X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/book_gpu.git/blobdiff_plain/c8b308b236018000b34d6d58d5bdc4cb8313111b..04efb8c2e1780f52f54ac8d64ee67c68eb104a36:/BookGPU/Chapters/chapter12/ch12.aux diff --git a/BookGPU/Chapters/chapter12/ch12.aux b/BookGPU/Chapters/chapter12/ch12.aux index f25a484..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 {11}Solving sparse linear systems with GMRES and CG methods on GPU clusters}{251}} +\@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}{{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}} +\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}{270} +\setcounter{page}{278} \setcounter{equation}{22} \setcounter{enumi}{4} \setcounter{enumii}{0} @@ -85,7 +85,7 @@ \setcounter{enumiv}{10} \setcounter{footnote}{0} \setcounter{mpfootnote}{0} -\setcounter{part}{1} +\setcounter{part}{5} \setcounter{chapter}{11} \setcounter{section}{5} \setcounter{subsection}{0} @@ -106,6 +106,13 @@ \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}