\relax
-\@writefile{toc}{\author{}{}}
+\@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}{249}}
+\@writefile{toc}{\contentsline {chapter}{\numberline {11}Solving sparse linear systems with GMRES and CG methods on GPU clusters}{259}}
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
-\@writefile{toc}{\contentsline {section}{\numberline {11.1}Introduction}{249}}
-\newlabel{sec:01}{{11.1}{249}}
-\@writefile{toc}{\contentsline {section}{\numberline {11.2}Krylov iterative methods}{250}}
-\newlabel{sec:02}{{11.2}{250}}
-\newlabel{eq:01}{{11.1}{250}}
-\newlabel{eq:02}{{11.2}{250}}
-\newlabel{eq:03}{{11.3}{250}}
-\newlabel{eq:11}{{11.4}{251}}
-\@writefile{toc}{\contentsline {subsection}{\numberline {11.2.1}CG method}{251}}
-\newlabel{sec:02.01}{{11.2.1}{251}}
-\newlabel{eq:04}{{11.5}{251}}
-\newlabel{eq:05}{{11.6}{251}}
-\newlabel{eq:06}{{11.7}{251}}
-\newlabel{eq:07}{{11.8}{251}}
-\newlabel{eq:08}{{11.9}{251}}
-\newlabel{eq:09}{{11.10}{251}}
-\@writefile{loa}{\contentsline {algocf}{\numberline {9}{\ignorespaces Left-preconditioned CG method\relax }}{252}}
-\newlabel{alg:01}{{9}{252}}
-\newlabel{eq:10}{{11.11}{252}}
-\@writefile{toc}{\contentsline {subsection}{\numberline {11.2.2}GMRES method}{253}}
-\newlabel{sec:02.02}{{11.2.2}{253}}
-\newlabel{eq:12}{{11.12}{253}}
-\newlabel{eq:13}{{11.13}{253}}
-\newlabel{eq:14}{{11.14}{253}}
-\newlabel{eq:15}{{11.15}{253}}
-\newlabel{eq:16}{{11.16}{253}}
-\newlabel{eq:17}{{11.17}{253}}
-\newlabel{eq:18}{{11.18}{253}}
-\newlabel{eq:19}{{11.19}{253}}
-\@writefile{loa}{\contentsline {algocf}{\numberline {10}{\ignorespaces Left-preconditioned GMRES method with restarts\relax }}{254}}
-\newlabel{alg:02}{{10}{254}}
-\@writefile{toc}{\contentsline {section}{\numberline {11.3}Parallel implementation on a GPU cluster}{255}}
-\newlabel{sec:03}{{11.3}{255}}
-\@writefile{toc}{\contentsline {subsection}{\numberline {11.3.1}Data partitioning}{255}}
-\newlabel{sec:03.01}{{11.3.1}{255}}
-\@writefile{lof}{\contentsline {figure}{\numberline {11.1}{\ignorespaces A data partitioning of the sparse matrix $A$, the solution vector $x$ and the right-hand side $b$ into four portions.\relax }}{256}}
-\newlabel{fig:01}{{11.1}{256}}
-\@writefile{toc}{\contentsline {subsection}{\numberline {11.3.2}GPU computing}{256}}
-\newlabel{sec:03.02}{{11.3.2}{256}}
-\@writefile{toc}{\contentsline {subsection}{\numberline {11.3.3}Data communications}{257}}
-\newlabel{sec:03.03}{{11.3.3}{257}}
-\@writefile{lof}{\contentsline {figure}{\numberline {11.2}{\ignorespaces Data exchanges between \textit {Node 1} and its neighbors \textit {Node 0}, \textit {Node 2} and \textit {Node 3}.\relax }}{258}}
-\newlabel{fig:02}{{11.2}{258}}
-\@writefile{lof}{\contentsline {figure}{\numberline {11.3}{\ignorespaces Columns reordering of a sparse sub-matrix.\relax }}{259}}
-\newlabel{fig:03}{{11.3}{259}}
-\@writefile{lof}{\contentsline {figure}{\numberline {11.4}{\ignorespaces General scheme of the GPU cluster of tests composed of six machines, each with two GPUs.\relax }}{260}}
-\newlabel{fig:04}{{11.4}{260}}
-\@writefile{toc}{\contentsline {section}{\numberline {11.4}Experimental results}{260}}
-\newlabel{sec:04}{{11.4}{260}}
