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351 \usepackage[utf8]{inputenc}
352 \usepackage[T1]{fontenc}
353 \usepackage{algorithm}
354 \usepackage{algpseudocode}
357 \usepackage{multirow}
358 \usepackage{graphicx}
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361 \algnewcommand\Input{\item[\algorithmicinput]}
363 \algnewcommand\algorithmicoutput{\textbf{Output:}}
364 \algnewcommand\Output{\item[\algorithmicoutput]}
366 \newtheorem{proposition}{Proposition}
371 % can use linebreaks \\ within to get better formatting as desired
372 \title{TSIRM: A Two-Stage Iteration with least-squares Residual Minimization algorithm to solve large sparse linear systems}
379 % author names and affiliations
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383 \author{\IEEEauthorblockN{Rapha\"el Couturier\IEEEauthorrefmark{1}, Lilia Ziane Khodja\IEEEauthorrefmark{2}, and Christophe Guyeux\IEEEauthorrefmark{1}}
384 \IEEEauthorblockA{\IEEEauthorrefmark{1} Femto-ST Institute, University of Franche-Comt\'e, France\\
385 Email: \{raphael.couturier,christophe.guyeux\}@univ-fcomte.fr}
386 \IEEEauthorblockA{\IEEEauthorrefmark{2} INRIA Bordeaux Sud-Ouest, France\\
387 Email: lilia.ziane@inria.fr}
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417 % use for special paper notices
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423 % make the title area
428 In this article, a two-stage iterative algorithm is proposed to improve the
429 convergence of Krylov based iterative methods, typically those of GMRES
430 variants. The principle of the proposed approach is to build an external
431 iteration over the Krylov method, and to frequently store its current residual
432 (at each GMRES restart for instance). After a given number of outer iterations,
433 a least-squares minimization step is applied on the matrix composed by the saved
434 residuals, in order to compute a better solution and to make new iterations if
435 required. It is proven that the proposal has the same convergence properties
436 than the inner embedded method itself. Experiments using up to 16,394 cores
437 also show that the proposed algorithm runs around 5 or 7 times faster than
442 Iterative Krylov methods; sparse linear systems; two stage iteration; least-squares residual minimization; PETSc
446 % For peer review papers, you can put extra information on the cover
448 % \ifCLASSOPTIONpeerreview
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538 % footnotes above bottom floats. This can be corrected via the \fnbelowfloat
539 % command of the stfloats package.
543 %%%*********************************************************
544 %%%*********************************************************
545 \section{Introduction}
547 % You must have at least 2 lines in the paragraph with the drop letter
548 % (should never be an issue)
550 Iterative methods have recently become more attractive than direct ones to solve
551 very large sparse linear systems\cite{Saad2003}. They are more efficient in a
552 parallel context, supporting thousands of cores, and they require less memory
553 and arithmetic operations than direct methods~\cite{bahicontascoutu}. This is
554 why new iterative methods are frequently proposed or adapted by researchers, and
555 the increasing need to solve very large sparse linear systems has triggered the
556 development of such efficient iterative techniques suitable for parallel
559 Most of the successful iterative methods currently available are based on
560 so-called ``Krylov subspaces''. They consist in forming a basis of successive
561 matrix powers multiplied by an initial vector, which can be for instance the
562 residual. These methods use vectors orthogonality of the Krylov subspace basis
563 in order to solve linear systems. The most known iterative Krylov subspace
564 methods are conjugate gradient and GMRES ones (Generalized Minimal RESidual).
567 However, iterative methods suffer from scalability problems on parallel
568 computing platforms with many processors, due to their need of reduction
569 operations, and to collective communications to achieve matrix-vector
570 multiplications. The communications on large clusters with thousands of cores
571 and large sizes of messages can significantly affect the performances of these
572 iterative methods. As a consequence, Krylov subspace iteration methods are often
573 used with preconditioners in practice, to increase their convergence and
574 accelerate their performances. However, most of the good preconditioners are
575 not scalable on large clusters.
577 In this research work, a two-stage algorithm based on two nested iterations
578 called inner-outer iterations is proposed. This algorithm consists in solving
579 the sparse linear system iteratively with a small number of inner iterations,
580 and restarting the outer step with a new solution minimizing some error
581 functions over some previous residuals. For further information on two-stage
582 iteration methods, interested readers are invited to
583 consult~\cite{Nichols:1973:CTS}. Two-stage algorithms are easy to parallelize on
584 large clusters. Furthermore, the least-squares minimization technique improves
585 its convergence and performances.
587 The present article is organized as follows. Related works are presented in
588 Section~\ref{sec:02}. Section~\ref{sec:03} details the two-stage algorithm using
589 a least-squares residual minimization, while Section~\ref{sec:04} provides
590 convergence results regarding this method. Section~\ref{sec:05} shows some
591 experimental results obtained on large clusters using routines of PETSc
592 toolkit. This research work ends by a conclusion section, in which the proposal
593 is summarized while intended perspectives are provided.
