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Shape optimization for piecewise parameter identification in inverse diffusion problems with a single boundary measurement

Manabu Machida, Hirofumi Notsu, Julius Fergy Tiongson Rabago

TL;DR

This work addresses inverse diffusion problems where the absorption coefficient $\mu$ is piecewise-constant and the forward model uses a Robin boundary condition $\alpha\partial_{\mathbf{n}}u+\frac{1}{\zeta}u=0$. It introduces a shape-optimization approach that jointly reconstructs the subdomain boundary $\partial\omega$ and the absorption values from a single boundary measurement by leveraging the Eulerian derivative of a least-squares misfit plus regularization. Key contributions include a rigorous shape-sensitivity framework, derivation of the shape gradient via adjoint states, a regularized well-posed formulation for the coefficient, and a Sobolev-gradient numerical algorithm capable of handling non-convex, non-smooth interfaces. The proposed method demonstrates robust performance across 2D radial problems, non-circular geometries, and noisy data, highlighting its potential for diffuse optical tomography and related inverse-geometry problems where data are highly limited.

Abstract

This paper explores the reconstruction of a space-dependent parameter in inverse diffusion problems, proposing a shape-optimization-based approach. We consider a Robin boundary condition, physically motivated in diffuse optical tomography to model partial reflection of light at tissue boundaries [Arr99, GFB83a]. This ensures well-posedness of the forward problem, while related inverse problems with Dirichlet or Neumann conditions have also been considered in previous studies [Mef21]. The main objective is to recover the absorption coefficient from a single boundary measurement. While conventional gradient-based methods rely on the Frechet derivative of a cost functional with respect to the unknown parameter, we also utilize its Eulerian derivative with respect to the unknown boundary interface for recovery. This non-conventional approach addresses parameter recovery when only a single boundary measurement can be obtained, providing a method for its reconstruction. Numerical experiments confirm the effectiveness of the proposed method, even for intricate and non-convex boundary interfaces.

Shape optimization for piecewise parameter identification in inverse diffusion problems with a single boundary measurement

TL;DR

This work addresses inverse diffusion problems where the absorption coefficient is piecewise-constant and the forward model uses a Robin boundary condition . It introduces a shape-optimization approach that jointly reconstructs the subdomain boundary and the absorption values from a single boundary measurement by leveraging the Eulerian derivative of a least-squares misfit plus regularization. Key contributions include a rigorous shape-sensitivity framework, derivation of the shape gradient via adjoint states, a regularized well-posed formulation for the coefficient, and a Sobolev-gradient numerical algorithm capable of handling non-convex, non-smooth interfaces. The proposed method demonstrates robust performance across 2D radial problems, non-circular geometries, and noisy data, highlighting its potential for diffuse optical tomography and related inverse-geometry problems where data are highly limited.

Abstract

This paper explores the reconstruction of a space-dependent parameter in inverse diffusion problems, proposing a shape-optimization-based approach. We consider a Robin boundary condition, physically motivated in diffuse optical tomography to model partial reflection of light at tissue boundaries [Arr99, GFB83a]. This ensures well-posedness of the forward problem, while related inverse problems with Dirichlet or Neumann conditions have also been considered in previous studies [Mef21]. The main objective is to recover the absorption coefficient from a single boundary measurement. While conventional gradient-based methods rely on the Frechet derivative of a cost functional with respect to the unknown parameter, we also utilize its Eulerian derivative with respect to the unknown boundary interface for recovery. This non-conventional approach addresses parameter recovery when only a single boundary measurement can be obtained, providing a method for its reconstruction. Numerical experiments confirm the effectiveness of the proposed method, even for intricate and non-convex boundary interfaces.

Paper Structure

This paper contains 33 sections, 13 theorems, 116 equations, 22 figures.

Table of Contents

  1. Introduction
  2. Recovery of the absorption coefficient
  3. Geometric and coefficient setting.
  4. Well-posedness of the state and continuity of the coefficient-to-parameter map
  5. Regularized problem: well-posedness and first-order optimality condition
  6. Shape recovery of the boundary interface
  7. Shape sensitivity analysis
  8. Notations and some definitions
  9. Lagrangian and Eulerian derivative of the states
  10. Computation of the shape gradient without the Eulerian derivative of the state
  11. Numerical Algorithm and Examples
  12. Forward problem
  13. Choice of regularization functional
  14. Choice of regularization parameter
  15. Numerical algorithm
  16. ...and 18 more sections

Key Result

Lemma 2.4

Let ${\alpha}, \zeta \in \mathbb{R}_{+}$, ${\mu}\in {\color{black}{\mathcal{A}}}$, and $f\in H^{-1}(\Omega)$. Then, there exists a unique weak solution $u \in V$ to Problem prob:weak_form_optimal_tomography.

Figures (22)

  • Figure 1: A radial problem in 2D with $\mu_{1}^{\ast} = 1.2$ and source function $f=1$
  • Figure 2: 2D radial problem with source function $f=1$ and varying $\mu_{1}^{\ast}$
  • Figure 3: Results for a 2D radial problem with $f(x) = \delta(x)$
  • Figure 4: Results for a 2D radial problem with $f(x) = \delta(x)$ and varying $\eta_{a}^{\ast}$
  • Figure 5: Results for a 2D radial problem with $f(x) = \delta(x)$ under noisy data, with $\eta^{\ast} = 0.2$ (left three plots) and $\eta^{\ast} = 5$ (right three plots)
  • ...and 17 more figures

Theorems & Definitions (36)

  • Lemma 2.4
  • Proposition 2.5
  • Proposition 2.7
  • Proposition 2.8
  • Proposition 2.10: Strict convexity of $J_{\rho}$
  • Remark 2.11
  • Theorem 2.12
  • proof
  • Remark 3.1
  • Theorem 3.2
  • ...and 26 more