%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%                          %%
%%       CHAPTER 13         %%
%%                          %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
\chapterauthor{}{}
\newcommand{\scalprod}[2]%
{\ensuremath{\langle #1 \, , #2 \rangle}}
\chapter{Solving sparse nonlinear systems of obstacle problems on GPU clusters}

%%--------------------------%%
%%       SECTION 1          %%
%%--------------------------%%
\section{Introduction}
\label{sec:01}
The obstacle problem is one kind of free boundary problems. It allows to model, for example, an elastic membrane covering a solid obstacle.
In this case, the objective is to find an equilibrium position of this membrane constrained to be above the obstacle and which tends to minimize
its surface and/or its energy. The study of such problems occurs in many applications, for example: fluid mechanics, bio-mathematics (tumour growth
process) or financial mathematics (American or European option pricing).

In this chapter, we focus on solutions of large obstacle problems defined in a three-dimensional domain. Particularly, the present study consists
in solving large nonlinear systems derived from the spatial discretization of these problems. Owing to the great size of such systems, in order to
reduce computation times, we proceed by solving them by parallel synchronous or asynchronous iterative algorithms. Moreover, we aim at harnessing
the computing power of GPUs to accelerate computations of these parallel algorithms. For this, we use an iterative method involving a projection
on a convex set, which is: the projected Richardson method. We choose this method among other iterative methods because it is easy to implement on
parallel computers and easy to adapt to GPU architectures.

In Section~\ref{sec:02}, we present the mathematical model of obstacle problems then, in Section~\ref{sec:03}, we describe the general principle of 
the parallel projected Richardson method. Next, in Section~\ref{sec:04}, we give the main key points of the parallel implementation of both synchronous
and asynchronous algorithms of the projected Richardson method on a GPU cluster. In Section~\ref{sec:05}, we present the performances of both parallel
algorithms obtained from simulations carried out on a CPU and GPU clusters. Finally, in Section~\ref{sec:06}, we use the read-black ordering technique
to improve the convergence and, thus, the execution times of the parallel projected Richardson algorithms on the GPU cluster. 


%%--------------------------%%
%%       SECTION 2          %%
%%--------------------------%%
\section{Obstacle problems}
\label{sec:02}
In this section, we present the mathematical model of obstacle problems defined in a three-dimensional domain.
This model is based on that presented in~\cite{ref1}.

%%*******************
%%*******************
\subsection{Mathematical model}
\label{sec:02.01}
An obstacle problem, arising for example in mechanics or financial derivatives, consists in solving a time dependent
nonlinear equation:
\begin{equation}
\left\{
\begin{array}{l}
\frac{\partial u}{\partial t}+b^t.\nabla u-\eta.\Delta u+c.u-f\geq 0\mbox{,~}u\geq\phi\mbox{,~a.e.w. in~}\lbrack 0,T\rbrack\times\Omega\mbox{,~}\eta>0,\\
(\frac{\partial u}{\partial t}+b^t.\nabla u-\eta.\Delta u+c.u-f)(u-\phi)=0\mbox{,~a.e.w. in~}\lbrack 0,T\rbrack\times\Omega,\\
u(0,x,y,z)=u_0(x,y,z),\\
\mbox{B.C. on~}u(t,x,y,z)\mbox{~defined on~}\partial\Omega,
\end{array}
\right.
\label{eq:01}
\end{equation}
where $u_0$ is the initial condition, $c\geq 0$, $b$ and $\eta$ are physical parameters, $T$ is the final time, $u=u(t,x,y,z)$
is an element of the solution vector $U$ to compute, $f$ is the right-hand right that could represent, for example, the external
forces, B.C. describes the boundary conditions on the boundary $\partial\Omega$ of the domain $\Omega$, $\phi$ models a constraint 
imposed to $u$, $\Delta$ is the Laplacian operator, $\nabla$ is the gradient operator, a.e.w. means almost every where and ``.''
defines the products between two scalars, a scalar and a vector or a matrix and a vector. In practice the boundary condition,
generally considered, is the Dirichlet condition (where $u$ is fixed on $\partial\Omega$) or the Neumann condition (where the
normal derivative of $u$ is fixed on $\partial\Omega$). 

The time dependent equation~(\ref{eq:01}) is numerically solved by considering an implicit or a semi-implicit time marching,
where at each time step $k$ a stationary nonlinear problem is solved:
\begin{equation}
\left\{
\begin{array}{l}
b^t.\nabla u-\eta.\Delta u+(c+\delta).u-g\geq 0\mbox{,~}u\geq\phi\mbox{,~a.e.w. in~}\lbrack 0,T\rbrack\times\Omega\mbox{,~}\eta>0, \\
(b^t.\nabla u-\eta.\Delta u+(c+\delta).u- g)(u-\phi)=0\mbox{,~a.e.w. in~}\lbrack 0,T\rbrack\times\Omega, \\
\mbox{B.C. on~}u(t,x,y,z)\mbox{~defined on~}\partial\Omega,
\end{array}
\right.
\label{eq:02}
\end{equation}
where $\delta=\frac{1}{k}$ is the inverse of the time step $k$, $g=f+\delta u^{prev}$ and $u^{prev}$ is the solution computed at the
previous time step. 


%%*******************
%%*******************
\subsection{Discretization}
\label{sec:02.02}
First, we note that the spatial discretization of the previous stationary problem~(\ref{eq:02}) does not provide a symmetric matrix,
because the convection-diffusion operator is not self-adjoint. Moreover, the fact that the operator is self-adjoint or not plays an
important role in the choice of the appropriate algorithm for solving nonlinear systems derived from the discretization of the obstacle
problem. Nevertheless, since the convection coefficients arising in the operator~(\ref{eq:02}) are constant, we can formulate the same 
problem by an self-adjoint operator by performing a classical change of variables. Then, we can replace the stationary convection-diffusion
problem:
\begin{equation}
b^{t}.\nabla v-\eta.\Delta v+(c+\delta).v=g\mbox{,~a.e.w. in~}\lbrack 0,T\rbrack\times\Omega\mbox{,~}c\geq 0\mbox{,~}\delta\geq~0,
\label{eq:03}
\end{equation}
by the following stationary diffusion operator:
\begin{equation}
-\eta.\Delta u+(\frac{\|b\|^{2}_{2}}{4\eta}+c+\delta).u=e^{-a}g=f,
\label{eq:04}
\end{equation}
where $b=\{b_{1},b_{2},b_{3}\}$, $\|b\|_{2}$ denotes the Euclidean norm of $b$ and $v=e^{-a}.u$ represents the general change of variables
such that $a=\frac{b^{t}(x,y,z)}{2\eta}$. Consequently, the numerical resolution of the diffusion problem (the self-adjoint operator~(\ref{eq:04}))
is done by optimization algorithms, in contrast, that of the convection-diffusion problem (non self-adjoint operator~(\ref{eq:03})) is
done by relaxation algorithms. In the case of our studied algorithm, the convergence is ensured by M-matrix property then, the performance
is linked to the magnitude of the spectral radius of the iteration matrix, which is independent of the condition number.

