Journée "Problème de Poisson"

Programme

lundi 26/01

09:30 10:00

Accueil

10:00 10:30

Pourquoi la résolution de l'equation de Poisson doit être précise et rapide pour les simulations plasmas froids ?

Francois Pechereau (Cerfacs) et Anne Bourdon (LPP, Ecole Polytechnique)

La résolution de l'équation de Poisson est particulièrement importante pour la simulation des plasmas froids à pression atmosphérique Ces plasmas sont étudiés pour des applications très variées comme la dépollution, la combustion assistée par plasma et les applications biomédicales. La simulation de ces décharges électriques nécessite de calculer la génération et la propagation d'ondes d'ionisation, avec un front de décharge dans lequel règne un champ électrique élevé (100kV/cm) du à une charge d'espace et un canal plasma quasi-neutre où le champ électrique est beaucoup plus faible. Dans les travaux que nous avons menés jusqu'à présent, nous réalisons des simulations 2D-axisymetriques avec un modèle fluide. Dans ce modèle, les équations de continuité des espèces chargées sont couplées à l'equation de Poisson. Pour simuler la propagation des décharges, il faut donc résoudre l'équation de Poisson à chaque pas dans le temps. Comme les termes sources des espèces et leurs paramètres de transport dépendent de façon très non-linéaire du champ électrique, la résolution de l'équation de Poisson doit également être très précise. Dans les travaux que nous avons menés jusqu'à présent, le pourcentage de temps passé dans la résolution de l'équation de Poisson peut varier entre 30 et 60% du temps total. Dans cet exposé, nous discuterons des différentes méthodes que nous avons utilisées pour résoudre l'équation de Poisson avec leur avantages et leurs inconvénients. Pour finir, nous présenterons les challenges actuels pour la simulation des décharges plasmas à pression atmosphérique et à basse pression.

10:30 11:00

Méthodes de projection pour la résolution des équations de Navier-Stokes pour les écoulements incompressibles

Bérengère Podvin (LIMSI)

La dérivation des équations de Navier-Stokes pour les écoulements de fluide incompressible conduit à la fomulation d'un problème de Poisson pour la pression, pour laquelle des conditions aux limites adaptées doivent être définies. On s'intéressera ici plus particulièrement aux méthodes de projection et à leur résolution numérique dans un contexte parallèle (O(100) processeurs). Celle-ci sera illustrée dans le cas d'un code spectral ainsi que dans le cas du code SUNFLUIDH développé pour des volumes finis au LIMSI.

11:00 11:30

Scalable Poisson solver for gyrokinetic simulation

Virginie Grandgirard (CEA)

Predicting the performance of fusion plasmas in terms of amplification factor, namely the ratio of the fusion power over the injected power, is among the key challenges in fusion plasma physics. In this perspective, turbulence and heat transport need being modeled within the most accurate theoretical framework, using first-principle non-linear simulation tools. The gyrokinetic equation for each species, coupled to Maxwell's equations is an appropriate self-consistent description of this problem. A new class of global full-f codes has recently emerged, solving the gyrokinetic equation for the entire distribution function on a large radial domain of the tokamak and using some prescribed external heat source. Such simulations are extremely challenging and require state-of-the-art high performance computing (HPC). The GYSELA code we develop at the IRFM institute is one of them. In the code the parallelization of Vlasov and Poisson equations are tightly coupled. Indeed, a large amount of distributed data are exchanged between these two parallel components. The Poisson solver is considered to get electric potential from ion density, assuming an adiabatic electron response within cylindrical surfaces. The numerical scheme of the Poisson solver relies on Fourier transform and a finite difference scheme. A brief overview of performances up to thousands cores will be given. Bottlenecks concerning parallel scalability and possible solutions will be discussed.

11:30 12:00

The gravitational field solving the Poisson equation for astrophysics purposes

Patrick Hennebelle (CEA)

In astrophysics, newtonian gravity is playing a fundamental role from the largest scales of our universe such as clusters of galaxies, up to solar systems. In many circumstances, self-gravitating fluids must then be considered and the Poisson equation must be solved at every time steps. Moreover, many systems present very different spatial scales and it is then necessary to use appropriate schemes with adaptive resolution. I will describe some of the methods used in astrophysics focussing more specifically on grid adaptive mesh refinement and the so called smooth particle hydrodynamics, which is a Lagrangian techniques. For each of them specific approaches have been developed to solve the Poisson equation.

12:00 14:00

Repas

14:00 14:45

On the design of parallel linear solvers for large scale problems

Mathieu Faverge (LABRI)

In this talk we will discuss our research activities on the design of parallel linear solvers for large scale problems that range from dense linear algebra, to parallel sparse direct solver and hybrid iterative-direct approaches. In particular we will describe the implementations designed on top of runtime systems that should provide both code and performance portabilities.

14:45 15:30

Improving multifrontal solvers by means of Block Low-Rank approximations for PDEs

Alfredo Buttari (CNRS et INPT-IRIT)

Matrices coming from elliptic Partial Differential Equations (PDEs) and in particular Poisson equations have been shown to have a low-rank property: well defined off-diagonal blocks of their Schur complements can be approximated by low-rank products. Given a suitable ordering of the matrix which gives to the blocks a geometrical meaning, such approximations can be computed using an SVD or a rank-revealing QR factorization. The resulting representation offers a substantial reduction of the memory requirement and gives efficient ways to perform many of the basic dense linear algebra operations. Several strategies, mostly based on hierarchical formats, have been proposed to exploit this property. We study a simple, non-hierarchical, low-rank format called Block Low-Rank (BLR), and explain how it can be used to reduce the memory footprint and the complexity of sparse direct solvers based on the multifrontal method. We present experimental results that show that even if BLR based factorizations are asymptotically less efficient than hierarchical approaches, they still deliver considerable gains. The BLR format is compatible with numerical pivoting, and its simplicity and flexibility make it easy to use in the context of a general purpose, algebraic solver. In this talk, we show how Block low-rank feature was incorportated in the general purpose solver MUMPS and provide experiments showing the gains obtained with respect to standard (full rank) MUMPS factorizations.

15:30 16:00

Pause

16:00 16:45

Communication avoiding and hiding in preconditioned Krylov solvers for Poisson problems

Wim Vanroose (Université d'Anvers)

Traditionally Krylov solvers have been designed to minimize the number of flops to arrive at a solution. However, currently communication and synchronization overhead is now the main concern due to the dramatic increases in concurrency on all hardware. This introduces severe communication bottlenecks in Krylov solvers most importantly the latency of global reductions and the limited bandwidth to the memory. In this talk we discuss how Krylov algorithms can be redesigned to remove some of these communication bottlenecks and leading to a class of so called pipelined Krylov solvers. These methods have a better scaling behaviour on modern hardware.

16:45 17:30

Robust solution of Poisson-like problems with aggregation-based AMG

Yvan Notay (Université Libre de Bruxelles)

The fast iterative solution of discrete Poisson-like problems requires methods of multigrid or multilevel type, because classical iterative scheme cannot efficiently damp smooth error components.

While there exists a plenty of such methods, their use often requires complicated coding, or, when a software code is available, a careful variant selection or parameter tuning. In this talk, we shall present a multilevel method based on the aggregation of the unknowns, which has the attractive feature that the related code offers stable performances while being used in a purely black box fashion. It also scales well in parallel and compare favorably with state-of-the-art competitors.