the nonlinear systems on a GPU cluster ($Solve()$ function)}
+\label{alg:02}
+\end{algorithm}
+
+As many other iterative methods, the algorithm of the projected Richardson method is based on algebraic functions operating on vectors
+and/or matrices, which are more efficient on parallel computers when they work on large vectors. Its parallel implementation on the GPU
+cluster is carried out so that the GPUs execute the vector operations as kernels and the CPUs execute the serial codes, supervise the
+kernel executions and the data exchanges with the neighboring nodes and supply the GPUs with data. Algorithm~\ref{alg:02} shows the
+main key points of the parallel iterative algorithm (function $Solve()$ in Algorithm~\ref{alg:01}). All the vector operations inside
+the main loop ({\bf repeat} ... {\bf until}) are executed by the GPU. We use the following functions of the CUBLAS library:
+\begin{itemize*}
+\item \verb+cublasDaxpy()+ to compute the difference between the solution vectors $U^{p}$ and $U^{p+1}$ computed in two successive relaxations
+$p$ and $p+1$ (line~$7$ in Algorithm~\ref{alg:02}),
+\item \verb+cublasDnrm2()+ to perform the Euclidean norm (line~$8$) and,
+\item \verb+cublasDcpy()+ for the data copy of a vector to another one in the GPU memory (lines~$3$ and~$9$).
+\end{itemize*}
+
+The dimensions of the grid and blocks of threads that execute a given kernel depend on the resources of the GPU multiprocessor and the
+resource requirements of the kernel. So, if $block$ defines the size of a thread block, which must not exceed the maximum size of a thread
+block, then the number of thread blocks in the grid, denoted by $grid$, can be computed according to the size of the local sub-problem
+as follows: \[grid = \frac{(NX\times ny\times nz)+block-1}{block}.\] However, when solving very large problems, the size of the thread
+grid can exceed the maximum number of thread blocks that can be executed on the GPUs (up-to $65.535$ thread blocks) and, thus, the kernel
+will fail to launch. Therefore, for each kernel, we decompose the three-dimensional sub-problem into $nz$ two-dimensional slices of size
+($NX\times ny$), as is shown in Figure~\ref{fig:02}. All slices of the same kernel are executed using {\bf for} loop by $NX\times ny$ parallel
+threads organized in a two-dimensional grid of two-dimensional thread blocks, as is shown in Listing~\ref{list:01}. Each thread is in charge
+of $nz$ discretization points (one from each slice), accessed in the GPU memory with a constant stride $(NX\times ny)$.
+
+\begin{figure}
+\centerline{\includegraphics[scale=0.30]{Chapters/chapter13/figures/splitGPU}}
+\caption{Decomposition of a sub-problem in a GPU into $nz$ slices.}
+\label{fig:02}
+\end{figure}
+
+\begin{center}
+\lstinputlisting[label=list:01,caption=Skeleton codes of a GPU kernel and a CPU function]{Chapters/chapter13/ex1.cu}
+\end{center}
+The function $Determine\_Bordering\_Vector\_Elements()$ (line~$5$ in Algorithm~\ref{alg:02}) determines the values of the vector
+elements shared at the boundaries with neighboring computing nodes. Its main operations are defined as follows:
+\begin{enumerate*}
+\item define the values associated to the bordering points needed by the neighbors,
+\item copy the values associated to the bordering points from the GPU to the CPU,
+\item exchange the values associated to the bordering points between the neighboring CPUs,
+\item copy the received values associated to the bordering points from the CPU to the GPU,
+\end{enumerate*}
+The first operation of this function is implemented as kernels to be performed by the GPU:
+\begin{itemize*}
+\item a kernel executed by $NX\times nz$ threads to define the values associated to the bordering vector elements along $y$-axis and,
+\item a kernel executed by $NX\times ny$ threads to define the values associated to the bordering vector elements along $z$-axis.  
+\end{itemize*}
+As mentioned before, we develop the \emph{synchronous} and \emph{asynchronous} algorithms of the projected Richardson method. Obviously,
+in this scope, the synchronous or asynchronous communications refer to the communications between the CPU cores (MPI processes) on the
+GPU cluster, in order to exchange the vector elements associated to subdomain boundaries. For the memory copies between a CPU core and
+its GPU, we use the synchronous communication routines of the CUBLAS library: \verb+cublasSetVector()+ and \verb+cublasGetVector()+
+in the synchronous algorithm and the asynchronous ones: \verb+cublasSetVectorAsync()+ and \verb+cublasGetVectorAsync()+ in the
+asynchronous algorithm. Moreover, we use the communication routines of the MPI library to carry out the data exchanges between the neighboring
+nodes. We use the following communication routines: \verb+MPI_Isend()+ and \verb+MPI_Irecv()+ to perform non-blocking sends and receptions,
+respectively. For the synchronous algorithm, we use the MPI routine \verb+MPI_Waitall()+ which puts the MPI process of a computing node
+in blocking status until all data exchanges with neighboring nodes (sends and receptions) are completed. In contrast, for the asynchronous
+algorithms, we use the MPI routine \verb+MPI_Test()+ which tests the completion of a data exchange (send or reception) without putting the
+MPI process in blocking status.   
+
+The function $Compute\_New\_Vector\_Elements()$ (line~$6$ in Algorithm~\ref{alg:02}) computes, at each iteration, the new elements
+of the iterate vector $U$. Its general code is presented in Listing~\ref{list:01} (CPU function). The iterations of the projected
+Richardson method, based on those of the Jacobi method, are defined as follows: 
+\begin{equation}
+\begin{array}{ll}
+u^{p+1}(x,y,z) =& \frac{1}{Center}(g(x,y,z) - (Center\cdot u^{p}(x,y,z) + \\
+& West\cdot u^{p}(x-h,y,z) + East\cdot u^{p}(x+h,y,z) + \\
+& South\cdot u^{p}(x,y-h,z) + North\cdot u^{p}(x,y+h,z) + \\
+& Rear\cdot u^{p}(x,y,z-h) + Front\cdot u^{p}(x,y,z+h))),  
+\end{array}
+\label{eq:17}
+\end{equation}
+where $u^{p}(x,y,z)$ is an element of the iterate vector $U$ computed at the iteration $p$ and $g(x,y,z)$ is a vector element of the
+right-hand side $G$. The scalars $Center$, $West$, $East$, $South$, $North$, $Rear$ and $Front$ define constant coefficients of the
+block matrix $A$. Figure~\ref{fig:03} shows the positions of these coefficients in a three-dimensional domain.  
+
+\begin{figure}
+\centerline{\includegraphics[scale=0.35]{Chapters/chapter13/figures/matrix}}
+\caption{Matrix constant coefficients in a three-dimensional domain.}
+\label{fig:03}
+\end{figure}
+
+The kernel implementations of the projected Richardson method on GPUs uses a perfect fine-grain multithreading parallelism. Since the 
+projected Richardson algorithm is implemented as a fixed point method, each kernel is executed by a large number of GPU threads such
+that each thread is in charge of the computation of one element of the iterate vector $U$. Moreover, this method uses the vector elements
+updates of the Jacobi method, which means that each thread $i$ computes the new value of its element $u_{i}^{p+1}$ independently of the
+new values $u_{j}^{p+1}$, where $j\neq i$, of those computed in parallel by other threads at the same iteration $p+1$. Listing~\ref{list:02}
+shows the GPU implementations of the main kernels of the projected Richardson method, which are: the matrix-vector multiplication
+(\verb+MV_Multiplication()+) and the vector elements updates (\verb+Vector_Updates()+). The codes of these kernels are based on
+that presented in Listing~\ref{list:01}.