595 %%%*********************************************************
596 %%%*********************************************************
600 %%%*********************************************************
601 %%%*********************************************************
602 \section{Related works}
604 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{}.
606 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.
608 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.
610 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 methods for multicore processors and multi-GPU machines~\cite{} {\bf MANQUE REF}.
612 Compared to all these works and to all the other works on Krylov iterative
613 method, the originality of our work is to build a second iteration over a Krylov
614 iterative method and to minimize the residuals with a least-squares method after
615 a given number of outer iteration.
617 %%%*********************************************************
618 %%%*********************************************************
622 %%%*********************************************************
623 %%%*********************************************************
624 \section{Two-stage iteration with least-squares residuals minimization algorithm}
626 A two-stage algorithm is proposed to solve large sparse linear systems of the
627 form $Ax=b$, where $A\in\mathbb{R}^{n\times n}$ is a sparse and square
628 nonsingular matrix, $x\in\mathbb{R}^n$ is the solution vector, and
629 $b\in\mathbb{R}^n$ is the right-hand side. As explained previously,
630 the algorithm is implemented as an
631 inner-outer iteration solver based on iterative Krylov methods. The main
632 key-points of the proposed solver are given in Algorithm~\ref{algo:01}.
633 It can be summarized as follows: the
634 inner solver is a Krylov based one. In order to accelerate its convergence, the
635 outer solver periodically applies a least-squares minimization on the residuals computed by the inner one. %Tsolver which does not required to be changed.
637 At each outer iteration, the sparse linear system $Ax=b$ is partially solved
638 using only $m$ iterations of an iterative method, this latter being initialized
639 with the last obtained approximation. GMRES method~\cite{Saad86}, or any of its
640 variants, can potentially be used as inner solver. The current approximation of
641 the Krylov method is then stored inside a $n \times s$ matrix $S$, which is
642 composed by the $s$ last solutions that have been computed during the inner
643 iterations phase. In the remainder, the $i$-th column vector of $S$ will be
646 At each $s$ iterations, another kind of minimization step is applied in order to
647 compute a new solution $x$. For that, the previous residuals of $Ax=b$ are computed by
648 the inner iterations with $(b-AS)$. The minimization of the residuals is obtained by
650 \underset{\alpha\in\mathbb{R}^{s}}{min}\|b-R\alpha\|_2
653 with $R=AS$. The new solution $x$ is then computed with $x=S\alpha$.
656 In practice, $R$ is a dense rectangular matrix belonging in $\mathbb{R}^{n\times s}$,
657 with $s\ll n$. In order to minimize~\eqref{eq:01}, a least-squares method such as
658 CGLS ~\cite{Hestenes52} or LSQR~\cite{Paige82} is used. Remark that these methods are more
659 appropriate than a single direct method in a parallel context.
665 \begin{algorithmic}[1]
666 \Input $A$ (sparse matrix), $b$ (right-hand side)
667 \Output $x$ (solution vector)\vspace{0.2cm}
668 \State Set the initial guess $x_0$
669 \For {$k=1,2,3,\ldots$ until convergence ($error<\epsilon_{tsirm}$)} \label{algo:conv}
670 \State $[x_k,error]=Solve(A,b,x_{k-1},max\_iter_{kryl})$ \label{algo:solve}
671 \State $S_{k \mod s}=x_k$ \label{algo:store} \Comment{update column ($k \mod s$) of $S$}
672 \If {$k \mod s=0$ {\bf and} $error>\epsilon_{kryl}$}
673 \State $R=AS$ \Comment{compute dense matrix} \label{algo:matrix_mul}
674 \State $\alpha=Least\_Squares(R,b,max\_iter_{ls})$ \label{algo:}
675 \State $x_k=S\alpha$ \Comment{compute new solution}
682 Algorithm~\ref{algo:01} summarizes the principle of the proposed method. The
683 outer iteration is inside the \emph{for} loop. Line~\ref{algo:solve}, the Krylov
684 method is called for a maximum of $max\_iter_{kryl}$ iterations. In practice,
685 we suggest to set this parameter equal to the restart number in the GMRES-like
686 method. Moreover, a tolerance threshold must be specified for the solver. In
687 practice, this threshold must be much smaller than the convergence threshold of
688 the TSIRM algorithm (\emph{i.e.}, $\epsilon_{tsirm}$). We also consider that
689 after the call of the $Solve$ function, we obtain the vector $x_k$ and the error
690 which is defined by $||Ax_k-b||_2$.
692 Line~\ref{algo:store},
693 $S_{k \mod s}=x_k$ consists in copying the solution $x_k$ into the column $k
694 \mod s$ of $S$. After the minimization, the matrix $S$ is reused with the new
695 values of the residuals. To solve the minimization problem, an iterative method
696 is used. Two parameters are required for that: the maximum number of iterations
697 and the threshold to stop the method.
