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%\chapterauthor{}{}
\chapterauthor{Lilia Ziane Khodja}{Femto-ST Institute, University of Franche-Comte, France}
\chapterauthor{Raphaël Couturier}{Femto-ST Institute, University of Franche-Comte, France}
\chapterauthor{Jacques Bahi}{Femto-ST Institute, University of Franche-Comte, France}

\chapter{Solving sparse linear systems with GMRES and CG methods on GPU clusters}
\label{ch12}

%%--------------------------%%
%%       SECTION 1          %%
%%--------------------------%%
\section{Introduction}
\label{ch12:sec:01}
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 are considered as the most expensive process in
terms of execution time 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 can be classified in two classes: the direct and iterative methods.
However, the iterative methods are often more suitable than their counterpart, direct
methods, to  solve these 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 to solve sparse linear systems with iterative
solutions. Nowadays, graphics processing units (GPUs) have become attractive to solve
these systems, due to their computing power and their ability to compute faster than
traditional CPUs.

In Section~\ref{ch12: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{ch12:sec:03},
we give the main key points of the parallel implementation of both methods on a cluster of
GPUs. Finally, in Section~\ref{ch12:sec:04}, we present the experimental results obtained on a
CPU cluster and on a GPU cluster, to solve large sparse linear systems.    


%%--------------------------%%
%%       SECTION 2          %%
%%--------------------------%%
\section{Krylov iterative methods}
\label{ch12:sec:02}
Let us consider the following system of $n$ linear equations\index{Sparse~linear~system}
in $\mathbb{R}$: 
\begin{equation}
Ax=b,
\label{ch12: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\index{Iterative~method} for solving the large sparse linear system~(\ref{ch12: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{ch12: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\index{Convergence}
is satisfied as follows:   
\begin{equation}
\|b-A\tilde{x}\| < \varepsilon,
\label{ch12:eq:03}
\end{equation}
where $\varepsilon<1$ is the required convergence tolerance threshold\index{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 subspace methods}~\cite{ch12:ref1}\index{Iterative~method!Krylov~subspace}.
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 subspace methods are usually used with preconditioners that allow to improve their
convergence. So, in what follows, the CG and GMRES methods are used to solve the left-preconditioned\index{Sparse~linear~system!Preconditioned}
sparse linear system:
\begin{equation}
M^{-1}Ax=M^{-1}b,
\label{ch12:eq:11}
\end{equation}
where $M$ is the preconditioning matrix.


%%****************%%
%%****************%%
\subsection{CG method}
\label{ch12:sec:02.01}
The conjugate gradient method was initially developed by Hestenes and Stiefel in 1952~\cite{ch12:ref2}.
It is one of the well known iterative method to solve large sparse linear systems. In addition, it
can be adapted to solve 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\index{Iterative~method!CG} is the computation of a sequence of approximate
solutions $\{x_k\}_{k\geq 0}$ in a Krylov subspace\index{Iterative~method!Krylov~subspace} of order $k$ as
follows: 
\begin{equation}
x_k \in x_0 + \mathcal{K}_k(A,r_0),
\label{ch12:eq:04}
\end{equation}
such that the Galerkin condition\index{Galerkin~condition} must be satisfied:
\begin{equation}
r_k \bot \mathcal{K}_k(A,r_0),
\label{ch12: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 subspace 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{ch12: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{ch12: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{ch12: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{ch12: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 subspace $\mathcal{K}_{k}$ 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{ch12:eq:08}) and~(\ref{ch12: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{ch12:eq:10}
\end{equation}

\begin{algorithm}[!t]
  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{ch12:alg:01}
\end{algorithm}

Algorithm~\ref{ch12:alg:01} shows the main key points of the preconditioned CG method. It allows
to solve the left-preconditioned\index{Sparse~linear~system!Preconditioned} sparse linear system~(\ref{ch12: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{ch12:eq:07}) and~(\ref{ch12:eq:08}), respectively. The CG method converges after, at
most, $n$ iterations. In practice, the CG algorithm stops when the tolerance threshold\index{Convergence!Tolerance~threshold}
$\varepsilon$ and/or the maximum number of iterations\index{Convergence!Maximum~number~of~iterations}
$maxiter$ are reached.


