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

Technical Summary: Shape Optimization for Piecewise Parameter Identification in Inverse Diffusion Problems

Problem Statement
This study addresses the inverse diffusion problem of reconstructing a space-dependent absorption coefficient ($\mu$) within a bounded domain $\Omega \subset \mathbb{R}^d$ ($d \in \{2, 3\}$). The physical model is governed by the steady-state diffusion equation with a Robin boundary condition, which models partial light reflection at tissue boundaries in diffuse optical tomography (DOT). The primary challenge lies in the ill-posedness of the problem, specifically the difficulty of recovering $\mu$ from a single boundary measurement ($u|_{\Sigma} = h$).

The authors focus on a specific geometric structure where the absorption coefficient is piecewise constant, defined as $\mu = \mu_0 \chi_{\Omega \setminus \omega} + \mu_1 \chi_{\omega}$, where $\omega$ is an unknown subdomain with a piecewise smooth boundary interface $\partial \omega$. This formulation shifts the problem from a standard parameter identification task to an inverse geometry problem, where both the value of the coefficient and the shape of the interface must be recovered.

Methodology
The paper proposes a shape-optimization-based approach that diverges from conventional gradient-based methods relying solely on the Fréchet derivative with respect to the parameter. Instead, the method utilizes the Eulerian derivative of the cost functional with respect to the unknown boundary interface.

1. Mathematical Framework:

   • The forward problem is formulated as a Robin-to-Dirichlet map.
   • The inverse problem is cast as a minimization of a regularized least-squares functional: $J_\rho(\mu, \omega) = \frac{1}{2}\|u(\mu, \omega) - h\|^2_{L^2(\partial \Omega)} + R(\mu, \rho)$.
   • Tikhonov regularization is employed to mitigate ill-posedness. The authors also explore a heuristic "balancing principle" for selecting the regularization parameter $\rho$ without prior knowledge of noise levels.

2. Shape Sensitivity Analysis:

   • The core theoretical contribution involves deriving the Eulerian derivative (shape gradient) of the cost functional with respect to the subdomain $\omega$.
   • The authors rigorously compute this derivative using the adjoint method. They introduce an adjoint state $p$ satisfying a specific PDE system.
   • A key theoretical result (Theorem 3.6) provides the expression for the shape gradient on the interface $\partial \omega$ without requiring the explicit computation of the Eulerian derivative of the state variable $u$. This derivation relies on mild regularity assumptions (Lipschitz boundaries) and utilizes the Hadamard boundary differentiation formula.

3. Numerical Algorithm:

   • An iterative descent scheme is developed within a finite element framework.
   • The algorithm updates both the absorption coefficient values and the boundary interface geometry simultaneously.
   • To ensure stability and suppress oscillations during the shape update, the method employs a Sobolev gradient descent scheme. This involves solving a vector-type Laplace equation to generate a smooth deformation field $\theta$ from the raw shape gradient.
   • The algorithm handles non-convex and complex geometries by moving mesh nodes based on the computed deformation field.

Key Contributions

• Single-Measurement Reconstruction: The paper demonstrates that accurate reconstruction of the absorption coefficient and its boundary interface is feasible using only a single boundary measurement, a scenario where conventional methods often fail or require multiple inputs.
• Shape Optimization for Inverse Diffusion: This is presented as the first study to approach this specific reconstruction problem from a shape optimization perspective, treating the interface as the primary variable for recovery.
• Rigorous Derivation: The authors provide a rigorous derivation of the shape gradient for the least-squares functional, explicitly handling the jump conditions across the interface and the Robin boundary conditions.
• Handling Non-Smooth Geometries: The numerical scheme is shown to be effective for reconstructing non-convex boundaries and interfaces with sharp edges (e.g., squares, T-shapes), despite the theoretical regularity assumptions requiring smoother boundaries.

Results
Numerical experiments were conducted in 2D using both exact and noisy data (Gaussian noise). The tests included:

• Radial Problems: Recovery of circular interfaces with varying absorption contrasts.
• Complex Geometries: Reconstruction of non-circular, non-convex interfaces (e.g., peanut shapes, inverted T-shapes, squares).
• Source Variations: Tests using constant source functions, point sources (Dirac delta), and multiple sources near the boundary.

The results indicate:

• The method successfully recovers both the magnitude of the absorption coefficient and the geometry of the interface.
• The approach is robust to noise, though reconstruction accuracy for concave regions and sharp vertices degrades slightly with high noise levels.
• The choice of the initial guess and the regularization parameter (specifically the balancing parameter $\beta$) significantly influences convergence and accuracy.
• The method performs well even when the source term violates strict regularity assumptions (e.g., point sources), suggesting practical applicability beyond theoretical constraints.

Significance and Claims
The authors claim that the proposed method offers a novel alternative to conventional approaches by avoiding the need for multiple measurements or boundary inputs. By utilizing shape optimization and the Eulerian derivative, the method achieves a precise delineation of the boundary interface, contrasting with conventional methods that often yield blurred or diffuse boundaries.

The paper emphasizes that while the problem is inherently ill-posed, the combination of shape optimization, Tikhonov regularization, and the balancing principle provides a stable and effective reconstruction strategy. The authors note that the method's success in reconstructing non-convex and non-smooth boundaries highlights its potential for complex reconstruction problems involving jump discontinuities, with potential applications extending beyond standard DOT to fluorescence DOT and other optical imaging techniques. The study concludes that the approach is practical and effective, particularly for scenarios where the absorption coefficient is piecewise constant.