699 Let us summarize the most important parameters of TSIRM:
701 \item $\epsilon_{tsirm}$: the threshold to stop the TSIRM method;
702 \item $max\_iter_{kryl}$: the maximum number of iterations for the Krylov method;
703 \item $s$: the number of outer iterations before applying the minimization step;
704 \item $max\_iter_{ls}$: the maximum number of iterations for the iterative least-squares method;
705 \item $\epsilon_{ls}$: the threshold used to stop the least-squares method.
709 The parallelization of TSIRM relies on the parallelization of all its
710 parts. More precisely, except the least-squares step, all the other parts are
711 obvious to achieve out in parallel. In order to develop a parallel version of
712 our code, we have chosen to use PETSc~\cite{petsc-web-page}. For
713 line~\ref{algo:matrix_mul} the matrix-matrix multiplication is implemented and
714 efficient since the matrix $A$ is sparse and since the matrix $S$ contains few
715 columns in practice. As explained previously, at least two methods seem to be
716 interesting to solve the least-squares minimization, CGLS and LSQR.
718 In the following we remind the CGLS algorithm. The LSQR method follows more or
719 less the same principle but it takes more place, so we briefly explain the parallelization of CGLS which is similar to LSQR.
723 \begin{algorithmic}[1]
724 \Input $A$ (matrix), $b$ (right-hand side)
725 \Output $x$ (solution vector)\vspace{0.2cm}
726 \State Let $x_0$ be an initial approximation
730 \State $\gamma=||s_0||^2_2$
731 \For {$k=1,2,3,\ldots$ until convergence ($\gamma<\epsilon_{ls}$)} \label{algo2:conv}
733 \State $\alpha_k=\gamma/||q_k||^2_2$
734 \State $x_k=x_{k-1}+\alpha_kp_k$
735 \State $r_k=r_{k-1}-\alpha_kq_k$
737 \State $\gamma_{old}=\gamma$
738 \State $\gamma=||s_k||^2_2$
739 \State $\beta_k=\gamma/\gamma_{old}$
740 \State $p_{k+1}=s_k+\beta_kp_k$
747 In each iteration of CGLS, there is two matrix-vector multiplications and some
748 classical operations: dot product, norm, multiplication and addition on vectors. All
749 these operations are easy to implement in PETSc or similar environment.
753 %%%*********************************************************
754 %%%*********************************************************
756 \section{Convergence results}
760 We can now claim that,
763 If $A$ is either a definite positive or a positive matrix and GMRES($m$) is used as solver, then the TSIRM algorithm is convergent.
765 Furthermore, let $r_k$ be the
766 $k$-th residue of TSIRM, then
767 we have the following boundaries:
769 \item when $A$ is positive:
771 ||r_k|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0|| ,
773 where $M$ is the symmetric part of $A$, $\alpha = \lambda_{min}(M)^2$ and $\beta = \lambda_{max}(A^T A)$;
774 \item when $A$ is positive definite:
776 \|r_k\| \leq \left( 1-\frac{\lambda_{\mathrm{min}}^2(1/2(A^T + A))}{ \lambda_{\mathrm{max}}(A^T A)} \right)^{km/2} \|r_0\|.
779 %In the general case, where A is not positive definite, we have
780 %$\|r_n\| \le \inf_{p \in P_n} \|p(A)\| \le \kappa_2(V) \inf_{p \in P_n} \max_{\lambda \in \sigma(A)} |p(\lambda)| \|r_0\|, .$
784 Let us first recall that the residue is under control when considering the GMRES algorithm on a positive definite matrix, and it is bounded as follows:
786 \|r_k\| \leq \left( 1-\frac{\lambda_{\mathrm{min}}^2(1/2(A^T + A))}{ \lambda_{\mathrm{max}}(A^T A)} \right)^{k/2} \|r_0\| .
788 Additionally, when $A$ is a positive real matrix with symmetric part $M$, then the residual norm provided at the $m$-th step of GMRES satisfies:
790 ||r_m|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{m}{2}} ||r_0|| ,
792 where $\alpha$ and $\beta$ are defined as in Proposition~\ref{prop:saad}, which proves
793 the convergence of GMRES($m$) for all $m$ under such assumptions regarding $A$.
794 These well-known results can be found, \emph{e.g.}, in~\cite{Saad86}.
796 We will now prove by a mathematical induction that, for each $k \in \mathbb{N}^\ast$,
797 $||r_k|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{mk}{2}} ||r_0||$ when $A$ is positive, and $\|r_k\| \leq \left( 1-\frac{\lambda_{\mathrm{min}}^2(1/2(A^T + A))}{ \lambda_{\mathrm{max}}(A^T A)} \right)^{km/2} \|r_0\|$ when $A$ is positive definite.
