Numerical solution by shape optimization method to an inverse shape problem in multi-dimensional advection-diffusion problem with space dependent coefficients

This work focuses on numerically solving a shape identification problem related to advection-diffusion processes with space-dependent coefficients using shape optimization techniques. Two boundary-type cost functionals are considered, and their corresponding variations with respect to shapes are derived using the adjoint method, employing the chain rule approach. This involves firstly utilizing the material derivative of the state system and secondly using its shape derivative. Subsequently, an alternating direction method of multipliers (ADMM) combined with the Sobolev-gradient-descent algorithm is applied to stably solve the shape reconstruction problem. Numerical experiments in two and three dimensions are conducted to demonstrate the feasibility of the methods.