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Numerical methods for diffusion coefficient recovery

Sahat Pandapotan Nainggolan, Julius Fergy Tiongson Rabago, Hirofumi Notsu

TL;DR

The paper tackles the inverse problem of recovering a spatially varying diffusion coefficient $\alpha$ in a stationary elliptic PDE from boundary Cauchy data. It introduces a gradient-weighted modification of the coupled complex-boundary method (CCBM) with an $H^1$-type misfit term and casts the reconstruction as a regularized optimization on a bounded, finite-dimensional coefficient set, proving Lipschitz continuity and differentiability of the forward map, as well as existence, stability, and convergence results under noisy measurements. Numerically, it employs a Sobolev gradient scheme and a projection step for piecewise-constant diffusion, showing improved stability and reduced high-frequency artifacts across smooth and heterogeneous cases, with favorable performance relative to classical boundary-based formulations. The accompanying results highlight the method’s potential for robust diffusion parameter recovery in both smooth and piecewise-constant media, and its applicability to related inverse problems involving piecewise-defined parameters.

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.

Numerical methods for diffusion coefficient recovery

TL;DR

The paper tackles the inverse problem of recovering a spatially varying diffusion coefficient in a stationary elliptic PDE from boundary Cauchy data. It introduces a gradient-weighted modification of the coupled complex-boundary method (CCBM) with an -type misfit term and casts the reconstruction as a regularized optimization on a bounded, finite-dimensional coefficient set, proving Lipschitz continuity and differentiability of the forward map, as well as existence, stability, and convergence results under noisy measurements. Numerically, it employs a Sobolev gradient scheme and a projection step for piecewise-constant diffusion, showing improved stability and reduced high-frequency artifacts across smooth and heterogeneous cases, with favorable performance relative to classical boundary-based formulations. The accompanying results highlight the method’s potential for robust diffusion parameter recovery in both smooth and piecewise-constant media, and its applicability to related inverse problems involving piecewise-defined parameters.

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 -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 -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.
Paper Structure (26 sections, 13 theorems, 114 equations, 23 figures, 6 tables, 2 algorithms)

This paper contains 26 sections, 13 theorems, 114 equations, 23 figures, 6 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations and the problem setting
  4. Continuity and differentiability of the forward map
  5. CCBM-based optimization reformulation of the inverse problem
  6. Lipschitz continuity of the modified CCBM cost functional
  7. Convexity and characterization of the minimizer
  8. Convergence Analysis under Noisy Measurements
  9. Convergence results
  10. Convergence proofs
  11. Proof of existence of a minimizer
  12. Proof of continuous dependence to the measured data
  13. Proof of convergence under vanishing noise
  14. Numerical Approximation via Sobolev Gradient Method
  15. Numerical Experiments
  16. ...and 11 more sections

Key Result

Lemma 2.4

Let Assumption ass:coefficients hold and let $\alpha \in \overline{\mathcal{A}}$, so that $\alpha \ge \underline{\alpha} > 0$ a.e. in $\varOmega$. Then the sesquilinear form $B(\cdot,\cdot;\alpha)$ is coercive on $H^{1}(\varOmega)$, i.e., there exists a constant $c_{B}>0$, independent of $\alpha$, s Consequently, for any $Q \in L^{2}(\varOmega)$ and $f,g \in H^{-1/2}(\varGamma)$, Problem prob:weak

Figures (23)

  • Figure 1: Exact diffusion coefficient profile (top row) and reconstruction results using exact measurements, without $H^{1}$ smoothing (second row) and with $H^{1}$ smoothing for different smoothing parameters $\mu$ (third to fifth rows).
  • Figure 2: Histories of the cost functional and gradient norm corresponding to Figure \ref{['fig:effect_of_mu']}. Left: $\mu=10^{-3}$; right: $\mu=10^{-6}$.
  • Figure 3: Boundary measurements at different noise levels $\delta$ with input data $g=1.0$.
  • Figure 4: Influence of the Tikhonov parameter $\rho$ on the reconstruction when $\delta = 0.0003$, without gradient smoothing.
  • Figure 5: Influence of the Tikhonov parameter $\rho$ on the reconstruction when $\delta = 0.0003$, with gradient smoothing ($\mu = 0.01$).
  • ...and 18 more figures

Theorems & Definitions (32)

  • Lemma 2.4: Coercivity, well-posedness, and stability
  • proof
  • Proposition 2.5
  • Proposition 2.6
  • Remark 2.7
  • Proposition 2.8
  • Proposition 2.11
  • proof
  • Proposition 2.12
  • proof
  • ...and 22 more