Next, the three-dimensional domain $\Omega\subset\mathbb{R}^{3}$ is set to $\Omega=\lbrack 0,1\rbrack^{3}$ and discretized with an uniform 
Cartesian mesh constituted by $M=m^3$ discretization points, where $m$ related to the spatial discretization step by $h=\frac{1}{m+1}$. This
is carried out by using a classical order 2 finite difference approximation of the Laplacian. So, the complete discretization of both stationary
boundary value problems~(\ref{eq:03}) and~(\ref{eq:04}) leads to the solution of a large discrete complementary problem of the following
form, when both Dirichlet or Neumann boundary conditions are used:
\begin{equation}
\left\{
\begin{array}{l}
\mbox{Find~}U^{*}\in\mathbb{R}^{M}\mbox{~such~that} \\
(A+\delta I)U^{*}-G\geq 0\mbox{,~}U^{*}\geq\bar{\Phi},\\
((A+\delta I)U^{*}-G)^{T}(U^{*}-\bar{\Phi})=0,\\
\end{array}
\right.
\label{eq:05}
\end{equation}
where $A$ is a matrix obtained after the spatial discretization by a finite difference method, $G$ is derived from the Euler first order implicit time
marching scheme and from the discretized right-hand side of the obstacle problem, $\delta$ is the inverse of the time step $k$ and $I$ is the identity
matrix. The matrix $A$ is symmetric when the self-adjoint operator is considered and nonsymmetric otherwise.

According to the chosen discretization scheme of the Laplacian, $A$ is an M-matrix (irreducibly diagonal dominant, see~\cite{ref2}) and, consequently,
the matrix $(A+\delta I)$ is also an M-matrix. This property is important to the convergence of iterative methods.


%%--------------------------%%
%%       SECTION 3          %%
%%--------------------------%%
\section{Parallel iterative method}
\label{sec:03}
Owing to the large size of the previous discrete complementary problem~(\ref{eq:05}), we will solve it by parallel synchronous or asynchronous iterative
algorithms (see~\cite{ref3,ref4,ref5}). In this chapter, we aim at harnessing the computing power of GPU clusters for solving these large nonlinear systems.
Then, we choose to use the projected Richardson iterative method for solving the diffusion problem~(\ref{eq:04}). Indeed, this method is based on the iterations
of the Jacobi method which are easy to parallelize on parallel computers and easy to adapt to GPU architectures. Then, according to the boundary value problem
formulation with a self-adjoint operator~(\ref{eq:04}), we can consider here the equivalent optimization problem and the fixed point mapping associated to
its solution.

Assume that $E=\mathbb{R}^{M}$ is a Hilbert space, in which $\scalprod{.}{.}$ is the scalar product and $\|.\|$ its associated norm. So, the general fixed
point problem to be solved is defined as follows:
\begin{equation}
\left\{
\begin{array}{l}
\mbox{Find~} U^{*} \in E \mbox{~such that} \\
U^{*} = F(U^{*}), \\
\end{array}
\right.
\label{eq:06}
\end{equation}
where $U\mapsto F(U)$ is an application from $E$ to $E$.

Let $K$ be a closed convex set defined by:
\begin{equation}
K = \{U | U \geq \Phi \mbox{~everywhere in~} E\},
\label{eq:07}
\end{equation}
where $\Phi$ is the discrete obstacle function. In fact, the obstacle problem~(\ref{eq:05}) is formulated as the following constrained optimization problem:
\begin{equation}
\left\{
\begin{array}{l}
\mbox{Find~} U^{*} \in K \mbox{~such that} \\
\forall V \in K, J(U^{*}) \leq J(V), \\
\end{array}
\right.
\label{eq:08}
\end{equation}
where the cost function is given by:
\begin{equation}
J(U) = \frac{1}{2}\scalprod{\mathcal{A}.U}{U} - \scalprod{G}{U},
\label{eq:09}
\end{equation}
in which $\scalprod{.}{.}$ denotes the scalar product in $E$, $\mathcal{A}=A+\delta I$ is a symmetric positive definite, $A$ is the discretization matrix
associated with the self-adjoint operator~(\ref{eq:04}) after change of variables.

For any $U\in E$, let $P_K(U)$ be the projection of $U$ on $K$. For any $\gamma\in\mathbb{R}$, $\gamma>0$, the fixed point mapping $F_{\gamma}$ of the projected
Richardson method is defined as follows:
\begin{equation}
U^{*} = F_{\gamma}(U^{*}) = P_K(U^{*} - \gamma(\mathcal{A}.U^{*} - G)).
\label{eq:10}
\end{equation}
In order to reduce the computation time, the large optimization problem is solved in a numerical way by using a parallel asynchronous algorithm of the projected
Richardson method on the convex set $K$. Particularly, we will consider an asynchronous parallel adaptation of the projected Richardson method~\cite{ref6}.

Let $\alpha\in\mathbb{N}$ be a positive integer. We consider that the space $E=\displaystyle\prod_{i=1}^{\alpha} E_i$ is a product of $\alpha$ subspaces $E_i$
where $i\in\{1,\ldots,\alpha\}$. Note that $E_i=\mathbb{R}^{m_i}$, where $\displaystyle\sum_{i=1}^{\alpha} m_{i}=M$, is also a Hilbert space in which $\scalprod{.}{.}_i$
denotes the scalar product and $|.|_i$ the associated norm, for all $i\in\{1,\ldots,\alpha\}$. Then, for all $u,v\in E$, $\scalprod{u}{v}=\displaystyle\sum_{i=1}^{\alpha}\scalprod{u_i}{v_i}_i$
is the scalar product on $E$.

