Inverse problems in fluid mechanics solved by game strategies

Résumé

This thesis aims to study the ability of theoretic game approaches to deal with ill-posed problems. The first part of the thesis is dedicated to the Stokes system’s linear problem, with the goal of detecting unknown geometric inclusions or pointwise sources in a stationary viscous fluid, using a single compatible pair of Dirichlet and Neumann data, available only on a partially accessible part of the boundary. Inverse geometric-or-source identification for the Cauchy-Stokes problem is severely ill-posed (in the sense of Hadamard) for both the inclusions or sources and the missing data reconstructions, and designing stable and efficient algorithms is challenging. To solve the joint completion/detection problem, we reformulate it as a three players Nash game. The two first players aim at recovering the missing data (Dirichlet and Neumann conditions prescribed over the inaccessible boundary), while the third player seeks to identify the shape and locations of the inclusions (in Chapter 2) or determine the source term (in Chapter 3). We then introduce new algorithms dedicated to the Nash equilibria, which is expected to approximate the original coupled problems’ solutions. We present different numerical experiments to illustrate the efficiency and robustness of our 3- player Nash game strategy. The extension of this work to another situation, such as identifying small objects, has been carried out (in Chapter 4). The second purpose of this thesis is to extend those results to the case of quasi-Newtonian fluid flow whose viscosity is assumed to be a nonlinear function that varies upon the imposed rate of deformation. The considered problem then is a nonlinear Cauchy type because of the non-linearity of the viscosity function. Two different iterative procedures, control-type and Nash game algorithms, are considered to solve it. From a computational point of view, the non-linearity needs some particular algorithms. We propose a novel one-shot algorithm to solve the nonlinear state equations during a recovery process, representing a different idea to treat the nonlinear Cauchy problems. Some numerical experiments are provided to demonstrate our algorithm’s efficiency in the noise-free and noisy data cases. A comparison between the one-shot scheme and the fixed-point method was performed. Finally, we introduce an algorithm to jointly recover the missing boundary data and the location and shape of the inclusions for nonlinear Stokes models based on the Game-Theoretic approach.

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