Coupled data recovery and shape identification : Nash games for the nonlinear Cauchy-Stokes case

Abstract

International audienceIn this work, we investigate nonlinear Cauchy-type problems arising in quasi-Newtonian Stokes flows, where the viscosity exhibits a nonlinear dependence on the deformation tensor, modeled by the Carreau law. To tackle the inherent ill-posedness of the Cauchy-Stokes problem, we propose three iterative methods, each reformulating the original problem into a sequence of well-posed mixed boundary value problems (BVPs). A classical control framework is employed to construct a control-type algorithm for the nonlinear inverse problem. Then, we introduce two novel algorithms based on a Nash game formulation; the second algorithm enables each player to linearize the adverse state equations, enhancing computational efficiency and convergence. We further extend this linearized Nash approach to simultaneously recover missing boundary data and identify the location and shape of unknown inclusions. Finite element simulations validate the robustness and effectiveness of the proposed methods