%%****************%%
%%****************%%
\subsection{GMRES method} 
\label{ch12:sec:02.02}
The iterative GMRES method was developed by Saad and Schultz in 1986~\cite{ch12:ref3} as a generalization
of the minimum residual method MINRES~\cite{ch12:ref4}\index{Iterative~method!MINRES}. Indeed, GMRES can
be applied for solving symmetric or nonsymmetric linear systems. 

The main principle of the GMRES method\index{Iterative~method!GMRES} 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 subspace\index{Iterative~method!Krylov~subspace} $\mathcal{K}_k$ 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{ch12:eq:12}
\end{equation} 
so that the Petrov-Galerkin condition\index{Petrov-Galerkin~condition} is satisfied:
\begin{equation}
\begin{array}{ll}
r_k \bot A \mathcal{K}_k(A, v_1).
\end{array}
\label{ch12:eq:13}
\end{equation}
GMRES uses the Arnoldi process~\cite{ch12:ref5}\index{Iterative~method!Arnoldi~process} to construct an
orthonormal basis $V_k$ for the Krylov subspace $\mathcal{K}_k$ and an upper Hessenberg matrix\index{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{ch12:eq:14}
\end{equation}
and
\begin{equation}
A V_k = V_{k+1} \bar{H}_k.
\label{ch12:eq:15}
\end{equation}

Then, at each iteration $k$, an approximate solution $x_k$ is computed in the Krylov subspace $\mathcal{K}_k$
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{ch12:eq:16}
\end{equation}
From both formulas~(\ref{ch12:eq:15}) and~(\ref{ch12: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{ch12: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{ch12: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{ch12:ref1,ch12: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{ch12: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]
  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{ch12:alg:02}
\end{algorithm}

Algorithm~\ref{ch12:alg:02} shows the key points of the GMRES method with restarts.
It solves the left-preconditioned\index{Sparse~linear~system!Preconditioned} sparse linear
system~(\ref{ch12:eq:11}), such that $M$ is the preconditioning matrix. At each iteration
$k$, GMRES uses the Arnoldi process\index{Iterative~method!Arnoldi~process} (defined from
line~$7$ to line~$17$) to construct a basis $V_m$ of $m$ orthogonal vectors and an upper
Hessenberg matrix\index{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 subspace 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\index{Convergence!Maximum~number~of~iterations} ($maxiter$)
is reached.


%%--------------------------%%
%%       SECTION 3          %%
%%--------------------------%%
\section{Parallel implementation on a GPU cluster}
\label{ch12:sec:03}
In this section, we present the parallel algorithms of both iterative CG\index{Iterative~method!CG}
and GMRES\index{Iterative~method!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\index{Computing~node} 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{ch12:sec:03.01}
The parallel solving of the large sparse linear system~(\ref{ch12: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 (row-by-row decomposition) on the data of the sparse linear systems to be solved.
Figure~\ref{ch12: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{ch12:fig:01}
\end{figure}


%%****************%%
%%****************%%
\subsection{GPU computing}
\label{ch12: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\index{CUBLAS} library (CUDA Basic Linear
Algebra Subroutines), developed by Nvidia~\cite{ch12: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 subspace 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  line~$12$ in
Algorithm~\ref{ch12:alg:01}. In the GMRES solver, we program a kernel for the scalar-vector
multiplication (lines~$7$ and~$15$ in Algorithm~\ref{ch12:alg:02}), a kernel to
solve the least-squares problem and a kernel to update the elements  of the solution
vector $x$.

