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<p>This work presents a shape optimization framework for geometric inverse problems governed by the advection–diffusion equation, based on the coupled complex boundary method (CCBM). The shape of an unknown inclusion is recovered by minimizing a cost functional defined from the imaginary part of a complex-valued state variable. We derive the associated shape derivative, provide explicit expressions for the gradient and second-order information, and examine the compactness properties of the Hessian at a critical shape. The optimization problem is solved using a Sobolev gradient method within a finite element framework. To enhance the reconstruction of inclusions with concave boundaries under noise and combined advection–diffusion effects, we introduce an ADMM-inspired scheme. An efficient adjoint-based implementation using partial gradients leads to a CCBM–ADMM algorithm. Numerical experiments in two and three dimensions demonstrate the accuracy and robustness of the proposed approach.</p>

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Enhanced Shape Recovery in Advection–Diffusion Problems Via a Novel ADMM-Based CCBM Optimization

  • 期刊论文

  • 发表日期：

    2026年4月21日

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  • Elmehdi Cherrat,
  • Lekbir Afraites,
  • Julius Fergy Tiongson Rabago

摘要

This work presents a shape optimization framework for geometric inverse problems governed by the advection–diffusion equation, based on the coupled complex boundary method (CCBM). The shape of an unknown inclusion is recovered by minimizing a cost functional defined from the imaginary part of a complex-valued state variable. We derive the associated shape derivative, provide explicit expressions for the gradient and second-order information, and examine the compactness properties of the Hessian at a critical shape. The optimization problem is solved using a Sobolev gradient method within a finite element framework. To enhance the reconstruction of inclusions with concave boundaries under noise and combined advection–diffusion effects, we introduce an ADMM-inspired scheme. An efficient adjoint-based implementation using partial gradients leads to a CCBM–ADMM algorithm. Numerical experiments in two and three dimensions demonstrate the accuracy and robustness of the proposed approach.