-\@writefile{lof}{\contentsline {figure}{\numberline {11.5}{\ignorespaces Sketches of sparse matrices chosen from the Davis's collection.\relax }}{261}}
-\newlabel{fig:05}{{11.5}{261}}
-\@writefile{lot}{\contentsline {table}{\numberline {11.1}{\ignorespaces Main characteristics of sparse matrices chosen from the Davis's collection.\relax }}{262}}
-\newlabel{tab:01}{{11.1}{262}}
-\@writefile{lot}{\contentsline {table}{\numberline {11.2}{\ignorespaces Performances of the parallel CG method on a cluster of 24 CPU cores vs. on a cluster of 12 GPUs.\relax }}{262}}
-\newlabel{tab:02}{{11.2}{262}}
-\@writefile{lot}{\contentsline {table}{\numberline {11.3}{\ignorespaces Performances of the parallel GMRES method on a cluster 24 CPU cores vs. on cluster of 12 GPUs.\relax }}{263}}
-\newlabel{tab:03}{{11.3}{263}}
-\newlabel{eq:20}{{11.20}{263}}
-\@writefile{lof}{\contentsline {figure}{\numberline {11.6}{\ignorespaces Parallel generation of a large sparse matrix by four computing nodes.\relax }}{264}}
-\newlabel{fig:06}{{11.6}{264}}
-\@writefile{lot}{\contentsline {table}{\numberline {11.4}{\ignorespaces Main characteristics of sparse banded matrices generated from those of the Davis's collection.\relax }}{265}}
-\newlabel{tab:04}{{11.4}{265}}
-\@writefile{lot}{\contentsline {table}{\numberline {11.5}{\ignorespaces Performances of the parallel CG method for solving linear systems associated to sparse banded matrices on a cluster of 24 CPU cores vs. on a cluster of 12 GPUs.\relax }}{265}}
-\newlabel{tab:05}{{11.5}{265}}
-\@writefile{toc}{\contentsline {section}{\numberline {11.5}Hypergraph partitioning}{265}}
-\newlabel{sec:05}{{11.5}{265}}
-\@writefile{lot}{\contentsline {table}{\numberline {11.6}{\ignorespaces Performances of the parallel GMRES method for solving linear systems associated to sparse banded matrices on a cluster of 24 CPU cores vs. on a cluster of 12 GPUs.\relax }}{266}}
-\newlabel{tab:06}{{11.6}{266}}
-\@writefile{lot}{\contentsline {table}{\numberline {11.7}{\ignorespaces Main characteristics of sparse five-bands matrices generated from those of the Davis's collection.\relax }}{266}}
-\newlabel{tab:07}{{11.7}{266}}
-\@writefile{lof}{\contentsline {figure}{\numberline {11.7}{\ignorespaces Parallel generation of a large sparse five-bands matrix by four computing nodes.\relax }}{267}}
-\newlabel{fig:07}{{11.7}{267}}
-\@writefile{lot}{\contentsline {table}{\numberline {11.8}{\ignorespaces Performances of parallel CG solver for solving linear systems associated to sparse five-bands matrices on a cluster of 24 CPU cores vs. on a cluster of 12 GPUs\relax }}{267}}
-\newlabel{tab:08}{{11.8}{267}}
-\@writefile{lot}{\contentsline {table}{\numberline {11.9}{\ignorespaces Performances of parallel GMRES solver for solving linear systems associated to sparse five-bands matrices on a cluster of 24 CPU cores vs. on a cluster of 12 GPUs\relax }}{268}}
-\newlabel{tab:09}{{11.9}{268}}
-\@writefile{lof}{\contentsline {figure}{\numberline {11.8}{\ignorespaces An example of the hypergraph partitioning of a sparse matrix decomposed between three computing nodes.\relax }}{269}}
-\newlabel{fig:08}{{11.8}{269}}
