%\include{frontmatter/Foreword}
\include{frontmatter/preface}
+\tableofcontents
\listoffigures
\listoftables
-\tableofcontents
\mainmatter
It is almost straightforward to parallelise this scheme for GPUs, by processing each subinterval $[x_i,x_{i+1}]$ independently in a separate thread. However, it is not known in advance whether an extra knot $t_i$ needs to be inserted or not, and therefore calculation of the position of the knot in the output sequence of knots ${t_i}$ is problematic for parallel implementation (for a sequential algorithm no such issue arises). To avoid serialisation, we decided to insert an additional knot in every interval $[x_i,x_{i+1}]$, but set $t_i=x_i$ when the extra knot is not actually needed. This way we know in advance the position of the output knots and the length of this sequence is $2(n-1)$, and therefore all calculations can now be performed independently. The price we pay is that some of the spline knots can coincide. However, this does not affect spline evaluation, as one of the coinciding knots is simply disregarded, and the spline coefficients are replicated (so for a double knot $t_i=t_{i+1}$, we have $\alpha_i=\alpha_{i+1}$, $\beta_i=\beta_{i+1}$, $\gamma_i=\gamma_{i+1}$). Our implementation is presented in Figures \ref{ch11:algcoef}-\ref{ch11:algcoef1}.
+\lstinputlisting[label=ch11:algcoef1,caption=Calculation of monotone spline knots and coefficients.]{Chapters/chapter11/code2.cu}
+
+
At the spline evaluation stage we need to compute $s(z_k)$ for a sequence of query values ${z_k}, k=1,\ldots,K$. For each $z_k$ we locate the interval $[t_i,t_{i+1}]$ containing $z_k$, using bisection algorithm presented in Figure \ref{ch11:algeval}, and then apply the appropriate coefficients of the quadratic function. This is also done in parallel.
The bisection algorithm could be implemented using texture memory (to cache the array \texttt{z}), but this is not shown in Figure \ref{ch11:algeval}.
+\pagebreak
\lstinputlisting[label=ch11:algcoef,caption=Implementation of the kernel for calculating spline knots and coefficients. Function fmax is used to avoid division by zero for data with coinciding abscissae.]{Chapters/chapter11/code1.cu}
%% \end{figure}
-\lstinputlisting[label=ch11:algcoef1,caption=Calculation of monotone spline knots and coefficients.]{Chapters/chapter11/code2.cu}
%% \begin{figure}[!hp]
%% \renewcommand{\baselinestretch}{1}
%% %\renewcommand{\baselinestretch}{1}
\begin{table}[!h]
\begin{center}
-\caption{The average CPU time (sec) of the serial PAVA, MLS and parallel MLS algorithms. } \label{ch11:table1}
\begin{tabular}{|r|r|r|r|}
-
+\hline
Data & PAVA & MLS & GPU MLS \\ \hline
monotone increasing $f$ & & & \\
$n=10 \times 10^6$ &1.9& 1.9& 3500 \\
$n=20 \times 10^6$ &3.5& 4.0&-- \\
$n=50 \times 10^6$ &11& 11& -- \\
-
+\hline
\end{tabular}
\end{center}
+\caption{The average CPU time (sec) of the serial PAVA, MLS and parallel MLS algorithms. }
+\label{ch11:table1}
\end{table}
%% %\renewcommand{\baselinestretch}{2}
\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}
+\chapter[Solving linear systems with GMRES and CG methods on GPU clusters]{Solving sparse linear systems with GMRES and CG methods on GPU clusters}
\label{ch12}
%%--------------------------%%
diagonal are filled with sub-copies (left-copy and right-copy in Figure~\ref{ch12:fig:06}) of the same
initial matrix.
-\begin{figure}
+\begin{figure}[htbp]
\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]
+\begin{table}[htbp]
\centering
\begin{tabular}{|c|c|c|c|}
\hline
\label{ch12:tab:04}
\end{table}
-\begin{table}
+\begin{table}[htbp]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
\chapterauthor{Alan Gray and Kevin Stratford}{EPCC, The University of Edinburgh}
-\chapter{Ludwig: multiple GPUs for a complex fluid lattice Boltzmann
+\chapter[Ludwig: multiple GPUs for a fluid lattice Boltzmann
+application]{Ludwig: multiple GPUs for a complex fluid lattice Boltzmann
application}
%\putbib[biblio]
%\newcommand{\fixme}[1]{{\bf #1}}
-\chapter[Numerical validation and performance optimization on GPUs in atomic physics]{Numerical validation and performance optimization on GPUs of an application in atomic physics}
+\chapter[Numerical validation on GPUs in atomic physics]{Numerical validation and performance optimization on GPUs of an application in atomic physics}
\label{chapter15}
\section{Introduction}\label{ch15:intro}
\chapterauthor{Guillaume Laville}{Femto-ST Institute, University of Franche-Comte, France}
\chapterauthor{Christophe Lang}{Femto-ST Institute, University of Franche-Comte, France}
\chapterauthor{Kamel Mazouzi}{Franche-Comte Computing Center, University of Franche-Comte, France}
-\chapterauthor{Nicolas Marilleau}{UMMISCO, Institut de Recherche pour le Développement (IRD), France}
+\chapterauthor{Nicolas Marilleau}{UMMISCO, Institut de Recherche pour le Developpement (IRD), France}
\chapterauthor{Bénédicte Herrmann}{Femto-ST Institute, University of Franche-Comte, France}
\chapterauthor{Laurent Philippe}{Femto-ST Institute, University of Franche-Comte, France}
\begin{figure}[!htb]
\begin{center}
\setlength\figureheight{0.35\textwidth}
- \setlength\figurewidth{0.37\textwidth}
+ \setlength\figurewidth{0.35\textwidth}
\subfigure[Performance scaling]{
% {\small\input{Chapters/chapter7/figures/PararealScaletestGTX590.tikz}}
- \includegraphics[width=0.5\textwidth]{Chapters/chapter7/figures/PararealScaletestGTX590_conv.pdf}
+ \includegraphics[width=0.47\textwidth]{Chapters/chapter7/figures/PararealScaletestGTX590_conv.pdf}
}
\subfigure[Speedup]{
% {\small\input{Chapters/chapter7/figures/PararealSpeedupGTX590.tikz}}
- \includegraphics[width=0.5\textwidth]{Chapters/chapter7/figures/PararealSpeedupGTX590_conv.pdf}
+ \includegraphics[width=0.47\textwidth]{Chapters/chapter7/figures/PararealSpeedupGTX590_conv.pdf}
}
\end{center}
\caption[Parareal absolute timings and parareal speedup.]{(a) Parareal absolute timings for an increasingly number of water waves traveling one wave length, each wave resolution is ($33\times 9$). (b) Parareal speedup for two to sixteen compute nodes compared to the purely sequential single GPU solver. Notice how insensitive the parareal scheme is to the size of the problem solved. Test environment 2.}\label{ch7:fig:DDPA_SPEEDUP}
