12-10-2014 04

[GMRES2stage.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index a4c9b268faf6197c47d7c288188adc6bf299b3de..4b49998c2f4f46fbdcd77fa9e43c559e4ee21488 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -601,7 +601,14 @@ is summarized while intended perspectives are provided.
 %%%*********************************************************
 \section{Related works}
 \label{sec:02} 
 %%%*********************************************************
 \section{Related works}
 \label{sec:02} 

-%Wherever Times is specified, Times Roman or Times New Roman may be used. If neither is available on your system, please use the font closest in appearance to Times. Avoid using bit-mapped fonts if possible. True-Type 1 or Open Type fonts are preferred. Please embed symbol fonts, as well, for math, etc.
+Krylov subspace iteration methods have increasingly become useful and successful techniques for solving linear and nonlinear systems and eigenvalue problems, especially since the increase development of the preconditioners~\cite{Saad2003,Meijerink77}. One reason of the popularity of these methods is their generality, simplicity and efficiency to solve systems of equations arising from very large and complex problems. %A Krylov method is based on a projection process onto a Krylov subspace spanned by vectors and it forms a sequence of approximations by minimizing the residual over the subspace formed~\cite{}.
+
+GMRES is one of the most widely used Krylov iterative method for solving sparse and large linear systems. It is developed by Saad and al.~\cite{Saad86} as a generalized method to deal with unsymmetric and non-Hermitian problems, and indefinite symmetric problems too. In its original version called full GMRES, it minimizes the residual over the current Krylov subspace until convergence in at most $n$ iterations, where $n$ is the size of the sparse matrix. It should be noted that full GMRES is too expensive in the case of large matrices since the required orthogonalization process per iteration grows quadratically with the number of iterations. For that reason, in practice GMRES is restarted after each $m\ll n$ iterations to avoid the storage of a large orthonormal basis. However, the convergence behavior of the restarted GMRES, called GMRES($m$), in many cases depends quite critically on the value of $m$~\cite{Huang89}. Therefore in most cases, a preconditioning technique is applied to the restarted GMRES method in order to improve its convergence.
+
+Van der Vorst in~\cite{Vorst94} has proposed variants of the GMRES algorithm in which a different preconditioner is applied in each iteration, so-called GMRESR family of nested methods. In fact, the GMRES method is effectively preconditioned with other iterative schemes, where the iterations of the GMRES method are called outer iterations while the iterations of the preconditioning process referred to as inner iterations.      
+
+%FGMRES , GMRESR, two-stage, communication avoiding 
+

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


@@ -654,10 +661,10 @@ appropriate than a single direct method in a parallel context.
   \Input $A$ (sparse matrix), $b$ (right-hand side)
   \Output $x$ (solution vector)\vspace{0.2cm}
   \State Set the initial guess $x_0$
   \Input $A$ (sparse matrix), $b$ (right-hand side)
   \Output $x$ (solution vector)\vspace{0.2cm}
   \State Set the initial guess $x_0$

-  \For {$k=1,2,3,\ldots$ until convergence (error$<\epsilon_{tsirm}$)} \label{algo:conv}
+  \For {$k=1,2,3,\ldots$ until convergence ($error<\epsilon_{tsirm}$)} \label{algo:conv}

     \State  $[x_k,error]=Solve(A,b,x_{k-1},max\_iter_{kryl})$   \label{algo:solve}
     \State  $[x_k,error]=Solve(A,b,x_{k-1},max\_iter_{kryl})$   \label{algo:solve}

-    \State $S_{k \mod s}=x_k$ \label{algo:store} \Comment{update column (k mod s) of S}
-    \If {$k \mod s=0$ {\bf and} error$>\epsilon_{kryl}$}
+    \State $S_{k \mod s}=x_k$ \label{algo:store} \Comment{update column ($k \mod s$) of $S$}
+    \If {$k \mod s=0$ {\bf and} $error>\epsilon_{kryl}$}

       \State $R=AS$ \Comment{compute dense matrix} \label{algo:matrix_mul}
             \State $\alpha=Least\_Squares(R,b,max\_iter_{ls})$ \label{algo:}
       \State $x_k=S\alpha$  \Comment{compute new solution}
       \State $R=AS$ \Comment{compute dense matrix} \label{algo:matrix_mul}
             \State $\alpha=Least\_Squares(R,b,max\_iter_{ls})$ \label{algo:}
       \State $x_k=S\alpha$  \Comment{compute new solution}


@@ -675,10 +682,10 @@ method. Moreover,  a tolerance  threshold must be  specified for the  solver. In
 practice, this threshold must be  much smaller than the convergence threshold of
 the  TSIRM algorithm  (\emph{i.e.}, $\epsilon_{tsirm}$).  We also  consider that
 after the call of the $Solve$ function, we obtain the vector $x_k$ and the error
 practice, this threshold must be  much smaller than the convergence threshold of
 the  TSIRM algorithm  (\emph{i.e.}, $\epsilon_{tsirm}$).  We also  consider that
 after the call of the $Solve$ function, we obtain the vector $x_k$ and the error

-which is defined by $||Ax^k-b||_2$.
+which is defined by $||Ax_k-b||_2$.