-\@writefile{lot}{\contentsline {table}{\numberline {11.10}{\ignorespaces Performances of the parallel CG solver using hypergraph partitioning for solving linear systems associated to sparse five-bands matrices on a cluster of 24 CPU cores vs. on a cluster of 12 GPU.\relax }}{270}}
-\newlabel{tab:10}{{11.10}{270}}
-\@writefile{lot}{\contentsline {table}{\numberline {11.11}{\ignorespaces Performances of the parallel GMRES solver using hypergraph partitioning for solving linear systems associated to sparse five-bands matrices on a cluster of 24 CPU cores vs. on a cluster of 12 GPU.\relax }}{271}}
-\newlabel{tab:11}{{11.11}{271}}
-\@writefile{lot}{\contentsline {table}{\numberline {11.12}{\ignorespaces The total communication volume between 12 GPU computing nodes without and with the hypergraph partitioning method.\relax }}{272}}
-\newlabel{tab:12}{{11.12}{272}}
-\@writefile{lof}{\contentsline {figure}{\numberline {11.9}{\ignorespaces Weak-scaling of the parallel CG and GMRES solvers on a GPU cluster for solving large sparse linear systems.\relax }}{273}}
-\newlabel{fig:09}{{11.9}{273}}
-\@writefile{toc}{\contentsline {section}{\numberline {11.6}Conclusion}{273}}
-\newlabel{sec:06}{{11.6}{273}}
-\@writefile{toc}{\contentsline {section}{Bibliography}{274}}
+\newlabel{ch12}{{11}{259}}
+\@writefile{toc}{\contentsline {section}{\numberline {11.1}Introduction}{259}}
+\newlabel{ch12:sec:01}{{11.1}{259}}
+\@writefile{toc}{\contentsline {section}{\numberline {11.2}Krylov iterative methods}{260}}
+\newlabel{ch12:sec:02}{{11.2}{260}}
+\newlabel{ch12:eq:01}{{11.1}{260}}
+\newlabel{ch12:eq:02}{{11.2}{260}}
+\newlabel{ch12:eq:03}{{11.3}{260}}
+\newlabel{ch12:eq:11}{{11.4}{261}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {11.2.1}CG method}{261}}
+\newlabel{ch12:sec:02.01}{{11.2.1}{261}}
+\newlabel{ch12:eq:04}{{11.5}{261}}
+\newlabel{ch12:eq:05}{{11.6}{261}}
+\newlabel{ch12:eq:06}{{11.7}{261}}
+\newlabel{ch12:eq:07}{{11.8}{261}}
+\newlabel{ch12:eq:08}{{11.9}{261}}
+\newlabel{ch12:eq:09}{{11.10}{261}}
+\@writefile{loa}{\contentsline {algocf}{\numberline {9}{\ignorespaces Left-preconditioned CG method\relax }}{262}}
+\newlabel{ch12:alg:01}{{9}{262}}
+\newlabel{ch12:eq:10}{{11.11}{262}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {11.2.2}GMRES method}{263}}
+\newlabel{ch12:sec:02.02}{{11.2.2}{263}}
+\newlabel{ch12:eq:12}{{11.12}{263}}
+\newlabel{ch12:eq:13}{{11.13}{263}}
+\newlabel{ch12:eq:14}{{11.14}{263}}
+\newlabel{ch12:eq:15}{{11.15}{263}}
+\newlabel{ch12:eq:16}{{11.16}{263}}
+\newlabel{ch12:eq:17}{{11.17}{263}}
+\newlabel{ch12:eq:18}{{11.18}{263}}
+\newlabel{ch12:eq:19}{{11.19}{263}}
+\@writefile{loa}{\contentsline {algocf}{\numberline {10}{\ignorespaces Left-preconditioned GMRES method with restarts\relax }}{264}}
+\newlabel{ch12:alg:02}{{10}{264}}
+\@writefile{toc}{\contentsline {section}{\numberline {11.3}Parallel implementation on a GPU cluster}{265}}
+\newlabel{ch12:sec:03}{{11.3}{265}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {11.3.1}Data partitioning}{265}}
+\newlabel{ch12:sec:03.01}{{11.3.1}{265}}
+\@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 }}{266}}
+\newlabel{ch12:fig:01}{{11.1}{266}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {11.3.2}GPU computing}{266}}
