fig

[GMRES2stage.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index ba4d0b0ef527043fdd891ff7f3fdc219e019cc01..f64255d6571c69d1107676c8dad1206ec7098332 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -354,6 +354,7 @@
 \usepackage{amsmath}
 \usepackage{amssymb}
 \usepackage{multirow}
 \usepackage{amsmath}
 \usepackage{amssymb}
 \usepackage{multirow}

+\usepackage{graphicx}

 
 \algnewcommand\algorithmicinput{\textbf{Input:}}
 \algnewcommand\Input{\item[\algorithmicinput]}
 
 \algnewcommand\algorithmicinput{\textbf{Input:}}
 \algnewcommand\Input{\item[\algorithmicinput]}


@@ -431,9 +432,9 @@ convergence of Krylov iterative methods,  typically those of GMRES variants. The
 principle of  our approach  is to  build an external  iteration over  the Krylov
 method  and to  save  the current  residual  frequently (for  example, for  each
 restart of GMRES). Then after a given number of outer iterations, a minimization
 principle of  our approach  is to  build an external  iteration over  the Krylov
 method  and to  save  the current  residual  frequently (for  example, for  each
 restart of GMRES). Then after a given number of outer iterations, a minimization

-step is applied on the matrix composed of the save residuals in order to compute
-a  better solution and  make a  new iteration  if necessary.  We prove  that our
-method  has the  same  convergence property  than  the inner  method used.  Some
+step  is applied  on the  matrix composed  of the  saved residuals  in  order to
+compute a better solution and make  a new iteration if necessary.  We prove that
+our method has  the same convergence property than the  inner method used.  Some

 experiments using up  to 16,394 cores show that compared  to GMRES our algorithm
 can be around 7 times faster.
 \end{abstract}
 experiments using up  to 16,394 cores show that compared  to GMRES our algorithm
 can be around 7 times faster.
 \end{abstract}


@@ -583,8 +584,7 @@ performances.
 The present paper is organized  as follows. In Section~\ref{sec:02} some related
 works are presented. Section~\ref{sec:03} presents our two-stage algorithm using
 a  least-square  residual  minimization.   Section~\ref{sec:04}  describes  some
 The present paper is organized  as follows. In Section~\ref{sec:02} some related
 works are presented. Section~\ref{sec:03} presents our two-stage algorithm using
 a  least-square  residual  minimization.   Section~\ref{sec:04}  describes  some

-convergence  results  on this  method.   In Section~\ref{sec:05},  parallization
-details  of  TSARM  are  given.  Section~\ref{sec:06}  shows  some  experimental
+convergence  results  on this  method.   Section~\ref{sec:05}  shows  some  experimental

 results  obtained on large  clusters of  our algorithm  using routines  of PETSc
 toolkit.  Finally Section~\ref{sec:06} concludes and gives some perspectives.
 %%%*********************************************************
 results  obtained on large  clusters of  our algorithm  using routines  of PETSc
 toolkit.  Finally Section~\ref{sec:06} concludes and gives some perspectives.
 %%%*********************************************************


@@ -680,18 +680,6 @@ To summarize, the important parameters of TSARM are:
 \item $\epsilon_{ls}$ the threshold to stop the least-square method
 \end{itemize}
 
 \item $\epsilon_{ls}$ the threshold to stop the least-square method
 \end{itemize}
 

-%%%*********************************************************
-%%%*********************************************************
-
-\section{Convergence results}
-\label{sec:04}
-
-
-
-%%%*********************************************************
-%%%*********************************************************
-\section{Parallelization}
-\label{sec:05}

 
 The  parallelisation  of  TSARM  relies   on  the  parallelization  of  all  its
 parts. More  precisely, except  the least-square step,  all the other  parts are
 
 The  parallelisation  of  TSARM  relies   on  the  parallelization  of  all  its
 parts. More  precisely, except  the least-square step,  all the other  parts are


@@ -733,10 +721,21 @@ In each iteration  of CGLS, there is two  matrix-vector multiplications and some
 classical operations:  dots, norm, multiplication  and addition on  vectors. All
 these operations are easy to implement in PETSc or similar environment.
 
 classical operations:  dots, norm, multiplication  and addition on  vectors. All
 these operations are easy to implement in PETSc or similar environment.
 

