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

Elmehdi Cherrat, Lekbir Afraites, Julius Fergy Tiongson Rabago

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.

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.

Paper Structure

This paper contains 22 sections, 10 theorems, 63 equations, 9 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. The problem setting
  3. Key assumptions and the inverse geometry problem
  4. The coupled complex boundary value problem
  5. Existence of shape solution
  6. Shape Sensitivity Analysis
  7. Shape perturbations and material derivative
  8. Shape derivatives of the shape functional
  9. Second-order shape derivative of the cost function at the critical shape
  10. Compactness of the Hessian at a critical shape
  11. Numerical Algorithms and Examples
  12. Conventional numerical scheme for shape optimization
  13. Numerical examples in 2D
  14. Numerical results and discussion in 2D
  15. Alternating Direction Method of Multipliers in shape optimization setting
  16. ...and 7 more sections

Key Result

Proposition 1

Problem prob:shape_optimization admits a solution in $\mathcal{O}_{ad}$.

Figures (9)

  • Figure 1: Conceptual model
  • Figure 2: Results of the numerical experiments under exact and noisy measurements with noise levels $\delta = 2\%, 5\%$: left, constant $\sigma \equiv 1$; right, spatially varying $\sigma(x) = 1.1 + \sin(\pi x_{1})\sin(\pi x_{2})$.
  • Figure 3: Cost and gradient norm histories for the L-block case: left, constant $\sigma \equiv 1$; right, spatially varying $\sigma(x) = 1.1 + \sin(\pi x_{1})\sin(\pi x_{2})$
  • Figure 4: Geometry and mesh of the exact cavities (first three columns) and an initial guess with radius $r = 0.5$.
  • Figure 5: Impact of adjoint and gradient selection on reconstructed shape
  • ...and 4 more figures

Theorems & Definitions (21)

  • Proposition 1
  • Lemma 2.2
  • proof
  • Remark 1
  • Proposition 2
  • proof
  • Proposition 3
  • Lemma 3.1
  • Proposition 4: Shape gradient of $J$
  • proof : Proof of Proposition \ref{['prop:shape_gradients']}
  • ...and 11 more