799 The base case is obvious, as for $k=1$, the TSIRM algorithm simply consists in applying GMRES($m$) once, leading to a new residual $r_1$ that follows the inductive hypothesis due, to the results recalled above.
801 Suppose now that the claim holds for all $m=1, 2, \hdots, k-1$, that is, $\forall m \in \{1,2,\hdots, k-1\}$, $||r_m|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0||$ in the positive case, and $\|r_k\| \leq \left( 1-\frac{\lambda_{\mathrm{min}}^2(1/2(A^T + A))}{ \lambda_{\mathrm{max}}(A^T A)} \right)^{km/2} \|r_0\|$ in the definite positive one.
802 We will show that the statement holds too for $r_k$. Two situations can occur:
804 \item If $k \not\equiv 0 ~(\textrm{mod}\ m)$, then the TSIRM algorithm consists in executing GMRES once. In that case and by using the inductive hypothesis, we obtain either $||r_k|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{m}{2}} ||r_{k-1}||\leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0||$ if $A$ is positive, or $\|r_k\| \leqslant \left( 1-\frac{\lambda_{\mathrm{min}}^2(1/2(A^T + A))}{ \lambda_{\mathrm{max}}(A^T A)} \right)^{m/2} \|r_{k-1}\|$ $\leqslant$ $\left( 1-\frac{\lambda_{\mathrm{min}}^2(1/2(A^T + A))}{ \lambda_{\mathrm{max}}(A^T A)} \right)^{km/2} \|r_{0}\|$ in the positive definite case.
805 \item Else, the TSIRM algorithm consists in two stages: a first GMRES($m$) execution leads to a temporary $x_k$ whose residue satisfies:
807 \item $||r_k|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{m}{2}} ||r_{k-1}||\leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0||$ in the positive case,
808 \item $\|r_k\| \leqslant \left( 1-\frac{\lambda_{\mathrm{min}}^2(1/2(A^T + A))}{ \lambda_{\mathrm{max}}(A^T A)} \right)^{m/2} \|r_{k-1}\|$ $\leqslant$ $\left( 1-\frac{\lambda_{\mathrm{min}}^2(1/2(A^T + A))}{ \lambda_{\mathrm{max}}(A^T A)} \right)^{km/2} \|r_{0}\|$ in the positive definite one,
810 and a least squares resolution.
811 Let $\operatorname{span}(S) = \left \{ {\sum_{i=1}^k \lambda_i v_i \Big| k \in \mathbb{N}, v_i \in S, \lambda _i \in \mathbb{R}} \right \}$ be the linear span of a set of real vectors $S$. So,\\
812 $\min_{\alpha \in \mathbb{R}^s} ||b-R\alpha ||_2 = \min_{\alpha \in \mathbb{R}^s} ||b-AS\alpha ||_2$
815 & = \min_{x \in span\left(S_{k-s+1}, S_{k-s+2}, \hdots, S_{k} \right)} ||b-AS\alpha ||_2\\
816 & = \min_{x \in span\left(x_{k-s+1}, x_{k-s}+2, \hdots, x_{k} \right)} ||b-AS\alpha ||_2\\
817 & \leqslant \min_{x \in span\left( x_{k} \right)} ||b-Ax ||_2\\
818 & \leqslant \min_{\lambda \in \mathbb{R}} ||b-\lambda Ax_{k} ||_2\\
819 & \leqslant ||b-Ax_{k}||_2\\
821 & \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0||, \textrm{ if $A$ is positive,}\\
822 & \leqslant \left( 1-\frac{\lambda_{\mathrm{min}}^2(1/2(A^T + A))}{ \lambda_{\mathrm{max}}(A^T A)} \right)^{km/2} \|r_{0}\|, \textrm{ if $A$ is}\\
823 & \textrm{positive definite,}
826 which concludes the induction and the proof.
829 %We can remark that, at each iterate, the residue of the TSIRM algorithm is lower
830 %than the one of the GMRES method.
832 %%%*********************************************************
833 %%%*********************************************************
834 \section{Experiments using PETSc}
838 In order to see the behavior of the proposal when considering only one processor, a first
839 comparison with GMRES or FGMRES and the new algorithm detailed previously has been experimented.
840 Matrices that have been used with their characteristics (names, fields, rows, and nonzero coefficients) are detailed in
841 Table~\ref{tab:01}. These latter, which are real-world applications matrices,
843 from the Davis collection, University of
844 Florida~\cite{Dav97}.
848 \begin{tabular}{|c|c|r|r|r|}
850 Matrix name & Field &\# Rows & \# Nonzeros \\\hline \hline
851 crashbasis & Optimization & 160,000 & 1,750,416 \\
852 parabolic\_fem & Comput. fluid dynamics & 525,825 & 2,100,225 \\
853 epb3 & Thermal problem & 84,617 & 463,625 \\
854 atmosmodj & Comput. fluid dynamics & 1,270,432 & 8,814,880 \\
855 bfwa398 & Electromagnetics pb & 398 & 3,678 \\
856 torso3 & 2D/3D problem & 259,156 & 4,429,042 \\
860 \caption{Main characteristics of the sparse matrices chosen from the Davis collection}
864 Chosen parameters are detailed below.
865 %The following parameters have been chosen for our experiments.
