X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/a559d16ba3585ae8fb9457c67cf66851a5d463c8..fd78ced113c137f5f83f7aea69a10961467aea3e:/paper.tex diff --git a/paper.tex b/paper.tex index c8e503d..16b2de5 100644 --- a/paper.tex +++ b/paper.tex @@ -601,29 +601,29 @@ is summarized while intended perspectives are provided. %%%********************************************************* \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 -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 -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 -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 -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 -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 -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. -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}. @@ -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 -$error$ 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 @@ -716,7 +716,7 @@ $error$ which is defined by $||Ax_k-b||_2$. 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; @@ -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 -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. @@ -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 -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. @@ -846,8 +846,11 @@ $\begin{array}{ll} 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} -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 -$\epsilon_{tsirm}=1e-10$. 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 @@ -920,7 +924,7 @@ torso3 & fgmres / sor & 37.70 & 565 & 34.97 & 510 \\ \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} @@ -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 -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} -\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 @@ -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 -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 -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.