sdjfo

[GMRES2stage.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index 7e4945e0f0127478c09f3c6d3ace9f2f6a8bab0e..16b2de588eaeb4977240ffb3598fae1e9dd79cf3 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -601,29 +601,29 @@ is summarized while intended perspectives are provided.
 %%%*********************************************************
 \section{Related works}
 \label{sec:02} 
 %%%*********************************************************
 \section{Related works}
 \label{sec:02} 

-Krylov subspace iteration methods have increasingly become useful and successful
-techniques  for  solving  linear,  nonlinear systems  and  eigenvalue  problems,
-especially      since       the      increase      development       of      the
+Krylov subspace iteration methods have increasingly become key
+techniques  for  solving  linear and nonlinear systems,  or  eigenvalue  problems,
+especially      since       the      increasing      development       of      

 preconditioners~\cite{Saad2003,Meijerink77}.  One reason  of  the popularity  of
 preconditioners~\cite{Saad2003,Meijerink77}.  One reason  of  the popularity  of

-these methods is their generality, simplicity and efficiency to solve systems of
+these methods is their generality, simplicity, and efficiency to solve systems of

 equations arising from very large and complex problems.
 
 GMRES is one of the most  widely used Krylov iterative method for solving sparse
 equations arising from very large and complex problems.
 
 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
+and large  linear systems. It  has been developed by  Saad \emph{et al.}~\cite{Saad86}  as a

 generalized  method to  deal with  unsymmetric and  non-Hermitian  problems, and
 generalized  method to  deal with  unsymmetric and  non-Hermitian  problems, and

-indefinite symmetric problems too. In its original version called full GMRES, it
+indefinite symmetric problems too. In its original version called full GMRES, this algorithm

 minimizes the residual over the  current Krylov subspace until convergence in at
 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
-noticed that full GMRES is too expensive in the case of large matrices since the
+most $n$ iterations,  where $n$ is the  size of the sparse matrix.  
+Full GMRES is however too much expensive in the case of large matrices, since the

 required orthogonalization  process per  iteration grows quadratically  with 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,
+number of iterations. For that reason, GMRES is restarted in practice 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
 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
+cases depends quite critically on  the $m$ value~\cite{Huang89}. Therefore in

 most cases, a preconditioning technique is applied to the restarted GMRES method
 in order to improve its convergence.
 
 most cases, a preconditioning technique is applied to the restarted GMRES method
 in order to improve its convergence.
 

-In order to enhance the robustness of Krylov iterative solvers, some techniques have been proposed allowing the use of different preconditioners, if necessary, within the iteration instead of restarting. Those techniques may lead to considerable savings in CPU time and memory requirements. 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 (or GMRES itself), where the iterations of the GMRES method are called outer iterations while the iterations of the preconditioning process referred to as inner iterations. Saad in~\cite{Saad:1993} has proposed FGMRES which is another variant of the GMRES algorithm using a variable preconditioner. In FGMRES the search directions are preconditioned whereas in GMRESR the residuals are preconditioned. However in practice the good preconditioners are those based on direct methods, as ILU preconditioners, which are not easy to parallelize and suffer from the scalability problems on large clusters of thousands of cores.  
+To enhance the robustness of Krylov iterative solvers, some techniques have been proposed allowing the use of different preconditioners, if necessary, within the iteration instead of restarting. Those techniques may lead to considerable savings in CPU time and memory requirements. Van der Vorst in~\cite{Vorst94} has for instance proposed variants of the GMRES algorithm in which a different preconditioner is applied in each iteration, leading to the so-called GMRESR family of nested methods. In fact, the GMRES method is effectively preconditioned with other iterative schemes (or GMRES itself), where the iterations of the GMRES method are called outer iterations while the iterations of the preconditioning process is referred to as inner iterations. Saad in~\cite{Saad:1993} has proposed FGMRES which is another variant of the GMRES algorithm using a variable preconditioner. In FGMRES the search directions are preconditioned whereas in GMRESR the residuals are preconditioned. However, in practice, good preconditioners are those based on direct methods, as ILU preconditioners, which are not easy to parallelize and suffer from the scalability problems on large clusters of thousands of cores.  

