new

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

diff --git a/BookGPU/Chapters/chapter12/ch12.tex b/BookGPU/Chapters/chapter12/ch12.tex
index a514438a7ecac26fe29d7b80526eb12ee933b892..5c0a5b2302b3adcd5086daa1d49126b7515da756 100755 (executable)
--- a/BookGPU/Chapters/chapter12/ch12.tex
+++ b/BookGPU/Chapters/chapter12/ch12.tex
@@ -217,7 +217,7 @@ r_k \bot A \mathcal{K}_k(A, v_1).
 \end{array}
 \label{ch12:eq:13}
 \end{equation}
 \end{array}
 \label{ch12:eq:13}
 \end{equation}

-GMRES uses the Arnoldi process~\cite{ch12:ref5}\index{iterative method!Arnoldi process} to construct an
+GMRES uses the Arnoldi iterations~\cite{ch12:ref5}\index{iterative method!Arnoldi iterations} to construct an

 orthonormal basis $V_k$ for the Krylov subspace $\mathcal{K}_k$ and an upper Hessenberg matrix\index{Hessenberg matrix}
 $\bar{H}_k$ of order $(k+1)\times k$:
 \begin{equation}
 orthonormal basis $V_k$ for the Krylov subspace $\mathcal{K}_k$ and an upper Hessenberg matrix\index{Hessenberg matrix}
 $\bar{H}_k$ of order $(k+1)\times k$:
 \begin{equation}


@@ -313,7 +313,7 @@ $V$ to $m$ orthogonal vectors.
 Algorithm~\ref{ch12:alg:02} shows the key points of the GMRES method with restarts.
 It solves the left-preconditioned\index{sparse linear system!preconditioned} sparse linear
 system~(\ref{ch12:eq:11}), such that $M$ is the preconditioning matrix. At each iteration
 Algorithm~\ref{ch12:alg:02} shows the key points of the GMRES method with restarts.
 It solves the left-preconditioned\index{sparse linear system!preconditioned} sparse linear
 system~(\ref{ch12:eq:11}), such that $M$ is the preconditioning matrix. At each iteration

-$k$, GMRES uses the Arnoldi process\index{iterative method!Arnoldi process} (defined from
+$k$, GMRES uses the Arnoldi iterations\index{iterative method!Arnoldi iterations} (defined from

 line~$7$ to line~$17$) to construct a basis $V_m$ of $m$ orthogonal vectors and an upper
 Hessenberg matrix\index{Hessenberg matrix} $\bar{H}_m$ of size $(m+1)\times m$. Then, it
 solves the linear least-squares problem of size $m$ to find the vector $y\in\mathbb{R}^{m}$
 line~$7$ to line~$17$) to construct a basis $V_m$ of $m$ orthogonal vectors and an upper
 Hessenberg matrix\index{Hessenberg matrix} $\bar{H}_m$ of size $(m+1)\times m$. Then, it
 solves the linear least-squares problem of size $m$ to find the vector $y\in\mathbb{R}^{m}$


@@ -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()+.
 
 


@@ -526,7 +526,7 @@ is managed by one MPI process and is composed of one CPU core and one GPU card.
 All tests are made on double-precision floating point operations. The parameters of both linear
 solvers are initialized as follows: the residual tolerance threshold $\varepsilon=10^{-12}$, the
 maximum number of iterations $maxiter=500$, the right-hand side $b$ is filled with $1.0$, and the
 All tests are made on double-precision floating point operations. The parameters of both linear
 solvers are initialized as follows: the residual tolerance threshold $\varepsilon=10^{-12}$, the
 maximum number of iterations $maxiter=500$, the right-hand side $b$ is filled with $1.0$, and the

-initial guess $x_0$ is filled with $0.0$. In addition, we limited the Arnoldi process\index{iterative method!Arnoldi process}
+initial guess $x_0$ is filled with $0.0$. In addition, we limited the Arnoldi iterations\index{iterative method!Arnoldi iterations}

 used in the GMRES method to $16$ iterations ($m=16$). For the sake of simplicity, we have chosen
 the preconditioner $M$ as the main diagonal of the sparse matrix $A$. Indeed, it allows us to easily
 compute the required inverse matrix $M^{-1}$, and it provides a relatively good preconditioning for
 used in the GMRES method to $16$ iterations ($m=16$). For the sake of simplicity, we have chosen
 the preconditioner $M$ as the main diagonal of the sparse matrix $A$. Indeed, it allows us to easily
 compute the required inverse matrix $M^{-1}$, and it provides a relatively good preconditioning for


@@ -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}