ch12

diff --git a/BookGPU/BookGPU.tex b/BookGPU/BookGPU.tex
index 0b7f5cfa4f392604a72701a70b1170064ed37991..76dc8f6af5f85ff78eba3908482f7b24f53316cb 100755 (executable)
--- a/BookGPU/BookGPU.tex
+++ b/BookGPU/BookGPU.tex
@@ -138,18 +138,19 @@
 
 \setcounter{page}{1}
 \part{This is a Part}
-%%\include{Chapters/chapter1/ch1}
-%% \include{Chapters/chapter2/ch2}
-%% \include{Chapters/chapter3/ch3}
-%% \include{Chapters/chapter5/ch5}
-%% \include{Chapters/chapter6/ch6}
-%% \include{Chapters/chapter7/ch7}
-%% \include{Chapters/chapter8/ch8}
-%% \include{Chapters/chapter9/ch9}
-%% \include{Chapters/chapter11/ch11}
-%% \include{Chapters/chapter14/ch14}
-%% \include{Chapters/chapter15/ch15}
-%% \include{Chapters/chapter16/ch16}
+\include{Chapters/chapter1/ch1}
+\include{Chapters/chapter2/ch2}
+\include{Chapters/chapter3/ch3}
+\include{Chapters/chapter5/ch5}
+\include{Chapters/chapter6/ch6}
+\include{Chapters/chapter7/ch7}
+\include{Chapters/chapter8/ch8}
+\include{Chapters/chapter9/ch9}
+\include{Chapters/chapter11/ch11}
+\include{Chapters/chapter12/ch12}
+\include{Chapters/chapter14/ch14}
+\include{Chapters/chapter15/ch15}
+\include{Chapters/chapter16/ch16}
  \include{Chapters/chapter18/ch18}
 
 \bibliographystyle{hep}

diff --git a/BookGPU/Chapters/chapter12/biblio12.bib b/BookGPU/Chapters/chapter12/biblio12.bib
new file mode 100644 (file)
index 0000000..8cc4ee7
--- /dev/null
+++ b/BookGPU/Chapters/chapter12/biblio12.bib
@@ -0,0 +1,163 @@
+@article{ref1,
+title = {Iterative methods for sparse linear systems},
+author = {Saad, Yousef},
+journal = {Society for Industrial and Applied Mathematics,  2nd edition},
+volume = {},
+number = {},
+pages = {},
+year = {2003},
+}
+
+@article{ref2,
+title = {Methods of conjugate gradients for solving linear systems},
+author = {Hestenes, Maqnus R. and Stiefel, Eduard},
+journal = {Journal of Research of the National Bureau of Standards},
+volume = {49},
+number = {6},
+pages = {409--436},
+year = {1952},
+}
+
+@article{ref3,
+title = {{GMRES}: a generalized minimal residual algorithm for solving nonsymmetric linear systems},
+author = {Saad, Yousef and Schultz, Martin H.},
+journal = {SIAM Journal on Scientific and Statistical Computing},
+volume = {7},
+number = {3},
+pages = {856--869},
+year = {1986},
+}
+
+@article{ref4,
+title = {Solution of sparse indefinite systems of linear equations},
+author = {Paige, Chris C. and Saunders, Michael A.},
+journal = {SIAM Journal on Numerical Analysis},
+volume = {12},
+number = {4},
+pages = {617--629},
+year = {1975},
+}
+
+@article{ref5,
+title = {The principle of minimized iteration in the solution of the matrix eigenvalue problem},
+author = {Arnoldi, Walter E.},
+journal = {Quarterly of Applied Mathematics},
+volume = {9},
+number = {17},
+pages = {17--29},
+year = {1951},
+}
+
+@article{ref6,
+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{ref7,
+title = {Efficient sparse matrix-vector multiplication on {CUDA}},
+author = {Bell, Nathan and Garland, Michael},
+journal = {NVIDIA Technical Report NVR-2008-004, NVIDIA Corporation},
+volume = {},
+number = {},
+pages = {},
+year = {2008},
+}
+
+@article{ref8,
+title = {{CUSP} {L}ibrary},
+author = {},
+journal = {},
+volume = {},
+number = {},
+pages = {},
+note = {\url{http://code.google.com/p/cusp-library/}},
+year = {},
+}
+
+@article{ref9,
+title = {{NVIDIA} {CUDA} {C} programming guide},
+author = {NVIDIA Corporation},
+journal = {},
+volume = {},
+number = {},
+pages = {},
+note = {Version 4.2},
+year = {2012},
+}
+
+@article{ref10,
+title = {The university of {F}lorida sparse matrix collection},
+author = {Davis, T. and Hu, Y.},
+journal = {},
+volume = {},
+number = {},
+pages = {},
+note = {\url{http://www.cise.ufl.edu/research/sparse/matrices/list_by_id.html}},
+year = {1997},
+}
+
+@article{ref11,
+title = {Hypergraph partitioning based decomposition for parallel sparse matrix-vector multiplication},