 
 Recently, communication-avoiding methods have been developed to reduce the communication overheads in Krylov subspace iterative solvers. On modern computer architectures, communications between processors are much slower than floating-point arithmetic operations on a given processor. Communication-avoiding techniques reduce either communications between processors or data movements between levels of the memory hierarchy, by reformulating the communication-bound kernels (more frequently SpMV kernels) and the orthogonalization operations within the Krylov iterative solver. Different works have studied the communication-avoiding techniques for the GMRES method, so-called CA-GMRES, on multicore processors and multi-GPU machines~\cite{Mohiyuddin2009,Hoemmen2010,Yamazaki2014}. 
 
 
 Recently, communication-avoiding methods have been developed to reduce the communication overheads in Krylov subspace iterative solvers. On modern computer architectures, communications between processors are much slower than floating-point arithmetic operations on a given processor. Communication-avoiding techniques reduce either communications between processors or data movements between levels of the memory hierarchy, by reformulating the communication-bound kernels (more frequently SpMV kernels) and the orthogonalization operations within the Krylov iterative solver. Different works have studied the communication-avoiding techniques for the GMRES method, so-called CA-GMRES, on multicore processors and multi-GPU machines~\cite{Mohiyuddin2009,Hoemmen2010,Yamazaki2014}. 
 


@@ -705,7 +705,7 @@ 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
 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 $||Ax_k-b||_2$.
+$error$, which is defined by $||Ax_k-b||_2$.

 
   Line~\ref{algo:store},  $S_{k \mod  s}=x_k$ consists  in copying  the solution
   $x_k$ into the  column $k \mod s$ of $S$.  After  the minimization, the matrix
 
   Line~\ref{algo:store},  $S_{k \mod  s}=x_k$ consists  in copying  the solution
   $x_k$ into the  column $k \mod s$ of $S$.  After  the minimization, the matrix


@@ -716,7 +716,7 @@ $error$ which is defined by $||Ax_k-b||_2$.
 
 Let us summarize the most important parameters of TSIRM:
 \begin{itemize}
 
 Let us summarize the most important parameters of TSIRM:
 \begin{itemize}

-\item $\epsilon_{tsirm}$: the threshold to stop the TSIRM method;
+\item $\epsilon_{tsirm}$: the threshold that stops the TSIRM method;

 \item $max\_iter_{kryl}$: the maximum number of iterations for the Krylov method;
 \item $s$: the number of outer iterations before applying the minimization step;
 \item $max\_iter_{ls}$: the maximum number of iterations for the iterative least-squares method;
 \item $max\_iter_{kryl}$: the maximum number of iterations for the Krylov method;
 \item $s$: the number of outer iterations before applying the minimization step;
 \item $max\_iter_{ls}$: the maximum number of iterations for the iterative least-squares method;


@@ -727,9 +727,9 @@ Let us summarize the most important parameters of TSIRM:
 The  parallelization  of  TSIRM  relies   on  the  parallelization  of  all  its
 parts. More  precisely, except  the least-squares step,  all the other  parts are
 obvious to  achieve out in parallel. In  order to develop a  parallel version of
 The  parallelization  of  TSIRM  relies   on  the  parallelization  of  all  its
 parts. More  precisely, except  the least-squares step,  all the other  parts are
 obvious to  achieve out in parallel. In  order to develop a  parallel version of

-our   code,   we   have   chosen  to   use   PETSc~\cite{petsc-web-page}.    For
-line~\ref{algo:matrix_mul} the  matrix-matrix multiplication is  implemented and
-efficient since the  matrix $A$ is sparse and since the  matrix $S$ contains few
+our   code,   we   have   chosen  to   use   PETSc~\cite{petsc-web-page}.    In
+line~\ref{algo:matrix_mul}, the  matrix-matrix multiplication is  implemented and
+efficient since the  matrix $A$ is sparse and the  matrix $S$ contains few

 columns in  practice. As explained  previously, at least  two methods seem  to be
 interesting to solve the least-squares minimization, CGLS and LSQR.
 
 columns in  practice. As explained  previously, at least  two methods seem  to be
 interesting to solve the least-squares minimization, CGLS and LSQR.
 


