\relax 
\@writefile{toc}{\author{Lilia Ziane Khodja}{}}
\@writefile{toc}{\author{Rapha\IeC {\"e}l Couturier}{}}
\@writefile{toc}{\author{Jacques Bahi}{}}
\@writefile{loa}{\addvspace {10\p@ }}
\@writefile{toc}{\contentsline {chapter}{\numberline {11}Solving sparse linear systems with GMRES and CG methods on GPU clusters}{251}}
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\newlabel{ch12}{{11}{251}}
\@writefile{toc}{\contentsline {section}{\numberline {11.1}Introduction}{251}}
\newlabel{ch12:sec:01}{{11.1}{251}}
\@writefile{toc}{\contentsline {section}{\numberline {11.2}Krylov iterative methods}{252}}
\newlabel{ch12:sec:02}{{11.2}{252}}
\newlabel{ch12:eq:01}{{11.1}{252}}
\newlabel{ch12:eq:02}{{11.2}{252}}
\newlabel{ch12:eq:03}{{11.3}{252}}
\newlabel{ch12:eq:11}{{11.4}{253}}
\@writefile{toc}{\contentsline {subsection}{\numberline {11.2.1}CG method}{253}}
\newlabel{ch12:sec:02.01}{{11.2.1}{253}}
\newlabel{ch12:eq:04}{{11.5}{253}}
\newlabel{ch12:eq:05}{{11.6}{253}}
\newlabel{ch12:eq:06}{{11.7}{253}}
\newlabel{ch12:eq:07}{{11.8}{253}}
\newlabel{ch12:eq:08}{{11.9}{253}}
\newlabel{ch12:eq:09}{{11.10}{253}}
\@writefile{loa}{\contentsline {algocf}{\numberline {9}{\ignorespaces Left-preconditioned CG method\relax }}{254}}
\newlabel{ch12:alg:01}{{9}{254}}
\newlabel{ch12:eq:10}{{11.11}{254}}
\@writefile{toc}{\contentsline {subsection}{\numberline {11.2.2}GMRES method}{255}}
\newlabel{ch12:sec:02.02}{{11.2.2}{255}}
\newlabel{ch12:eq:12}{{11.12}{255}}
\newlabel{ch12:eq:13}{{11.13}{255}}
\newlabel{ch12:eq:14}{{11.14}{255}}
\newlabel{ch12:eq:15}{{11.15}{255}}
\newlabel{ch12:eq:16}{{11.16}{255}}
\newlabel{ch12:eq:17}{{11.17}{255}}
\newlabel{ch12:eq:18}{{11.18}{255}}
\newlabel{ch12:eq:19}{{11.19}{255}}
\@writefile{loa}{\contentsline {algocf}{\numberline {10}{\ignorespaces Left-preconditioned GMRES method with restarts\relax }}{256}}
\newlabel{ch12:alg:02}{{10}{256}}
\@writefile{toc}{\contentsline {section}{\numberline {11.3}Parallel implementation on a GPU cluster}{257}}
\newlabel{ch12:sec:03}{{11.3}{257}}
\@writefile{toc}{\contentsline {subsection}{\numberline {11.3.1}Data partitioning}{257}}
\newlabel{ch12:sec:03.01}{{11.3.1}{257}}
\@writefile{lof}{\contentsline {figure}{\numberline {11.1}{\ignorespaces A data partitioning of the sparse matrix $A$, the solution vector $x$ and the right-hand side $b$ into four portions.\relax }}{258}}
\newlabel{ch12:fig:01}{{11.1}{258}}
\@writefile{toc}{\contentsline {subsection}{\numberline {11.3.2}GPU computing}{258}}
\newlabel{ch12:sec:03.02}{{11.3.2}{258}}
\@writefile{toc}{\contentsline {subsection}{\numberline {11.3.3}Data communications}{259}}
\newlabel{ch12:sec:03.03}{{11.3.3}{259}}
\@writefile{lof}{\contentsline {figure}{\numberline {11.2}{\ignorespaces Data exchanges between \textit  {Node 1} and its neighbors \textit  {Node 0}, \textit  {Node 2} and \textit  {Node 3}.\relax }}{260}}
\newlabel{ch12:fig:02}{{11.2}{260}}
\@writefile{lof}{\contentsline {figure}{\numberline {11.3}{\ignorespaces Columns reordering of a sparse sub-matrix.\relax }}{261}}
\newlabel{ch12:fig:03}{{11.3}{261}}
\@writefile{toc}{\contentsline {section}{\numberline {11.4}Experimental results}{262}}
\newlabel{ch12:sec:04}{{11.4}{262}}
