Organizing committee

Caroline Japhet – Yvon Maday - Olga Mula

Presentation of the day

The purpose of the event of the 7th of June is to discuss recent developments in various aspects of parallel in time methods, bringing together mathematicians and computational scientists who are working on this subject.

This day is organized by the ANR project CINE-PARA (ANR-15-CE23-0019), the CEREMADE team of University Paris-Dauphine, the LJLL, and LAGA.

The presentations are 40 minutes long, including 10 minutes for questions and discussion. They will take place in the Room P-504 (take the elevators or the stairs in the main building, 5th floor) at University Paris-Dauphine.
Click here for the access map to University Paris-Dauphine

Schedule

• 8h45-9h00: Welcome
• 9h00-9h40 : Upanshu Sharma (CERMICS, École des Ponts ParisTech)
    Parareal algorithm in the context of molecular dynamics. — Abstract

  The central ingredients of the Parareal algorithm are a fine and a coarse model for the system under consideration, for instance, integrators with fine and coarse time-steps for differential equations. In this talk I will discuss the utility of the parareal method in the context of molecular dynamics, by choosing the coarse model to be a simplification of the potential energy landscape which drives the fine simulation.

  This work is in collaboration with Frédéric Legoll and Tony Lelièvre (CERMICS, École des Ponts ParisTech and INRIA Paris)

• 9h40-10h20 : Van Thanh Nguyen (ALPINES, INRIA Paris)
    Another interpretation of Parareal as a two-level domain decomposition preconditioner. — Abstract

  In the literature, three variants of Parareal have been interpreted. The first approach just uses Parareal terminology. The second one is based on geometric multigrid, and the third one is based on algebraic multigrid. The Parareal algorithm is a two-level method, it comes from the idea of domain decomposition methods, but instead of decomposing in space, it decomposes the time direction. Starting from that point of view, we derive a new approach based on the two-level domain decomposition preconditioner in the temporal domain. We also present numerical tests to verify the convergence results and compare to the Parareal algorithm. Running the fine solver is often the most expensive task in the Parareal algorithm. Despite of being computed in parallel, the performance of the fine solver is often very costly in terms of both execution time and memory consumption, especially for the higher dimension problems. In this work, we also propose a new strategy using the idea of recycling Krylov subspaces of last iterations to reduce the number of iterations for solving the next system as well as the total number of matrix-vector products required to solve all systems. We present numerical tests and the comparison to the normal Krylov subspace solver like GMRES-DR both in case of preconditioned and non- preconditioned.

  This work is in collaboration with Laura Grigori (ALPINES, INRIA)

                -Coffee break-

• 10h35-11h15 : Duc Quang Bui (LAGA, Université Paris 13)
    Optimized Schwarz Waveform relaxation method for time-dependent Stokes equation. — Abstract

  In this contribution, we introduce some first observations about the Optimized Schwarz Waveform Relaxation method applied to the time-dependent, incompressible Stokes equation. In particular, some simple examples show that the current algorithm does not always converge for the pressure, so we suggest a slight modification. A convergence result of the new OSWR algorithm is given for general domain decomposition. Numerical experiments with FreeFEM++ for two-dimensional problems are presented to illustrate the performance of the method.

  This work is in collaboration with Caroline Japhet (Univ. Paris 13) and Pascal Omnes (CEA Saclay, DM2S-STMF)

• 11h15-11h55 : Katia Ait Ameur (LJLL, Sorbonne Université & CEA Saclay)
    Parareal algorithm for two-phase flows simulation. — Abstract

  In the field of nuclear energy, computations of complex two-phase flows are required for the design and safety studies of nuclear reactors. System codes are dedicated to the thermal-hydraulic analysis of nuclear reactors at system scale by simulating the whole reactor. We are here interested in the Cathare code developed by CEA [1]. Typical cases involve up to a million of numerical time iterations, computing the approximate solution during long physical simulation times. A space domain decomposition method has already been implemented and to improve the response time, we will consider a strategy of time domain decomposition, based on the parareal method [3]. The Cathare time discretization is based on a multistep time scheme. In this work, we derive a strategy to adapt the parareal algorithm to multistep schemes that is not intrusive in the code. We propose a variant of the parareal algorithm for time dependent problems involving a multistep time scheme in the coarse and/or fine propagators. This choice can potentially bring higher approximation orders than plain one-step methods but the initialization of each time window needs to be appropriately chosen. Here, we explore some possible initializations and demonstrate their relevance on a Dahlquist test equation followed by numerical results on an advection-diffusion equation and on an industrial test case with an application on the Cathare code [2].

