:Equipes-Projets ALPINES Rapport de recherche n°8597 -September 2014 -46 pagesRésumé : Dans cet article, nous présentons deux nouveau methodes iterative pour la résolution des systèmes linéaires d'équations de très grande taille en minimisant les communications. Ces deux methodes sont basées sur l'enrichissement de l'espace de Krylov en décomposant le domaine de A. Thus it is possible to search for the solution of the system Ax " b in K t,k pA, r 0 q instead of K k pA, r 0 q. Moreover, we show in this paper that the enlarged Krylov projection subspace methods lead to faster convergence in terms of iterations and parallelizable algorithms with less communication, with respect to Krylov methods. In this paper we focus on Conjugate Gradient (CG) [16], a Krylov projection method for symmetric (Hermitian) positive definite matrices. We discuss two new versions of Conjugate Gradient (section 3). The first method, multiple search direction with orthogonalization CG (MSDO-CG), is an adapted version of MSD-CG [14] with the A-orthonormalization of the search directions to obtain a projection method that guarentees convergence at least as fast as CG. The second projection method that we propose here, long recurrence enlarged CG (LRE-CG), is similar to GMRES in that we build an orthonormal basis for the enlarged Krylov subspace rather than finding search directions. Then, we use the whole basis to update the solution and the residual. Both methods converge faster than CG in terms of iterations, but LRE-CG converges faster than MSDO-CG since it uses the whole basis to update the solution rather than only t search directions. And the more subdomains are introduced or the larger t is, the faster is the convergence of both methods with respect to CG in terms of iterations. For example, for t " 64 the MSDO-CG and LRE-CG methods converge in 75% up to 98% less iteration with respect to CG for the different test matrices. But increasing t also means increasing the memory requirements. Thus, in practice, t should be relatively small, depending on the available memory, on the size of the matrix, and on the number of iterations needed for convergence, as explained in section 4. We also present the parallel algorithms along with their expected performance based on the estimated run times, and the preconditioned versions with their convergence behavior.

No abstract

Abstract. In this paper we present a communication avoiding ILU0 preconditioner for solving large linear systems of equations by using iterative Krylov subspace methods. Recent research has focused on communication avoiding Krylov subspace methods based on so-called s-step methods. However, there are not many communication avoiding preconditioners yet, and this represents a serious limitation of these methods. Our preconditioner allows us to perform s iterations of the iterative method with no communication, through ghosting some of the input data and performing redundant computation. To avoid communication, an alternating reordering algorithm is introduced for structured and well partitioned unstructured matrices, which requires the input matrix to be ordered by using a graph partitioning technique such as k-way or nested dissection. We show that the reordering does not affect the convergence rate of the ILU0 preconditioned system as compared to kway or nested dissection ordering, while it reduces data movement and is expected to reduce the time needed to solve a linear system. In addition to communication avoiding Krylov subspace methods, our preconditioner can be used with classical methods such as GMRES to reduce communication. In the parallel case, the input matrix is distributed over processors, and each iteration involves multiplying the input matrix with a vector, followed by an orthogonalization process. Both these operations require communication among processors. Since A is usually very sparse, the communication dominates the overall cost of the iterative methods when the number of processors is increased to a large number. While the matrix-vector product can be performed by using pointto-point communication routines between subsets of processors, the orthogonalization step requires the usage of collective communication routines, and these routines are known to scale poorly. More generally, on current machines the cost of communication, the movement of data, is much higher than the cost of arithmetic operations, and this gap is expected to continue to increase exponentially. As a result, communication is often the bottleneck in numerical algorithms.In a quest to address the communication problem, recent research has focused on reformulating linear algebra operations such that the movement of data is significantly reduced or even minimized, as in the case of dense matrix factorizations [12,20,3]. Such algorithms are referred to as communication avoiding. The communication avoiding Krylov subspace methods [29,24,6] are based on s-step methods

No abstract

-In basin and reservoir simulations, the most expensive and time consuming phase is solving systems of linear equations using Krylov subspace methods such as BiCGStab. For this reason, we explore the possibility of using communication avoiding Krylov subspace methods (s-step BiCGStab), that speedup of the convergence time on modern-day architectures, by restructuring the algorithms to reduce communication. We introduce some variants of s-step BiCGStab with better numerical stability for the targeted systems.Résumé -Méthodes s-step BiCGStab appliquées en Géosciences -Dans les simulateurs d'écoulement en milieu poreux, comme les simulateurs de réservoir et de bassin, la résolution de système linéaire constitue l'étape la plus consommatrice en temps de calcul et peut même représenter jusqu'à 80 % du temps de la simulation. Ceci montre que la performance de ces simulateurs dépend fortement de l'efficacité des solveurs linéaires. En même temps, les machines parallèles modernes disposent d'un grand nombre de processeurs et d'unités de calcul massivement parallèle. Dans cet article, nous proposons de nouveaux algorithmes BiCGStab, basés sur l'algorithme à moindre communication nommé s-step, permettant d'éviter un certain nombre de communication afin d'exploiter pleinement les architectures hautement parallèles.

In a recent work [1], two of the authors have formulated the non-linear space-time Hasegawa-Mima plasma equation as a coupled system of two linear PDEs, a solution of which is a pair (u, w), with w = (I − ∆)u. The first equation is of hyperbolic type and the second of elliptic type. Variational frames for obtaining weak solutions to the initial value Hasegawa-Mima problem with periodic boundary conditions were also derived. Using the Fourier basis in the space variables, existence of solutions were obtained. Implementation of algorithms based on Fourier series leads to systems of dense matrices. In this paper, we use a finite element space-domain approach to semi-discretize the coupled variational Hasegawa-Mima model, obtaining global existence of solutions in H 2 on any time interval [0, T ], ∀T . In the sequel, full-discretization using an implicit time scheme on the semi-discretized system leads to a nonlinear full space-time discrete system with a nonrestrictive condition on the time step. Tests on a semi-linear version of the implicit nonlinear full-discrete system are conducted for several initial data, assessing the efficiency of our approach.Acknowledgements: The authors would like to express their gratitude to Prof. Ghassan Antar, AUB Physics Department, for the several discussions held on the physics of the Hasegawa-Mima wave phenomena, providing us suitable testing initial conditions and for his positive comments on the overall numerical results.

In [1], the non-linear space-time Hasegawa-Mima plasma equation is formulated as a coupled system of two linear PDEs, a solution of which is a pair (u, w), with w = (I −∆)u. The first equation is of hyperbolic type and the second of elliptic type. Variational frames for obtaining weak solutions to the initial value Hasegawa-Mima problem with periodic boundary conditions were also derived. In a more recent work [2], a numerical approach consisting of a finite element space-domain combined with an Euler-implicit time scheme was used to discretize the coupled variational Hasegawa-Mima model.A semi-linear version of this implicit nonlinear scheme was tested for several types of initial conditions. This semi-linear scheme proved to lack efficiency for long time, which necessitates imposing a cap on the magnitude of the solution.To circumvent this difficulty, in this paper, we use Newton-type methods (Newton, Chord and an introduced Modified Newton method) to solve numerically the fully-implicit non-linear scheme. Testing these methods in FreeFEM++ indicates significant improvements as no cap needs to be imposed for long time. In the sequel, we demonstrate the validity of these methods by proving several results, in particular the convergence of the implemented methods.

No abstract