+
+\lstinputlisting[label=list:02,caption=GPU kernels of the projected Richardson method]{Chapters/chapter13/ex2.cu}
+
+\begin{figure}
+\centerline{\includegraphics[scale=0.3]{Chapters/chapter13/figures/points3D}}
+\caption{Computation of a vector element with the projected Richardson method.}
+\label{fig:04}
+\end{figure}
+
+Each kernel is executed by $NX\times ny$ GPU threads so that $nz$ slices of $(NX\times ny)$ vector elements are computed in
+a {\bf for} loop. In this case, each thread is in charge of one vector element from each slice (in total $nz$ vector elements
+along $z$-axis). We can notice from the formula~(\ref{eq:17}) that the computation of a vector element $u^{p+1}(x,y,z)$, by
+a thread at iteration $p+1$, requires seven vector elements computed at the previous iteration $p$: two vector elements in
+each dimension plus the vector element at the intersection of the three axises $x$, $y$ and $z$ (see Figure~\ref{fig:04}). 
+So, to reduce the memory accesses to the high-latency global memory, the vector elements of the current slice can be stored
+in the low-latency shared memories of thread blocks, as is described in~\cite{ref9}. Nevertheless, the fact that the computation
+of a vector element requires only two elements in each dimension does not allow to maximize the data reuse from the shared memories.
+The computation of a slice involves in total $(bx+2)\times(by+2)$ accesses to the global memory per thread block, to fill the
+required vector elements in the shared memory where $bx$ and $by$ are the dimensions of a thread block. Then, in order to optimize
+the memory accesses on GPUs, the elements of the iterate vector $U$ are filled in the cache texture memory (see~\cite{ref10}).
+In new GPU generations as Fermi or Kepler, the global memory accesses are always cached in L1 and L2 caches. For example, for
+a given kernel, we can favour the use of the L1 cache to that of the shared memory by using the function \verb+cudaFuncSetCacheConfig(Kernel,cudaFuncCachePreferL1)+.
+So, the initial access to the global memory loads the vector elements required by the threads of a block into the cache memory
+(texture or L1/L2 caches). Then, all the following memory accesses read from this cache memory. In Listing~\ref{list:02}, the
+function \verb+fetch_double(v,i)+ is used to read from the texture memory the $i^{th}$ element of the double-precision vector
+\verb+v+ (see Listing~\ref{list:03}). Moreover, the seven constant coefficients of matrix $A$ can be stored in the constant memory
+but, since they are reused $nz$ times by each thread, it is more interesting to fill them on the low-latency registers of each thread.    
+
+\lstinputlisting[label=list:03,caption=Memory access to the cache texture memory]{Chapters/chapter13/ex3.cu}
+
+The function $Convergence()$ (line~$11$ in Algorithm~\ref{alg:02}) allows to detect the convergence of the parallel iterative algorithm
+and is based on the tolerance threshold $\varepsilon$ and the maximum number of relaxations $MaxRelax$. We take into account the number 
+of relaxations since that of iterations cannot be computed in the asynchronous case. Indeed, a relaxation is the update~(\ref{eq:13}) of
+a local iterate vector $U_i$ according to $F_i$. Then, counting the number of relaxations is possible in both synchronous and asynchronous
+cases. On the other hand, an iteration is the update of at least all vector components with $F_i$.
+
+In the synchronous algorithm, the global convergence is detected when the maximal value of the absolute error, $error$, is sufficiently small
+and/or the maximum number of relaxations, $MaxRelax$, is reached, as follows:
+$$
+\begin{array}{l}
+error=\|U^{p}-U^{p+1}\|_{2}; \\
+AllReduce(error,\hspace{0.1cm}maxerror,\hspace{0.1cm}MAX); \\
+\text{if}((maxerror<\varepsilon)\hspace{0.2cm}\text{or}\hspace{0.2cm}(p\geq MaxRelax)) \\
+conv \leftarrow true;
+\end{array}
+$$
+where the function $AllReduce()$ uses the MPI reduction subroutine \verb+MPI_Allreduce()+ to compute the maximal value, $maxerror$, among the
+local absolute errors, $error$, of all computing nodes and $p$ (in Algorithm~\ref{alg:02}) is used as a counter of the local relaxations carried
+out by a computing node. In the asynchronous algorithms, the global convergence is detected when all computing nodes locally converge. For this,
+we use a token ring architecture around which a boolean token travels, in one direction, from a computing node to another. Starting from node $0$,
+the boolean token is set to $true$ by node $i$ if the local convergence is reached or to $false$ otherwise and, then, it is sent to node $i+1$.
+Finally, the global convergence is detected when node $0$ receives from its neighbor node $S-1$, in the ring architecture, a token set to $true$.
+In this case, node $0$ sends a stop message (end of parallel solving) to all computing nodes in the cluster.
+
+
+%%--------------------------%%
+%%       SECTION 5          %%
+%%--------------------------%%
+\section{Experimental tests on a GPU cluster}
+\label{sec:05}
+The GPU cluster of tests, that we used in this chapter, is an $20Gbps$ Infiniband network of six machines. Each machine is a Quad-Core Xeon
+E5530 CPU running at $2.4$GHz. It provides a RAM memory of $12$GB with a memory bandwidth of $25.6$GB/s and it is equipped with two Nvidia
+Tesla C1060 GPUs. A Tesla GPU contains in total $240$ cores running at $1.3$GHz. It provides $4$GB of global memory with a memory bandwidth
+of $102$GB/s, accessible by all its cores and also by the CPU through the PCI-Express 16x Gen 2.0 interface with a throughput of $8$GB/s.
+Hence, the memory copy operations between the GPU and the CPU are about $12$ times slower than those of the Tesla GPU memory. We have performed
+our simulations on a cluster of $24$ CPU cores and on a cluster of $12$ GPUs. Figure~\ref{fig:05} describes the components of the GPU cluster
+of tests.
+
+\begin{figure}
+\centerline{\includegraphics[scale=0.25]{Chapters/chapter13/figures/cluster}}
+\caption{GPU cluster of tests composed of 12 computing nodes (six machines, each with two GPUs.}
+\label{fig:05}
+\end{figure}
+
+Linux cluster version 2.6.39 OS is installed on CPUs. C programming language is used for coding the parallel algorithms of the methods on both
+GPU cluster and CPU cluster. CUDA version 4.0~\cite{ref12} is used for programming GPUs, using CUBLAS library~\cite{ref8} to deal with vector
+operations in GPUs and, finally, MPI functions of OpenMPI 1.3.3 are used to carry out the synchronous and asynchronous communications between
+CPU cores. Indeed, in our experiments, a computing node is managed by a MPI process and it is composed of one CPU core and one GPU card.