Let $U\in E$, we consider the following decomposition of $U$ and the corresponding decomposition of $F_\gamma$ into $\alpha$ blocks:
\begin{equation}
\begin{array}{rcl}
U    & = & (U_1,\ldots,U_{\alpha}), \\
F_{\gamma}(U) & = & (F_{1,\gamma}(U),\ldots,F_{\alpha,\gamma}(U)). \\
\end{array}
\label{eq:11}
\end{equation}
Assume that the convex set $K=\displaystyle\prod_{i=1}^{\alpha}K_{i}$, such that $\forall i\in\{1,\ldots,\alpha\},K_i\subset E_i$ and $K_i$ is a closed convex set.
Let also $G=(G_1,\ldots,G_{\alpha})\in E$ and, for any $U\in E$, $P_K(U)=(P_{K_1}(U_1),\ldots,P_{K_{\alpha}}(U_{\alpha}))$ is the projection of $U$ on $K$ where $\forall i\in\{1,\ldots,\alpha\},P_{K_i}$
is the projector from $E_i$ onto $K_i$. So, the fixed point mapping of the projected Richardson method~(\ref{eq:10}) can be written in the following way:
\begin{equation}
\forall U\in E\mbox{,~}\forall i\in\{1,\ldots,\alpha\}\mbox{,~}F_{i,\gamma}(U) = P_{K_i}(U_i - \gamma(\mathcal{A}_i.U - G_i)).
\label{eq:12}
\end{equation}
Note that $\displaystyle\mathcal{A}_i.U= \sum_{j=1}^{\alpha}\mathcal{A}_{i,j}.U_j$, where $\mathcal{A}_{i,j}$ denote block matrices of $\mathcal{A}$.

The parallel asynchronous iterations of the projected Richardson method for solving the obstacle problem~(\ref{eq:08}) are defined as follows: let $U^0\in E,U^0=(U^0_1,\ldots,U^0_\alpha)$ be
the initial solution, then for all $p\in\mathbb{N}$, the iterate $U^{p+1}=(U^{p+1}_1,\ldots,U^{p+1}_{\alpha})$ is recursively defined by:
\begin{equation}
U_i^{p+1} = 
\left\{
\begin{array}{l}
F_{i,\gamma}(U_1^{\rho_1(p)}, \ldots, U_{\alpha}^{\rho_{\alpha}(p)}) \mbox{~if~} i\in s(p), \\
U_i^p \mbox{~otherwise}, \\
\end{array}
\right.
\label{eq:13}
\end{equation}
where
\begin{equation}
\left\{
\begin{array}{l}
\forall p\in\mathbb{N}, s(p)\subset\{1,\ldots,\alpha\}\mbox{~and~} s(p)\ne\emptyset, \\
\forall i\in\{1,\ldots,\alpha\},\{p \ | \ i \in s(p)\}\mbox{~is denombrable},
\end{array}
\right.
\label{eq:14}
\end{equation}
and $\forall j\in\{1,\ldots,\alpha\}$,
\begin{equation}
\left\{
\begin{array}{l}
\forall p\in\mathbb{N}, \rho_j(p)\in\mathbb{N}, 0\leq\rho_j(p)\leq p\mbox{~and~}\rho_j(p)=p\mbox{~if~} j\in s(p),\\
\displaystyle\lim_{p\to\infty}\rho_j(p) = +\infty.\\
\end{array}
\right.
\label{eq:15}
\end{equation}

The previous asynchronous scheme of the projected Richardson method models computations that are carried out in parallel
without order nor synchronization (according to the behavior of the parallel iterative method) and describes a subdomain
method without overlapping. It is a general model that takes into account all possible situations of parallel computations
and non-blocking message passing. So, the synchronous iterative scheme is defined by:
\begin{equation}
\forall j\in\{1,\ldots,\alpha\} \mbox{,~} \forall p\in\mathbb{N} \mbox{,~} \rho_j(p)=p.
\label{eq:16}
\end{equation}
The values of $s(p)$ and $\rho_j(p)$ are defined dynamically and not explicitly by the parallel asynchronous or synchronous
execution of the algorithm. Particularly, it enables one to consider distributed computations whereby processors compute at
their own pace according to their intrinsic characteristics and computational load. The parallelism between the processors is
well described by the set $s(p)$ which contains at each step $p$ the index of the components relaxed by each processor on a
parallel way while the use of delayed components in~(\ref{eq:13}) permits one to model nondeterministic behavior and does not
imply inefficiency of the considered distributed scheme of computation. Note that, according to~\cite{ref7}, theoretically,
each component of the vector must be relaxed an infinity of time. The choice of the relaxed components to be used in the
computational process may be guided by any criterion and, in particular, a natural criterion is to pick-up the most recently
available values of the components computed by the processors. Furthermore, the asynchronous iterations are implemented by
means of non-blocking MPI communication subroutines (asynchronous communications).

The important property ensuring the convergence of the parallel projected Richardson method, both  synchronous and asynchronous
algorithms, is the fact that $\mathcal{A}$ is an M-matrix. Moreover, the convergence proceeds from a result of~\cite{ref6}.
Indeed, there exists a value $\gamma_0>0$, such that $\forall\gamma\in ]0,\gamma_0[$, the parallel iterations~(\ref{eq:13}), 
(\ref{eq:14}) and~(\ref{eq:15}), associated to the fixed point mapping $F_\gamma$~(\ref{eq:12}), converge to the unique solution
$U^{*}$ of the discretized problem. 


%%--------------------------%%
%%       SECTION 4          %%
%%--------------------------%%
\section{Parallel implementation on a GPU cluster}
\label{sec:04}
In this section, we give the main key points of the parallel implementation of the projected Richardson method, both synchronous
and asynchronous versions, on a GPU cluster, for solving the nonlinear systems derived from the discretization of large obstacle
problems. More precisely, each nonlinear system is solved iteratively using the whole cluster. We use a heteregeneous CUDA/MPI
programming. Indeed, the communication of data, at each iteration between the GPU computing nodes, can be either synchronous
or asynchronous using the MPI communication subroutines, whereas inside each GPU node, a CUDA parallelization is performed.

\begin{figure}[!h]
\centerline{\includegraphics[scale=0.30]{Chapters/chapter13/figures/splitCPU}}
\caption{Data partitioning of a problem to be solved among $S=3\times 4$ computing nodes.}
\label{fig:01}
\end{figure}

Let $S$ denote the number of computing nodes on the GPU cluster, where a computing node is composed of CPU core holding one MPI
process and a GPU card. So, before starting computations, the obstacle problem of size $(NX\times NY\times NZ)$ is split into $S$
parallelepipedic sub-problems, each for a node (MPI process, GPU), as is shown in Figure~\ref{fig:01}. Indeed, the $NY$ and $NZ$
dimensions (according to the $y$ and $z$ axises) of the three-dimensional problem are, respectively, split into $Sy$ and $Sz$ parts,
such that $S=Sy\times Sz$. In this case, each computing node has at most four neighboring nodes. This kind of the data partitioning
reduces the data exchanges at subdomain boundaries compared to a naive $z$-axis-wise partitioning.

