{"ID":2879066,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2508.16898","arxiv_id":"2508.16898","title":"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 [Afr22, Rab23, Rab25, RAN25, RN24], 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 state-of-the-art 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 todevelop a more efficient CCBM-ADMM scheme. The accuracy and robustness of the proposed computational approach are validated through various numerical experiments.","short_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 [Afr22, Rab23, Rab25, RAN25, RN24], we aim to recover the shape of an unknown inclusion via shape opt...","url_abs":"https://arxiv.org/abs/2508.16898","url_pdf":"https://arxiv.org/pdf/2508.16898v2","authors":"[\"Elmehdi Cherrat\",\"Lekbir Afraites\",\"Julius Fergy Tiongson Rabago\"]","published":"2025-08-23T05:02:23Z","proceeding":"math.NA","tasks":"[\"math.NA\",\"math.OC\"]","methods":"[\"Diffusion Model\"]","has_code":false}