867 the restart of GMRES is performed every 30 iterations, we have chosen to stop
868 the GMRES every 30 iterations (\emph{i.e.} $max\_iter_{kryl}=30$). $s$ is set to 8. CGLS is
869 chosen to minimize the least-squares problem with the following parameters:
870 $\epsilon_{ls}=1e-40$ and $max\_iter_{ls}=20$. The external precision is set to
871 $\epsilon_{tsirm}=1e-10$. Those experiments have been performed on a Intel(R)
872 Core(TM) i7-3630QM CPU @ 2.40GHz with the version 3.5.1 of PETSc.
875 In Table~\ref{tab:02}, some experiments comparing the solving of the linear
876 systems obtained with the previous matrices with a GMRES variant and with out 2
877 stage algorithm are given. In the second column, it can be noticed that either
878 GRMES or FGMRES (Flexible GMRES)~\cite{Saad:1993} is used to solve the linear
879 system. According to the matrices, different preconditioner is used. With
880 TSIRM, the same solver and the same preconditionner are used. This Table shows
881 that TSIRM can drastically reduce the number of iterations to reach the
882 convergence when the number of iterations for the normal GMRES is more or less
883 greater than 500. In fact this also depends on tow parameters: the number of
884 iterations to stop GMRES and the number of iterations to perform the
890 \begin{tabular}{|c|c|r|r|r|r|}
893 \multirow{2}{*}{Matrix name} & Solver / & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSIRM CGLS} \\
895 & precond & Time & \# Iter. & Time & \# Iter. \\\hline \hline
897 crashbasis & gmres / none & 15.65 & 518 & 14.12 & 450 \\
898 parabolic\_fem & gmres / ilu & 1009.94 & 7573 & 401.52 & 2970 \\
899 epb3 & fgmres / sor & 8.67 & 600 & 8.21 & 540 \\
900 atmosmodj & fgmres / sor & 104.23 & 451 & 88.97 & 366 \\
901 bfwa398 & gmres / none & 1.42 & 9612 & 0.28 & 1650 \\
902 torso3 & fgmres / sor & 37.70 & 565 & 34.97 & 510 \\
906 \caption{Comparison of (F)GMRES and TSIRM with (F)GMRES in sequential with some matrices, time is expressed in seconds.}
915 In order to perform larger experiments, we have tested some example applications
916 of PETSc. Those applications are available in the ksp part which is suited for
917 scalable linear equations solvers:
919 \item ex15 is an example which solves in parallel an operator using a finite
920 difference scheme. The diagonal is equal to 4 and 4 extra-diagonals
921 representing the neighbors in each directions are equal to -1. This example is
922 used in many physical phenomena, for example, heat and fluid flow, wave
924 \item ex54 is another example based on 2D problem discretized with quadrilateral
925 finite elements. For this example, the user can define the scaling of material
926 coefficient in embedded circle called $\alpha$.
928 For more technical details on these applications, interested readers are invited
929 to read the codes available in the PETSc sources. Those problems have been
930 chosen because they are scalable with many cores which is not the case of other
931 problems that we have tested.
933 In the following larger experiments are described on two large scale
934 architectures: Curie and Juqeen. Both these architectures are supercomputer
935 composed of 80,640 cores for Curie and 458,752 cores for Juqueen. Those machines
936 are respectively hosted by GENCI in France and Jülich Supercomputing Centre in
937 Germany. They belongs with other similar architectures of the PRACE initiative (
938 Partnership for Advanced Computing in Europe) which aims at proposing high
939 performance supercomputing architecture to enhance research in Europe. The Curie
940 architecture is composed of Intel E5-2680 processors at 2.7 GHz with 2Gb memory
941 by core. The Juqueen architecture is composed of IBM PowerPC A2 at 1.6 GHz with
942 1Gb memory per core. Both those architecture are equiped with a dedicated high
946 In many situations, using preconditioners is essential in order to find the
947 solution of a linear system. There are many preconditioners available in PETSc.
948 For parallel applications all the preconditioners based on matrix factorization
949 are not available. In our experiments, we have tested different kinds of
950 preconditioners, however as it is not the subject of this paper, we will not
951 present results with many preconditioners. In practise, we have chosen to use a
952 multigrid (mg) and successive over-relaxation (sor). For more details on the
953 preconditioner in PETSc please consult~\cite{petsc-web-page}.
