last version

[book_gpu.git] / BookGPU / Chapters / chapter12 / ch12.tex

diff --git a/BookGPU/Chapters/chapter12/ch12.tex b/BookGPU/Chapters/chapter12/ch12.tex
index 0269fd28a3957ccd5faaf8dec9ee15c92eee3703..4bc95a689ec0ba128acfc043de19955b2f3931c5 100755 (executable)
--- a/BookGPU/Chapters/chapter12/ch12.tex
+++ b/BookGPU/Chapters/chapter12/ch12.tex
@@ -19,7 +19,7 @@
 \label{ch12:sec:01}
 Sparse linear systems are used to model many scientific and industrial problems,
 such as the environmental simulations or the industrial processing of the complex or
 \label{ch12:sec:01}
 Sparse linear systems are used to model many scientific and industrial problems,
 such as the environmental simulations or the industrial processing of the complex or

-non-Newtonian fluids. Moreover, the resolution of these problems often involves the
+nonNewtonian fluids. Moreover, the resolution of these problems often involves the

 solving of such linear systems that are considered the most expensive process in
 terms of execution time and memory space. Therefore, solving sparse linear systems
 must be as efficient as possible in order to deal with problems of ever increasing
 solving of such linear systems that are considered the most expensive process in
 terms of execution time and memory space. Therefore, solving sparse linear systems
 must be as efficient as possible in order to deal with problems of ever increasing


@@ -457,8 +457,8 @@ nodes\index{neighboring node} over the GPU cluster must exchange between them th
 elements necessary to compute this multiplication. First, each computing node determines, in its
 local subvector, the vector elements needed by other nodes. Then, the neighboring nodes exchange
 between them these shared vector elements. The data exchanges are implemented by using the MPI
 elements necessary to compute this multiplication. First, each computing node determines, in its
 local subvector, the vector elements needed by other nodes. Then, the neighboring nodes exchange
 between them these shared vector elements. The data exchanges are implemented by using the MPI

-point-to-point communication routines: blocking\index{MPI subroutines!blocking} sends with \verb+MPI_Send()+
-and nonblocking\index{MPI subroutines!nonblocking} receives with \verb+MPI_Irecv()+. Figure~\ref{ch12:fig:02}
+point-to-point communication routines: blocking\index{MPI!blocking} sends with \verb+MPI_Send()+
+and nonblocking\index{MPI!nonblocking} receives with \verb+MPI_Irecv()+. Figure~\ref{ch12:fig:02}

 shows an example of data exchanges between \textit{Node 1} and its neighbors \textit{Node 0}, \textit{Node 2},
 and \textit{Node 3}. In this example, the iterate matrix $A$ split between these four computing
 nodes is that presented in Figure~\ref{ch12:fig:01}.
 shows an example of data exchanges between \textit{Node 1} and its neighbors \textit{Node 0}, \textit{Node 2},
 and \textit{Node 3}. In this example, the iterate matrix $A$ split between these four computing
 nodes is that presented in Figure~\ref{ch12:fig:01}.


@@ -491,7 +491,7 @@ cluster. Consequently, the vector elements to be exchanged must be copied from t
 and vice versa before and after the synchronization operation between CPUs. We have used the CUBLAS\index{CUBLAS}
 communication subroutines to perform the data transfers between a CPU core and its GPU: \verb+cublasGetVector()+
 and \verb+cublasSetVector()+. Finally, in addition to the data exchanges, GPU nodes perform reduction operations
 and vice versa before and after the synchronization operation between CPUs. We have used the CUBLAS\index{CUBLAS}
 communication subroutines to perform the data transfers between a CPU core and its GPU: \verb+cublasGetVector()+
 and \verb+cublasSetVector()+. Finally, in addition to the data exchanges, GPU nodes perform reduction operations

-to compute in parallel the dot products and Euclidean norms. This is implemented by using the MPI global communication\index{MPI subroutines!global}
+to compute in parallel the dot products and Euclidean norms. This is implemented by using the MPI global communication\index{MPI!global}

 \verb+MPI_Allreduce()+.
 
 
 \verb+MPI_Allreduce()+.
 
 


@@ -548,8 +548,10 @@ which are the number of rows, the total number of nonzero values, and the maxima
 the present chapter, the bandwidth of a sparse matrix is defined as the number of matrix columns separating
 the first and the last nonzero value on a matrix row.
 
 the present chapter, the bandwidth of a sparse matrix is defined as the number of matrix columns separating
 the first and the last nonzero value on a matrix row.
 

+

 \begin{table}
 \centering
 \begin{table}
 \centering

+\begin{small}

 \begin{tabular}{|c|c|c|c|c|}
 \hline
 {\bf Matrix Type}             & {\bf Matrix Name} & {\bf \# Rows} & {\bf \# Nonzeros} & {\bf Bandwidth} \\ \hline \hline
 \begin{tabular}{|c|c|c|c|c|}
 \hline
 {\bf Matrix Type}             & {\bf Matrix Name} & {\bf \# Rows} & {\bf \# Nonzeros} & {\bf Bandwidth} \\ \hline \hline


@@ -578,10 +580,12 @@ the first and the last nonzero value on a matrix row.
 