+author = {Catalyurek, U. and Aykanat, C.},
+journal = {{IEEE} {T}ransactions on {P}arallel and {D}istributed {S}ystems},
+volume = {10},
+number = {7},
+pages = {673--693},
+note = {},
+year = {1999},
+}
+
+@article{ref12,
+title = {{hMETIS}: A hypergraph partitioning package},
+author = {Karypis, G. and Kumar, V.},
+journal = {},
+volume = {},
+number = {},
+pages = {},
+note = {},
+year = {1998},
+}
+
+@article{ref13,
+title = {{PaToH}: partitioning tool for hypergraphs},
+author = {Catalyurek, U. and Aykanat, C.},
+journal = {},
+volume = {},
+number = {},
+pages = {},
+note = {},
+year = {1999},
+}
+
+@article{ref14,
+title = {Parallel hypergraph partitioning for scientific computing},
+author = {Devine, K.D. and Boman, E.G. and Heaphy, R.T. and Bisseling, R.H and Catalyurek, U.V.},
+journal = {In Proceedings of the 20th international conference on Parallel and distributed processing, IPDPS’06},
+volume = {},
+number = {},
+pages = {124–124},
+note = {},
+year = {2006},
+}
+
+@article{ref15,
+title = {{PHG} - parallel hypergraph and graph partitioning with {Z}oltan},
+author = {},
+journal = {},
+volume = {},
+number = {},
+pages = {},
+note = {\url{http://www.cs.sandia.gov/Zoltan/ug_html/ug_alg_phg.html}},
+year = {},
+}
+
+
+
+
+

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new file mode 100644 (file)
index 0000000..e5e7bb9
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+++ b/BookGPU/Chapters/chapter12/bibunits.sty
@@ -0,0 +1,314 @@
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diff --git a/BookGPU/Chapters/chapter12/ch12.aux b/BookGPU/Chapters/chapter12/ch12.aux
new file mode 100644 (file)
index 0000000..0cc343d
--- /dev/null
+++ b/BookGPU/Chapters/chapter12/ch12.aux
@@ -0,0 +1,133 @@
+\relax 
+\@writefile{toc}{\author{}{}}
+\@writefile{loa}{\addvspace {10\p@ }}
+\@writefile{toc}{\contentsline {chapter}{\numberline {11}Solving sparse linear systems with GMRES and CG methods on GPU clusters}{249}}
+\@writefile{lof}{\addvspace {10\p@ }}
+\@writefile{lot}{\addvspace {10\p@ }}
+\@writefile{toc}{\contentsline {section}{\numberline {11.1}Introduction}{249}}
+\newlabel{sec:01}{{11.1}{249}}
+\@writefile{toc}{\contentsline {section}{\numberline {11.2}Krylov iterative methods}{250}}
+\newlabel{sec:02}{{11.2}{250}}
+\newlabel{eq:01}{{11.1}{250}}
+\newlabel{eq:02}{{11.2}{250}}
+\newlabel{eq:03}{{11.3}{250}}
+\newlabel{eq:11}{{11.4}{251}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {11.2.1}CG method}{251}}
+\newlabel{sec:02.01}{{11.2.1}{251}}
+\newlabel{eq:04}{{11.5}{251}}
+\newlabel{eq:05}{{11.6}{251}}
+\newlabel{eq:06}{{11.7}{251}}
+\newlabel{eq:07}{{11.8}{251}}
+\newlabel{eq:08}{{11.9}{251}}
+\newlabel{eq:09}{{11.10}{251}}
+\@writefile{loa}{\contentsline {algocf}{\numberline {9}{\ignorespaces Left-preconditioned CG method\relax }}{252}}
+\newlabel{alg:01}{{9}{252}}
+\newlabel{eq:10}{{11.11}{252}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {11.2.2}GMRES method}{253}}
+\newlabel{sec:02.02}{{11.2.2}{253}}
+\newlabel{eq:12}{{11.12}{253}}
+\newlabel{eq:13}{{11.13}{253}}
+\newlabel{eq:14}{{11.14}{253}}
+\newlabel{eq:15}{{11.15}{253}}
+\newlabel{eq:16}{{11.16}{253}}
+\newlabel{eq:17}{{11.17}{253}}
+\newlabel{eq:18}{{11.18}{253}}
+\newlabel{eq:19}{{11.19}{253}}
+\@writefile{loa}{\contentsline {algocf}{\numberline {10}{\ignorespaces Left-preconditioned GMRES method with restarts\relax }}{254}}
+\newlabel{alg:02}{{10}{254}}
+\@writefile{toc}{\contentsline {section}{\numberline {11.3}Parallel implementation on a GPU cluster}{255}}
+\newlabel{sec:03}{{11.3}{255}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {11.3.1}Data partitioning}{255}}
+\newlabel{sec:03.01}{{11.3.1}{255}}
+\@writefile{lof}{\contentsline {figure}{\numberline {11.1}{\ignorespaces A data partitioning of the sparse matrix $A$, the solution vector $x$ and the right-hand side $b$ into four portions.\relax }}{256}}
