\@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{toc}{\contentsline {chapter}{\numberline {12}Solving sparse linear systems with GMRES and CG methods on GPU clusters}{293}}
\@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{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{toc}{\contentsline {section}{\numberline {11.4}Experimental results}{262}}
-\newlabel{ch12:sec:04}{{11.4}{262}}
-\@writefile{lof}{\contentsline {figure}{\numberline {11.5}{\ignorespaces Sketches of sparse matrices chosen from the Davis's 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's collection.\relax }}{264}}
-\newlabel{ch12:tab:01}{{11.1}{264}}
-\@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 }}{265}}
-\newlabel{ch12:tab:03}{{11.3}{265}}
-\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's collection.\relax }}{267}}
-\newlabel{ch12:tab:04}{{11.4}{267}}
-\@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{toc}{\contentsline {section}{\numberline {11.5}Hypergraph partitioning}{267}}
-\newlabel{ch12:sec: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 }}{268}}
-\newlabel{ch12:tab:06}{{11.6}{268}}
-\@writefile{lot}{\contentsline {table}{\numberline {11.7}{\ignorespaces Main characteristics of sparse five-bands matrices generated from those of the Davis's collection.\relax }}{268}}
-\newlabel{ch12:tab:07}{{11.7}{268}}
-\@writefile{lof}{\contentsline {figure}{\numberline {11.7}{\ignorespaces Parallel generation of a large sparse five-bands matrix by four computing nodes.\relax }}{269}}
-\newlabel{ch12:fig:07}{{11.7}{269}}
-\@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 }}{269}}
-\newlabel{ch12:tab:08}{{11.8}{269}}
-\@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 }}{270}}
-\newlabel{ch12:tab:09}{{11.9}{270}}
-\@writefile{lof}{\contentsline {figure}{\numberline {11.8}{\ignorespaces An example of the hypergraph partitioning of a sparse matrix decomposed between three computing nodes.\relax }}{271}}
-\newlabel{ch12:fig:08}{{11.8}{271}}
-\@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 }}{272}}
-\newlabel{ch12:tab:10}{{11.10}{272}}
-\@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 }}{273}}
-\newlabel{ch12:tab:11}{{11.11}{273}}
-\@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 }}{274}}
-\newlabel{ch12:tab:12}{{11.12}{274}}
-\newlabel{ch12:fig:09.01}{{11.9(a)}{275}}
-\newlabel{sub@ch12:fig:09.01}{{(a)}{275}}
-\newlabel{ch12:fig:09.02}{{11.9(b)}{275}}
-\newlabel{sub@ch12:fig:09.02}{{(b)}{275}}
-\@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 }}{275}}
-\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {Sparse band matrices}}}{275}}
-\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {Sparse five-bands matrices}}}{275}}
-\newlabel{ch12:fig:09}{{11.9}{275}}
-\@writefile{toc}{\contentsline {section}{\numberline {11.6}Conclusion}{275}}
-\newlabel{ch12:sec:06}{{11.6}{275}}
-\@writefile{toc}{\contentsline {section}{Bibliography}{276}}
+\newlabel{ch12}{{12}{293}}
+\@writefile{toc}{\contentsline {section}{\numberline {12.1}Introduction}{293}}
+\newlabel{ch12:sec:01}{{12.1}{293}}
+\@writefile{toc}{\contentsline {section}{\numberline {12.2}Krylov iterative methods}{294}}
+\newlabel{ch12:sec:02}{{12.2}{294}}
+\newlabel{ch12:eq:01}{{12.1}{294}}
+\newlabel{ch12:eq:02}{{12.2}{294}}
+\newlabel{ch12:eq:03}{{12.3}{294}}
+\newlabel{ch12:eq:11}{{12.4}{295}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {12.2.1}CG method}{295}}
+\newlabel{ch12:sec:02.01}{{12.2.1}{295}}
+\newlabel{ch12:eq:04}{{12.5}{295}}
+\newlabel{ch12:eq:05}{{12.6}{295}}
+\newlabel{ch12:eq:06}{{12.7}{295}}
+\newlabel{ch12:eq:07}{{12.8}{295}}
+\newlabel{ch12:eq:08}{{12.9}{295}}
+\newlabel{ch12:eq:09}{{12.10}{295}}
+\@writefile{loa}{\contentsline {algocf}{\numberline {14}{\ignorespaces Left-preconditioned CG method\relax }}{296}}
+\newlabel{ch12:alg:01}{{14}{296}}
+\newlabel{ch12:eq:10}{{12.11}{296}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {12.2.2}GMRES method}{297}}
+\newlabel{ch12:sec:02.02}{{12.2.2}{297}}
+\newlabel{ch12:eq:12}{{12.12}{297}}
+\newlabel{ch12:eq:13}{{12.13}{297}}
+\newlabel{ch12:eq:14}{{12.14}{297}}
+\newlabel{ch12:eq:15}{{12.15}{297}}
+\newlabel{ch12:eq:16}{{12.16}{297}}
+\newlabel{ch12:eq:17}{{12.17}{297}}
+\newlabel{ch12:eq:18}{{12.18}{297}}
+\newlabel{ch12:eq:19}{{12.19}{297}}