959 \begin{tabular}{|r|r|r|r|r|r|r|r|r|}
962 nb. cores & precond & \multicolumn{2}{c|}{FGMRES} & \multicolumn{2}{c|}{TSIRM CGLS} & \multicolumn{2}{c|}{TSIRM LSQR} & best gain \\
964 & & Time & \# Iter. & Time & \# Iter. & Time & \# Iter. & \\\hline \hline
965 2,048 & mg & 403.49 & 18,210 & 73.89 & 3,060 & 77.84 & 3,270 & 5.46 \\
966 2,048 & sor & 745.37 & 57,060 & 87.31 & 6,150 & 104.21 & 7,230 & 8.53 \\
967 4,096 & mg & 562.25 & 25,170 & 97.23 & 3,990 & 89.71 & 3,630 & 6.27 \\
968 4,096 & sor & 912.12 & 70,194 & 145.57 & 9,750 & 168.97 & 10,980 & 6.26 \\
969 8,192 & mg & 917.02 & 40,290 & 148.81 & 5,730 & 143.03 & 5,280 & 6.41 \\
970 8,192 & sor & 1,404.53 & 106,530 & 212.55 & 12,990 & 180.97 & 10,470 & 7.76 \\
971 16,384 & mg & 1,430.56 & 63,930 & 237.17 & 8,310 & 244.26 & 7,950 & 6.03 \\
972 16,384 & sor & 2,852.14 & 216,240 & 418.46 & 21,690 & 505.26 & 23,970 & 6.82 \\
976 \caption{Comparison of FGMRES and TSIRM with FGMRES for example ex15 of PETSc with two preconditioners (mg and sor) with 25,000 components per core on Juqueen (threshold 1e-3, restart=30, s=12), time is expressed in seconds.}
981 Table~\ref{tab:03} shows the execution times and the number of iterations of
982 example ex15 of PETSc on the Juqueen architecture. Different numbers of cores
983 are studied ranging from 2,048 up-to 16,383 with the two preconditioners {\it mg} and {\it sor}. For those experiments, the number of components (or unknowns of the
984 problems) per core is fixed to 25,000, also called weak scaling. This
985 number can seem relatively small. In fact, for some applications that need a lot
986 of memory, the number of components per processor requires sometimes to be
991 In Table~\ref{tab:03}, we can notice that TSIRM is always faster than FGMRES. The last
992 column shows the ratio between FGMRES and the best version of TSIRM according to
993 the minimization procedure: CGLS or LSQR. Even if we have computed the worst
994 case between CGLS and LSQR, it is clear that TSIRM is always faster than
995 FGMRES. For this example, the multigrid preconditioner is faster than SOR. The
996 gain between TSIRM and FGMRES is more or less similar for the two
997 preconditioners. Looking at the number of iterations to reach the convergence,
998 it is obvious that TSIRM allows the reduction of the number of iterations. It
999 should be noticed that for TSIRM, in those experiments, only the iterations of
1000 the Krylov solver are taken into account. Iterations of CGLS or LSQR were not
1001 recorded but they are time-consuming. In general each $max\_iter_{kryl}*s$ which
1002 corresponds to 30*12, there are $max\_iter_{ls}$ which corresponds to 15.
1004 \begin{figure}[htbp]
1006 \includegraphics[width=0.45\textwidth]{nb_iter_sec_ex15_juqueen}
1007 \caption{Number of iterations per second with ex15 and the same parameters than in Table~\ref{tab:03} (weak scaling)}
1012 In Figure~\ref{fig:01}, the number of iterations per second corresponding to
1013 Table~\ref{tab:03} is displayed. It can be noticed that the number of
1014 iterations per second of FMGRES is constant whereas it decreases with TSIRM with
1015 both preconditioners. This can be explained by the fact that when the number of
1016 cores increases the time for the least-squares minimization step also increases but, generally,
1017 when the number of cores increases, the number of iterations to reach the
1018 threshold also increases, and, in that case, TSIRM is more efficient to reduce
1019 the number of iterations. So, the overall benefit of using TSIRM is interesting.