\begin{algorithm}[!t]
%\SetLine
%\linesnumbered
Initialization of the parameters of the sub-problem\;
Allocate and fill the data in the global memory GPU\;
\For{$i=1$ {\bf to} $NbSteps$}{
   $G = \frac{1}{k}.U^0 + F$\;
   Solve($A$, $U^0$, $G$, $U$, $\varepsilon$, $MaxRelax$)\;
   $U^0 = U$\;
}
Copy the solution $U$ back from GPU memory\;
\caption{Parallel solving of the obstacle problem on a GPU cluster}
\label{alg:01}
\end{algorithm}

All the computing nodes of the GPU cluster execute in parallel the same Algorithm~\ref{alg:01} but on different three-dimensional
sub-problems of size $(NX\times ny\times nz)$. This algorithm gives the main key points for solving an obstacle problem defined in
a three-dimensional domain, where $A$ is the discretization matrix, $G$ is the right-hand side and $U$ is the solution vector. After
the initialization step, all the data generated from the partitioning operation are copied from the CPU memories to the GPU global
memories, to be processed on the GPUs. Next, the algorithm uses $NbSteps$ time steps to solve the global obstacle problem. In fact,
it uses a parallel algorithm adapted to GPUs of the projected Richardson iterative method for solving the nonlinear systems of the
obstacle problem. This function is defined by {\it Solve()} in Algorithm~\ref{alg:01}. At every time step, the initial guess $U^0$
for the iterative algorithm is set to the solution found at the previous time step. Moreover, the right-hand side $G$ is computed
as follows: \[G = \frac{1}{k}.U^{prev} + F\] where $k$ is the time step, $U^{prev}$ is the solution computed in the previous time
step and each element $f(x, y, z)$ of the vector $F$ is computed as follows:
\begin{equation}
f(x,y,z)=\cos(2\pi x)\cdot\cos(4\pi y)\cdot\cos(6\pi z).
\label{eq:18}
\end{equation}
Finally, the solution $U$ of the obstacle problem is copied back from the GPU global memories to the CPU memories. We use the
communication subroutines of the CUBLAS library~\cite{ref8} (CUDA Basic Linear Algebra Subroutines) for the memory allocations in
the GPU (\verb+cublasAlloc()+) and the data transfers between the CPU and its GPU: \verb+cublasSetVector()+ and \verb+cublasGetVector()+. 

\begin{algorithm}[!t]
%  \SetLine
%  \linesnumbered
  $p = 0$\;
  $conv = false$\;
  $U = U^{0}$\;
  \Repeat{$(conv=true)$}{
    Determine\_Bordering\_Vector\_Elements($U$)\;
    Compute\_New\_Vector\_Elements($A$, $G$, $U$)\;
    $tmp = U^{0} - U$\;
    $error = \|tmp\|_{2}$\;
    $U^{0} = U$\;
    $p = p + 1$\;
    $conv$ = Convergence($error$, $p$, $\varepsilon$, $MaxRelax$)\;
  }
\caption{Parallel iterative solving of the nonlinear systems on a GPU cluster ($Solve()$ function)}
\label{alg:02}
\end{algorithm}

As many other iterative methods, the algorithm of the projected Richardson method is based on algebraic functions operating on vectors
and/or matrices, which are more efficient on parallel computers when they work on large vectors. Its parallel implementation on the GPU
cluster is carried out so that the GPUs execute the vector operations as kernels and the CPUs execute the serial codes, supervise the
kernel executions and the data exchanges with the neighboring nodes and supply the GPUs with data. Algorithm~\ref{alg:02} shows the
main key points of the parallel iterative algorithm (function $Solve()$ in Algorithm~\ref{alg:01}). All the vector operations inside
the main loop ({\bf repeat} ... {\bf until}) are executed by the GPU. We use the following functions of the CUBLAS library:
\begin{itemize}
\item \verb+cublasDaxpy()+ to compute the difference between the solution vectors $U^{p}$ and $U^{p+1}$ computed in two successive relaxations
$p$ and $p+1$ (line~$7$ in Algorithm~\ref{alg:02}),
\item \verb+cublasDnrm2()+ to perform the Euclidean norm (line~$8$) and,
\item \verb+cublasDcpy()+ for the data copy of a vector to another one in the GPU memory (lines~$3$ and~$9$).
\end{itemize}

The dimensions of the grid and blocks of threads that execute a given kernel depend on the resources of the GPU multiprocessor and the
resource requirements of the kernel. So, if $block$ defines the size of a thread block, which must not exceed the maximum size of a thread
block, then the number of thread blocks in the grid, denoted by $grid$, can be computed according to the size of the local sub-problem
as follows: \[grid = \frac{(NX\times ny\times nz)+block-1}{block}.\] However, when solving very large problems, the size of the thread
grid can exceed the maximum number of thread blocks that can be executed on the GPUs (up-to $65.535$ thread blocks) and, thus, the kernel
will fail to launch. Therefore, for each kernel, we decompose the three-dimensional sub-problem into $nz$ two-dimensional slices of size
($NX\times ny$), as is shown in Figure~\ref{fig:02}. All slices of the same kernel are executed using {\bf for} loop by $NX\times ny$ parallel
threads organized in a two-dimensional grid of two-dimensional thread blocks, as is shown in Listing~\ref{list:01}. Each thread is in charge
of $nz$ discretization points (one from each slice), accessed in the GPU memory with a constant stride $(NX\times ny)$.

\begin{figure}
\centerline{\includegraphics[scale=0.30]{Chapters/chapter13/figures/splitGPU}}
\caption{Decomposition of a sub-problem in a GPU into $nz$ slices.}
\label{fig:02}
\end{figure}