  This work is in collaboration with Yvon Maday (Sorbonne Université & IUF) and Marc Tajchman (CEA Saclay)

  [1] D. Bestion, The physical closure laws in the CATHARE code, Nuclear Engineering and Design, vol. 124, 1990.

  [2] G.F. Hewitt, J.M. Delhaye, N. Zuber, Multiphase science and technology, vol. 6, 1991.

  [3] J.-L. Lions, Y. Maday, G. Turinici, Résolution par un schéma en temps ”pararéel”, C. R. Acad. Sci. Paris, 2001.

• 11h55-12h35 : Julien Salomon (INRIA-ANGE & Laboratoire J-L. Lions, Sorbonne-Université)
    A time parallelization method for Luenberger observers. — Abstract

  The coupling of parallelization procedures in time with control, optimization and inverse problem solving procedures has been the subject of many contributions over the past ten years. A common point of the proposed approaches is that they often apply to problems where the time interval is fixed a priori, which makes it possible to break it down in a fairly simple way. In the case of data assimilation procedures, this characteristic no longer exists in general, since data can arrive uninterrupted. To begin to answer this problem, we present in this talk a first algorithm to parallelize in time a Luenberger observer. Our approach consists in introducing a sequence of data processing windows, on which time parallelization is performed and in defining a stopping criterion guaranteeing the preservation of the convergence rate of the observer under consideration.

-Lunch-

• 13h40-14h20 : Florent Hédin ( CERMICS, École des Ponts ParisTech & INRIA Paris)
    gen.parRep: Generalized Parallel Replica dynamics for the long time simulation of metastable biochemical systems. — Abstract

  Metastability is one of the major encountered obstacles when performing long molecular dynamics simulations, and many methods were developed to address this challenge. The “Parallel Replica”(ParRep) dynamics is known for allowing to simulate very long trajectories of metastable Langevin dynamics in the materials science community, but it relies on assumptions that can hardly be transposed to the world of biochemical simulations. The later developed “Generalized ParRep” variant solves those issues, but it was not applied to significant systems of interest so far.

  In this talk, we will present the program gen.parRep, the first publicly available implementation of the Generalized Parallel Replica method (BSD 3-Clause license), targeting frequently encountered metastable biochemical systems, such as conformational equilibria or dissociation of protein–ligand complexes. It will be shown that the resulting C++ implementation exhibits a strong linear scalability, providing up to 70% of the maximum possible speedup on several hundreds of CPUs.

  This work is in collaboration with Tony Lelièvre (CERMICS, École des Ponts ParisTech and INRIA Paris)

• 14h20-15h00 : Yvon Maday (LJLL, Sorbonne Université & IUF)
    Quelques éléments pour améliorer l’efficacité de l’algorithme pararéel.... and more. 

  TBA.

• 15h00-16h00 : Discussion

Poster

                                                                     

ANR Partners

ANR, Agence Nationale de le Recherche

ANR project CINE-PARA (ANR-15-CE23-0019), partners :

• CEREMADE, UMR 7534 Centre de Recherche en Mathématiques de la Décision, Université Paris-Dauphine
• DM2S CEA-Saclay, Département de modélisation des systèmes et structures, CEA Paris-Saclay
• Project-team ALPINES, Institut National de Recherche en Informatique et Automatique, Inria -Centre de Paris
• LAGA, Laboratoire Analyse Géométrie et Applications, Université Paris 13
• LJLL, Laboratoire Jacques-Louis Lions, UPMC Sorbonne Université
• ENPC/NAVIER, Laboratoire Navier, Ecole des Ponts ParisTech

Organization and sponsors