X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/book_gpu.git/blobdiff_plain/beb4b32a38394d2c847ba5a733a4906781d95881..8e13f9e62b8323f091076223183e5436646a87e7:/BookGPU/Chapters/chapter12/ch12.aux?ds=inline diff --git a/BookGPU/Chapters/chapter12/ch12.aux b/BookGPU/Chapters/chapter12/ch12.aux index d3b6ca3..97242d0 100644 --- a/BookGPU/Chapters/chapter12/ch12.aux +++ b/BookGPU/Chapters/chapter12/ch12.aux @@ -3,125 +3,99 @@ \@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 {10}Solving sparse linear systems with GMRES and CG methods on GPU clusters}{215}} \@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}{{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}} \@setckpt{Chapters/chapter12/ch12}{ -\setcounter{page}{278} -\setcounter{equation}{25} +\setcounter{page}{234} +\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{chapter}{11} -\setcounter{section}{6} +\setcounter{part}{5} +\setcounter{chapter}{10} +\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{parentequation}{8} \setcounter{subfigure}{0} \setcounter{lofdepth}{1} \setcounter{subtable}{0} @@ -132,6 +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}