+ 
+All experimental results of the parallel projected Richardson algorithms are obtained from simulations made in double precision data. The obstacle
+problems to be solved are defined in constant three-dimensional domain $\Omega\subset\mathbb{R}^{3}$. The numerical values of the parameters of the
+obstacle problems are: $\eta=0.2$, $c=1.1$, $f$ is computed by formula~(\ref{eq:18}) and final time $T=0.02$. Moreover, three time steps ($NbSteps=3$)
+are computed with $k=0.0066$. As the discretization matrix is constant along the time steps, the convergence properties of the iterative algorithms
+do not change. Thus, the performance characteristics obtained with three time steps will still be valid for more time steps. The initial function
+$u(0,x,y,z)$ of the obstacle problem~(\ref{eq:01}) is set to $0$, with a constraint $u\geq\phi=0$. The relaxation parameter $\gamma$ used by the
+projected Richardson method is computed automatically thanks to the diagonal entries of the discretization matrix. The formula and its proof can be
+found in~\cite{ref11}, Section~2.3. The convergence tolerance threshold $\varepsilon$ is set to $1e$-$04$ and the maximum number of relaxations is
+limited to $10^{6}$ relaxations. Finally, the number of threads per block is set to $256$ threads, which gives, in general, good performances for
+most GPU applications. We have performed some tests for the execution configurations and we have noticed that the best configuration of the $256$
+threads per block is an organization into two dimensions of sizes $(64,4)$. 
+
+\begin{table}[!h]
+\centering
+\begin{tabular}{|c|c|c|c|c|c|}
+\hline
+\multirow{2}{*}{\bf Pb. size} & \multicolumn{2}{c|}{\bf Synchronous} & \multicolumn{2}{c|}{\bf Asynchronous} & \multirow{2}{*}{\bf Gain\%} \\ \cline{2-5}
+
+                              & $\mathbf{T_{cpu}}$ & {\bf \#relax.}  & $\mathbf{T_{cpu}}$ & {\bf \#relax.}  &  \\ \hline \hline
+
+$256^{3}$                     & $575.22$           & $198,288$        & $539.25$          & $198,613$        & $6.25$ \\ \hline \hline
+
+$512^{3}$                     & $19,250.25$        & $750,912$        & $18,237.14$       & $769,611$        & $5.26$ \\ \hline \hline 
+
+$768^{3}$                     & $206,159.44$       & $1,635,264$      & $183,582.60$      & $1,577,004$      & $10.95$ \\ \hline \hline
+
+$800^{3}$                     & $222,108.09$       & $1,769,232$      & $188,790.04$      & $1,701,735$      & $15.00$ \\ \hline
+\end{tabular}
+\vspace{0.5cm}
+\caption{Execution times in seconds of the parallel projected Richardson method implemented on a cluster of 24 CPU cores.}
+\label{tab:01}
+\end{table}
+
+\begin{table}[!h]
+\centering
+\begin{tabular}{|c|c|c|c|c|c|c|c|}
+\hline
+\multirow{2}{*}{\bf Pb. size} & \multicolumn{3}{c|}{\bf Synchronous}                   & \multicolumn{3}{c|}{\bf Asynchronous}                   & \multirow{2}{*}{\bf Gain\%}  \\ \cline{2-7}
+
+                              & $\mathbf{T_{gpu}}$ & {\bf \#relax.}  & $\mathbf{\tau}$ & $\mathbf{T_{gpu}}$ & {\bf \#relax.}  & $\mathbf{\tau}$ &           \\  \hline \hline
+
+$256^{3}$                     & $29.67$            & $100,692$        &  $19.39$        & $18.00$           & $94,215$         & $29.96$         & $39.33$  \\ \hline \hline
+
+$512^{3}$                     & $521.83$           & $381,300$        & $36.89$         & $425.15$          & $347,279$        & $42.89$         & $18.53$ \\ \hline \hline 
+
+$768^{3}$                     & $4,112.68$         & $831,144$        & $50.13$         & $3,313.87$        & $750,232$        & $55.40$         & $19.42$ \\ \hline \hline 
+
+$800^{3}$                     & $3,950.87$         & $899,088$        & $56.22$         & $3,636.57$        & $834,900$        & $51.91$         & $7.95$ \\ \hline 
+\end{tabular}
+\vspace{0.5cm}
+\caption{Execution times in seconds of the parallel projected Richardson method implemented on a cluster of 12 GPUs.}
+\label{tab:02}
+\end{table}
+
+The performance measures that we took into account are the execution times and the number of relaxations performed by the parallel iterative algorithms,
+both synchronous and asynchronous versions, on the GPU and CPU clusters. These algorithms are used for solving nonlinear systems derived from the discretization
+of obstacle problems of sizes $256^{3}$, $512^{3}$, $768^{3}$ and $800^{3}$. In Table~\ref{tab:01} and Table~\ref{tab:02}, we show the performances
+of the parallel synchronous and asynchronous algorithms of the projected Richardson method implemented, respectively, on a cluster of $24$ CPU cores
+and on a cluster of $12$ GPUs. In these tables, the execution time defines the time spent by the slowest computing node and the number of relaxations
+is computed as the summation of those carried out by all computing nodes.
+
+In the sixth column of Table~\ref{tab:01} and in the eighth column of Table~\ref{tab:02}, we give the gains in $\%$ obtained by using an
+asynchronous algorithm compared to a synchronous one. We can notice that the asynchronous version on CPU and GPU clusters is slightly faster
+than the synchronous one for both methods. Indeed, the cluster of tests is composed of local and homogeneous nodes communicating via low-latency
+connections. So, in the case of distant and/or heterogeneous nodes (or even with geographically distant clusters) the asynchronous version
+would be faster than the synchronous one. However, the gains obtained on the GPU cluster are better than those obtained on the CPU cluster.
+In fact, the computation times are reduced by accelerating the computations on GPUs while the communication times still unchanged.  
+
+The fourth and seventh columns of Table~\ref{tab:02} show the relative gains obtained by executing the parallel algorithms on the cluster
+of $12$ GPUs instead on the cluster of $24$ CPU cores. We compute the relative gain $\tau$ as a ratio between the execution time $T_{cpu}$
+spent on the CPU cluster over that $T_{gpu}$ spent on the GPU cluster: \[\tau=\frac{T_{cpu}}{T_{gpu}}.\] We can see from these ratios that
+solving large obstacle problems is faster on the GPU cluster than on the CPU cluster. Indeed, the GPUs are more efficient than their
+counterpart CPUs to execute large data-parallel operations. In addition, the projected Richardson method is implemented as a fixed point-based
+iteration and uses the Jacobi vector updates that allow a well thread-parallelization on GPUs, such that each GPU thread is in charge
+of one vector component at a time without being dependent on other vector components computed by other threads. Then, this allow to exploit
+at best the high performance computing of the GPUs by using all the GPU resources and avoiding the idle cores.
+
+Finally, the number of relaxations performed by the parallel synchronous algorithm is different in the CPU and GPU versions, because the number
+of computing nodes involved in the GPU cluster and in the CPU cluster is different. In the CPU case, $24$ computing nodes ($24$ CPU cores) are
+considered, whereas in the GPU case, $12$ computing nodes ($12$ GPUs) are considered. As the number of relaxations depends on the domain decomposition,
+consequently it also depends on the number of computing nodes.
+
+
+%%--------------------------%%
+%%       SECTION 6          %%
+%%--------------------------%%
+\section{Red-Black ordering technique}
+\label{sec:06}
+As is well-known, the Jacobi method is characterized by a slow convergence rate compared to some iterative methods (for example Gauss-Seidel method).
+So, in this section, we present some solutions to reduce the execution time and the number of relaxations and, more specifically, to speed up the
+convergence of the parallel projected Richardson method on the GPU cluster. We propose to use the point red-black ordering technique to accelerate
+the convergence. This technique is often used to increase the parallelism of iterative methods for solving linear systems~\cite{ref13,ref14,ref15}.
