Enhanced shape recovery in advection--diffusion problems via a novel ADMM-based CCBM optimization

Abstract: This work proposes a novel shape optimization framework for geometric inverse problems governed by the advection-diffusion equation, based on the coupled complex boundary method (CCBM). Building on recent developments [2, 45, 46, 47, 51], we aim to recover the shape of an unknown inclusion via shape optimization driven by a cost functional constructed from the imaginary part of the complex-valued state variable over the entire domain. We rigorously derive the associated shape derivative in variational form and provide explicit expressions for the gradient and second-order information. Optimization is carried out using a Sobolev gradient method within a finite element framework. To address difficulties in reconstructing obstacles with concave boundaries, particularly under measurement noise and the combined effects of advection and diffusion, we introduce a numerical scheme inspired by the Alternating Direction Method of Multipliers (ADMM). In addition to implementing this non-conventional approach, we demonstrate how the adjoint method can be efficiently applied and utilize partial gradients to develop a more efficient CCBM-ADMM scheme. The accuracy and robustness of the proposed computational approach are validated through various numerical experiments.