Bayesian inference for the fractional Calder\'on problem with a single measurement

TL;DR

This paper demonstrates that in the fractional Calderón problem with a single measurement, the Bayesian posterior concentrates around the true parameter as data increases, providing convergence rates despite the challenges posed by fractional elliptic regularity.

Contribution

It establishes posterior concentration and convergence rates for the fractional Calderón problem using Bayesian methods with Gaussian priors, addressing stability and regularity challenges.

Findings

Posterior concentrates around the true parameter with increasing measurements.

Derived explicit convergence rates for the posterior mean reconstruction.

Addressed stability estimates for fractional elliptic problems.

Abstract

This paper investigates the consistency of a posterior distribution in the single-measurement fractional Calder\'on problem with additive Gaussian noise. We consider a Bayesian framework with rescaled and Gaussian sieve priors, using a collection of noisy, discrete observations taken from a suitable exterior domain. Our main result shows that the posterior distribution concentrates around the true parameter as the number of measurements increases. Furthermore, we establish tight convergence rates for the reconstruction error of the posterior mean. A central technical challenge is to obtain refined stability estimates for both the forward and inverse problems. In particular, the required forward estimates are delicate to obtain because the fractional elliptic problems do not enjoy as strong regularity theory as their classical counterparts.