+We apply it to the projected Richardson method as a compromise between the Jacobi and Gauss-Seidel iterative methods. 
+
+The general principle of the red-black technique is as follows. Let $t$ be the summation of the integer $x$-, $y$- and $z$-coordinates of a vector
+element $u(x,y,z)$ on a three-dimensional domain: $t=x+y+z$. As is shown in Figure~\ref{fig:06.01}, the red-black ordering technique consists in the
+parallel computing of the red vector elements having even value $t$ by using the values of the black ones then, the parallel computing of the black
+vector elements having odd values $t$ by using the new values of the red ones.
+
+\begin{figure}
+\centering
+  \mbox{\subfigure[Red-black ordering on x, y and z axises]{\includegraphics[width=2.3in]{Chapters/chapter13/figures/rouge-noir}\label{fig:06.01}}\quad
+        \subfigure[Red-black ordering on y axis]{\includegraphics[width=2.3in]{Chapters/chapter13/figures/rouge-noir-y}\label{fig:06.02}}}
+\caption{Red-black ordering for computing the iterate vector elements in a three-dimensional space.}
+\end{figure}
+
+This technique can be implemented on the GPU in two different manners:
+\begin{itemize*}
+\item among all launched threads ($NX\times ny$ threads), only one thread out of two computes its red or black vector element at a time or,
+\item all launched threads (on average half of $NX\times ny$ threads) compute the red vector elements first and, then, the black ones.
+\end{itemize*}
+However, in both solutions, for each memory transaction, only half of the memory segment addressed by a half-warp is used. So, the computation of the
+red and black vector elements leads to use twice the initial number of memory transactions. Then, we apply the point red-black ordering accordingly to
+the $y$-coordinate, as is shown in Figure~\ref{fig:06.02}. In this case, the vector elements having even $y$-coordinate are computed in parallel using
+the values of those having odd $y$-coordinate and then vice-versa. Moreover, in the GPU implementation of the parallel projected Richardson method (Section~\ref{sec:04}),
+we have shown that a sub-problem of size $(NX\times ny\times nz)$ is decomposed into $nz$ grids of size $(NX\times ny)$. Then, each kernel is executed
+in parallel by $NX\times ny$ GPU threads, so that each thread is in charge of $nz$ vector elements along $z$-axis (one vector element in each grid of
+the sub-problem). So, we propose to use the new values of the vector elements computed in grid $i$ to compute those of the vector elements in grid $i+1$.
+Listing~\ref{list:04} describes the kernel of the matrix-vector multiplication and the kernel of the vector elements updates of the parallel projected
+Richardson method using the red-black ordering technique.
+
+\lstinputlisting[label=list:04,caption=GPU kernels of the projected Richardson method using the red-black technique]{Chapters/chapter13/ex4.cu}
+
+Finally, we exploit the concurrent executions between the host functions and the GPU kernels provided by the GPU hardware and software. In fact, the kernel
+launches are asynchronous (when this environment variable is not disabled on the GPUs), such that the control is returned to the host (MPI process) before
+the GPU has completed the requested task (kernel)~\cite{ref12}. Therefore, all the kernels necessary to update the local vector elements, $u(x,y,z)$ where
+$0<y<(ny-1)$ and $0<z<(nz-1)$, are executed first. Then, the values associated to the bordering vector elements are exchanged between the neighbors. Finally,
+the values of the vector elements associated to the bordering vector elements are updated. In this case, the computation of the local vector elements is
+performed concurrently with the data exchanges between neighboring CPUs and this in both synchronous and asynchronous cases.
+
+\begin{table}[!h]
+\centering
+\begin{tabular}{|c|c|c|c|c|c|}
+\hline
+\multirow{2}{*}{\bf Pb. size} & \multicolumn{2}{c|}{\bf Synchronous} & \multicolumn{2}{c|}{\bf Asynchronous} & \multirow{2}{*}{\bf Gain\%}  \\ \cline{2-5}
+
+                              & $\mathbf{T_{gpu}}$ & {\bf \#relax.}   & $\mathbf{T_{gpu}}$ & {\bf \#relax.}   &           \\  \hline \hline
+
+$256^{3}$                     & $18.37$            & $71,988$         & $12.58$           & $67,638$         & $31.52$  \\ \hline \hline
+
+$512^{3}$                     & $349.23$           & $271,188$        & $289.41$          & $246,036$        & $17.13$ \\ \hline \hline 
+
+$768^{3}$                     & $2,773.65$         & $590,652$        & $2,222.22$        & $532,806$        & $19.88$ \\ \hline \hline 
+
+$800^{3}$                     & $2,748.23$         & $638,916$        & $2,502.61$        & $592,525$        & $8.92$ \\ \hline 
+\end{tabular}
+\vspace{0.5cm}
+\caption{Execution times in seconds of the parallel projected Richardson method using read-black ordering technique implemented on a cluster of 12 GPUs.}
+\label{tab:03}
+\end{table}
+
+In Table~\ref{tab:03}, we report the execution times and the number of relaxations performed on a cluster of $12$ GPUs by the parallel projected Richardson
+algorithms; it can be noted that the performances of the projected Richardson are improved by using the point read-black ordering. We compare the performances
+of the parallel projected Richardson method with and without this later ordering (Tables~\ref{tab:02} and~\ref{tab:03}). We can notice that both parallel synchronous
+and asynchronous algorithms are faster when they use the red-black ordering. Indeed, we can see in Table~\ref{tab:03} that the execution times of these algorithms
+are reduced, on average, by $32\%$ compared to those shown in Table~\ref{tab:02}.
+
+\begin{figure}
+\centerline{\includegraphics[scale=0.9]{Chapters/chapter13/figures/scale}}
+\caption{Weak scaling of both synchronous and asynchronous algorithms of the projected Richardson method using red-black ordering technique.}
+\label{fig:07}
+\end{figure}
+
+In Figure~\ref{fig:07}, we study the ratio between the computation time and the communication time of the parallel projected Richardson algorithms on a GPU cluster.
+The experimental tests are carried out on a cluster composed of one to ten Tesla GPUs. We have focused on the weak scaling of both parallel, synchronous and asynchronous,
+algorithms using the red-black ordering technique. For this, we have fixed the size of a sub-problem to $256^{3}$ per computing node (a CPU core and a GPU). Then,
+Figure~\ref{fig:07} shows the number of relaxations performed, on average, per second by a computing node. We can see from this figure that the efficiency of the
+asynchronous algorithm is almost stable, while that of the synchronous algorithm decreases (down to $81\%$ in this example) with the increasing of the number of
+computing nodes on the cluster. This is due to the fact that the ratio between the time of the computation over that of the communication is reduced when the computations
+are performed on GPUs. Indeed, GPUs compute faster than CPUs and communications are more time consuming. In this context, asynchronous algorithms are more scalable
+than synchronous ones. So, with large scale GPU clusters, synchronous algorithms might be more penalized by communications, as can be deduced from Figure~\ref{fig:07}.
+That is why we think that asynchronous iterative algorithms are all the more interesting in this case.
