2 \@writefile{toc}{\author{Lilia Ziane Khodja}{}}
3 \@writefile{toc}{\author{Ming Chau}{}}
4 \@writefile{toc}{\author{Rapha\IeC {\"e}l Couturier}{}}
5 \@writefile{toc}{\author{Pierre Spit\IeC {\'e}ri}{}}
6 \@writefile{toc}{\author{Jacques Bahi}{}}
7 \@writefile{loa}{\addvspace {10\p@ }}
8 \@writefile{toc}{\contentsline {chapter}{\numberline {12}Solving sparse nonlinear systems of obstacle problems on GPU clusters}{277}}
9 \@writefile{lof}{\addvspace {10\p@ }}
10 \@writefile{lot}{\addvspace {10\p@ }}
11 \newlabel{ch13}{{12}{277}}
12 \@writefile{toc}{\contentsline {section}{\numberline {12.1}Introduction}{277}}
13 \newlabel{ch13:sec:01}{{12.1}{277}}
14 \@writefile{toc}{\contentsline {section}{\numberline {12.2}Obstacle problems}{278}}
15 \newlabel{ch13:sec:02}{{12.2}{278}}
16 \@writefile{toc}{\contentsline {subsection}{\numberline {12.2.1}Mathematical model}{278}}
17 \newlabel{ch13:sec:02.01}{{12.2.1}{278}}
18 \newlabel{ch13:eq:01}{{12.1}{278}}
19 \newlabel{ch13:eq:02}{{12.2}{278}}
20 \@writefile{toc}{\contentsline {subsection}{\numberline {12.2.2}Discretization}{279}}
21 \newlabel{ch13:sec:02.02}{{12.2.2}{279}}
22 \newlabel{ch13:eq:03}{{12.3}{279}}
23 \newlabel{ch13:eq:04}{{12.4}{279}}
24 \newlabel{ch13:eq:05}{{12.5}{279}}
25 \@writefile{toc}{\contentsline {section}{\numberline {12.3}Parallel iterative method}{280}}
26 \newlabel{ch13:sec:03}{{12.3}{280}}
27 \newlabel{ch13:eq:06}{{12.6}{280}}
28 \newlabel{ch13:eq:07}{{12.7}{280}}
29 \newlabel{ch13:eq:08}{{12.8}{280}}
30 \newlabel{ch13:eq:09}{{12.9}{280}}
31 \newlabel{ch13:eq:10}{{12.10}{281}}
32 \newlabel{ch13:eq:11}{{12.11}{281}}
33 \newlabel{ch13:eq:12}{{12.12}{281}}
34 \newlabel{ch13:eq:13}{{12.13}{282}}
35 \newlabel{ch13:eq:14}{{12.14}{282}}
36 \newlabel{ch13:eq:15}{{12.15}{282}}
37 \newlabel{ch13:eq:16}{{12.16}{282}}
38 \@writefile{toc}{\contentsline {section}{\numberline {12.4}Parallel implementation on a GPU cluster}{283}}
39 \newlabel{ch13:sec:04}{{12.4}{283}}
40 \@writefile{lof}{\contentsline {figure}{\numberline {12.1}{\ignorespaces Data partitioning of a problem to be solved among $S=3\times 4$ computing nodes.\relax }}{283}}
41 \newlabel{ch13:fig:01}{{12.1}{283}}
42 \@writefile{loa}{\contentsline {algocf}{\numberline {11}{\ignorespaces Parallel solving of the obstacle problem on a GPU cluster\relax }}{284}}
43 \newlabel{ch13:alg:01}{{11}{284}}
44 \newlabel{ch13:eq:18}{{12.17}{284}}
45 \@writefile{loa}{\contentsline {algocf}{\numberline {12}{\ignorespaces Parallel iterative solving of the nonlinear systems on a GPU cluster ($Solve()$ function)\relax }}{285}}
46 \newlabel{ch13:alg:02}{{12}{285}}
47 \@writefile{lof}{\contentsline {figure}{\numberline {12.2}{\ignorespaces Decomposition of a sub-problem in a GPU into $nz$ slices.\relax }}{286}}
48 \newlabel{ch13:fig:02}{{12.2}{286}}
49 \newlabel{ch13:list:01}{{12.1}{286}}
50 \@writefile{lol}{\contentsline {lstlisting}{\numberline {12.1}Skeleton codes of a GPU kernel and a CPU function}{286}}
51 \@writefile{lof}{\contentsline {figure}{\numberline {12.3}{\ignorespaces Matrix constant coefficients in a three-dimensional domain.\relax }}{288}}
52 \newlabel{ch13:fig:03}{{12.3}{288}}
53 \newlabel{ch13:eq:17}{{12.18}{288}}
54 \newlabel{ch13:list:02}{{12.2}{288}}
55 \@writefile{lol}{\contentsline {lstlisting}{\numberline {12.2}GPU kernels of the projected Richardson method}{288}}
56 \@writefile{lof}{\contentsline {figure}{\numberline {12.4}{\ignorespaces Computation of a vector element with the projected Richardson method.\relax }}{290}}
57 \newlabel{ch13:fig:04}{{12.4}{290}}
58 \newlabel{ch13:list:03}{{12.3}{290}}