+\@writefile{loa}{\contentsline {algocf}{\numberline {15}{\ignorespaces Left-preconditioned GMRES method with restarts\relax }}{298}}
+\newlabel{ch12:alg:02}{{15}{298}}
+\@writefile{toc}{\contentsline {section}{\numberline {12.3}Parallel implementation on a GPU cluster}{299}}
+\newlabel{ch12:sec:03}{{12.3}{299}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {12.3.1}Data partitioning}{299}}
+\newlabel{ch12:sec:03.01}{{12.3.1}{299}}
+\@writefile{lof}{\contentsline {figure}{\numberline {12.1}{\ignorespaces A data partitioning of the sparse matrix $A$, the solution vector $x$ and the right-hand side $b$ into four portions.\relax }}{300}}
+\newlabel{ch12:fig:01}{{12.1}{300}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {12.3.2}GPU computing}{300}}
+\newlabel{ch12:sec:03.02}{{12.3.2}{300}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {12.3.3}Data communications}{301}}
+\newlabel{ch12:sec:03.03}{{12.3.3}{301}}
+\@writefile{lof}{\contentsline {figure}{\numberline {12.2}{\ignorespaces Data exchanges between \textit {Node 1} and its neighbors \textit {Node 0}, \textit {Node 2} and \textit {Node 3}.\relax }}{302}}
+\newlabel{ch12:fig:02}{{12.2}{302}}
+\@writefile{lof}{\contentsline {figure}{\numberline {12.3}{\ignorespaces Columns reordering of a sparse sub-matrix.\relax }}{303}}
+\newlabel{ch12:fig:03}{{12.3}{303}}
+\@writefile{toc}{\contentsline {section}{\numberline {12.4}Experimental results}{304}}
+\newlabel{ch12:sec:04}{{12.4}{304}}
+\@writefile{lof}{\contentsline {figure}{\numberline {12.4}{\ignorespaces General scheme of the GPU cluster of tests composed of six machines, each with two GPUs.\relax }}{304}}
+\newlabel{ch12:fig:04}{{12.4}{304}}
+\@writefile{lof}{\contentsline {figure}{\numberline {12.5}{\ignorespaces Sketches of sparse matrices chosen from the Davis collection.\relax }}{305}}
+\newlabel{ch12:fig:05}{{12.5}{305}}
+\@writefile{lot}{\contentsline {table}{\numberline {12.1}{\ignorespaces Main characteristics of sparse matrices chosen from the Davis collection.\relax }}{305}}
+\newlabel{ch12:tab:01}{{12.1}{305}}
+\@writefile{lot}{\contentsline {table}{\numberline {12.2}{\ignorespaces Performances of the parallel CG method on a cluster of 24 CPU cores vs. on a cluster of 12 GPUs.\relax }}{306}}
+\newlabel{ch12:tab:02}{{12.2}{306}}
+\@writefile{lot}{\contentsline {table}{\numberline {12.3}{\ignorespaces Performances of the parallel GMRES method on a cluster 24 CPU cores vs. on cluster of 12 GPUs.\relax }}{306}}
+\newlabel{ch12:tab:03}{{12.3}{306}}
+\newlabel{ch12:eq:20}{{12.20}{307}}
+\@writefile{lof}{\contentsline {figure}{\numberline {12.6}{\ignorespaces Parallel generation of a large sparse matrix by four computing nodes.\relax }}{308}}
+\newlabel{ch12:fig:06}{{12.6}{308}}
+\@writefile{lot}{\contentsline {table}{\numberline {12.4}{\ignorespaces Main characteristics of sparse banded matrices generated from those of the Davis collection.\relax }}{308}}
+\newlabel{ch12:tab:04}{{12.4}{308}}
+\@writefile{lot}{\contentsline {table}{\numberline {12.5}{\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 }}{309}}
+\newlabel{ch12:tab:05}{{12.5}{309}}
+\@writefile{toc}{\contentsline {section}{\numberline {12.5}Conclusion}{309}}
+\newlabel{ch12:sec:05}{{12.5}{309}}
+\@writefile{lot}{\contentsline {table}{\numberline {12.6}{\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 }}{310}}
+\newlabel{ch12:tab:06}{{12.6}{310}}
+\@writefile{toc}{\contentsline {section}{Bibliography}{310}}
\@setckpt{Chapters/chapter12/ch12}{
-\setcounter{page}{278}
-\setcounter{equation}{25}
-\setcounter{enumi}{4}
+\setcounter{page}{312}
+\setcounter{equation}{22}
+\setcounter{enumi}{2}
\setcounter{enumii}{0}
\setcounter{enumiii}{0}
-\setcounter{enumiv}{15}
+\setcounter{enumiv}{10}
\setcounter{footnote}{0}
\setcounter{mpfootnote}{0}
-\setcounter{part}{1}
-\setcounter{chapter}{11}
-\setcounter{section}{6}
+\setcounter{part}{5}
+\setcounter{chapter}{12}
+\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{lstnumber}{50}
\setcounter{ContinuedFloat}{0}
\setcounter{AlgoLine}{29}
-\setcounter{algocfline}{10}
-\setcounter{algocfproc}{10}
-\setcounter{algocf}{10}
+\setcounter{algocfline}{15}
+\setcounter{algocfproc}{15}
+\setcounter{algocf}{15}
+\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}