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+\@writefile{toc}{\contentsline {subsection}{\numberline {11.3.2}GPU computing}{256}}
+\newlabel{sec:03.02}{{11.3.2}{256}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {11.3.3}Data communications}{257}}
+\newlabel{sec:03.03}{{11.3.3}{257}}
+\@writefile{lof}{\contentsline {figure}{\numberline {11.2}{\ignorespaces Data exchanges between \textit  {Node 1} and its neighbors \textit  {Node 0}, \textit  {Node 2} and \textit  {Node 3}.\relax }}{258}}
+\newlabel{fig:02}{{11.2}{258}}
+\@writefile{lof}{\contentsline {figure}{\numberline {11.3}{\ignorespaces Columns reordering of a sparse sub-matrix.\relax }}{259}}
+\newlabel{fig:03}{{11.3}{259}}
+\@writefile{lof}{\contentsline {figure}{\numberline {11.4}{\ignorespaces General scheme of the GPU cluster of tests composed of six machines, each with two GPUs.\relax }}{260}}
+\newlabel{fig:04}{{11.4}{260}}
+\@writefile{toc}{\contentsline {section}{\numberline {11.4}Experimental results}{260}}
+\newlabel{sec:04}{{11.4}{260}}
+\@writefile{lof}{\contentsline {figure}{\numberline {11.5}{\ignorespaces Sketches of sparse matrices chosen from the Davis's collection.\relax }}{261}}
+\newlabel{fig:05}{{11.5}{261}}
+\@writefile{lot}{\contentsline {table}{\numberline {11.1}{\ignorespaces Main characteristics of sparse matrices chosen from the Davis's collection.\relax }}{262}}
+\newlabel{tab:01}{{11.1}{262}}
+\@writefile{lot}{\contentsline {table}{\numberline {11.2}{\ignorespaces Performances of the parallel CG method on a cluster of 24 CPU cores vs. on a cluster of 12 GPUs.\relax }}{262}}
+\newlabel{tab:02}{{11.2}{262}}
+\@writefile{lot}{\contentsline {table}{\numberline {11.3}{\ignorespaces Performances of the parallel GMRES method on a cluster 24 CPU cores vs. on cluster of 12 GPUs.\relax }}{263}}
+\newlabel{tab:03}{{11.3}{263}}
+\newlabel{eq:20}{{11.20}{263}}
+\@writefile{lof}{\contentsline {figure}{\numberline {11.6}{\ignorespaces Parallel generation of a large sparse matrix by four computing nodes.\relax }}{264}}
+\newlabel{fig:06}{{11.6}{264}}
+\@writefile{lot}{\contentsline {table}{\numberline {11.4}{\ignorespaces Main characteristics of sparse banded matrices generated from those of the Davis's collection.\relax }}{265}}
+\newlabel{tab:04}{{11.4}{265}}
+\@writefile{lot}{\contentsline {table}{\numberline {11.5}{\ignorespaces 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.\relax }}{265}}
+\newlabel{tab:05}{{11.5}{265}}
+\@writefile{toc}{\contentsline {section}{\numberline {11.5}Hypergraph partitioning}{265}}
+\newlabel{sec:05}{{11.5}{265}}
+\@writefile{lot}{\contentsline {table}{\numberline {11.6}{\ignorespaces 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.\relax }}{266}}
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+\@writefile{lot}{\contentsline {table}{\numberline {11.7}{\ignorespaces Main characteristics of sparse five-bands matrices generated from those of the Davis's collection.\relax }}{266}}
+\newlabel{tab:07}{{11.7}{266}}
+\@writefile{lof}{\contentsline {figure}{\numberline {11.7}{\ignorespaces Parallel generation of a large sparse five-bands matrix by four computing nodes.\relax }}{267}}
+\newlabel{fig:07}{{11.7}{267}}
+\@writefile{lot}{\contentsline {table}{\numberline {11.8}{\ignorespaces 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\relax }}{267}}
+\newlabel{tab:08}{{11.8}{267}}
+\@writefile{lot}{\contentsline {table}{\numberline {11.9}{\ignorespaces 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\relax }}{268}}
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diff --git a/BookGPU/Chapters/chapter12/ch12.tex b/BookGPU/Chapters/chapter12/ch12.tex
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+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%                          %%
+%%       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}
+
+\begin{algorithm}[!t]
+  %\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}
+
+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.