\begin{center}
\lstinputlisting[label=list:01,caption=Skeleton codes of a GPU kernel and a CPU function]{Chapters/chapter13/ex1.cu}
\end{center}
The function $Determine\_Bordering\_Vector\_Elements()$ (line~$5$ in Algorithm~\ref{alg:02}) determines the values of the vector
elements shared at the boundaries with neighboring computing nodes. Its main operations are defined as follows:
\begin{enumerate}
\item define the values associated to the bordering points needed by the neighbors,
\item copy the values associated to the bordering points from the GPU to the CPU,
\item exchange the values associated to the bordering points between the neighboring CPUs,
\item copy the received values associated to the bordering points from the CPU to the GPU,
\end{enumerate}
The first operation of this function is implemented as kernels to be performed by the GPU:
\begin{itemize}
\item a kernel executed by $NX\times nz$ threads to define the values associated to the bordering vector elements along $y$-axis and,
\item a kernel executed by $NX\times ny$ threads to define the values associated to the bordering vector elements along $z$-axis.  
\end{itemize}
As mentioned before, we develop the \emph{synchronous} and \emph{asynchronous} algorithms of the projected Richardson method. Obviously,
in this scope, the synchronous or asynchronous communications refer to the communications between the CPU cores (MPI processes) on the
GPU cluster, in order to exchange the vector elements associated to subdomain boundaries. For the memory copies between a CPU core and
its GPU, we use the synchronous communication routines of the CUBLAS library: \verb+cublasSetVector()+ and \verb+cublasGetVector()+
in the synchronous algorithm and the asynchronous ones: \verb+cublasSetVectorAsync()+ and \verb+cublasGetVectorAsync()+ in the
asynchronous algorithm. Moreover, we use the communication routines of the MPI library to carry out the data exchanges between the neighboring
nodes. We use the following communication routines: \verb+MPI_Isend()+ and \verb+MPI_Irecv()+ to perform non-blocking sends and receptions,
respectively. For the synchronous algorithm, we use the MPI routine \verb+MPI_Waitall()+ which puts the MPI process of a computing node
in blocking status until all data exchanges with neighboring nodes (sends and receptions) are completed. In contrast, for the asynchronous
algorithms, we use the MPI routine \verb+MPI_Test()+ which tests the completion of a data exchange (send or reception) without putting the
MPI process in blocking status.   

The function $Compute\_New\_Vector\_Elements()$ (line~$6$ in Algorithm~\ref{alg:02}) computes, at each iteration, the new elements
of the iterate vector $U$. Its general code is presented in Listing~\ref{list:01} (CPU function). The iterations of the projected
Richardson method, based on those of the Jacobi method, are defined as follows: 
\begin{equation}
\begin{array}{ll}
u^{p+1}(x,y,z) =& \frac{1}{Center}(g(x,y,z) - (Center\cdot u^{p}(x,y,z) + \\
& West\cdot u^{p}(x-h,y,z) + East\cdot u^{p}(x+h,y,z) + \\
& South\cdot u^{p}(x,y-h,z) + North\cdot u^{p}(x,y+h,z) + \\
& Rear\cdot u^{p}(x,y,z-h) + Front\cdot u^{p}(x,y,z+h))),  
\end{array}
\label{eq:17}
\end{equation}
where $u^{p}(x,y,z)$ is an element of the iterate vector $U$ computed at the iteration $p$ and $g(x,y,z)$ is a vector element of the
right-hand side $G$. The scalars $Center$, $West$, $East$, $South$, $North$, $Rear$ and $Front$ define constant coefficients of the
block matrix $A$. Figure~\ref{fig:03} shows the positions of these coefficients in a three-dimensional domain.  

\begin{figure}
\centerline{\includegraphics[scale=0.35]{Chapters/chapter13/figures/matrix}}
\caption{Matrix constant coefficients in a three-dimensional domain.}
\label{fig:03}
\end{figure}

The kernel implementations of the projected Richardson method on GPUs uses a perfect fine-grain multithreading parallelism. Since the 
projected Richardson algorithm is implemented as a fixed point method, each kernel is executed by a large number of GPU threads such
that each thread is in charge of the computation of one element of the iterate vector $U$. Moreover, this method uses the vector elements
updates of the Jacobi method, which means that each thread $i$ computes the new value of its element $u_{i}^{p+1}$ independently of the
new values $u_{j}^{p+1}$, where $j\neq i$, of those computed in parallel by other threads at the same iteration $p+1$. Listing~\ref{list:02}
shows the GPU implementations of the main kernels of the projected Richardson method, which are: the matrix-vector multiplication
(\verb+MV_Multiplication()+) and the vector elements updates (\verb+Vector_Updates()+). The codes of these kernels are based on
that presented in Listing~\ref{list:01}.

\lstinputlisting[label=list:02,caption=GPU kernels of the projected Richardson method]{Chapters/chapter13/ex2.cu}

\begin{figure}
\centerline{\includegraphics[scale=0.3]{Chapters/chapter13/figures/points3D}}
\caption{Computation of a vector element with the projected Richardson method.}
\label{fig:04}
\end{figure}

Each kernel is executed by $NX\times ny$ GPU threads so that $nz$ slices of $(NX\times ny)$ vector elements are computed in
a {\bf for} loop. In this case, each thread is in charge of one vector element from each slice (in total $nz$ vector elements
along $z$-axis). We can notice from the formula~(\ref{eq:17}) that the computation of a vector element $u^{p+1}(x,y,z)$, by
a thread at iteration $p+1$, requires seven vector elements computed at the previous iteration $p$: two vector elements in
each dimension plus the vector element at the intersection of the three axises $x$, $y$ and $z$ (see Figure~\ref{fig:04}). 
So, to reduce the memory accesses to the high-latency global memory, the vector elements of the current slice can be stored
in the low-latency shared memories of thread blocks, as is described in~\cite{ref9}. Nevertheless, the fact that the computation
of a vector element requires only two elements in each dimension does not allow to maximize the data reuse from the shared memories.
The computation of a slice involves in total $(bx+2)\times(by+2)$ accesses to the global memory per thread block, to fill the
required vector elements in the shared memory where $bx$ and $by$ are the dimensions of a thread block. Then, in order to optimize
the memory accesses on GPUs, the elements of the iterate vector $U$ are filled in the cache texture memory (see~\cite{ref10}).
In new GPU generations as Fermi or Kepler, the global memory accesses are always cached in L1 and L2 caches. For example, for
a given kernel, we can favour the use of the L1 cache to that of the shared memory by using the function \verb+cudaFuncSetCacheConfig(Kernel,cudaFuncCachePreferL1)+.
So, the initial access to the global memory loads the vector elements required by the threads of a block into the cache memory
(texture or L1/L2 caches). Then, all the following memory accesses read from this cache memory. In Listing~\ref{list:02}, the
function \verb+fetch_double(v,i)+ is used to read from the texture memory the $i^{th}$ element of the double-precision vector
\verb+v+ (see Listing~\ref{list:03}). Moreover, the seven constant coefficients of matrix $A$ can be stored in the constant memory
but, since they are reused $nz$ times by each thread, it is more interesting to fill them on the low-latency registers of each thread.    

\lstinputlisting[label=list:03,caption=Memory access to the cache texture memory]{Chapters/chapter13/ex3.cu}

The function $Convergence()$ (line~$11$ in Algorithm~\ref{alg:02}) allows to detect the convergence of the parallel iterative algorithm
and is based on the tolerance threshold $\varepsilon$ and the maximum number of relaxations $MaxRelax$. We take into account the number 
of relaxations since that of iterations cannot be computed in the asynchronous case. Indeed, a relaxation is the update~(\ref{eq:13}) of
a local iterate vector $U_i$ according to $F_i$. Then, counting the number of relaxations is possible in both synchronous and asynchronous
cases. On the other hand, an iteration is the update of at least all vector components with $F_i$.