1026 \begin{table*}[htbp]
1028 \begin{tabular}{|r|r|r|r|r|r|r|r|r|}
1031 nb. cores & threshold & \multicolumn{2}{c|}{FGMRES} & \multicolumn{2}{c|}{TSIRM CGLS} & \multicolumn{2}{c|}{TSIRM LSQR} & best gain \\
1033 & & Time & \# Iter. & Time & \# Iter. & Time & \# Iter. & \\\hline \hline
1034 2,048 & 8e-5 & 108.88 & 16,560 & 23.06 & 3,630 & 22.79 & 3,630 & 4.77 \\
1035 2,048 & 6e-5 & 194.01 & 30,270 & 35.50 & 5,430 & 27.74 & 4,350 & 6.99 \\
1036 4,096 & 7e-5 & 160.59 & 22,530 & 35.15 & 5,130 & 29.21 & 4,350 & 5.49 \\
1037 4,096 & 6e-5 & 249.27 & 35,520 & 52.13 & 7,950 & 39.24 & 5,790 & 6.35 \\
1038 8,192 & 6e-5 & 149.54 & 17,280 & 28.68 & 3,810 & 29.05 & 3,990 & 5.21 \\
1039 8,192 & 5e-5 & 785.04 & 109,590 & 76.07 & 10,470 & 69.42 & 9,030 & 11.30 \\
1040 16,384 & 4e-5 & 718.61 & 86,400 & 98.98 & 10,830 & 131.86 & 14,790 & 7.26 \\
1044 \caption{Comparison of FGMRES and TSIRM with FGMRES algorithms for ex54 of Petsc (both with the MG preconditioner) with 25,000 components per core on Curie (restart=30, s=12), time is expressed in seconds.}
1050 In Table~\ref{tab:04}, some experiments with example ex54 on the Curie
1051 architecture are reported. For this application, we fixed $\alpha=0.6$. As it
1052 can be seen in that Table, the size of the problem has a strong influence on the
1053 number of iterations to reach the convergence. That is why we have preferred to
1054 change the threshold. If we set it to $1e-3$ as with the previous application,
1055 only one iteration is necessray to reach the convergence. So Table~\ref{tab:04}
1056 shows the results of differents executions with differents number of cores and
1057 differents thresholds. As with the previous example, we can observe that TSIRM
1058 is faster than FGMRES. The ratio greatly depends on the number of iterations for
1059 FMGRES to reach the threshold. The greater the number of iterations to reach the
1060 convergence is, the better the ratio between our algorithm and FMGRES is. This
1061 experiment is also a weak scaling with approximately $25,000$ components per
1062 core. It can also be observed that the difference between CGLS and LSQR is not
1063 significant. Both can be good but it seems not possible to know in advance which
1064 one will be the best.
1066 Table~\ref{tab:05} show a strong scaling experiment with the exemple ex54 on the
1067 Curie architecture. So in this case, the number of unknownws is fixed to
1068 $204,919,225$ and the number of cores ranges from $512$ to $8192$ with the power
1069 of two. The threshold is fixed to $5e-5$ and only the $mg$ preconditioner has
1070 been tested. Here again we can see that TSIRM is faster that FGMRES. Efficiecy
1071 of each algorithms is reported. It can be noticed that FGMRES is more efficient
1072 than TSIRM except with $8,192$ cores and that its efficiency is greater that one
1073 whereas the efficiency of TSIRM is lower than one. Nevertheless, the ratio of
1074 TSIRM with any version of the least-squares method is always faster. With
1075 $8,192$ cores when the number of iterations is far more important for FGMRES, we
1076 can see that it is only slightly more important for TSIRM.
1078 In Figure~\ref{fig:02} we report the number of iterations per second for
1079 experiments reported in Table~\ref{tab:05}. This Figure highlights that the
1080 number of iterations per seconds is more of less the same for FGMRES and TSIRM
1081 with a little advantage for FGMRES. It can be explained by the fact that, as we
1082 have previously explained, that the iterations of the least-sqaure steps are not
1083 taken into account with TSIRM.
1085 \begin{table*}[htbp]
1087 \begin{tabular}{|r|r|r|r|r|r|r|r|r|r|r|}
1090 nb. cores & \multicolumn{2}{c|}{FGMRES} & \multicolumn{2}{c|}{TSIRM CGLS} & \multicolumn{2}{c|}{TSIRM LSQR} & best gain & \multicolumn{3}{c|}{efficiency} \\
1091 \cline{2-7} \cline{9-11}
1092 & Time & \# Iter. & Time & \# Iter. & Time & \# Iter. & & FGMRES & TS CGLS & TS LSQR\\\hline \hline
1093 512 & 3,969.69 & 33,120 & 709.57 & 5,790 & 622.76 & 5,070 & 6.37 & 1 & 1 & 1 \\
1094 1024 & 1,530.06 & 25,860 & 290.95 & 4,830 & 307.71 & 5,070 & 5.25 & 1.30 & 1.21 & 1.01 \\
1095 2048 & 919.62 & 31,470 & 237.52 & 8,040 & 194.22 & 6,510 & 4.73 & 1.08 & .75 & .80\\
1096 4096 & 405.60 & 28,380 & 111.67 & 7,590 & 91.72 & 6,510 & 4.42 & 1.22 & .79 & .84 \\
1097 8192 & 785.04 & 109,590 & 76.07 & 10,470 & 69.42 & 9,030 & 11.30 & .32 & .58 & .56 \\
1102 \caption{Comparison of FGMRES and TSIRM with FGMRES 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, threshold 5e-5), time is expressed in seconds.}
1107 \begin{figure}[htbp]
1109 \includegraphics[width=0.45\textwidth]{nb_iter_sec_ex54_curie}
1110 \caption{Number of iterations per second with ex54 and the same parameters than in Table~\ref{tab:05} (strong scaling)}
1115 Concerning the experiments some other remarks are interesting.
1117 \item We can tested other examples of PETSc (ex29, ex45, ex49). For all these
1118 examples, we also obtained similar gain between GMRES and TSIRM but those
1119 examples are not scalable with many cores. In general, we had some problems
1120 with more than $4,096$ cores.