+
+
+%%%*********************************************************
+%%%*********************************************************
+
+\section{Convergence results}
+\label{sec:04}
+
+
+
+

 %%%*********************************************************
 %%%*********************************************************
 \section{Experiments using petsc}
 %%%*********************************************************
 %%%*********************************************************
 \section{Experiments using petsc}

-\label{sec:06}
+\label{sec:05}

 
 
 In order to see the influence of our algorithm with only one processor, we first
 
 
 In order to see the influence of our algorithm with only one processor, we first


@@ -793,7 +792,7 @@ minimization.
 \begin{tabular}{|c|c|r|r|r|r|} 
 \hline
 
 \begin{tabular}{|c|c|r|r|r|r|} 
 \hline
 

- \multirow{2}{*}{Matrix name}  & Solver /   & \multicolumn{2}{c|}{gmres variant} & \multicolumn{2}{c|}{2 stage CGLS} \\ 
+ \multirow{2}{*}{Matrix name}  & Solver /   & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSARM CGLS} \\ 

 \cline{3-6}
        &  precond             & Time  & \# Iter.  & Time  & \# Iter.  \\\hline \hline
 
 \cline{3-6}
        &  precond             & Time  & \# Iter.  & Time  & \# Iter.  \\\hline \hline
 


@@ -814,14 +813,29 @@ torso3             & fgmres / sor  & 37.70 & 565 & 34.97 & 510 \\
 
 
 
 
 
 

-Larger experiments ....
+
+In the following we describe the applications of PETSc we have experimented. Those applications are available in the ksp part which is suited for  scalable linear equations solvers:
+\begin{itemize}
+\item ex15  is an example  which solves in  parallel an operator using  a  finite  difference  scheme.  The  diagonal is  equals  to  4  and  4
+  extra-diagonals  representing the  neighbors in  each directions  is  equal to
+  -1. This  example is used in many  physical phenomena , for  exemple, heat and
+  fluid flow, wave propagation...
+\item ex54 is another example based on 2D problem discretized  with quadrilateral finite elements. For this example, the user can define the scaling of material coefficient in embedded circle, it is called $\alpha$.
+\end{itemize}
+For more technical details on  these applications, interested reader are invited
+to  read the  codes available  in the  PETSc sources.   Those problem  have been
+chosen because they  are scalable with many cores. We  have tested other problem
+but they are not scalable with many cores.
+
+
+

 
 \begin{table*}
 \begin{center}
 \begin{tabular}{|r|r|r|r|r|r|r|r|r|} 
 \hline
 
 
 \begin{table*}
 \begin{center}
 \begin{tabular}{|r|r|r|r|r|r|r|r|r|} 
 \hline
 

-  nb. cores & precond   & \multicolumn{2}{c|}{gmres variant} & \multicolumn{2}{c|}{2 stage CGLS} &  \multicolumn{2}{c|}{2 stage LSQR} & best gain \\ 
+  nb. cores & precond   & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSARM CGLS} &  \multicolumn{2}{c|}{TSARM LSQR} & best gain \\ 

 \cline{3-8}
              &                       & Time  & \# Iter.  & Time  & \# Iter. & Time  & \# Iter. & \\\hline \hline
   2,048      & mg                    & 403.49   & 18,210    & 73.89  & 3,060   & 77.84  & 3,270  & 5.46 \\
 \cline{3-8}
              &                       & Time  & \# Iter.  & Time  & \# Iter. & Time  & \# Iter. & \\\hline \hline
   2,048      & mg                    & 403.49   & 18,210    & 73.89  & 3,060   & 77.84  & 3,270  & 5.46 \\


@@ -841,12 +855,23 @@ Larger experiments ....
 \end{table*}
 
 
 \end{table*}
 
 

+\begin{figure}
+\centering
+  \includegraphics[width=0.45\textwidth]{nb_iter_sec_ex15_juqueen}
+\caption{Number of iterations per second with ex15 and the same parameters than in Table~\ref{tab:03}}
+\label{fig:01}
+\end{figure}
+
+
+
+
+

 \begin{table*}
 \begin{center}
 \begin{tabular}{|r|r|r|r|r|r|r|r|r|} 
 \hline
 
 \begin{table*}
 \begin{center}
 \begin{tabular}{|r|r|r|r|r|r|r|r|r|} 
 \hline
 

-  nb. cores & threshold   & \multicolumn{2}{c|}{gmres variant} & \multicolumn{2}{c|}{2 stage CGLS} &  \multicolumn{2}{c|}{2 stage LSQR} & best gain \\ 
+  nb. cores & threshold   & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSARM CGLS} &  \multicolumn{2}{c|}{TSARM LSQR} & best gain \\ 