+
+
+%%--------------------------%%
+%%       SECTION 7          %%
+%%--------------------------%%
+\section{Conclusion}
+\label{sec:07}
+Our main contribution, in this chapter, is the parallel implementation of an asynchronous iterative method on GPU clusters for solving large scale nonlinear
+systems derived from the spatial discretization of three-dimensional obstacle problems. For this, we have implemented both synchronous and asynchronous algorithms of the
+Richardson iterative method using a projection on a convex set. Indeed, this method uses point-based iterations of the Jacobi method that are very easy to parallelize on
+parallel computers. We have shown that its adapted parallel algorithms to GPU architectures allows to exploit at best the computing power of the GPUs and to accelerate the
+resolution of large nonlinear systems. Consequently, the experimental results have shown that solving nonlinear systems of large obstacle problems with this method is about
+fifty times faster on a cluster of $12$ GPUs than on a cluster of $24$ CPU cores. Moreover, we have applied to this projected Richardson method the red-black ordering technique
+which allows it to improve its convergence rate. Thus, the execution times of both parallel algorithms performed on the cluster of $12$ GPUs are reduced on average of $32\%$.
+
+Afterwards, the experiments have shown that the asynchronous version is slightly more efficient than the synchronous one. In fact, the computations are accelerated by using GPUs
+while the communication times still unchanged. In addition, we have studied the weak-scaling in the synchronous and asynchronous cases, which has confirmed that the ratio between
+the computations and the communications are reduced when using a cluster of GPUs. We highlight that asynchronous iterative algorithms are more scalable than synchronous ones.
+Therefore, we can conclude that asynchronous iterations are well suited to tackle scalability issues on GPU clusters.
+
+In future works, we plan to perform experiments on large scale GPU clusters and on geographically distant GPU clusters, because we expect that asynchronous versions would
+be faster and more scalable on such architectures. Furthermore, we want to study the performance behavior and the scalability of other numerical algorithms which support,
+if possible, the model of asynchronous iterations.
+
+\putbib[Chapters/chapter13/biblio13]
+
diff --git a/BookGPU/Chapters/chapter13/ex1.cu b/BookGPU/Chapters/chapter13/ex1.cu
new file mode 100644
index 0000000..546d8ef
--- /dev/null
+++ b/BookGPU/Chapters/chapter13/ex1.cu
@@ -0,0 +1,30 @@
+/* GPU kernel */
+__global__ void kernel(..., int n, int nx, int ny, int slices, int stride, ...)
+{ 
+	int tx = blockIdx.x * blockDim.x + threadIdx.x;//x-coordinate
+	int ty = blockIdx.y * blockDim.y + threadIdx.y;//y-coordinate
+	int tid = tx + ty * nx;                        //thread ID
+	for(int i=0; i<slices; i++){
+		if((tx<nx) && (ty<ny) && (tid<n)){
+			... 
+		}
+		tid += stride;
+	}
+}
+
+/* CPU function */
+void Function(...)
+{
+	int n = NX * ny * nz; //size of the sub-problem
+	int slices = nz;
+	int stride = NX * ny;
+	int bx = 64, by = 4;
+	int gx = (NX + bx - 1) / bx;
+	int gy = (ny + by - 1) / by;
+	dim3 block(bx,by);   //dimensions of a thread block
+	dim3 grid(gx,gy);    //dimensions of the grid
+	...
+	kernel<<<grid,block>>>(..., n, NX, ny, slices, stride, ...);
+	...
+}
+
diff --git a/BookGPU/Chapters/chapter13/ex2.cu b/BookGPU/Chapters/chapter13/ex2.cu
new file mode 100644
index 0000000..f5d30ce
--- /dev/null
+++ b/BookGPU/Chapters/chapter13/ex2.cu
@@ -0,0 +1,48 @@
+/* Kernel of the matrix-vector multiplication */
+__global__ void MV_Multiplication (..., double* U, double* Y)
+{
+	...
+	//Matrix coefficients filled in registers
+
+	for(int tz=0; tz<slices; tz++){
+		if((tx<NX) && (ty<ny) && (tid<n)){
+			double sum = Center * fetch_double(U, tid);
+			if(tx != 0)      sum += West  * fetch_double(U,tid-1);
+			if(tx != (NX-1)) sum += East  * fetch_double(U,tid+1);
+			if(ty != 0)      sum += South * fetch_double(U,tid-NX);
+			if(ty != (ny-1)) sum += North * fetch_double(U,tid+NX);
+			if(tz != 0)      sum += Rear  * fetch_double(U,tid-NX*ny);
+			if(tz != (nz-1)) sum += Front * fetch_double(U,tid+NX*ny);
+			Y[tid] = sum;
+		}
+		tid += stride;
+	}
+}
+
+/* Kernel of the vector elements updates */
+__global__ void Vector_Updates(..., double* G,double* Y,double* U)
+{
+	...
+	//Matrix coefficient filled in a register: Center
+
+	for(int tz=0; tz<slices; tz++){
+		if((tx<NX) && (ty<ny) && (tid<n)){
+			double var = (G[tid]-Y[tid]) / Center + fetch_double(U,tid);
+			if(var < 0.0) var = 0.0; //projection
+			U[tid] = var;
+		}
+		tid += stride;
+	}
+}
+
+/* Function to be executed by the CPU */
+void Computation_New_Vector_Elements(double*A,double* G,double* U)
+{
+	double *Y;
+	//Allocate a GPU memory space for the vector Y
+	//Configure the kernel execution: grid and block
+	//Elements of vector U filled in the texture memory
+
+	MV_Multiplication<<<grid,block>>>(..., U, Y);
+	Vector_Updates<<<grid,block>>>(..., G, Y, U);
+}
diff --git a/BookGPU/Chapters/chapter13/ex3.cu b/BookGPU/Chapters/chapter13/ex3.cu
new file mode 100644
index 0000000..6090907
--- /dev/null
+++ b/BookGPU/Chapters/chapter13/ex3.cu
@@ -0,0 +1,8 @@
+texture<int2,1> v;
+
+static __inline__
+__device__ double fetch_double(texture<int2,1> v, int i)
+{
+	int2 w = tex1Dfetch(v, i);
+	return __hiloint2double(w.y, w.x);
+}
diff --git a/BookGPU/Chapters/chapter13/ex4.cu b/BookGPU/Chapters/chapter13/ex4.cu
new file mode 100644
index 0000000..738ca3d
--- /dev/null
+++ b/BookGPU/Chapters/chapter13/ex4.cu
@@ -0,0 +1,54 @@
+/* Kernel of the matrix-vector multiplication */
+__global__ void MV_Multiplication(...,double*G,double*U,double*Y)
+{
+	...
+	//Matrix coefficients filled in registers:
+	//Center, West, East, Front
+
+	for(int tz=0; tz<slices; tz++){
+		if((tx<NX) && (ty<ny) && (tid<n)){
+			double sum = G[tid] - Center * fetch_double(U,tid);
+			if(tx != 0)      sum -= West * fetch_double(U,tid-1);
+			if(tx != (NX-1)) sum -= East * fetch_double(U,tid+1);
+			if(tz != (nz-1)) sum -= Front * fetch_double(U,tid+NX*ny);
+			Y[tid] = sum;
+		}
+		tid += stride;
+	}
+}
+
+/* Kernel of the vector elements updates */
+__global__ void Vector_Updates(..., int rb, double* Y, double* U)
+{
+	...