1121 \item We have tested many iterative solvers available in PETSc. In fast, it is
1122 possible to use most of them with TSIRM. From our point of view, the condition
1123 to use a solver inside TSIRM is that the solver must have a restart
1124 feature. More precisely, the solver must support to be stoped and restarted
1125 without decrease its converge. That is why with GMRES we stop it when it is
1126 naturraly restarted (i.e. with $m$ the restart parameter). The Conjugate
1127 Gradient (CG) and all its variants do not have ``restarted'' version in PETSc,
1128 so they are not efficient. They will converge with TSIRM but not quickly
1129 because if we compare a normal CG with a CG for which we stop it each 16
1130 iterations for example, the normal CG will be for more efficient. Some
1131 restarted CG or CG variant versions exist and may be interested to study in
1134 %%%*********************************************************
1135 %%%*********************************************************
1139 %%%*********************************************************
1140 %%%*********************************************************
1141 \section{Conclusion}
1143 %The conclusion goes here. this is more of the conclusion
1144 %%%*********************************************************
1145 %%%*********************************************************
1147 A novel two-stage iterative algorithm has been proposed in this article,
1148 in order to accelerate the convergence Krylov iterative methods.
1149 Our TSIRM proposal acts as a merger between Krylov based solvers and
1150 a least-squares minimization step.
1151 The convergence of the method has been proven in some situations, while
1152 experiments up to 16,394 cores have been led to verify that TSIRM runs
1153 5 or 7 times faster than GMRES.
1156 For future work, the authors' intention is to investigate other kinds of
1157 matrices, problems, and inner solvers. The influence of all parameters must be
1158 tested too, while other methods to minimize the residuals must be regarded. The
1159 number of outer iterations to minimize should become adaptative to improve the
1160 overall performances of the proposal. Finally, this solver will be implemented
1161 inside PETSc. This would be very interesting because it would allow us to test
1162 all the non-linear examples and compare our algorithm with the other algorithm
1163 implemented in PETSc.
1166 % conference papers do not normally have an appendix
1170 % use section* for acknowledgement
1171 %%%*********************************************************
1172 %%%*********************************************************
1173 \section*{Acknowledgment}
1174 This paper is partially funded by the Labex ACTION program (contract
1175 ANR-11-LABX-01-01). We acknowledge PRACE for awarding us access to resources
1176 Curie and Juqueen respectively based in France and Germany.
1180 % trigger a \newpage just before the given reference
1181 % number - used to balance the columns on the last page
1182 % adjust value as needed - may need to be readjusted if
1183 % the document is modified later
1184 %\IEEEtriggeratref{8}
1185 % The "triggered" command can be changed if desired:
1186 %\IEEEtriggercmd{\enlargethispage{-5in}}
1188 % references section
1190 % can use a bibliography generated by BibTeX as a .bbl file
1191 % BibTeX documentation can be easily obtained at:
1192 % http://www.ctan.org/tex-archive/biblio/bibtex/contrib/doc/
1193 % The IEEEtran BibTeX style support page is at:
1194 % http://www.michaelshell.org/tex/ieeetran/bibtex/
1195 \bibliographystyle{IEEEtran}
1196 % argument is your BibTeX string definitions and bibliography database(s)
1197 \bibliography{biblio}
1199 % <OR> manually copy in the resultant .bbl file
1200 % set second argument of \begin to the number of references
1201 % (used to reserve space for the reference number labels box)
1202 %% \begin{thebibliography}{1}
1204 %% \bibitem{saad86} Y.~Saad and M.~H.~Schultz, \emph{GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems}, SIAM Journal on Scientific and Statistical Computing, 7(3):856--869, 1986.
1206 %% \bibitem{saad96} Y.~Saad, \emph{Iterative Methods for Sparse Linear Systems}, PWS Publishing, New York, 1996.
1208 %% \bibitem{hestenes52} M.~R.~Hestenes and E.~Stiefel, \emph{Methods of conjugate gradients for solving linear system}, Journal of Research of National Bureau of Standards, B49:409--436, 1952.
1210 %% \bibitem{paige82} C.~C.~Paige and A.~M.~Saunders, \emph{LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares}, ACM Trans. Math. Softw. 8(1):43--71, 1982.
1211 %% \end{thebibliography}