+
+%%****************%%
+%%****************%%
+\subsection{GMRES method} 
+\label{sec:02.02}
+The iterative GMRES method 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 nonsymmetric 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.
+
+\begin{algorithm}[!t]
+  %\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}
+
+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.
+
+%%--------------------------%%
+%%       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 $(\frac{n}{p},n)$ 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}). 
+
+\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}
+\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 performances of solving 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 clusters 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 $10^{-10}$ to $10^{-26}$. 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}
+
+\begin{table}[!h]
+\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}
+
+We have used the parallel CG and GMRES algorithms for solving sparse linear systems of $25$ million 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}[!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  & $1.625s$             & $0.401s$              & $4.05$          & $14$                & $5.73e$-$11$     & $5.20e$-$18$ \\
+
+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}[!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    & $3.597s$             & $0.514s$              & $6.99$          & $21$                & $2.11e$-$14$     & $8.67e$-$18$ \\
+
+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. 
+
+\begin{figure}
+\centerline{\includegraphics[scale=0.22]{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}[!h]
+\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}
+
+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 unknown values.   
+
+\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    & $6.041s$     & $3.338s$      & $1.81$ & $30$ & $6.77e$-$11$ & $3.25e$-$19$ \\
+
+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 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~\cite{ref14}. We have initialized the parameters of the partitioning operation 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. 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 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}
+\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 to avoid the partitioning of the sparse matrices at each resolution of the linear systems.
+
+\begin{figure}[!t]
+\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}
+
+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 experiments 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 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.
+
+
+
+%%--------------------------%%
+%%       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 are 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/chapter12/ch12.tex~ b/BookGPU/Chapters/chapter12/ch12.tex~
new file mode 100755 (executable)
index 0000000..a3deec0
--- /dev/null
+++ b/BookGPU/Chapters/chapter12/ch12.tex~
@@ -0,0 +1,1078 @@
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%                          %%
+%%       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}
+
+\begin{algorithm}[!t]
+  \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}
+
+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.
+
+%%****************%%
+%%****************%%
+\subsection{GMRES method} 
+\label{sec:02.02}
+The iterative GMRES method 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 nonsymmetric 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.
+
+\begin{algorithm}[!t]
+  \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}
+
+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.
+
+%%--------------------------%%
+%%       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 $(\frac{n}{p},n)$ 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}). 
+
+\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}
+\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 performances of solving 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 clusters 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 $10^{-10}$ to $10^{-26}$. 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}
+
+\begin{table}[!h]
+\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}
+
+We have used the parallel CG and GMRES algorithms for solving sparse linear systems of $25$ million 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}[!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  & $1.625s$             & $0.401s$              & $4.05$          & $14$                & $5.73e$-$11$     & $5.20e$-$18$ \\
+
+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}[!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    & $3.597s$             & $0.514s$              & $6.99$          & $21$                & $2.11e$-$14$     & $8.67e$-$18$ \\
+
+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. 
+
+\begin{figure}
+\centerline{\includegraphics[scale=0.22]{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}[!h]
+\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}
+
+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 unknown values.   
+
+\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    & $6.041s$     & $3.338s$      & $1.81$ & $30$ & $6.77e$-$11$ & $3.25e$-$19$ \\
+
+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 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~\cite{ref14}. We have initialized the parameters of the partitioning operation 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. 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 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}
+\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 to avoid the partitioning of the sparse matrices at each resolution of the linear systems.
+
+\begin{figure}[!t]
+\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}
+
+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 experiments 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 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.