59 \@writefile{lol}{\contentsline {lstlisting}{\numberline {12.3}Memory access to the cache texture memory}{290}}
60 \@writefile{toc}{\contentsline {section}{\numberline {12.5}Experimental tests on a GPU cluster}{291}}
61 \newlabel{ch13:sec:05}{{12.5}{291}}
62 \@writefile{lof}{\contentsline {figure}{\numberline {12.5}{\ignorespaces GPU cluster of tests composed of 12 computing nodes (six machines, each with two GPUs.\relax }}{293}}
63 \newlabel{ch13:fig:05}{{12.5}{293}}
64 \@writefile{lot}{\contentsline {table}{\numberline {12.1}{\ignorespaces Execution times in seconds of the parallel projected Richardson method implemented on a cluster of 24 CPU cores.\relax }}{293}}
65 \newlabel{ch13:tab:01}{{12.1}{293}}
66 \@writefile{lot}{\contentsline {table}{\numberline {12.2}{\ignorespaces Execution times in seconds of the parallel projected Richardson method implemented on a cluster of 12 GPUs.\relax }}{294}}
67 \newlabel{ch13:tab:02}{{12.2}{294}}
68 \@writefile{toc}{\contentsline {section}{\numberline {12.6}Red-Black ordering technique}{294}}
69 \newlabel{ch13:sec:06}{{12.6}{294}}
70 \newlabel{ch13:list:04}{{12.4}{295}}
71 \@writefile{lol}{\contentsline {lstlisting}{\numberline {12.4}GPU kernels of the projected Richardson method using the red-black technique}{295}}
72 \newlabel{ch13:fig:06.01}{{12.6(a)}{296}}
73 \newlabel{sub@ch13:fig:06.01}{{(a)}{296}}
74 \newlabel{ch13:fig:06.02}{{12.6(b)}{296}}
75 \newlabel{sub@ch13:fig:06.02}{{(b)}{296}}
76 \@writefile{lof}{\contentsline {figure}{\numberline {12.6}{\ignorespaces Red-Black ordering for computing the iterate vector elements in a three-dimensional space.\relax }}{296}}
77 \@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {Red-Black ordering on x, y and z axises}}}{296}}
78 \@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {Red-Black ordering on y axis}}}{296}}
79 \@writefile{lot}{\contentsline {table}{\numberline {12.3}{\ignorespaces Execution times in seconds of the parallel projected Richardson method using read-black ordering technique implemented on a cluster of 12 GPUs.\relax }}{297}}
80 \newlabel{ch13:tab:03}{{12.3}{297}}
81 \@writefile{lof}{\contentsline {figure}{\numberline {12.7}{\ignorespaces Weak scaling of both synchronous and asynchronous algorithms of the projected Richardson method using red-black ordering technique.\relax }}{298}}
82 \newlabel{ch13:fig:07}{{12.7}{298}}
83 \@writefile{toc}{\contentsline {section}{\numberline {12.7}Conclusion}{298}}
84 \newlabel{ch13:sec:07}{{12.7}{298}}
85 \@writefile{toc}{\contentsline {section}{Bibliography}{299}}
86 \@setckpt{Chapters/chapter13/ch13}{
87 \setcounter{page}{301}
88 \setcounter{equation}{18}
90 \setcounter{enumii}{0}
91 \setcounter{enumiii}{0}
92 \setcounter{enumiv}{15}
93 \setcounter{footnote}{0}
94 \setcounter{mpfootnote}{0}
96 \setcounter{chapter}{12}
97 \setcounter{section}{7}
98 \setcounter{subsection}{0}
99 \setcounter{subsubsection}{0}
100 \setcounter{paragraph}{0}
101 \setcounter{subparagraph}{0}
102 \setcounter{figure}{7}
103 \setcounter{table}{3}
104 \setcounter{numauthors}{0}
105 \setcounter{parentequation}{46}
106 \setcounter{subfigure}{0}
107 \setcounter{lofdepth}{1}
108 \setcounter{subtable}{0}
109 \setcounter{lotdepth}{1}
110 \setcounter{lstnumber}{55}
111 \setcounter{ContinuedFloat}{0}
112 \setcounter{AlgoLine}{12}
113 \setcounter{algocfline}{12}
114 \setcounter{algocfproc}{12}
115 \setcounter{algocf}{12}
116 \setcounter{proposition}{1}
117 \setcounter{theorem}{0}
118 \setcounter{exercise}{0}
119 \setcounter{example}{0}
120 \setcounter{definition}{0}
121 \setcounter{proof}{1}
122 \setcounter{lstlisting}{4}