Section: New Results

Numerical algorithms

Hybrid algebraic sparse linear solvers

Participants : Jocelyne Erhel, David Imberti.

Grants and projects: EXA2CT 8.3.1 , C2S@EXA 8.2.3

Publications: [17]

Abstract: Sparse linear systems arise in computational science and engineering. The goal is to reduce the memory requirements and the computational cost, by means of high performance computing algorithms. Krylov methods combined with Domain Decomposition are very efficient. Numerical results show the benefits of our methodology.

GMRES and Polynomial Equivalence

Participant : David Imberti.

Grants and projects: EXA2CT 8.3.1 , C2S@EXA 8.2.3

Publications: in preparation.

Abstract: We have established a theoretical link between GMRES and the much simpler problem of polynomial evaluation along with some algebraic structures to describe the most important elements of the GMRES algorithm. We use these structures to show the connection between sequential GMRES and Horner’s Rule, s-step GMRES and Dorn’s rule, and predict future possible GMRES-like algorithms.

Variables s-step GMRES

Participant : David Imberti.

Grants and projects: EXA2CT 8.3.1 , C2S@EXA 8.2.3

Publications: in preparation.

Abstract: We introduce a new variation on s-step GMRES in order to improve its stability, reduce the number of iterations necessary to ensure convergence, and thereby improve parallel performance. In doing so, we develop a new block variant that allows us to express the stability difficulties in s-step GMRES more fully. We use the algebraic structures previous established via the polynomial equivalence to support an intuitive choice for the variation in the s-step procedure, and reinforce its utility in some communication cost estimates.

FGMRES dynamics

Participant : David Imberti.

Grants and projects: EXA2CT 8.3.1 , C2S@EXA 8.2.3

Publications: in preparation.

Abstract: The FGMRES algorithm has met with varying success and we detail theoretical relationships between FGMRES and GMRES including a geometric mean conjecture. Further, we build on the current literature regarding GMRES convergence with an analysis of the dynamical properties of FGMRES.

RPM Coupling Factors

Participant : David Imberti.

Grants and projects: EXA2CT 8.3.1 , C2S@EXA 8.2.3

Publications: in preparation.

Abstract: We have improved the Recursive Projection Method (RPM) with a subspace version that effectively utilizes parallelism. Furthermore, we include a discussion, numerical experiments, and suggestions for the heretofor neglected coupling factor in RPM, and how they influence convergence of the algorithm.

Hastings-Metropolis Algorithm on Markov Chains for Small-Probability Estimation

Participant : Lionel Lenôtre.

Grants: H2MNO4 8.2.1

Publications: [13]

Abstract: Shielding studies in neutron transport, with Monte Carlo codes, yield challenging problems of small-probability estimation. The particularity of these studies is that the small probability to estimate is formulated in terms of the distribution of a Markov chain, instead of that of a random vector in more classical cases. Thus, it is not straightforward to adapt classical statistical methods, for estimating small probabilities involving random vectors, to these neutron-transport problems. A recent interacting-particle method for small-probability estimation, relying on the Hastings-Metropolis algorithm, is presented. It is shown how to adapt the Hastings-Metropolis algorithm when dealing with Markov chains. A convergence result is also shown. Then, the practical implementation of the resulting method for small-probability estimation is treated in details, for a Monte Carlo shielding study. Finally, it is shown, for this study, that the proposed interacting-particle method considerably outperforms a simple Monte Carlo method, when the probability to estimate is small.

A Strategy for the Parallel Implementations of Stochastic Lagrangian Methods

Participant : Lionel Lenôtre.

Grants: H2MNO4 8.2.1

Software: PALMTREE 5.3.1

Publications: [34]

Abstract: We present some investigations on the parallelization of a stochastic Lagrangian simulation. For the self sufficiency of this work, we start by recalling the stochastic methods used to solve Parabolic Partial Differential Equations with a few physical remarks. Then, we exhibit different object-oriented ideas for such methods. In order to clearly illustrate these ideas, we give an overview of the library PALMTREE that we developed. After these considerations, we discuss the importance of the management of random numbers and argue for the choice of a particular strategy. To support our point, we show some numerical experiments of this approach, and display a speedup curve of PALMTREE. Then, we discuss the problem in managing the parallelization scheme. Finally, we analyze the parallelization of hybrid simulation for a system of Partial Differential Equations. We use some works done in hydrogeology to demonstrate the power of such a concept to avoid numerical diffusion in the solution of Fokker-Planck Equations and investigate the problem of parallelizing scheme under the constraint entailed by domain decomposition. We conclude with a presentation of the latest design that was created for PALMTREE and give a sketch of the possible work to get a powerful parallelized scheme.