\@writefile{lof}{\contentsline {figure}{\numberline {11.4}{\ignorespaces General scheme of the GPU cluster of tests composed of six machines, each with two GPUs.\relax }}{262}}
\newlabel{ch12:fig:04}{{11.4}{262}}
\@writefile{lof}{\contentsline {figure}{\numberline {11.5}{\ignorespaces Sketches of sparse matrices chosen from the Davis collection.\relax }}{263}}
\newlabel{ch12:fig:05}{{11.5}{263}}
\@writefile{lot}{\contentsline {table}{\numberline {11.1}{\ignorespaces Main characteristics of sparse matrices chosen from the Davis collection.\relax }}{263}}
\newlabel{ch12:tab:01}{{11.1}{263}}
\@writefile{lot}{\contentsline {table}{\numberline {11.2}{\ignorespaces Performances of the parallel CG method on a cluster of 24 CPU cores vs. on a cluster of 12 GPUs.\relax }}{264}}
\newlabel{ch12:tab:02}{{11.2}{264}}
\@writefile{lot}{\contentsline {table}{\numberline {11.3}{\ignorespaces Performances of the parallel GMRES method on a cluster 24 CPU cores vs. on cluster of 12 GPUs.\relax }}{264}}
\newlabel{ch12:tab:03}{{11.3}{264}}
\newlabel{ch12:eq:20}{{11.20}{265}}
\@writefile{lof}{\contentsline {figure}{\numberline {11.6}{\ignorespaces Parallel generation of a large sparse matrix by four computing nodes.\relax }}{266}}
\newlabel{ch12:fig:06}{{11.6}{266}}
\@writefile{lot}{\contentsline {table}{\numberline {11.4}{\ignorespaces Main characteristics of sparse banded matrices generated from those of the Davis collection.\relax }}{266}}
\newlabel{ch12:tab:04}{{11.4}{266}}
\@writefile{lot}{\contentsline {table}{\numberline {11.5}{\ignorespaces Performances of the parallel CG method for solving linear systems associated to sparse banded matrices on a cluster of 24 CPU cores vs. on a cluster of 12 GPUs.\relax }}{267}}
\newlabel{ch12:tab:05}{{11.5}{267}}
\@writefile{lot}{\contentsline {table}{\numberline {11.6}{\ignorespaces Performances of the parallel GMRES method for solving linear systems associated to sparse banded matrices on a cluster of 24 CPU cores vs. on a cluster of 12 GPUs.\relax }}{267}}
\newlabel{ch12:tab:06}{{11.6}{267}}
\@writefile{toc}{\contentsline {section}{\numberline {11.5}Conclusion}{267}}
\newlabel{ch12:sec:05}{{11.5}{267}}
\@writefile{toc}{\contentsline {section}{Bibliography}{268}}
\@setckpt{Chapters/chapter12/ch12}{
\setcounter{page}{270}
\setcounter{equation}{22}
\setcounter{enumi}{4}
\setcounter{enumii}{0}
\setcounter{enumiii}{0}
\setcounter{enumiv}{10}
\setcounter{footnote}{0}
\setcounter{mpfootnote}{0}
\setcounter{part}{1}
\setcounter{chapter}{11}
\setcounter{section}{5}
\setcounter{subsection}{0}
\setcounter{subsubsection}{0}
\setcounter{paragraph}{0}
\setcounter{subparagraph}{0}
\setcounter{figure}{6}
\setcounter{table}{6}
\setcounter{numauthors}{0}
\setcounter{parentequation}{46}
\setcounter{subfigure}{0}
\setcounter{lofdepth}{1}
\setcounter{subtable}{0}
\setcounter{lotdepth}{1}
\setcounter{lstnumber}{50}
\setcounter{ContinuedFloat}{0}
\setcounter{AlgoLine}{29}
\setcounter{algocfline}{10}
\setcounter{algocfproc}{10}
\setcounter{algocf}{10}
\setcounter{proposition}{1}
\setcounter{theorem}{0}
\setcounter{exercise}{0}
\setcounter{example}{0}
\setcounter{definition}{0}
\setcounter{proof}{1}
\setcounter{lstlisting}{0}
}