+	double val; //vector component computed in previous grid
+	//Matrix coefficients filled in registers: 
+	//Center, South, North, Rear
+
+	for(int tz=0; tz<slices; tz++){
+		if((tx<NX) && (ty<ny) && (tid<n) && ((ty&1)==rb)){
+			double sum = Y[tid] - South * fetch_double(U,tid-NX) - 
+				           North * fetch_double(U,tid+NX); 
+			//val: computed in previous grid 
+			if(tz != 0) sum -= Rear * val; 
+			sum = (sum / Center) + fetch_double(U,tid);
+			if(sum < 0) sum = 0; //projection
+			U[tid] = val = sum; //update of U
+		}
+		tid += stride;
+	}
+}
+
+/* CPU Function */
+void Computation_New_Vector_Components(double*A,double*G,double*U)
+{
+	double *Y;
+	int red=0, black=1;
+	//Allocate a GPU memory space for the vector Y
+	//Configure the kernel execution: grid and block
+	//Elements of vector U filled in the texture memory
+
+	MV_Multiplication<<<grid,block>>>(..., G, U, Y);
+	Vector_Updates<<<grid,block>>>(..., red, Y, U);
+	Vector_Updates<<<grid,block>>>(..., black, Y, U);
+}
diff --git a/BookGPU/Chapters/chapter13/figures/bib.bib b/BookGPU/Chapters/chapter13/figures/bib.bib
new file mode 100644
index 0000000..0262701
--- /dev/null
+++ b/BookGPU/Chapters/chapter13/figures/bib.bib
@@ -0,0 +1,358 @@
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+}
+
+@article{MS85a,
+ author = {Miellou, J. C. and Spiteri, P.},
+ title = {Un {C}rit\`ere de {C}onvergence pour des  {M}\'ethodes {G}\'en\'erales de {P}oint {F}ixe},
+ journal = {M2AN},
+ volume = {19},
+ number = {},
+ pages = {645--669},
+ year = {1985},
+}
+
+@article{MS85b,
+ author = {Miellou, J. C. and Spiteri, P.},
+ title = {Two {C}riteria for the {C}onvergence of {A}synchronous {I}terations},
+ journal = {in Computers and computing, P. Chenin et al. ed., Wiley Masson},
+ volume = {},
+ number = {},
+ pages = {91--95},
+ year = {1985},
+}
+
+@article{Miellou86,
+ author = {Miellou, J. C.},
+ title = {Asynchronous {I}terations and {O}rder {I}ntervals, {P}arallel {A}lgorithms},
+ journal = {(M. Cosnard et al. eds) Amsterdam: North-Holland},
+ volume = {},
+ number = {},
+ pages = {85--96},
+ year = {1986},
+}
+
+@article{GS91b,
+ author = {Giraud, L. and Spiteri, P.},
+ title = {R\'esolution {P}arall\`ele de {P}robl\`emes aux {L}imites non {L}in\'eaires},
+ journal = {M2 AN},
+ volume = {25},
+ number = {},
+ pages = {579--606},
+ year = {1991},
+}
+
+@article{MSE98,
+ author = {Miellou, J. C. and El Baz, D. and Spiteri, P.},
+ title = {A {N}ew {C}lass of {A}synchronous {I}terative {A}lgorithms with {O}rder {I}nterval},
+ journal = {Mathematics of Computation},
+ volume = {67},
+ number = {221},
+ pages = {237--255},
+ year = {1998},
+}
+
+@article{CM69,
+ author = {Chazan, D. and Miranker, W.},
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+ volume = {2},
+ number = {},
+ pages = {199--222},
+ year = {1969},
+}
+
+@article{M75a,
+ author = {Miellou, J. C.},
+ title = {Algorithmes de {R}elaxation {C}haotique \`a {R}etards},
+ journal = {RAIRO Analyse num\'erique},
+ volume = {R1},
+ number = {},
+ pages = {55--82},
+ year = {1975},
+}
+
+@article{Baudet78,
+ author = {Baudet, G.},
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+ journal = {Journal Assoc. Comput. Mach.},
+ volume = {25},
+ number = {},
+ pages = {226--244},
+ year = {1978},
+}
+
+@article{BT89,
+ author = {Bertsekas, D. and Tsitsiklis, J.},
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+ pages = {},
+ year = {1989},
+}
+
+@article{Bahi07,
+ author = {Bahi, J. M. and Contassot-Vivier, S. and Couturier, R.},
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+ journal = {Chapman \& Hall/CRC, Numerical Analysis \& Scientific Computating 1 (2007)},
+ volume = {},
+ number = {},
+ pages = {},
+ year = {2007},
+}
+
+@article{Nvi10,
+ author = {Nvidia},
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+ pages = {},
+ year = {2011},
+}
+
+@article{Nvid10,
+ author = {Nvidia},
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+ pages = {},
+ year = {2010},
+}
+
+@article{ref10,
+ author = {Micikevicius, P},
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+ pages = {79--84},
+ year = {2009},
+}
+
+@article{ref11,
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+ year = {2009},
+}
+
+@article{ref12,
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+}
+
+@article{ref13,
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+
+@article{ref15,
+ author = {Micikevicius, P.},
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+}
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+ number = {},
+ pages = {},
+  year = {1978},
+}
+
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+%%CreationDate: Mon Jan 14 14:35:51 2013
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+%%DocumentData: Clean7Bit
+%%LanguageLevel: 2
+%%EndComments
+%%BeginProlog
+%%EndProlog
+%%Page: 1 1
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+} if
+save
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+     /Width 113
+     /Height 113
+     /ImageMatrix [ 113 0 0 -113 0 113 ]
+     /DataSource Data
+     /BitsPerComponent 8
+     /Decode [0 1 0 1 0 1]
+  >> image
+  Data closefile
+  RawData flushfile
+  showpage
+  restore
+} exec
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+%
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+%
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+0 0 0 setrgbcolor
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+
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+%!PS-Adobe-3.0 EPSF-3.0
+%%Creator: jpeg2ps 3.2 by Thomas Merz
+%%Title: /home/ziane/Desktop/Revision_PC_07_01_2013/rouge.jpg
+%%CreationDate: Mon Jan 14 14:35:51 2013
+%%BoundingBox: 0 0 113 113
+%%DocumentData: Clean7Bit
+%%LanguageLevel: 2
+%%EndComments
+%%BeginProlog
+%%EndProlog
+%%Page: 1 1
+/languagelevel where {pop languagelevel 2 lt}{true} ifelse {
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+} if
+save
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+/Data RawData << >> /DCTDecode filter def
+/DeviceRGB setcolorspace
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+     /Width 113
+     /Height 113
+     /ImageMatrix [ 113 0 0 -113 0 113 ]
+     /DataSource Data
+     /BitsPerComponent 8
+     /Decode [0 1 0 1 0 1]
+  >> image
+  Data closefile
+  RawData flushfile
+  showpage
+  restore
+} exec
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+%
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+%
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+0 0 0 setrgbcolor
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+%!PS-Adobe-3.0 EPSF-3.0
+%%Creator: jpeg2ps 3.2 by Thomas Merz
+%%Title: /home/ziane/Desktop/Revision_PC_07_01_2013/rouge.jpg
+%%CreationDate: Mon Jan 14 14:35:51 2013
+%%BoundingBox: 0 0 113 113
+%%DocumentData: Clean7Bit
+%%LanguageLevel: 2
+%%EndComments
+%%BeginProlog
+%%EndProlog
+%%Page: 1 1
+/languagelevel where {pop languagelevel 2 lt}{true} ifelse {
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+} if
+save
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+     /Width 113
+     /Height 113
+     /ImageMatrix [ 113 0 0 -113 0 113 ]
+     /DataSource Data
+     /BitsPerComponent 8
+     /Decode [0 1 0 1 0 1]
+  >> image
+  Data closefile
+  RawData flushfile
+  showpage
+  restore
+} exec
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+%%LanguageLevel: 2
+%%EndComments
+%%BeginProlog
+%%EndProlog
+%%Page: 1 1
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+     /DataSource Data
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+} exec
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+0 0 0 setrgbcolor
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+%%CreationDate: Mon Jan 14 14:35:51 2013
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+%%DocumentData: Clean7Bit