The least-squares problem in the GMRES method is solved by performing a QR factorization
on the Hessenberg matrix\index{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\index{Iterative~method!CG} and GMRES\index{Iterative~method!GMRES}
methods is the sparse matrix-vector multiplication (SpMV)\index{SpMV~multiplication},
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 sparse
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 its performances even more. One of the
most efficient compressed storage formats\index{Compressed~storage~format} of sparse
matrices on GPUs is the HYB\index{Compressed~storage~format!HYB} format~\cite{ch12: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\index{Compressed~storage~format!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\index{Compressed~storage~format!COO}
which is insensitive to the matrix structure. Consequently, we use the HYB kernel~\cite{ch12: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{ch12:sec:03.03}
All the computing nodes of the GPU cluster execute in parallel the same iterative solver
(Algorithm~\ref{ch12:alg:01} or Algorithm~\ref{ch12:alg:02}) adapted to GPUs, but on their
own portions of the sparse linear system\index{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{ch12: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\index{Neighboring~node}.

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{ch12: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 subspace (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\index{SpMV~multiplication}, the neighboring
nodes\index{Neighboring~node} 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\index{MPI~subroutines!Blocking} sends with \verb+MPI_Send()+
and nonblocking\index{MPI~subroutines!Nonblocking} receives with \verb+MPI_Irecv()+. Figure~\ref{ch12: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{ch12: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{ch12: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{ch12: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{ch12:fig:03} shows a reordering of a sparse sub-matrix (sub-matrix of
\textit{Node 1}). 

\begin{figure}
\centerline{\includegraphics[scale=0.35]{Chapters/chapter12/figures/reorder}}
\caption{Columns reordering of a sparse sub-matrix.}
\label{ch12:fig:03}
\end{figure}

A GPU cluster\index{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 a 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 CUBLAS\index{CUBLAS}
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\index{MPI~subroutines!Global}
\verb+MPI_Allreduce()+.



%%--------------------------%%
%%       SECTION 4          %%
%%--------------------------%%
\section{Experimental results}
\label{ch12: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{ch12:fig:04} shows the general scheme of the GPU cluster\index{GPU~cluster}
that we used in the experimental tests.

Linux cluster version 2.6.39 OS is installed on CPUs. C programming language is used to code
the parallel algorithms of both methods on the GPU cluster. CUDA version 4.0~\cite{ch12:ref9}
is used to program GPUs, using CUBLAS library~\cite{ch12: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.

\begin{figure}[!h]
\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{ch12:fig:04}
\end{figure}

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\index{Iterative~method!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.

\begin{figure}
\centerline{\includegraphics[scale=0.30]{Chapters/chapter12/figures/matrices}}
\caption{Sketches of sparse matrices chosen from the Davis collection.}
\label{ch12: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}
\caption{Main characteristics of sparse matrices chosen from the Davis collection.}
\label{ch12:tab:01}
\end{table}

To get more realistic results, we have tested the CG and GMRES algorithms on sparse matrices of the Davis
collection~\cite{ch12:ref10}, that arise in a wide spectrum of real-world applications. We have chosen six
symmetric sparse matrices and six nonsymmetric ones from this collection. In Figure~\ref{ch12:fig:05},
we show the structures of these matrices and in Table~\ref{ch12: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{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{ch12:tab:02}
\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    & $0.234s$           & $0.124s$            & $1.88$        & $21$           & $2.10e$-$14$     & $3.47e$-$18$ \\

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{ch12:tab:03}
\end{center}
\end{table}

Tables~\ref{ch12:tab:02} and~\ref{ch12:tab:03} show the performances of the parallel
CG and GMRES solvers, respectively, for solving linear systems associated to the sparse
matrices presented in Tables~\ref{ch12: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{ch12:tab:02}
only shows 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\index{Relative~gain}, 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{ch12:eq:20}
\end{equation}
In addition, Tables~\ref{ch12:tab:02} and~\ref{ch12: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 it 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  the parallel
computations further.

