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

1. Problem Statement

The paper addresses a geometric inverse problem governed by the advection–diffusion equation. The objective is to reconstruct the shape of an unknown, open, connected, and bounded inclusion (an obstacle or sink, denoted as $\omega$) within a known flow domain $D$.

• Mathematical Model: The physical process is described by the equation $-\text{div}(\sigma\nabla u) + \mathbf{b} \cdot \nabla u = 0$ in the annular domain $\Omega = D \setminus \omega$.
• Boundary Conditions:
  • Unknown Boundary ($\Gamma = \partial\omega$): Homogeneous Dirichlet condition ($u=0$).
  • Known Boundary ($\Sigma = \partial D$): Overdetermined Cauchy data is provided: Dirichlet data $u=f$ and Neumann data $\sigma \partial_n u = g$.
• Challenge: The problem is ill-posed. Standard least-squares approaches often suffer from numerical instability, particularly when:
  • The coefficients (diffusion $\sigma$ and advection $\mathbf{b}$) vary spatially.
  • The target shape is non-convex (containing concavities).
  • The input data contains measurement noise.

2. Methodology

The authors propose a unified framework combining the Coupled Complex Boundary Method (CCBM) with the Alternating Direction Method of Multipliers (ADMM) and Sobolev Gradient Descent.

A. Coupled Complex Boundary Method (CCBM)

Instead of minimizing the misfit of boundary data directly (which is unstable), CCBM encodes the Cauchy data into a complex-valued Robin boundary condition on the known boundary $\Sigma$:
$$\sigma \partial_n u + i u = g + i f$$
Here, the real part corresponds to the Neumann data and the imaginary part to the Dirichlet data.

• Cost Functional: The inverse problem is transformed into minimizing the $L^2$ norm of the imaginary part ($u_2$) of the complex solution over the entire domain:
  $$J(\Omega) = \frac{1}{2} \int_{\Omega} |u_2|^2 \, dx$$
  If $u_2 \equiv 0$ in $\Omega$, the solution satisfies the original overdetermined system.

B. Shape Sensitivity Analysis

The authors rigorously derive the shape derivative (gradient) and shape Hessian (second-order derivative) of the cost functional.

• Adjoint Method: They introduce an adjoint state $p$ to compute the gradient efficiently.
• Hessian Compactness: They prove that the shape Hessian is a compact operator, confirming the inherent ill-posedness of the problem and the need for regularization.

C. ADMM-Based Optimization Framework

To handle non-convex shapes and noisy data without explicit perimeter regularization, the authors embed CCBM into an ADMM framework.

• Reformulation: They introduce an auxiliary variable $v$ and an inequality constraint ($a \le u_1 \le b$) to reformulate the problem.
• Augmented Lagrangian: The problem is solved by minimizing an augmented Lagrangian functional iteratively:
  1. $\omega$-subproblem: Update the shape $\Omega$ using a gradient descent step.
  2. $v$-subproblem: Update the auxiliary variable via projection onto the admissible set.
  3. Multiplier Update: Update the Lagrange multiplier.
• Sobolev Gradient Descent (SGBD): To ensure mesh quality and stability, the shape gradient is smoothed using an $H^1$-Riesz representation (Sobolev gradient) rather than a raw $L^2$ gradient.

D. Partial Gradient Strategy

A key innovation is the use of "partial gradients." Instead of computing the full, complex adjoint system for every term in the cost functional, the authors propose splitting the adjoint variables. They demonstrate that using a specific subset of adjoint terms (Method 3 in their experiments) yields higher accuracy and lower computational cost.

3. Key Contributions

1. Extension of CCBM to Advection-Diffusion: The framework is generalized to handle spatially varying diffusion ($\sigma$) and advection ($\mathbf{b}$) coefficients, which are more realistic for environmental applications than constant-coefficient models.
2. ADMM Integration for Shape Optimization: The paper introduces a novel ADMM scheme specifically adapted for CCBM. Unlike standard ADMM which uses real variables, this handles complex-valued state variables and introduces inequality constraints to stabilize the reconstruction of concave boundaries.
3. Partial Gradient Scheme: The authors propose and validate a "partial gradient" approach that reduces computational overhead while improving reconstruction accuracy for complex geometries.
4. Rigorous Mathematical Analysis: The paper provides a complete theoretical foundation, including existence proofs for the optimal shape, well-posedness of the complex state equation, and detailed shape sensitivity analysis (first and second-order derivatives).

4. Numerical Results

The method was validated using Finite Element Methods (FEM) in both 2D and 3D using the FreeFem++ software.

• Test Cases: Various geometries were tested, including ellipses, dumbbells, peanuts, and L-blocks (non-convex shapes with corners).
• Noise Robustness: Experiments included synthetic data with Gaussian noise levels up to 7%.
• Performance Comparison:
  • Conventional CCBM: Struggled to recover deep concavities and non-convex features, especially under noise.
  • CCBM-ADMM: Successfully reconstructed complex, non-convex shapes (including the L-block) with high accuracy, even with 5% noise.
  • Efficiency: The "partial gradient" method (Method 3) was found to be the most efficient, reducing CPU time by ~30% compared to other gradient formulations while achieving the lowest cost functional values.
• 3D Validation: The 3D experiments confirmed that the method remains robust for spatially varying coefficients and complex 3D geometries without requiring explicit perimeter regularization.

5. Significance

This work represents a significant advancement in geometric inverse problems for transport phenomena.

• Practical Impact: It offers a robust tool for environmental engineering, such as identifying pollutant sources or underground obstacles in groundwater flow, where data is often noisy and geometries are complex.
• Theoretical Impact: It demonstrates that ADMM can be effectively coupled with complex-valued boundary methods to solve ill-posed shape optimization problems, bypassing the need for traditional perimeter regularization which can sometimes distort the solution.
• Methodological Impact: The "partial gradient" strategy provides a new heuristic for optimizing computational cost in high-dimensional shape optimization problems involving complex PDE constraints.

In conclusion, the proposed CCBM–ADMM–SGBD framework provides a unified, stable, and efficient strategy for recovering complex inclusions in advection–diffusion systems, outperforming conventional methods in scenarios involving noise and non-convex geometries.