+%%LanguageLevel: 2
+%%EndComments
+%%BeginProlog
+%%EndProlog
+%%Page: 1 1
+/languagelevel where {pop languagelevel 2 lt}{true} ifelse {
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+} if
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+     /Height 113
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+     /DataSource Data
+     /BitsPerComponent 8
+     /Decode [0 1 0 1 0 1]
+  >> image
+  Data closefile
+  RawData flushfile
+  showpage
+  restore
+} exec
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+%%EndComments
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+%%EndProlog
+%%Page: 1 1
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+%%LanguageLevel: 2
+%%EndComments
+%%BeginProlog
+%%EndProlog
+%%Page: 1 1
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+} exec
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+0 0 0 setrgbcolor
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+%%EndComments
+%%BeginProlog
+%%EndProlog
+%%Page: 1 1
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+     /DataSource Data
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+  >> image
+  Data closefile
+  RawData flushfile
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+} exec
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+%%LanguageLevel: 2
+%%EndComments
+%%BeginProlog
+%%EndProlog
+%%Page: 1 1
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+} exec
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+%%EndComments
+%%BeginProlog
+%%EndProlog
+%%Page: 1 1
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+} if
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+     /DataSource Data
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+  >> image
+  Data closefile
+  RawData flushfile
+  showpage
+  restore
+} exec
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+%%EndComments
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+%%Page: 1 1
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+%%LanguageLevel: 2
+%%EndComments
+%%BeginProlog
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+%%Page: 1 1
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+} exec
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+%%EndComments
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+%%CreationDate: Mon Jan 14 14:35:51 2013
+%%BoundingBox: 0 0 113 113
+%%DocumentData: Clean7Bit
+%%LanguageLevel: 2
+%%EndComments
+%%BeginProlog
+%%EndProlog
+%%Page: 1 1
+/languagelevel where {pop languagelevel 2 lt}{true} ifelse {
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+} if
+save
+/RawData currentfile /ASCII85Decode filter def
+/Data RawData << >> /DCTDecode filter def
+/DeviceRGB setcolorspace
+{ << /ImageType 1
+     /Width 113
+     /Height 113
+     /ImageMatrix [ 113 0 0 -113 0 113 ]
+     /DataSource Data
+     /BitsPerComponent 8
+     /Decode [0 1 0 1 0 1]
+  >> image
+  Data closefile
+  RawData flushfile
+  showpage
+  restore
+} exec
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+0 0 0 setrgbcolor
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+%!PS-Adobe-3.0 EPSF-3.0
+%%Creator: jpeg2ps 3.2 by Thomas Merz
+%%Title: /home/ziane/Desktop/Revision_PC_07_01_2013/rouge.jpg
+%%CreationDate: Mon Jan 14 14:35:51 2013
+%%BoundingBox: 0 0 113 113
+%%DocumentData: Clean7Bit
+%%LanguageLevel: 2
+%%EndComments
+%%BeginProlog
+%%EndProlog
+%%Page: 1 1
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+     /Width 113
+     /Height 113
+     /ImageMatrix [ 113 0 0 -113 0 113 ]
+     /DataSource Data
+     /BitsPerComponent 8
+     /Decode [0 1 0 1 0 1]
+  >> image
+  Data closefile
+  RawData flushfile
+  showpage
+  restore
+} exec
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+%%EndComments
+%%BeginProlog
+%%EndProlog
+%%Page: 1 1
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+  >> image
+  Data closefile
+  RawData flushfile
+  showpage
+  restore
+} exec
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+0 0 0 setrgbcolor
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+%%CreationDate: Mon Jan 14 14:35:51 2013
+%%BoundingBox: 0 0 96 96
+%%DocumentData: Clean7Bit
+%%LanguageLevel: 2
+%%EndComments
+%%BeginProlog
+%%EndProlog
+%%Page: 1 1
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+  >> image
+  Data closefile
+  RawData flushfile
+  showpage
+  restore
+} exec
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+%%EndComments
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+%%EndProlog
+%%Page: 1 1
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+%
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+%
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+0 0 0 setrgbcolor
+% Begin Imported JPEG File: /home/ziane/Desktop/Revision_PC_07_01_2013/rouge.jpg
+
+n gs
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+113 -113 tr
+90 rot
+0 0 tr
+sa
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+113 113 sc
+%!PS-Adobe-3.0 EPSF-3.0
+%%Creator: jpeg2ps 3.2 by Thomas Merz
+%%Title: /home/ziane/Desktop/Revision_PC_07_01_2013/rouge.jpg
+%%CreationDate: Mon Jan 14 14:35:51 2013
+%%BoundingBox: 0 0 113 113
+%%DocumentData: Clean7Bit
+%%LanguageLevel: 2
+%%EndComments
+%%BeginProlog
+%%EndProlog
+%%Page: 1 1
+/languagelevel where {pop languagelevel 2 lt}{true} ifelse {
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+} if
+save
+/RawData currentfile /ASCII85Decode filter def
+/Data RawData << >> /DCTDecode filter def
+/DeviceRGB setcolorspace
+{ << /ImageType 1
+     /Width 113
+     /Height 113
+     /ImageMatrix [ 113 0 0 -113 0 113 ]
+     /DataSource Data
+     /BitsPerComponent 8
+     /Decode [0 1 0 1 0 1]
+  >> image
+  Data closefile
+  RawData flushfile
+  showpage
+  restore
+} exec
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+} exec
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+%%CreationDate: Mon Jan 14 14:35:51 2013
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+%%LanguageLevel: 2
+%%EndComments
+%%BeginProlog
+%%EndProlog
+%%Page: 1 1
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+     /ImageMatrix [ 113 0 0 -113 0 113 ]
+     /DataSource Data
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+} exec
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+%%Title: /home/ziane/Desktop/Revision_PC_07_01_2013/rouge.jpg
+%%CreationDate: Mon Jan 14 14:37:45 2013
+%%BoundingBox: 0 0 113 113
+%%DocumentData: Clean7Bit
+%%LanguageLevel: 2
+%%EndComments
+%%BeginProlog
+%%EndProlog
+%%Page: 1 1
+/languagelevel where {pop languagelevel 2 lt}{true} ifelse {
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+} if
+save
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+/Data RawData << >> /DCTDecode filter def
+/DeviceRGB setcolorspace
+{ << /ImageType 1
+     /Width 113
+     /Height 113
+     /ImageMatrix [ 113 0 0 -113 0 113 ]
+     /DataSource Data
+     /BitsPerComponent 8
+     /Decode [0 1 0 1 0 1]
+  >> image
+  Data closefile
+  RawData flushfile
+  showpage
+  restore
+} exec
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+0 0 0 setrgbcolor
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+%!PS-Adobe-3.0 EPSF-3.0
+%%Creator: jpeg2ps 3.2 by Thomas Merz
+%%Title: /home/ziane/Desktop/Revision_PC_07_01_2013/rouge.jpg
+%%CreationDate: Mon Jan 14 14:37:45 2013
+%%BoundingBox: 0 0 113 113
+%%DocumentData: Clean7Bit
+%%LanguageLevel: 2
+%%EndComments
+%%BeginProlog
+%%EndProlog
+%%Page: 1 1
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+     /Width 113
+     /Height 113
+     /ImageMatrix [ 113 0 0 -113 0 113 ]
+     /DataSource Data
+     /BitsPerComponent 8
+     /Decode [0 1 0 1 0 1]
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+  Data closefile
+  RawData flushfile