In the synchronous algorithm, the global convergence is detected when the maximal value of the absolute error, $error$, is sufficiently small
and/or the maximum number of relaxations, $MaxRelax$, is reached, as follows:
$$
\begin{array}{l}
error=\|U^{p}-U^{p+1}\|_{2}; \\
AllReduce(error,\hspace{0.1cm}maxerror,\hspace{0.1cm}MAX); \\
\text{if}((maxerror<\varepsilon)\hspace{0.2cm}\text{or}\hspace{0.2cm}(p\geq MaxRelax)) \\
conv \leftarrow true;
\end{array}
$$
where the function $AllReduce()$ uses the MPI reduction subroutine \verb+MPI_Allreduce()+ to compute the maximal value, $maxerror$, among the
local absolute errors, $error$, of all computing nodes and $p$ (in Algorithm~\ref{alg:02}) is used as a counter of the local relaxations carried
out by a computing node. In the asynchronous algorithms, the global convergence is detected when all computing nodes locally converge. For this,
we use a token ring architecture around which a boolean token travels, in one direction, from a computing node to another. Starting from node $0$,
the boolean token is set to $true$ by node $i$ if the local convergence is reached or to $false$ otherwise and, then, it is sent to node $i+1$.
Finally, the global convergence is detected when node $0$ receives from its neighbor node $S-1$, in the ring architecture, a token set to $true$.
In this case, node $0$ sends a stop message (end of parallel solving) to all computing nodes in the cluster.


%%--------------------------%%
%%       SECTION 5          %%
%%--------------------------%%
\section{Experimental tests on a GPU cluster}
\label{sec:05}
The GPU cluster of tests, that we used in this chapter, is an $20Gbps$ Infiniband network of six machines. Each machine is a Quad-Core Xeon
E5530 CPU running at $2.4$GHz. It provides a RAM memory of $12$GB with a memory bandwidth of $25.6$GB/s and it is equipped with two Nvidia
Tesla C1060 GPUs. A Tesla GPU contains in total $240$ cores running at $1.3$GHz. It provides $4$GB of global memory with a memory bandwidth
of $102$GB/s, accessible by all its cores and also by the CPU through the PCI-Express 16x Gen 2.0 interface with a throughput of $8$GB/s.
Hence, the memory copy operations between the GPU and the CPU are about $12$ times slower than those of the Tesla GPU memory. We have performed
our simulations on a cluster of $24$ CPU cores and on a cluster of $12$ GPUs. Figure~\ref{fig:05} describes the components of the GPU cluster
of tests.

\begin{figure}
\centerline{\includegraphics[scale=0.25]{Chapters/chapter13/figures/cluster}}
\caption{GPU cluster of tests composed of 12 computing nodes (six machines, each with two GPUs.}
\label{fig:05}
\end{figure}

Linux cluster version 2.6.39 OS is installed on CPUs. C programming language is used for coding the parallel algorithms of the methods on both
GPU cluster and CPU cluster. CUDA version 4.0~\cite{ref12} is used for programming GPUs, using CUBLAS library~\cite{ref8} to deal with vector
operations in GPUs and, finally, MPI functions of OpenMPI 1.3.3 are used to carry out the synchronous and asynchronous communications between
CPU cores. Indeed, in our experiments, a computing node is managed by a MPI process and it is composed of one CPU core and one GPU card.
 
All experimental results of the parallel projected Richardson algorithms are obtained from simulations made in double precision data. The obstacle
problems to be solved are defined in constant three-dimensional domain $\Omega\subset\mathbb{R}^{3}$. The numerical values of the parameters of the
obstacle problems are: $\eta=0.2$, $c=1.1$, $f$ is computed by formula~(\ref{eq:18}) and final time $T=0.02$. Moreover, three time steps ($NbSteps=3$)
are computed with $k=0.0066$. As the discretization matrix is constant along the time steps, the convergence properties of the iterative algorithms
do not change. Thus, the performance characteristics obtained with three time steps will still be valid for more time steps. The initial function
$u(0,x,y,z)$ of the obstacle problem~(\ref{eq:01}) is set to $0$, with a constraint $u\geq\phi=0$. The relaxation parameter $\gamma$ used by the
projected Richardson method is computed automatically thanks to the diagonal entries of the discretization matrix. The formula and its proof can be
found in~\cite{ref11}, Section~2.3. The convergence tolerance threshold $\varepsilon$ is set to $1e$-$04$ and the maximum number of relaxations is
limited to $10^{6}$ relaxations. Finally, the number of threads per block is set to $256$ threads, which gives, in general, good performances for
most GPU applications. We have performed some tests for the execution configurations and we have noticed that the best configuration of the $256$
threads per block is an organization into two dimensions of sizes $(64,4)$. 

\begin{table}[!h]
\centering
\begin{tabular}{|c|c|c|c|c|c|}
\hline
\multirow{2}{*}{\bf Pb. size} & \multicolumn{2}{c|}{\bf Synchronous} & \multicolumn{2}{c|}{\bf Asynchronous} & \multirow{2}{*}{\bf Gain\%} \\ \cline{2-5}

                              & $\mathbf{T_{cpu}}$ & {\bf \#relax.}  & $\mathbf{T_{cpu}}$ & {\bf \#relax.}  &  \\ \hline \hline

$256^{3}$                     & $575.22$           & $198,288$        & $539.25$          & $198,613$        & $6.25$ \\ \hline \hline

$512^{3}$                     & $19,250.25$        & $750,912$        & $18,237.14$       & $769,611$        & $5.26$ \\ \hline \hline 

$768^{3}$                     & $206,159.44$       & $1,635,264$      & $183,582.60$      & $1,577,004$      & $10.95$ \\ \hline \hline

$800^{3}$                     & $222,108.09$       & $1,769,232$      & $188,790.04$      & $1,701,735$      & $15.00$ \\ \hline
\end{tabular}
\vspace{0.5cm}
\caption{Execution times in seconds of the parallel projected Richardson method implemented on a cluster of 24 CPU cores.}
\label{tab:01}
\end{table}

\begin{table}[!h]
\centering
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
\multirow{2}{*}{\bf Pb. size} & \multicolumn{3}{c|}{\bf Synchronous}                   & \multicolumn{3}{c|}{\bf Asynchronous}                   & \multirow{2}{*}{\bf Gain\%}  \\ \cline{2-7}