+\newlabel{ch12:sec:03.02}{{11.3.2}{266}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {11.3.3}Data communications}{267}}
+\newlabel{ch12:sec:03.03}{{11.3.3}{267}}
+\@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 }}{268}}
+\newlabel{ch12:fig:02}{{11.2}{268}}
+\@writefile{lof}{\contentsline {figure}{\numberline {11.3}{\ignorespaces Columns reordering of a sparse sub-matrix.\relax }}{269}}
+\newlabel{ch12:fig:03}{{11.3}{269}}
+\@writefile{toc}{\contentsline {section}{\numberline {11.4}Experimental results}{270}}
+\newlabel{ch12:sec:04}{{11.4}{270}}
+\@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 }}{270}}
+\newlabel{ch12:fig:04}{{11.4}{270}}
+\@writefile{lof}{\contentsline {figure}{\numberline {11.5}{\ignorespaces Sketches of sparse matrices chosen from the Davis collection.\relax }}{271}}
+\newlabel{ch12:fig:05}{{11.5}{271}}
+\@writefile{lot}{\contentsline {table}{\numberline {11.1}{\ignorespaces Main characteristics of sparse matrices chosen from the Davis collection.\relax }}{271}}
+\newlabel{ch12:tab:01}{{11.1}{271}}
+\@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 }}{272}}
+\newlabel{ch12:tab:02}{{11.2}{272}}
+\@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 }}{272}}
+\newlabel{ch12:tab:03}{{11.3}{272}}
+\newlabel{ch12:eq:20}{{11.20}{273}}
+\@writefile{lof}{\contentsline {figure}{\numberline {11.6}{\ignorespaces Parallel generation of a large sparse matrix by four computing nodes.\relax }}{274}}
+\newlabel{ch12:fig:06}{{11.6}{274}}
+\@writefile{lot}{\contentsline {table}{\numberline {11.4}{\ignorespaces Main characteristics of sparse banded matrices generated from those of the Davis collection.\relax }}{274}}
+\newlabel{ch12:tab:04}{{11.4}{274}}
+\@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 }}{275}}
+\newlabel{ch12:tab:05}{{11.5}{275}}
+\@writefile{toc}{\contentsline {section}{\numberline {11.5}Conclusion}{275}}
+\newlabel{ch12:sec:05}{{11.5}{275}}
+\@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 }}{276}}
+\newlabel{ch12:tab:06}{{11.6}{276}}
+\@writefile{toc}{\contentsline {section}{Bibliography}{276}}
\@setckpt{Chapters/chapter12/ch12}{
-\setcounter{page}{276}
-\setcounter{equation}{25}
+\setcounter{page}{278}
+\setcounter{equation}{22}
\setcounter{enumi}{4}
\setcounter{enumii}{0}
\setcounter{enumiii}{0}
-\setcounter{enumiv}{15}
+\setcounter{enumiv}{10}
\setcounter{footnote}{0}
\setcounter{mpfootnote}{0}
-\setcounter{part}{1}
+\setcounter{part}{5}
\setcounter{chapter}{11}
-\setcounter{section}{6}
+\setcounter{section}{5}
\setcounter{subsection}{0}
\setcounter{subsubsection}{0}
\setcounter{paragraph}{0}
\setcounter{subparagraph}{0}
-\setcounter{figure}{9}
-\setcounter{table}{12}
+\setcounter{figure}{6}
+\setcounter{table}{6}
\setcounter{numauthors}{0}
\setcounter{parentequation}{46}
\setcounter{subfigure}{0}
\setcounter{algocfline}{10}
\setcounter{algocfproc}{10}
\setcounter{algocf}{10}
+\setcounter{nprt@mantissa@digitsbefore}{0}
+\setcounter{nprt@mantissa@digitsafter}{0}
+\setcounter{nprt@exponent@digitsbefore}{0}
+\setcounter{nprt@exponent@digitsafter}{0}
+\setcounter{nprt@digitsfirstblock}{0}
+\setcounter{nprt@blockcnt}{0}
+\setcounter{nprt@cntprint}{0}
\setcounter{proposition}{1}
\setcounter{theorem}{0}
\setcounter{exercise}{0}