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 collection~\cite{ch12:ref10}
as an initial matrix to build 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 build its sparse
sub-matrix. In the first experimental tests, we focused on sparse matrices having a banded structure,
because they are those arising the most in the majority 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 collection. Then, it puts all these copies on the main diagonal of the global matrix
(see Figure~\ref{ch12: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{ch12: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{ch12: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 collection.}
\label{ch12: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{ch12:tab:01}. Their main characteristics are given in Table~\ref{ch12:tab:04}.
Tables~\ref{ch12:tab:05} and~\ref{ch12: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 in a GPU cluster,
are better than those of the GMRES method for solving large symmetric linear systems. In fact, the
CG method is characterized by a better convergence\index{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}
\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{ch12:tab:05}
\end{center}
\end{table}

\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|} 
\hline
{\bf Matrix}      & $\mathbf{Time_{cpu}}$ & $\mathbf{Time_{gpu}}$ & $\mathbf{\tau}$ & $\mathbf{\# iter.}$ & $\mathbf{prec.}$ & $\mathbf{\Delta}$   \\ \hline \hline

2cubes\_sphere    & $3.597s$             & $0.514s$              & $6.99$          & $21$                & $2.11e$-$14$     & $8.67e$-$18$ \\

ecology2          & $2.549s$             & $0.288s$              & $8.83$          & $21$                & $4.88e$-$13$     & $2.08e$-$14$ \\

finan512          & $2.660s$             & $0.377s$              & $7.05$          & $17$                & $3.22e$-$12$     & $8.82e$-$14$ \\

G3\_circuit       & $3.139s$             & $0.480s$              & $6.53$          & $22$                & $1.04e$-$12$     & $5.00e$-$15$ \\

shallow\_water2   & $2.195s$             & $0.253s$              & $8.68$          & $17$                & $5.54e$-$21$     & $7.92e$-$24$ \\

thermal2          & $3.206s$             & $0.463s$              & $6.93$          & $21$                & $8.89e$-$12$     & $3.33e$-$16$ \\ \hline \hline

cage13            & $5.560s$             & $0.663s$              & $8.39$          & $26$                & $3.29e$-$11$     & $1.59e$-$14$ \\

crashbasis        & $25.802s$            & $3.511s$              & $7.35$          & $135$               & $6.81e$-$11$     & $4.61e$-$15$ \\

FEM\_3D\_thermal2 & $13.281s$            & $1.572s$              & $8.45$          & $64$                & $3.88e$-$09$     & $1.82e$-$12$ \\

language          & $12.553s$            & $1.760s$              & $7.13$          & $89$                & $2.11e$-$10$     & $1.60e$-$10$ \\

poli\_large       & $8.515s$             & $1.053s$              & $8.09$          & $69$                & $5.05e$-$11$     & $6.59e$-$12$ \\

torso3            & $31.463s$            & $3.681s$              & $8.55$          & $175$               & $2.69e$-$10$     & $2.66e$-$14$ \\ \hline
\end{tabular}
\caption{Performances of the parallel GMRES method for solving linear systems associated to sparse banded matrices on a cluster of 24 CPU cores vs. 
on a cluster of 12 GPUs.}
\label{ch12:tab:06}
\end{center}
\end{table}

%%--------------------------%%
%%       SECTION 5          %%
%%--------------------------%%
\section{Conclusion}
\label{ch12:sec:05}
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 subspace iterative methods: the CG and GMRES methods.
The first method is well-known for its efficiency to solve symmetric
linear systems and the second one is used, particularly, to solve
nonsymmetric linear systems. 

We have presented 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, show 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.

In our future works, we plan to test the parallel algorithms of CG and GMRES methods, adapted
to GPUs, for solving large linear systems associated to sparse matrices of different structures.
For example, the matrices having large bandwidths, which can lead to many data dependencies
between the computing nodes and, thus, degrade the performances of both algorithms. So in
this case, it would be interesting to study the different data partitioning techniques, in
order to minimize the dependencies between the computing nodes and thus to reduce the total
communication volume. This may improve the performances of both algorithms implemented on
a GPU cluster. Moreover, 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]

