H.D.R : Paul MYCEK : “Hierarchical methods for deterministic and stochastic partial differential equations.”
You tube link : https://youtube.com/live/Lon0ptR1GgU?feature=share
Hierarchical methods have become essential tools for the efficient numerical simulation of physical phenomena with ever-growing fidelity on modern high-performance supercomputers.
Well-established methods, such as multigrid and domain decomposition methods, owing to their scaling capabilities, have proven powerful to solve certain classes of large-scale problems in a parallel fashion on supercomputers.
On the other hand, the propagation of uncertainties in numerical simulations has received increasing attention in recent years.
For computationally demanding simulations of highly non-linear problems (w.r.t. uncertain input parameters), advanced uncertainty quantification methods need to be designed.
In this manuscript, we present methodological and algorithmic ideas to address specific questions related to hierarchical methods.
First, we present resilient domain decomposition approaches for the numerical solution of deterministic and stochastic partial differential equations, as well as surrogate-assisted Monte Carlo strategies for the propagation of uncertainties in domain decomposition solvers.
Second, we introduce novel multigrid solvers for the linear systems arising from hybridizable high-order discretizations.
Third, we propose hierarchical approaches, namely multilevel and multifidelity methods, for the propagation of uncertainties in expensive numerical simulators.
Finally, we also propose ideas for the efficient numerical solving of sequences of systems of linear equations that typically arise when solving time-dependent and/or non-linear problems, or when sampling stochastic partial differential equations.
Committee members :
Laurent Debreu, Director of research, Inria, Referee
Christophe Prud’homme, Professor, Univ. Strasbourg, Referee
Robert Scheichl, Professor, Univ. Heidelberg, Referee
Nathalie Bartoli, Director of research, Onera, Examiner
Grégory Pinon, Professor, Univ. Le Havre Normandie, Examiner
Carmen Rodrigo, Associate professor, Univ. Zaragoza, Examiner
Ulrich Rüde, Professor, FAU Erlangen-Nürnberg, Examiner
Elisabeth Ullmann, Professor, TU Munich, Examiner
Luc Giraud, Director of research, Inria, Invited member