Elliptic Bayesian Inverse Problems on Metric Graphs

TL;DR

This paper addresses Bayesian inverse problems on metric graphs, focusing on recovering diffusion coefficients of elliptic equations using noisy data, and demonstrates stability and accurate reconstruction with uncertainty quantification.

Contribution

It establishes well-posedness and stability of elliptic inverse problems on metric graphs, integrating Gaussian process priors and providing numerical validation.

Findings

Proved stability of elliptic and fractional elliptic forward models.

Achieved accurate reconstruction of diffusion coefficients.

Demonstrated effective uncertainty quantification.

Abstract

This paper studies the formulation, well-posedness, and numerical solution of Bayesian inverse problems on metric graphs, in which the edges represent one-dimensional wires connecting vertices. We focus on the inverse problem of recovering the diffusion coefficient of a (fractional) elliptic equation on a metric graph from noisy measurements of the solution. Well-posedness hinges on both stability of the forward model and an appropriate choice of prior. We establish the stability of elliptic and fractional elliptic forward models using recent regularity theory for differential equations on metric graphs. For the prior, we leverage modern Gaussian Whittle--Mat\'ern process models on metric graphs with sufficiently smooth sample paths. Numerical results demonstrate accurate reconstruction and effective uncertainty quantification.