Slides Schedule & Book of Abstracts Sponsors Participants Simulations
Thursday
- 09:00-10:00 : Iain Smears
Time-parallel iterative solvers for parabolic evolution equations: an inf-sup theoretic approach. — AbstractM. Neumüller¹, I. Smears²
¹ Institute of Computational Mathematics, Johannes Kepler University Linz, 4040 Linz, Austria
² Department of Mathematics, University College London, 25 Gordon Street, London WC1E 6BT, United Kingdom
Many parallel-in-time methods can be viewed as iterative solvers for a large time-global nonsymmetric linear system arising from the discretization of a time-dependent equation. The nonsymmetry of these systems represents a key challenge in the analysis, since the available theory for iterative methods for nonsymmetric systems is much more limited than for their symmetric counterparts.
In this talk [1], we will show how the underlying inf-sup theory of continuous and discretized parabolic problems provides an effective approach to the construction and rigorous analysis of parallel-in-time solvers. In particular, we consider the implicit Euler discretization of a general linear parabolic evolution equation with time-dependent self-adjoint spatial operators. We first show that the discrete system admits a similar inf-sup condition as for the underlying continuous operator. We use this to show that the standard nonsymmetric time-global system can be equivalently reformulated as an original symmetric saddle-point system that remains inf-sup stable in the same norms. The essential idea is that the mapping from trial functions to their optimal test functions in the inf-sup condition defines a left-preconditioner that symmetrizes the system in a stable way.
We then propose and analyse an inexact Uzawa method for the saddle- point reformulation based on an efficient parallel-in-time preconditioner. The preconditioners is non-intrusive and easy to implement in practice, since it simply combines existing spatial preconditioners with parallel Fast Fourier Transforms (FFT) in time. We prove robust spectral bounds, leading to convergence rates that are independent of the number of time-steps, final time, or spatial mesh sizes. The theoretical parallel complexity of the method then grows only logarithmically with respect to the number of time- steps, owing to the parallel FFT. Numerical experiments of large-scale parallel computations, with up to 131 072 processors and more than 2 billion unknowns, show the effectiveness of the method, along with its good weak and strong scaling properties.
References
[1] M. Neumüller and I. Smears. Time-parallel iterative solvers for parabolic evolution equations. ArXiv e-print 1802.08126, 2018.
- 10:30-11:00 : Sarah Ali Hassan
A posteriori stopping criteria for space-time domain decomposition for porous media flow.
— AbstractE. Ahmed¹²³ , S. Ali Hassan¹ , C. Japhet², M. Kern¹, M. Vohralík¹
¹ SERENA, Inria Paris and CERMICS (ENPC)
² LAGA, Université Paris 13, Sorbonne Paris Cité
³ Department of Mathematics, University of Bergen
In this work we develop a posteriori error estimates and stopping criteria for global-in-time domain decomposition methods with optimized Robin or Ventcell transmission conditions on the interface between different subdomains. The heat equation using the mixed finite element method in space and the discontinuous Galerkin scheme in time is first analysed. Then we consider a two-phase flow problem in a porous medium composed of two different rock types, so that the capillary pressure field is discontinuous at the interface between the rocks. Numerical approximation is achieved by a cell centered finite volume scheme, using the Matlab Reservoir Simu- lation Toolbox. We bound the error between the exact solution of the partial differential equation and the approximate numerical solution at each iteration of the space-time domain decomposition algorithm. Different error components (domain decomposition, space discretization, time discretization, Newton linearization) are distinguished, which allows us to define efficient stopping criteria for the DD algorithm. The estimates are based on a reconstruction technique for pressure and flux respectively in the spaces H¹ and H(div) [1, 2, 3]. For the flux, local Neumann problems in small bands arround the interfaces extracted from the subdomains are solved. Numerical experiments illustrate the theoretical findings.
References
[1] Sarah Ali Hassan, Caroline Japhet, Michel Kern, and Martin Vohralík. A posteriori stopping criteria for optimized Schwarz domain decomposition algorithms in mixed formulations. Comput. Methods Appl. Math, 2018, to appear.
[2] Sarah Ali Hassan, Caroline Japhet, and Martin Vohralík. A posteriori stopping criteria for space-time domain decomposition for the heat equation in mixed formulations. HAL Preprint 01586862, submitted for publication, 2017.
[3] Elyes Ahmed, Sarah Ali Hassan, Caroline Japhet, Michel Kern, and Martin Vohralík. A posteriori error estimates and stopping criteria for space-time domain decomposition for two-phase flow between different rock types. HAL Preprint 01540956, submitted for publication, 2017. - 11:00-11:30 : Katia Aït Ameur
Parareal algorithm for two phase flows simulation. — AbstractK. Aït Ameur¹², Y. Maday¹ and M. Tajchman²
¹ Sorbonne Université, UPMC
² CEA Saclay
In the nuclear energy domain, system codes are dedicated to the thermal-hydraulics analysis of nuclear reactors for safety studies. We are here interested in the Cathare code developed by CEA, EDF, AREVA-NP and IRSN.
Cathare solves the 6-equations/two-fluid model by considering a set of conservation laws (mass, momentum and energy) for each phase liquid and vapor. The discretization is based on a finite volumes method on staggered grid in space and on a fully implicit time integration method. Typical cases involve up to a million of numerical time iterations, computing the approximate solution during long physical simulation times. On the other hand, the discretization level is kept intentionally at a coarse level to be able to handle whole systems simulations.
Cathare is used in a simulator of a reactor submitted to accidental events. This platform is dedicated to train operators and prepare crisis management exercises thus requiring real-time response of the code. To optimize the response time, we consider a strategy of time domain decomposition, complementing the current space domain decomposition.
This strategy is based on the parareal method, introduced in [1], that provides a strategy for "parallel-in-time" computations and offers the potential for an increased level of parallelism.
Here we apply the parareal algorithm to the resolution of an oscillating manometer. This test case is proposed in [2] for system codes to test the ability of each numerical scheme to preserve system mass and to retain the gas-liquid interface.
References
[1] G. Turinici J.-L. Lions, Y. Maday. Résolution par un schéma en temps "pararéel". 2001.
[2] N. Zuber G.F. Hewitt, J.M. Delhaye. Multiphase science and technology, volume 6. 1991. - 11:30-12:00 : Olga Mula
A scalable adaptive parareal algorithm with online stopping criterion. — AbstractY. Maday¹²³ and O. Mula⁴
¹ Sorbonne Universités, UPMC Univ Paris 06 and CNRS UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France
² Institut Universitaire de France
³ Division of Applied Mathematics, Brown University, Providence RI, USA
⁴ Université Paris-Dauphine, PSL Research University, CNRS, UMR 7534, CEREMADE, 75016 Paris, France
In this talk, we present a new vision of the parareal in time algorithm which carries potential to overcome the classical scalability limitations of domain decomposition schemes for the time domain. For this, we reformulate the algorithm in an infinite dimensional functional space where the realization of each iteration involves approximations of increasing accuracy of time dependent subproblems formulated on small subdomains of the original time interval. We rigorously prove that the resulting parareal algorithm presents a near-optimal convergence rate and has a parallel efficiency which is significantly superior to the traditional version. In some cases, it can even be close to the ideal value of one. We illustrate our findings for two different problems where, in addition, we explore the potential benefits of reusing information from previous parareal iterations. - 12:00-12:30 : Discussion
-Excursion in the Laber Bay and on the Perharidy peninsula-