PhD in Applied Mathematics: Implicit solvers with time-step coresolution and warm start for evolution problems in geosciences

The numerical discretization of physical models representing evolutionary problems in many fields leads to nonlinear algebraic systems of very large dimensions. The process of solving these systems, which must be carried out robustly and rapidly at each time-step in order to increase software efficiency, comes up against a number of obstacles. In this thesis, we propose to examine two of the most notable ones, for which we have possible solutions.

The first obstacle involves the time-step size. In an implicit scheme, a large time-step makes it possible to go faster, at the cost of making the system stiffer and therefore more difficult for the solver to converge. On the other hand, a small time-step makes it easier to solve the system, at the expense of a greater number of time-steps required, and therefore the overall cost. At present, there is no ideal strategy for managing time-steps. For some ODEs, where error estimates can be devised, there are proven adaptivity techniques. For PDEs, however, we must content ourselves with various heuristics to adjust the time-step during simulation. In recent years, a new approach has emerged, inspired by continuation methods, whereby the time-step becomes an unknown in its own right. All the unknowns are updated simultaneously by an iterative method, hence the term co-solution. The advantage is that any intermediate iterate can be recovered as a backup solution in the event of non-convergence. The disadvantage, however, lies in the additional equation to be correctly prescribed. This is where we seek to go further than the current state of the art by considering the optimality conditions of an optimization problem.

The second obstacle relates to the initial point for the solver. Although the value of the unknowns at the previous time-step is a natural choice, there are instances where this is not advisable. Indeed, when certain equations are complementarity relations and the system is solved by an interior point method, it is essential to start from a strictly interior point. Then, the choice of the previous state, which lies on the boundary of the admissible domain, turns out to be detrimental. To remedy this, we recommend transposing the warm-start ideas of linear programming to the non-linear framework, as well as to that of time-step co-resolution. The success of this endeavor would enable us to deploy interior point methods on a realistic scale.

Keywords: nonlinear systems, Newton’s method, continuation method, coresolution, linear programming, constrained optimization

Academic supervisor: Mounir HADDOU (INSA Rennes)

Other supervisors: Quang Huy TRAN (IFPEN) and Ibtihel BEN GHARBIA

Doctoral School: ED MATISSE 601 (Université de Rennes)