@@ -764,7 +764,7 @@ the parallelization of CGLS which is  similar to LSQR.
 
 
 In each iteration  of CGLS, there is two  matrix-vector multiplications and some
 
 
 In each iteration  of CGLS, there is two  matrix-vector multiplications and some

-classical  operations:  dot  product,   norm,  multiplication  and  addition  on
+classical  operations:  dot  product,   norm,  multiplication,  and  addition  on

 vectors.  All  these  operations are  easy  to  implement  in PETSc  or  similar
 environment.  It should be noticed that LSQR follows the same principle, it is a
 little bit longer but it performs more or less the same operations.
 vectors.  All  these  operations are  easy  to  implement  in PETSc  or  similar
 environment.  It should be noticed that LSQR follows the same principle, it is a
 little bit longer but it performs more or less the same operations.


@@ -846,8 +846,11 @@ $\begin{array}{ll}
 which concludes the induction and the proof.
 \end{proof}
 
 which concludes the induction and the proof.
 \end{proof}
 

-%We can remark that, at each iterate, the residue of the TSIRM algorithm is lower 
-%than the one of the GMRES method.
+Remark that a similar proposition can be formulated at each time
+the given solver satisfies an inequality of the form $||r_n|| \leqslant \mu^n ||r_0||$,
+with $|\mu|<1$. Furthermore, it is \emph{a priori} possible in some particular cases 
+regarding $A$, 
+that the proposed TSIRM converges while the GMRES($m$) does not.

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


@@ -881,21 +884,22 @@ torso3             & 2D/3D problem & 259,156 & 4,429,042 \\
 \label{tab:01}
 \end{center}
 \end{table}
 \label{tab:01}
 \end{center}
 \end{table}

-Chosen parameters  are detailed below.   As by default  the restart of  GMRES is
-performed  every 30  iterations,  we have  chosen  to stop  the  GMRES every  30
-iterations (\emph{i.e.} $max\_iter_{kryl}=30$).  $s$ is set to 8. CGLS is chosen
-to   minimize  the   least-squares  problem   with  the   following  parameters:
+Chosen parameters  are detailed below.   
+We have  stopped  the  GMRES every  30
+iterations (\emph{i.e.}, $max\_iter_{kryl}=30$), which is the default 
+setting of GMRES.  $s$, for its part, has been set to 8. CGLS 
+ minimizes  the   least-squares  problem   with  parameters

 $\epsilon_{ls}=1e-40$ and $max\_iter_{ls}=20$.  The external precision is set to
 $\epsilon_{ls}=1e-40$ and $max\_iter_{ls}=20$.  The external precision is set to

-$\epsilon_{tsirm}=1e-10$.  Those  experiments have been performed  on a Intel(R)
-Core(TM) i7-3630QM CPU @ 2.40GHz with the version 3.5.1 of PETSc.
+$\epsilon_{tsirm}=1e-10$.  These  experiments have been performed  on an Intel(R)
+Core(TM) i7-3630QM CPU @ 2.40GHz with the 3.5.1 version  of PETSc.

