Numerical methods for diffusion coefficient recovery

TL;DR

This paper introduces a modified boundary method for more stable and accurate numerical reconstruction of spatially varying diffusion coefficients in elliptic equations, demonstrating improved robustness through theoretical analysis and extensive numerical experiments.

Contribution

It develops a gradient-weighted modification of the coupled complex-boundary method with theoretical guarantees and practical algorithms for stable diffusion coefficient recovery.

Findings

Modified CCBM yields more stable reconstructions.

Reduces high-frequency artifacts in numerical results.

Demonstrates robustness across various noise levels and boundary conditions.

Abstract

We revisit the inverse problem of reconstructing a spatially varying diffusion coefficient in stationary elliptic equations from boundary Cauchy data. From a theoretical perspective, we introduce a gradient-weighted modification of the coupled complex-boundary method (CCBM) incorporating an \(H^1\)-type term, and formulate the reconstruction as a regularized optimization problem over bounded admissible coefficients. We establish continuity and differentiability of the forward map, Lipschitz continuity of the modified cost functional, existence of minimizers, stability with respect to noisy data, and convergence under vanishing noise. From a numerical perspective, reconstructions are computed using a Sobolev-gradient descent scheme and evaluated through extensive numerical experiments across a range of noise levels, boundary inputs, and coefficient structures. In the reported tests, for sufficiently large but not excessive $H^1$-weights, the modified CCBM is observed to yield more stable reconstructions and to reduce certain high-frequency artifacts. Across the numerical scenarios considered in this study, the method often demonstrates favorable stability and robustness properties relative to several classical boundary-based formulations, although performance remains problem- and parameter-dependent. A projection-based extension further supports stable recovery of piecewise-constant diffusion coefficients in multi-subregion test cases. Our results indicate that, as long as all subdomains share a portion of the boundary, the proposed CCBM-based Tikhonov regularization approach with a pick-a-point strategy enables stable and reliable reconstruction of diffusion parameters.