                               & torso3            & $259,156$     & $4,429,042$  & $216,854$  \\ \hline
 \end{tabular}
 
                               & torso3            & $259,156$     & $4,429,042$  & $216,854$  \\ \hline
 \end{tabular}

+\end{small}

 \caption{Main characteristics of sparse matrices chosen from the University of Florida collection.}
 \label{ch12:tab:01}
 \end{table}
 
 \caption{Main characteristics of sparse matrices chosen from the University of Florida collection.}
 \label{ch12:tab:01}
 \end{table}
 

+

 \begin{table}[!h]
 \begin{center}
 \begin{tabular}{|c|c|c|c|c|c|c|} 
 \begin{table}[!h]
 \begin{center}
 \begin{tabular}{|c|c|c|c|c|c|c|} 


@@ -607,6 +611,7 @@ thermal2          & $1.172s$           & $0.622s$            & $1.88$        & $
 
 \begin{table}[!h]
 \begin{center}
 
 \begin{table}[!h]
 \begin{center}

+\begin{small}

 \begin{tabular}{|c|c|c|c|c|c|c|} 
 \hline
 {\bf Matrix}     & $\mathbf{Time_{cpu}}$ & $\mathbf{Time_{gpu}}$ & $\mathbf{\tau}$  & $\mathbf{\#~Iter.}$ & $\mathbf{Prec.}$     & $\mathbf{\Delta}$   \\ \hline \hline
 \begin{tabular}{|c|c|c|c|c|c|c|} 
 \hline
 {\bf Matrix}     & $\mathbf{Time_{cpu}}$ & $\mathbf{Time_{gpu}}$ & $\mathbf{\tau}$  & $\mathbf{\#~Iter.}$ & $\mathbf{Prec.}$     & $\mathbf{\Delta}$   \\ \hline \hline


@@ -635,6 +640,7 @@ poli\_large       & $0.097s$           & $0.095s$            & $1.02$        & $
 
 torso3            & $4.242s$           & $2.030s$            & $2.09$        & $175$          & $2.69e$-$10$     & $1.78e$-$14$ \\ \hline
 \end{tabular}
 
 torso3            & $4.242s$           & $2.030s$            & $2.09$        & $175$          & $2.69e$-$10$     & $1.78e$-$14$ \\ \hline
 \end{tabular}

+\end{small}

 \caption{Performances of the parallel GMRES method on a cluster 24 CPU cores vs. on cluster of 12 GPUs.}
 \label{ch12:tab:03}
 \end{center}
 \caption{Performances of the parallel GMRES method on a cluster 24 CPU cores vs. on cluster of 12 GPUs.}
 \label{ch12:tab:03}
 \end{center}


@@ -742,7 +748,7 @@ are better than those of the GMRES method for solving large symmetric linear sys
 CG method is characterized by a better convergence\index{convergence} rate and a shorter execution
 time of an iteration than those of the GMRES method. Moreover, an iteration of the parallel GMRES
 method requires more data exchanges between computing nodes compared to the parallel CG method.
 CG method is characterized by a better convergence\index{convergence} rate and a shorter execution
 time of an iteration than those of the GMRES method. Moreover, an iteration of the parallel GMRES
 method requires more data exchanges between computing nodes compared to the parallel CG method.

-
+\clearpage

 \begin{table}[!h]
 \begin{center}
 \begin{tabular}{|c|c|c|c|c|c|c|} 
 \begin{table}[!h]
 \begin{center}
 \begin{tabular}{|c|c|c|c|c|c|c|} 


@@ -769,6 +775,7 @@ on a cluster of 12 GPUs.}
 
 \begin{table}[!h]
 \begin{center}
 
 \begin{table}[!h]
 \begin{center}

+\begin{small}

 \begin{tabular}{|c|c|c|c|c|c|c|} 
 \hline
 {\bf Matrix}      & $\mathbf{Time_{cpu}}$ & $\mathbf{Time_{gpu}}$ & $\mathbf{\tau}$ & $\mathbf{\#~Iter.}$ & $\mathbf{Prec.}$ & $\mathbf{\Delta}$   \\ \hline \hline
 \begin{tabular}{|c|c|c|c|c|c|c|} 
 \hline
 {\bf Matrix}      & $\mathbf{Time_{cpu}}$ & $\mathbf{Time_{gpu}}$ & $\mathbf{\tau}$ & $\mathbf{\#~Iter.}$ & $\mathbf{Prec.}$ & $\mathbf{\Delta}$   \\ \hline \hline


@@ -797,6 +804,7 @@ poli\_large       & $8.515s$             & $1.053s$              & $8.09$
 
 torso3            & $31.463s$            & $3.681s$              & $8.55$          & $175$               & $2.69e$-$10$     & $2.66e$-$14$ \\ \hline
 \end{tabular}
 
 torso3            & $31.463s$            & $3.681s$              & $8.55$          & $175$               & $2.69e$-$10$     & $2.66e$-$14$ \\ \hline
 \end{tabular}

+\end{small}

 \caption{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.}
 \label{ch12:tab:06}
 \caption{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.}
 \label{ch12:tab:06}