 
   Line~\ref{algo:store},
 
   Line~\ref{algo:store},

-$S_{k \mod  s}=x^k$ consists in  copying the solution  $x_k$ into the  column $k
+$S_{k \mod  s}=x_k$ consists in  copying the solution  $x_k$ into the  column $k

 \mod s$ of $S$.   After the minimization, the matrix $S$ is  reused with the new
 values of the residuals.  To solve the minimization problem, an iterative method
 is used. Two parameters are required  for that: the maximum number of iterations
 \mod s$ of $S$.   After the minimization, the matrix $S$ is  reused with the new
 values of the residuals.  To solve the minimization problem, an iterative method
 is used. Two parameters are required  for that: the maximum number of iterations


@@ -1051,6 +1058,24 @@ core. It can also  be observed that the difference between CGLS  and LSQR is not
 significant. Both can be good but it seems not possible to know in advance which
 one will be the best.
 
 significant. Both can be good but it seems not possible to know in advance which
 one will be the best.
 

+Table~\ref{tab:05} show a strong scaling experiment with the exemple ex54 on the
+Curie  architecture. So  in  this case,  the  number of  unknownws  is fixed  to
+$204,919,225$ and the number of cores ranges from $512$ to $8192$ with the power
+of two.  The  threshold is fixed to $5e-5$ and only  the $mg$ preconditioner has
+been tested. Here  again we can see that TSIRM is  faster that FGMRES. Efficiecy
+of each algorithms is reported. It  can be noticed that FGMRES is more efficient
+than TSIRM except with $8,192$ cores and that its efficiency is greater that one
+whereas the  efficiency of TSIRM is  lower than one. Nevertheless,  the ratio of
+TSIRM  with any  version  of the  least-squares  method is  always faster.  With
+$8,192$ cores when the number of iterations is far more important for FGMRES, we
+can see that it is only slightly more important for TSIRM.
+
+In  Figure~\ref{fig:02}  we report  the  number  of  iterations per  second  for
+experiments  reported in  Table~\ref{tab:05}.  This Figure  highlights that  the
+number of iterations per  seconds is more of less the same  for FGMRES and TSIRM
+with a little advantage for FGMRES. It  can be explained by the fact that, as we
+have previously explained, that the iterations of the least-sqaure steps are not
+taken into account with TSIRM.

 
 \begin{table*}[htbp]
 \begin{center}
 
 \begin{table*}[htbp]
 \begin{center}


@@ -1081,6 +1106,26 @@ one will be the best.
 \label{fig:02}
 \end{figure}
 
 \label{fig:02}
 \end{figure}
 

+
+Concerning the  experiments some  other remarks are  interesting.
+\begin{itemize}
+\item We  can tested other examples of  PETSc (ex29, ex45, ex49).  For all these
+  examples,  we also obtained  similar gain  between GMRES  and TSIRM  but those
+  examples are  not scalable with many  cores. In general, we  had some problems
+  with more than $4,096$ cores.
+\item We have tested many iterative  solvers available in PETSc.  In fast, it is
+  possible to use most of them with TSIRM. From our point of view, the condition
+  to  use  a  solver inside  TSIRM  is  that  the  solver  must have  a  restart
+  feature. More  precisely, the solver must  support to be  stoped and restarted
+  without decrease its  converge. That is why  with GMRES we stop it  when it is
+  naturraly  restarted (i.e.  with  $m$ the  restart parameter).   The Conjugate
+  Gradient (CG) and all its variants do not have ``restarted'' version in PETSc,
+  so they  are not  efficient.  They  will converge with  TSIRM but  not quickly
+  because if  we compare  a normal CG  with a CG  for which  we stop it  each 16
+  iterations  for example,  the  normal CG  will  be for  more efficient.   Some
+  restarted CG  or CG variant versions exist  and may be interested  to study in
+  future works.
+\end{itemize}

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


@@ -1103,13 +1148,14 @@ experiments up to 16,394 cores have been led to verify that TSIRM runs
 5 or  7 times  faster than GMRES.
 
 
 5 or  7 times  faster than GMRES.
 
 

-For future work, the authors' intention is to investigate 
-other kinds of matrices, problems, and inner solvers. The 
-influence of all parameters must be tested too, while 
-other methods to minimize the residuals must be regarded.
-The number of outer iterations to minimize should become 
-adaptative to improve the overall performances of the proposal.
-Finally, this solver will be implemented inside PETSc.
+For  future  work, the  authors'  intention is  to  investigate  other kinds  of
+matrices, problems, and  inner solvers. The influence of  all parameters must be
+tested too, while other methods to minimize the residuals must be regarded.  The
+number of outer  iterations to minimize should become  adaptative to improve the
+overall performances of the proposal.   Finally, this solver will be implemented
+inside PETSc. This  would be very interesting because it would  allow us to test
+all the non-linear  examples and compare our algorithm  with the other algorithm
+implemented in PETSc.

 
 
 % conference papers do not normally have an appendix
 
 
 % conference papers do not normally have an appendix