X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/book_gpu.git/blobdiff_plain/8e13f9e62b8323f091076223183e5436646a87e7..04efb8c2e1780f52f54ac8d64ee67c68eb104a36:/BookGPU/Chapters/chapter12/ch12.aux?ds=sidebyside diff --git a/BookGPU/Chapters/chapter12/ch12.aux b/BookGPU/Chapters/chapter12/ch12.aux index 97242d0..2252156 100644 --- a/BookGPU/Chapters/chapter12/ch12.aux +++ b/BookGPU/Chapters/chapter12/ch12.aux @@ -3,81 +3,81 @@ \@writefile{toc}{\author{Rapha\IeC {\"e}l Couturier}{}} \@writefile{toc}{\author{Jacques Bahi}{}} \@writefile{loa}{\addvspace {10\p@ }} -\@writefile{toc}{\contentsline {chapter}{\numberline {10}Solving sparse linear systems with GMRES and CG methods on GPU clusters}{215}} +\@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@ }} -\newlabel{ch12}{{10}{215}} -\@writefile{toc}{\contentsline {section}{\numberline {10.1}Introduction}{215}} -\newlabel{ch12:sec:01}{{10.1}{215}} -\@writefile{toc}{\contentsline {section}{\numberline {10.2}Krylov iterative methods}{216}} -\newlabel{ch12:sec:02}{{10.2}{216}} -\newlabel{ch12:eq:01}{{10.1}{216}} -\newlabel{ch12:eq:02}{{10.2}{216}} -\newlabel{ch12:eq:03}{{10.3}{216}} -\newlabel{ch12:eq:11}{{10.4}{217}} -\@writefile{toc}{\contentsline {subsection}{\numberline {10.2.1}CG method}{217}} -\newlabel{ch12:sec:02.01}{{10.2.1}{217}} -\newlabel{ch12:eq:04}{{10.5}{217}} -\newlabel{ch12:eq:05}{{10.6}{217}} -\newlabel{ch12:eq:06}{{10.7}{217}} -\newlabel{ch12:eq:07}{{10.8}{217}} -\newlabel{ch12:eq:08}{{10.9}{217}} -\newlabel{ch12:eq:09}{{10.10}{217}} -\@writefile{loa}{\contentsline {algocf}{\numberline {9}{\ignorespaces Left-preconditioned CG method\relax }}{218}} -\newlabel{ch12:alg:01}{{9}{218}} -\newlabel{ch12:eq:10}{{10.11}{218}} -\@writefile{toc}{\contentsline {subsection}{\numberline {10.2.2}GMRES method}{219}} -\newlabel{ch12:sec:02.02}{{10.2.2}{219}} -\newlabel{ch12:eq:12}{{10.12}{219}} -\newlabel{ch12:eq:13}{{10.13}{219}} -\newlabel{ch12:eq:14}{{10.14}{219}} -\newlabel{ch12:eq:15}{{10.15}{219}} -\newlabel{ch12:eq:16}{{10.16}{219}} -\newlabel{ch12:eq:17}{{10.17}{219}} -\newlabel{ch12:eq:18}{{10.18}{219}} -\newlabel{ch12:eq:19}{{10.19}{219}} -\@writefile{loa}{\contentsline {algocf}{\numberline {10}{\ignorespaces Left-preconditioned GMRES method with restarts\relax }}{220}} -\newlabel{ch12:alg:02}{{10}{220}} -\@writefile{toc}{\contentsline {section}{\numberline {10.3}Parallel implementation on a GPU cluster}{221}} -\newlabel{ch12:sec:03}{{10.3}{221}} -\@writefile{toc}{\contentsline {subsection}{\numberline {10.3.1}Data partitioning}{221}} -\newlabel{ch12:sec:03.01}{{10.3.1}{221}} -\@writefile{lof}{\contentsline {figure}{\numberline {10.1}{\ignorespaces A data partitioning of the sparse matrix $A$, the solution vector $x$ and the right-hand side $b$ into four portions.\relax }}{222}} -\newlabel{ch12:fig:01}{{10.1}{222}} -\@writefile{toc}{\contentsline {subsection}{\numberline {10.3.2}GPU computing}{222}} -\newlabel{ch12:sec:03.02}{{10.3.2}{222}} -\@writefile{toc}{\contentsline {subsection}{\numberline {10.3.3}Data communications}{223}} -\newlabel{ch12:sec:03.03}{{10.3.3}{223}} -\@writefile{lof}{\contentsline {figure}{\numberline {10.2}{\ignorespaces Data exchanges between \textit {Node 1} and its neighbors \textit {Node 0}, \textit {Node 2} and \textit {Node 3}.