+
+
+
+%%--------------------------%%
+%%       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 are 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]
+

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diff --git a/BookGPU/Chapters/chapter12/figures/cluster.fig b/BookGPU/Chapters/chapter12/figures/cluster.fig
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+%***********************************************************************
+%*                                                                     *
+%* Object: Image decoding PS-routine                    Date: 01.02.93 *
+%* Author: Evgeni CHERNYAEV (chernaev@vxcern.cern.ch)                  *
+%*                                                                     *
+%* Function: Display a run-length encoded color image.                 *
+%*           The image is displayed in color on viewers and printers   *
+%*           that support color Postscript, otherwise it is displayed  *
+%*           as grayscale.                                             *
+%*                                                                     *
+%***********************************************************************
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+%***********************************************************************
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+%***********************************************************************
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diff --git a/BookGPU/Chapters/chapter12/figures/generation_1.fig.bak b/BookGPU/Chapters/chapter12/figures/generation_1.fig.bak
new file mode 100644 (file)
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+\@writefile{toc}{\contentsline {section}{\numberline {15.2}Basic Recalls}{336}}
+\newlabel{section:BASIC RECALLS}{{15.2}{336}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {15.2.1}Devaney's Chaotic Dynamical Systems}{337}}
+\@writefile{toc}{\contentsline {section}{\numberline {15.3}Toward Efficiency and Improvement for CI PRNG}{337}}
+\newlabel{sec:efficient PRNG}{{15.3}{337}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {15.3.1}First Efficient Implementation of a PRNG based on Chaotic Iterations}{337}}
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+\@writefile{lol}{\contentsline {lstlisting}{\numberline {15.1}C code of the sequential PRNG based on chaotic iterations}{337}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {15.3.2}Efficient PRNGs based on Chaotic Iterations on GPU}{337}}
+\newlabel{sec:efficient PRNG gpu}{{15.3.2}{337}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {15.3.3}Naive Version for GPU}{338}}
+\@writefile{loa}{\contentsline {algocf}{\numberline {14}{\ignorespaces Main kernel of the GPU ``naive'' version of the PRNG based on chaotic iterations\relax }}{338}}
+\newlabel{algo:gpu_kernel}{{14}{338}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {15.3.4}Improved Version for GPU}{339}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {15.3.5}Chaos Evaluation of the Improved Version}{339}}
+\newlabel{IR}{{15}{340}}
+\@writefile{loa}{\contentsline {algocf}{\numberline {15}{\ignorespaces Main kernel for the chaotic iterations based PRNG GPU efficient version\relax }}{340}}
+\newlabel{algo:gpu_kernel2}{{15}{340}}
+\@writefile{toc}{\contentsline {section}{\numberline {15.4}Experiments}{340}}
+\newlabel{sec:experiments}{{15.4}{340}}
+\@writefile{lof}{\contentsline {figure}{\numberline {15.1}{\ignorespaces Quantity of pseudorandom numbers generated per second with the xorlike-based PRNG\relax }}{341}}
+\newlabel{fig:time_xorlike_gpu}{{15.1}{341}}
+\@writefile{lof}{\contentsline {figure}{\numberline {15.2}{\ignorespaces Quantity of pseudorandom numbers generated per second using the BBS-based PRNG\relax }}{342}}
+\newlabel{fig:time_bbs_gpu}{{15.2}{342}}
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diff --git a/BookGPU/Makefile b/BookGPU/Makefile
index d23f8c598282bb8e83b88226d12c78ef1f008faa..a842e943fbf4a7cfdfb24aabe67ff9fbb7309121 100644 (file)
--- a/BookGPU/Makefile
+++ b/BookGPU/Makefile
@@ -5,17 +5,18 @@ all:
        pdflatex ${BOOK}
        pdflatex ${BOOK}
        bibtex bu1
-#      bibtex bu2
-#      bibtex bu3
-#      bibtex bu4
-#      bibtex bu5
-#      bibtex bu6
-#      bibtex bu7
-#      bibtex bu8
-#      bibtex bu9
-# #don't put chapter 14, refs are included     
-#      bibtex bu11     
-#      bibtex bu12
+       bibtex bu2
+       bibtex bu3
+       bibtex bu4
+       bibtex bu5
+       bibtex bu6
+       bibtex bu7
+       bibtex bu8
+       bibtex bu9
+       bibtex bu10
+#don't put chapter 14, refs are included       
+       bibtex bu12     
+       bibtex bu13
 
        makeindex  ${BOOK}.idx
        pdflatex ${BOOK}