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+} exec
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+%%EndComments
+%%BeginProlog
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+%%Page: 1 1
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+0 0 0 setrgbcolor
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+%%DocumentData: Clean7Bit
+%%LanguageLevel: 2
+%%EndComments
+%%BeginProlog
+%%EndProlog
+%%Page: 1 1
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+} if
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+     /DataSource Data
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+  >> image
+  Data closefile
+  RawData flushfile
+  showpage
+  restore
+} exec
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+%%EndComments
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+%%EndProlog
+%%Page: 1 1
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+%%LanguageLevel: 2
+%%EndComments
+%%BeginProlog
+%%EndProlog
+%%Page: 1 1
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+%%EndComments
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+     /DataSource Data
+     /BitsPerComponent 8
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+  RawData flushfile
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+  restore
+} exec
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+%%CreationDate: Mon Jan 14 14:37:45 2013
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+%%DocumentData: Clean7Bit
+%%LanguageLevel: 2
+%%EndComments
+%%BeginProlog
+%%EndProlog
+%%Page: 1 1
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+     /ImageMatrix [ 113 0 0 -113 0 113 ]
+     /DataSource Data
+     /BitsPerComponent 8
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+  Data closefile
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+} exec
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+%%EndComments
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+%%EndProlog
+%%Page: 1 1
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+%%EndComments
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+%%EndProlog
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+  RawData flushfile
+  showpage
+  restore
+} exec
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+%%EndComments
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+%%Page: 1 1
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+} exec
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+%%EndComments
+%%BeginProlog
+%%EndProlog
+%%Page: 1 1
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+     /DataSource Data
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+  >> image
+  Data closefile
+  RawData flushfile
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+  restore
+} exec
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+%%Page: 1 1
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+%%CreationDate: Mon Jan 14 14:37:45 2013
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+%%DocumentData: Clean7Bit
+%%LanguageLevel: 2
+%%EndComments
+%%BeginProlog
+%%EndProlog
+%%Page: 1 1
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+} if
+save
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+     /Width 113
+     /Height 113
+     /ImageMatrix [ 113 0 0 -113 0 113 ]
+     /DataSource Data
+     /BitsPerComponent 8
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+  >> image
+  Data closefile
+  RawData flushfile
+  showpage
+  restore
+} exec
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+%%CreationDate: Mon Jan 14 14:37:45 2013
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+%%DocumentData: Clean7Bit
+%%LanguageLevel: 2
+%%EndComments
+%%BeginProlog
+%%EndProlog
+%%Page: 1 1
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+} if
+save
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+/Data RawData << >> /DCTDecode filter def
+/DeviceRGB setcolorspace
+{ << /ImageType 1
+     /Width 113
+     /Height 113
+     /ImageMatrix [ 113 0 0 -113 0 113 ]
+     /DataSource Data
+     /BitsPerComponent 8
+     /Decode [0 1 0 1 0 1]
+  >> image
+  Data closefile
+  RawData flushfile
+  showpage
+  restore
+} exec
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+%
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+%
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+0 0 0 setrgbcolor
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+
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+%!PS-Adobe-3.0 EPSF-3.0
+%%Creator: jpeg2ps 3.2 by Thomas Merz
+%%Title: /home/ziane/Desktop/Revision_PC_07_01_2013/rouge.jpg
+%%CreationDate: Mon Jan 14 14:37:45 2013
+%%BoundingBox: 0 0 113 113
+%%DocumentData: Clean7Bit
+%%LanguageLevel: 2
+%%EndComments
+%%BeginProlog
+%%EndProlog
+%%Page: 1 1
+/languagelevel where {pop languagelevel 2 lt}{true} ifelse {
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+} if
+save
+/RawData currentfile /ASCII85Decode filter def
+/Data RawData << >> /DCTDecode filter def
+/DeviceRGB setcolorspace
+{ << /ImageType 1
+     /Width 113
+     /Height 113
+     /ImageMatrix [ 113 0 0 -113 0 113 ]
+     /DataSource Data
+     /BitsPerComponent 8
+     /Decode [0 1 0 1 0 1]
+  >> image
+  Data closefile
+  RawData flushfile
+  showpage
+  restore
+} exec
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+%%CreationDate: Mon Jan 14 14:37:45 2013
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+%%LanguageLevel: 2
+%%EndComments
+%%BeginProlog
+%%EndProlog
+%%Page: 1 1
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+     /Width 113
+     /Height 113
+     /ImageMatrix [ 113 0 0 -113 0 113 ]
+     /DataSource Data
+     /BitsPerComponent 8
+     /Decode [0 1 0 1 0 1]
+  >> image
+  Data closefile
+  RawData flushfile
+  showpage
+  restore
+} exec
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+%
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+%
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+0 0 0 setrgbcolor
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+
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+%%Creator: jpeg2ps 3.2 by Thomas Merz
+%%Title: /home/ziane/Desktop/Revision_PC_07_01_2013/rouge.jpg
+%%CreationDate: Mon Jan 14 14:37:45 2013
+%%BoundingBox: 0 0 113 113
+%%DocumentData: Clean7Bit
+%%LanguageLevel: 2
+%%EndComments
+%%BeginProlog
+%%EndProlog
+%%Page: 1 1
+/languagelevel where {pop languagelevel 2 lt}{true} ifelse {
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+} if
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+     /Height 113
+     /ImageMatrix [ 113 0 0 -113 0 113 ]
+     /DataSource Data
+     /BitsPerComponent 8
+     /Decode [0 1 0 1 0 1]
+  >> image
+  Data closefile
+  RawData flushfile
+  showpage
+  restore
+} exec
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+%%DocumentData: Clean7Bit
+%%LanguageLevel: 2
+%%EndComments
+%%BeginProlog
+%%EndProlog
+%%Page: 1 1
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+     /DataSource Data
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+  Data closefile
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+} exec
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+%%LanguageLevel: 2
+%%EndComments
+%%BeginProlog
+%%EndProlog
+%%Page: 1 1
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+  Data closefile
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+} exec
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diff --git a/BookGPU/Makefile b/BookGPU/Makefile
index a842e94..1e1a613 100644
--- a/BookGPU/Makefile
+++ b/BookGPU/Makefile
@@ -14,9 +14,10 @@ all:
 	bibtex bu8
 	bibtex bu9
 	bibtex bu10
+	bibtex bu11
 #don't put chapter 14, refs are included	
-	bibtex bu12	
-	bibtex bu13
+	bibtex bu13	
+	bibtex bu14
 
 	makeindex  ${BOOK}.idx
 	pdflatex ${BOOK}