                              & $\mathbf{T_{gpu}}$ & {\bf \#relax.}  & $\mathbf{\tau}$ & $\mathbf{T_{gpu}}$ & {\bf \#relax.}  & $\mathbf{\tau}$ &           \\  \hline \hline

$256^{3}$                     & $29.67$            & $100,692$        &  $19.39$        & $18.00$           & $94,215$         & $29.96$         & $39.33$  \\ \hline \hline

$512^{3}$                     & $521.83$           & $381,300$        & $36.89$         & $425.15$          & $347,279$        & $42.89$         & $18.53$ \\ \hline \hline 

$768^{3}$                     & $4,112.68$         & $831,144$        & $50.13$         & $3,313.87$        & $750,232$        & $55.40$         & $19.42$ \\ \hline \hline 

$800^{3}$                     & $3,950.87$         & $899,088$        & $56.22$         & $3,636.57$        & $834,900$        & $51.91$         & $7.95$ \\ \hline 
\end{tabular}
\vspace{0.5cm}
\caption{Execution times in seconds of the parallel projected Richardson method implemented on a cluster of 12 GPUs.}
\label{tab:02}
\end{table}

The performance measures that we took into account are the execution times and the number of relaxations performed by the parallel iterative algorithms,
both synchronous and asynchronous versions, on the GPU and CPU clusters. These algorithms are used for solving nonlinear systems derived from the discretization
of obstacle problems of sizes $256^{3}$, $512^{3}$, $768^{3}$ and $800^{3}$. In Table~\ref{tab:01} and Table~\ref{tab:02}, we show the performances
of the parallel synchronous and asynchronous algorithms of the projected Richardson method implemented, respectively, on a cluster of $24$ CPU cores
and on a cluster of $12$ GPUs. In these tables, the execution time defines the time spent by the slowest computing node and the number of relaxations
is computed as the summation of those carried out by all computing nodes.

In the sixth column of Table~\ref{tab:01} and in the eighth column of Table~\ref{tab:02}, we give the gains in $\%$ obtained by using an
asynchronous algorithm compared to a synchronous one. We can notice that the asynchronous version on CPU and GPU clusters is slightly faster
than the synchronous one for both methods. Indeed, the cluster of tests is composed of local and homogeneous nodes communicating via low-latency
connections. So, in the case of distant and/or heterogeneous nodes (or even with geographically distant clusters) the asynchronous version
would be faster than the synchronous one. However, the gains obtained on the GPU cluster are better than those obtained on the CPU cluster.
In fact, the computation times are reduced by accelerating the computations on GPUs while the communication times still unchanged.  

The fourth and seventh columns of Table~\ref{tab:02} show the relative gains obtained by executing the parallel algorithms on the cluster
of $12$ GPUs instead on the cluster of $24$ CPU cores. We compute the relative gain $\tau$ as a ratio between the execution time $T_{cpu}$
spent on the CPU cluster over that $T_{gpu}$ spent on the GPU cluster: \[\tau=\frac{T_{cpu}}{T_{gpu}}.\] We can see from these ratios that
solving large obstacle problems is faster on the GPU cluster than on the CPU cluster. Indeed, the GPUs are more efficient than their
counterpart CPUs to execute large data-parallel operations. In addition, the projected Richardson method is implemented as a fixed point-based
iteration and uses the Jacobi vector updates that allow a well thread-parallelization on GPUs, such that each GPU thread is in charge
of one vector component at a time without being dependent on other vector components computed by other threads. Then, this allow to exploit
at best the high performance computing of the GPUs by using all the GPU resources and avoiding the idle cores.

Finally, the number of relaxations performed by the parallel synchronous algorithm is different in the CPU and GPU versions, because the number
of computing nodes involved in the GPU cluster and in the CPU cluster is different. In the CPU case, $24$ computing nodes ($24$ CPU cores) are
considered, whereas in the GPU case, $12$ computing nodes ($12$ GPUs) are considered. As the number of relaxations depends on the domain decomposition,
consequently it also depends on the number of computing nodes.


%%--------------------------%%
%%       SECTION 6          %%
%%--------------------------%%
\section{Red-Black ordering technique}
\label{sec:06}
As is well-known, the Jacobi method is characterized by a slow convergence rate compared to some iterative methods (for example Gauss-Seidel method).
So, in this section, we present some solutions to reduce the execution time and the number of relaxations and, more specifically, to speed up the
convergence of the parallel projected Richardson method on the GPU cluster. We propose to use the point red-black ordering technique to accelerate
the convergence. This technique is often used to increase the parallelism of iterative methods for solving linear systems~\cite{ref13,ref14,ref15}.
We apply it to the projected Richardson method as a compromise between the Jacobi and Gauss-Seidel iterative methods. 

The general principle of the red-black technique is as follows. Let $t$ be the summation of the integer $x$-, $y$- and $z$-coordinates of a vector
element $u(x,y,z)$ on a three-dimensional domain: $t=x+y+z$. As is shown in Figure~\ref{fig:06.01}, the red-black ordering technique consists in the
parallel computing of the red vector elements having even value $t$ by using the values of the black ones then, the parallel computing of the black
vector elements having odd values $t$ by using the new values of the red ones.

\begin{figure}
\centering
  \mbox{\subfigure[Red-black ordering on x, y and z axises]{\includegraphics[width=2.3in]{Chapters/chapter13/figures/rouge-noir}\label{fig:06.01}}\quad
        \subfigure[Red-black ordering on y axis]{\includegraphics[width=2.3in]{Chapters/chapter13/figures/rouge-noir-y}\label{fig:06.02}}}
\caption{Red-black ordering for computing the iterate vector elements in a three-dimensional space.}
\end{figure}

This technique can be implemented on the GPU in two different manners:
\begin{itemize}
\item among all launched threads ($NX\times ny$ threads), only one thread out of two computes its red or black vector element at a time or,
\item all launched threads (on average half of $NX\times ny$ threads) compute the red vector elements first and, then, the black ones.
\end{itemize}
However, in both solutions, for each memory transaction, only half of the memory segment addressed by a half-warp is used. So, the computation of the
red and black vector elements leads to use twice the initial number of memory transactions. Then, we apply the point red-black ordering accordingly to
the $y$-coordinate, as is shown in Figure~\ref{fig:06.02}. In this case, the vector elements having even $y$-coordinate are computed in parallel using
the values of those having odd $y$-coordinate and then vice-versa. Moreover, in the GPU implementation of the parallel projected Richardson method (Section~\ref{sec:04}),
we have shown that a sub-problem of size $(NX\times ny\times nz)$ is decomposed into $nz$ grids of size $(NX\times ny)$. Then, each kernel is executed
in parallel by $NX\times ny$ GPU threads, so that each thread is in charge of $nz$ vector elements along $z$-axis (one vector element in each grid of
the sub-problem). So, we propose to use the new values of the vector elements computed in grid $i$ to compute those of the vector elements in grid $i+1$.
Listing~\ref{list:04} describes the kernel of the matrix-vector multiplication and the kernel of the vector elements updates of the parallel projected
Richardson method using the red-black ordering technique.

