Conception et analyse d'algorithmes parallèles en temps pour l'accélération de simulations numériques d'équations d'évolution

摘要

This thesis presents algorithms allowing parallelization in the time direction in the simulation of systems, which are governed by partial differential equations. These methods are applied in three application fields, and complex models. extbf{1. Parabolic Optimal Control:} We develop two parallel algorithms (SITPOC and PITPOC). These two algorithms are based on a general method of time domain decomposition of optimal control problems. A convergence result for SITPOC is obtained. Moreover, a matrix interpretation for both algorithms is also given. extbf{2. Kinetics of the population of neutrons in a nuclear reactor:} We design a parallel in time solver gathering all the variables associated with a nuclear reactor. Our method is adapted to various possible scenarios used in model reduction. The results of this solver are comparable with those obtained by the MINOS code of the CEA. The time parallelization is based on a parareal in time scheme for the time resolution. We consider several physical models of the kinetics of the neutrons. We simulate these models with the parareal in time algorithm in which the coarse solver corresponds to a reduced physical model. This reduction allows an important acceleration of the computational time. extbf{3. Pulse sequence design in nuclear magnetic resonance and quantum information:} This chapter presents preliminary work on a time-parallel method for the resolution of an optimal control problem related to magnetic nuclear resonance. Our method produces an important acceleration compared with the nonparallel version. Moreover, the control fields computed with our method are smooth, which allow a simpler experimental implementation. Numerical tests prove the efficiency of our approach. On academic examples and without optimizing the code, we obtain significant improvements.