\relax }}{224}} -\newlabel{ch12:fig:02}{{10.2}{224}} -\@writefile{lof}{\contentsline {figure}{\numberline {10.3}{\ignorespaces Columns reordering of a sparse sub-matrix.\relax }}{225}} -\newlabel{ch12:fig:03}{{10.3}{225}} -\@writefile{toc}{\contentsline {section}{\numberline {10.4}Experimental results}{226}} -\newlabel{ch12:sec:04}{{10.4}{226}} -\@writefile{lof}{\contentsline {figure}{\numberline {10.4}{\ignorespaces General scheme of the GPU cluster of tests composed of six machines, each with two GPUs.\relax }}{226}} -\newlabel{ch12:fig:04}{{10.4}{226}} -\@writefile{lof}{\contentsline {figure}{\numberline {10.5}{\ignorespaces Sketches of sparse matrices chosen from the Davis collection.\relax }}{227}} -\newlabel{ch12:fig:05}{{10.5}{227}} -\@writefile{lot}{\contentsline {table}{\numberline {10.1}{\ignorespaces Main characteristics of sparse matrices chosen from the Davis collection.\relax }}{227}} -\newlabel{ch12:tab:01}{{10.1}{227}} -\@writefile{lot}{\contentsline {table}{\numberline {10.2}{\ignorespaces Performances of the parallel CG method on a cluster of 24 CPU cores vs. on a cluster of 12 GPUs.\relax }}{228}} -\newlabel{ch12:tab:02}{{10.2}{228}} -\@writefile{lot}{\contentsline {table}{\numberline {10.3}{\ignorespaces Performances of the parallel GMRES method on a cluster 24 CPU cores vs. on cluster of 12 GPUs.\relax }}{228}} -\newlabel{ch12:tab:03}{{10.3}{228}} -\newlabel{ch12:eq:20}{{10.20}{229}} -\@writefile{lof}{\contentsline {figure}{\numberline {10.6}{\ignorespaces Parallel generation of a large sparse matrix by four computing nodes.\relax }}{230}} -\newlabel{ch12:fig:06}{{10.6}{230}} -\@writefile{lot}{\contentsline {table}{\numberline {10.4}{\ignorespaces Main characteristics of sparse banded matrices generated from those of the Davis collection.\relax }}{230}} -\newlabel{ch12:tab:04}{{10.4}{230}} -\@writefile{lot}{\contentsline {table}{\numberline {10.5}{\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 }}{231}} -\newlabel{ch12:tab:05}{{10.5}{231}} -\@writefile{toc}{\contentsline {section}{\numberline {10.5}Conclusion}{231}} -\newlabel{ch12:sec:05}{{10.5}{231}} -\@writefile{lot}{\contentsline {table}{\numberline {10.6}{\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 }}{232}} -\newlabel{ch12:tab:06}{{10.6}{232}} -\@writefile{toc}{\contentsline {section}{Bibliography}{232}} +\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}{234} +\setcounter{page}{278} \setcounter{equation}{22} \setcounter{enumi}{4} \setcounter{enumii}{0} @@ -86,7 +86,7 @@ \setcounter{footnote}{0} \setcounter{mpfootnote}{0} \setcounter{part}{5} -\setcounter{chapter}{10} +\setcounter{chapter}{11} \setcounter{section}{5} \setcounter{subsection}{0} \setcounter{subsubsection}{0} @@ -95,7 +95,7 @@ \setcounter{figure}{6} \setcounter{table}{6} \setcounter{numauthors}{0} -\setcounter{parentequation}{8} +\setcounter{parentequation}{46} \setcounter{subfigure}{0} \setcounter{lofdepth}{1} \setcounter{subtable}{0}