\lstinputlisting[label=list:04,caption=GPU kernels of the projected Richardson method using the red-black technique]{Chapters/chapter13/ex4.cu}

Finally, we exploit the concurrent executions between the host functions and the GPU kernels provided by the GPU hardware and software. In fact, the kernel
launches are asynchronous (when this environment variable is not disabled on the GPUs), such that the control is returned to the host (MPI process) before
the GPU has completed the requested task (kernel)~\cite{ref12}. Therefore, all the kernels necessary to update the local vector elements, $u(x,y,z)$ where
$0<y<(ny-1)$ and $0<z<(nz-1)$, are executed first. Then, the values associated to the bordering vector elements are exchanged between the neighbors. Finally,
the values of the vector elements associated to the bordering vector elements are updated. In this case, the computation of the local vector elements is
performed concurrently with the data exchanges between neighboring CPUs and this in both synchronous and asynchronous cases.

\begin{table}[!h]
\centering
\begin{tabular}{|c|c|c|c|c|c|}
\hline
\multirow{2}{*}{\bf Pb. size} & \multicolumn{2}{c|}{\bf Synchronous} & \multicolumn{2}{c|}{\bf Asynchronous} & \multirow{2}{*}{\bf Gain\%}  \\ \cline{2-5}

                              & $\mathbf{T_{gpu}}$ & {\bf \#relax.}   & $\mathbf{T_{gpu}}$ & {\bf \#relax.}   &           \\  \hline \hline

$256^{3}$                     & $18.37$            & $71,988$         & $12.58$           & $67,638$         & $31.52$  \\ \hline \hline

$512^{3}$                     & $349.23$           & $271,188$        & $289.41$          & $246,036$        & $17.13$ \\ \hline \hline 

$768^{3}$                     & $2,773.65$         & $590,652$        & $2,222.22$        & $532,806$        & $19.88$ \\ \hline \hline 

$800^{3}$                     & $2,748.23$         & $638,916$        & $2,502.61$        & $592,525$        & $8.92$ \\ \hline 
\end{tabular}
\vspace{0.5cm}
\caption{Execution times in seconds of the parallel projected Richardson method using read-black ordering technique implemented on a cluster of 12 GPUs.}
\label{tab:03}
\end{table}

In Table~\ref{tab:03}, we report the execution times and the number of relaxations performed on a cluster of $12$ GPUs by the parallel projected Richardson
algorithms; it can be noted that the performances of the projected Richardson are improved by using the point read-black ordering. We compare the performances
of the parallel projected Richardson method with and without this later ordering (Tables~\ref{tab:02} and~\ref{tab:03}). We can notice that both parallel synchronous
and asynchronous algorithms are faster when they use the red-black ordering. Indeed, we can see in Table~\ref{tab:03} that the execution times of these algorithms
are reduced, on average, by $32\%$ compared to those shown in Table~\ref{tab:02}.

\begin{figure}
\centerline{\includegraphics[scale=0.9]{Chapters/chapter13/figures/scale}}
\caption{Weak scaling of both synchronous and asynchronous algorithms of the projected Richardson method using red-black ordering technique.}
\label{fig:07}
\end{figure}

In Figure~\ref{fig:07}, we study the ratio between the computation time and the communication time of the parallel projected Richardson algorithms on a GPU cluster.
The experimental tests are carried out on a cluster composed of one to ten Tesla GPUs. We have focused on the weak scaling of both parallel, synchronous and asynchronous,
algorithms using the red-black ordering technique. For this, we have fixed the size of a sub-problem to $256^{3}$ per computing node (a CPU core and a GPU). Then,
Figure~\ref{fig:07} shows the number of relaxations performed, on average, per second by a computing node. We can see from this figure that the efficiency of the
asynchronous algorithm is almost stable, while that of the synchronous algorithm decreases (down to $81\%$ in this example) with the increasing of the number of
computing nodes on the cluster. This is due to the fact that the ratio between the time of the computation over that of the communication is reduced when the computations
are performed on GPUs. Indeed, GPUs compute faster than CPUs and communications are more time consuming. In this context, asynchronous algorithms are more scalable
than synchronous ones. So, with large scale GPU clusters, synchronous algorithms might be more penalized by communications, as can be deduced from Figure~\ref{fig:07}.
That is why we think that asynchronous iterative algorithms are all the more interesting in this case.


%%--------------------------%%
%%       SECTION 7          %%
%%--------------------------%%
\section{Conclusion}
\label{sec:07}
Our main contribution, in this chapter, is the parallel implementation of an asynchronous iterative method on GPU clusters for solving large scale nonlinear
systems derived from the spatial discretization of three-dimensional obstacle problems. For this, we have implemented both synchronous and asynchronous algorithms of the
Richardson iterative method using a projection on a convex set. Indeed, this method uses point-based iterations of the Jacobi method that are very easy to parallelize on
parallel computers. We have shown that its adapted parallel algorithms to GPU architectures allows to exploit at best the computing power of the GPUs and to accelerate the
resolution of large nonlinear systems. Consequently, the experimental results have shown that solving nonlinear systems of large obstacle problems with this method is about
fifty times faster on a cluster of $12$ GPUs than on a cluster of $24$ CPU cores. Moreover, we have applied to this projected Richardson method the red-black ordering technique
which allows it to improve its convergence rate. Thus, the execution times of both parallel algorithms performed on the cluster of $12$ GPUs are reduced on average of $32\%$.

Afterwards, the experiments have shown that the asynchronous version is slightly more efficient than the synchronous one. In fact, the computations are accelerated by using GPUs
while the communication times still unchanged. In addition, we have studied the weak-scaling in the synchronous and asynchronous cases, which has confirmed that the ratio between
the computations and the communications are reduced when using a cluster of GPUs. We highlight that asynchronous iterative algorithms are more scalable than synchronous ones.
Therefore, we can conclude that asynchronous iterations are well suited to tackle scalability issues on GPU clusters.

In future works, we plan to perform experiments on large scale GPU clusters and on geographically distant GPU clusters, because we expect that asynchronous versions would
be faster and more scalable on such architectures. Furthermore, we want to study the performance behavior and the scalability of other numerical algorithms which support,
if possible, the model of asynchronous iterations.

\putbib[Chapters/chapter13/biblio13]

