X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/book_gpu.git/blobdiff_plain/390b92667ac7e43b5bbd34c7fd7b8c8314a05b3d..b7e61e1f68e950462bff7221fe17c38d2ce7b3c0:/BookGPU/Chapters/chapter12/ch12.aux diff --git a/BookGPU/Chapters/chapter12/ch12.aux b/BookGPU/Chapters/chapter12/ch12.aux index 82783b4..f4f6c5c 100644 --- a/BookGPU/Chapters/chapter12/ch12.aux +++ b/BookGPU/Chapters/chapter12/ch12.aux @@ -3,90 +3,90 @@ \@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}{259}} +\@writefile{toc}{\contentsline {chapter}{\numberline {12}Solving sparse linear systems with GMRES and CG methods on GPU clusters}{295}} \@writefile{lof}{\addvspace {10\p@ }} \@writefile{lot}{\addvspace {10\p@ }} -\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{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 }}{275}} -\newlabel{ch12:tab:06}{{11.6}{275}} -\@writefile{toc}{\contentsline {section}{\numberline {11.5}Conclusion}{275}} -\newlabel{ch12:sec:05}{{11.5}{275}} -\@writefile{toc}{\contentsline {section}{Bibliography}{276}} +\newlabel{ch12}{{12}{295}} +\@writefile{toc}{\contentsline {section}{\numberline {12.1}Introduction}{295}} +\newlabel{ch12:sec:01}{{12.1}{295}} +\@writefile{toc}{\contentsline {section}{\numberline {12.2}Krylov iterative methods}{296}} +\newlabel{ch12:sec:02}{{12.2}{296}} +\newlabel{ch12:eq:01}{{12.1}{296}} +\newlabel{ch12:eq:02}{{12.2}{296}} +\newlabel{ch12:eq:03}{{12.3}{296}} +\newlabel{ch12:eq:11}{{12.4}{297}} +\@writefile{toc}{\contentsline {subsection}{\numberline {12.2.1}CG method}{297}} +\newlabel{ch12:sec:02.01}{{12.2.1}{297}} +\newlabel{ch12:eq:04}{{12.5}{297}} +\newlabel{ch12:eq:05}{{12.6}{297}} +\newlabel{ch12:eq:06}{{12.7}{297}} +\newlabel{ch12:eq:07}{{12.8}{297}} +\newlabel{ch12:eq:08}{{12.9}{297}} +\newlabel{ch12:eq:09}{{12.10}{297}} +\@writefile{loa}{\contentsline {algocf}{\numberline {14}{\ignorespaces Left-preconditioned CG method\relax }}{298}} +\newlabel{ch12:alg:01}{{14}{298}} +\newlabel{ch12:eq:10}{{12.11}{298}} +\@writefile{toc}{\contentsline {subsection}{\numberline {12.2.2}GMRES method}{299}} +\newlabel{ch12:sec:02.02}{{12.2.2}{299}} +\newlabel{ch12:eq:12}{{12.12}{299}} +\newlabel{ch12:eq:13}{{12.13}{299}} +\newlabel{ch12:eq:14}{{12.14}{299}} +\newlabel{ch12:eq:15}{{12.15}{299}} +\newlabel{ch12:eq:16}{{12.16}{299}} +\newlabel{ch12:eq:17}{{12.17}{299}} +\newlabel{ch12:eq:18}{{12.18}{299}} +\newlabel{ch12:eq:19}{{12.19}{299}} +\@writefile{loa}{\contentsline {algocf}{\numberline {15}{\ignorespaces Left-preconditioned GMRES method with restarts\relax }}{300}} +\newlabel{ch12:alg:02}{{15}{300}} +\@writefile{toc}{\contentsline {section}{\numberline {12.3}Parallel implementation on a GPU cluster}{301}} +\newlabel{ch12:sec:03}{{12.3}{301}} +\@writefile{toc}{\contentsline {subsection}{\numberline {12.3.1}Data partitioning}{301}} +\newlabel{ch12:sec:03.01}{{12.3.1}{301}} +\@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 }}{302}} +\newlabel{ch12:fig:01}{{12.1}{302}} +\@writefile{toc}{\contentsline {subsection}{\numberline {12.3.2}GPU computing}{302}} +\newlabel{ch12:sec:03.02}{{12.3.2}{302}} +\@writefile{toc}{\contentsline {subsection}{\numberline {12.3.3}Data communications}{303}} +\newlabel{ch12:sec:03.03}{{12.3.3}{303}} +\@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 }}{304}} +\newlabel{ch12:fig:02}{{12.2}{304}} +\@writefile{lof}{\contentsline {figure}{\numberline {12.3}{\ignorespaces Columns reordering of a sparse sub-matrix.\relax }}{305}} +\newlabel{ch12:fig:03}{{12.3}{305}} +\@writefile{toc}{\contentsline {section}{\numberline {12.4}Experimental results}{306}} +\newlabel{ch12:sec:04}{{12.4}{306}} +\@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 }}{306}} +\newlabel{ch12:fig:04}{{12.4}{306}} +\@writefile{lof}{\contentsline {figure}{\numberline {12.5}{\ignorespaces Sketches of sparse matrices chosen from the Davis collection.\relax }}{307}} +\newlabel{ch12:fig:05}{{12.5}{307}} +\@writefile{lot}{\contentsline {table}{\numberline {12.1}{\ignorespaces Main characteristics of sparse matrices chosen from the Davis collection.\relax }}{307}} +\newlabel{ch12:tab:01}{{12.1}{307}} +\@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 }}{308}} +\newlabel{ch12:tab:02}{{12.2}{308}} +\@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 }}{308}} +\newlabel{ch12:tab:03}{{12.3}{308}} +\newlabel{ch12:eq:20}{{12.20}{309}} +\@writefile{lof}{\contentsline {figure}{\numberline {12.6}{\ignorespaces Parallel generation of a large sparse matrix by four computing nodes.\relax }}{310}} +\newlabel{ch12:fig:06}{{12.6}{310}} +\@writefile{lot}{\contentsline {table}{\numberline {12.4}{\ignorespaces Main characteristics of sparse banded matrices generated from those of the Davis collection.\relax }}{310}} +\newlabel{ch12:tab:04}{{12.4}{310}} +\@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 }}{311}} +\newlabel{ch12:tab:05}{{12.5}{311}} +\@writefile{toc}{\contentsline {section}{\numberline {12.5}Conclusion}{311}} +\newlabel{ch12:sec:05}{{12.5}{311}} +\@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 }}{312}} +\newlabel{ch12:tab:06}{{12.6}{312}} +\@writefile{toc}{\contentsline {section}{Bibliography}{312}} \@setckpt{Chapters/chapter12/ch12}{ -\setcounter{page}{278} +\setcounter{page}{314} \setcounter{equation}{22} -\setcounter{enumi}{4} +\setcounter{enumi}{2} \setcounter{enumii}{0} \setcounter{enumiii}{0} \setcounter{enumiv}{10} \setcounter{footnote}{0} \setcounter{mpfootnote}{0} \setcounter{part}{5} -\setcounter{chapter}{11} +\setcounter{chapter}{12} \setcounter{section}{5} \setcounter{subsection}{0} \setcounter{subsubsection}{0} @@ -103,9 +103,16 @@ \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}