 
 
 
 

-In  Table~\ref{tab:02}, some  experiments comparing  the solving  of  the linear
-systems obtained with the previous matrices  with a GMRES variant and with TSIRM
-are given. In the  second column, it can be noticed that  either GMRES or FGMRES
-(Flexible GMRES)~\cite{Saad:1993} is used to solve the linear system.  According
-to  the matrices,  different preconditioners  are  used.  With  TSIRM, the  same
-solver and the  same preconditionner are used.  This Table  shows that TSIRM can
+Experiments comparing 
+a GMRES variant with TSIRM in the resolution of linear systems are given in  Table~\ref{tab:02}. 
+The  second column describes whether GMRES or FGMRES
+(Flexible GMRES~\cite{Saad:1993}) has been used for linear systems solving.  
+Different preconditioners  have been used according to the matrices.  With  TSIRM, the  same
+solver and the  same preconditionner are used.  This table  shows that TSIRM can

 drastically reduce  the number of iterations  to reach the  convergence when the
 number of iterations for  the normal GMRES is more or less  greater than 500. In
 fact this also depends on two parameters: the number of iterations to stop GMRES
 drastically reduce  the number of iterations  to reach the  convergence when the
 number of iterations for  the normal GMRES is more or less  greater than 500. In
 fact this also depends on two parameters: the number of iterations to stop GMRES


@@ -920,7 +924,7 @@ torso3             & fgmres / sor  & 37.70 & 565 & 34.97 & 510 \\
 \hline
 
 \end{tabular}
 \hline
 
 \end{tabular}

-\caption{Comparison of (F)GMRES and TSIRM with (F)GMRES in sequential with some matrices, time is expressed in seconds.}
+\caption{Comparison between sequential standalone (F)GMRES and TSIRM with (F)GMRES (time in seconds).}

 \label{tab:02}
 \end{center}
 \end{table}
 \label{tab:02}
 \end{center}
 \end{table}


@@ -930,10 +934,10 @@ torso3             & fgmres / sor  & 37.70 & 565 & 34.97 & 510 \\
 
 
 In order to perform larger experiments, we have tested some example applications
 
 
 In order to perform larger experiments, we have tested some example applications

-of  PETSc. Those  applications are  available in  the \emph{ksp}  part  which is
+of  PETSc. Those  applications are  available in  the \emph{ksp}  part,  which is

 suited for scalable linear equations solvers:
 \begin{itemize}
 suited for scalable linear equations solvers:
 \begin{itemize}

-\item ex15  is an example  which solves in  parallel an operator using  a finite
+\item ex15  is an example  that solves in  parallel an operator using  a finite

   difference  scheme.   The  diagonal  is  equal to  4  and  4  extra-diagonals
   representing the neighbors in each directions  are equal to -1. This example is
   used  in many  physical phenomena, for  example, heat  and fluid  flow, wave
   difference  scheme.   The  diagonal  is  equal to  4  and  4  extra-diagonals
   representing the neighbors in each directions  are equal to -1. This example is
   used  in many  physical phenomena, for  example, heat  and fluid  flow, wave


@@ -1138,12 +1142,12 @@ Concerning the  experiments some  other remarks are  interesting.
   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.
   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
+\item We have tested many iterative  solvers available in PETSc.  In fact, 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
   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
+  feature. More  precisely, the solver must  support to be  stopped and restarted

   without decrease its  converge. That is why  with GMRES we stop it  when it is
   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
+  naturally  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
   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


@@ -1174,11 +1178,13 @@ experiments up to 16,394 cores have been led to verify that TSIRM runs
 
 
 For  future  work, the  authors'  intention is  to  investigate  other kinds  of
 
 
 For  future  work, the  authors'  intention is  to  investigate  other kinds  of

-matrices, problems, and  inner solvers. The influence of  all parameters must be
+matrices, problems, and  inner solvers. In particular, the possibility 
+to obtain a convergence of TSIRM in situations where the GMRES is divergent will be
+investigated. 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
 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
+inside PETSc, which would be of interest as it would  allow us to test

 all the non-linear  examples and compare our algorithm  with the other algorithm
 implemented in PETSc.
 
 all the non-linear  examples and compare our algorithm  with the other algorithm
 implemented in PETSc.