 \cline{3-8}
              &                       & Time  & \# Iter.  & Time  & \# Iter. & Time  & \# Iter. & \\\hline \hline
   2,048      & 8e-5                  & 108.88 & 16,560  & 23.06  &  3,630  & 22.79  & 3,630   & 4.77 \\
 \cline{3-8}
              &                       & Time  & \# Iter.  & Time  & \# Iter. & Time  & \# Iter. & \\\hline \hline
   2,048      & 8e-5                  & 108.88 & 16,560  & 23.06  &  3,630  & 22.79  & 3,630   & 4.77 \\


@@ -854,7 +879,7 @@ Larger experiments ....
   4,096      & 7e-5                  & 160.59 & 22,530  & 35.15  &  5,130  & 29.21  & 4,350   & 5.49 \\
   4,096      & 6e-5                  & 249.27 & 35,520  & 52.13  &  7,950  & 39.24  & 5,790   & 6.35 \\
   8,192      & 6e-5                  & 149.54 & 17,280  & 28.68  &  3,810  & 29.05  & 3,990  & 5.21 \\
   4,096      & 7e-5                  & 160.59 & 22,530  & 35.15  &  5,130  & 29.21  & 4,350   & 5.49 \\
   4,096      & 6e-5                  & 249.27 & 35,520  & 52.13  &  7,950  & 39.24  & 5,790   & 6.35 \\
   8,192      & 6e-5                  & 149.54 & 17,280  & 28.68  &  3,810  & 29.05  & 3,990  & 5.21 \\

-  8,192      & 5e-5                  & 792.11 & 109,590 & 76.83  &  10,470  & 65.20  & 9,030  & 12.14 \\
+  8,192      & 5e-5                  & 785.04 & 109,590 & 76.07  &  10,470  & 69.42 & 9,030  & 11.30 \\

   16,384     & 4e-5                  & 718.61 & 86,400 & 98.98  &  10,830  & 131.86  & 14,790  & 7.26 \\
 \hline
 
   16,384     & 4e-5                  & 718.61 & 86,400 & 98.98  &  10,830  & 131.86  & 14,790  & 7.26 \\
 \hline
 


@@ -863,6 +888,33 @@ Larger experiments ....
 \label{tab:04}
 \end{center}
 \end{table*}
 \label{tab:04}
 \end{center}
 \end{table*}

+
+
+
+
+
+\begin{table*}
+\begin{center}
+\begin{tabular}{|r|r|r|r|r|r|r|r|r|r|r|} 
+\hline
+
+  nb. cores   & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSARM CGLS} &  \multicolumn{2}{c|}{TSARM LSQR} & best gain & \multicolumn{3}{c|}{efficiency} \\ 
+\cline{2-7} \cline{9-11}
+                    & Time  & \# Iter.  & Time  & \# Iter. & Time  & \# Iter. &   & GMRES & TS CGLS & TS LSQR\\\hline \hline
+   512              & 3,969.69 & 33,120 & 709.57 & 5,790  & 622.76 & 5,070  & 6.37  &   1    &    1    &     1     \\
+   1024             & 1,530.06  & 25,860 & 290.95 & 4,830  & 307.71 & 5,070 & 5.25  &  1.30  &    1.21  &   1.01     \\
+   2048             & 919.62    & 31,470 & 237.52 & 8,040  & 194.22 & 6,510 & 4.73  & 1.08   &    .75   &   .80\\
+   4096             & 405.60    & 28,380 & 111.67 & 7,590  & 91.72  & 6,510 & 4.42  & 1.22   &  .79     &   .84 \\
+   8192             & 785.04   & 109,590 & 76.07  & 10,470 & 69.42 & 9,030  & 11.30 &   .32  &   .58    &  .56 \\
+
+\hline
+
+\end{tabular}
+\caption{Comparison of FGMRES  and 2 stage FGMRES algorithms for ex54 of Petsc (both with the MG preconditioner) with 204,919,225 components on Curie with different number of cores (restart=30, s=12, threshol 5e-5),  time is expressed in seconds.}
+\label{tab:05}
+\end{center}
+\end{table*}
+

 %%%*********************************************************
 %%%*********************************************************
 
 %%%*********************************************************
 %%%*********************************************************
 


@@ -871,7 +923,7 @@ Larger experiments ....
 %%%*********************************************************
 %%%*********************************************************
 \section{Conclusion}
 %%%*********************************************************
 %%%*********************************************************
 \section{Conclusion}

-\label{sec:07}
+\label{sec:06}

 %The conclusion goes here. this is more of the conclusion
 %%%*********************************************************
 %%%*********************************************************
 %The conclusion goes here. this is more of the conclusion
 %%%*********************************************************
 %%%*********************************************************