Bayesian inverse problems with unknown operators

TL;DR

This paper develops a Bayesian framework for linear inverse problems with unknown operators, providing theoretical guarantees on posterior contraction rates and an adaptive empirical Bayes method, demonstrated through numerical examples.

Contribution

It introduces a Bayesian approach for inverse problems with unknown operators, establishing contraction rates and an adaptive procedure for unknown smoothness.

Findings

Posterior contraction rates match optimal convergence up to logs.

Empirical Bayes method adapts to unknown smoothness.

Numerical examples demonstrate practical effectiveness.

Abstract

We consider the Bayesian approach to linear inverse problems when the underlying operator depends on an unknown parameter. Allowing for finite dimensional as well as infinite dimensional parameters, the theory covers several models with different levels of uncertainty in the operator. Using product priors, we prove contraction rates for the posterior distribution which coincide with the optimal convergence rates up to logarithmic factors. In order to adapt to the unknown smoothness, an empirical Bayes procedure is constructed based on Lepski's method. The procedure is illustrated in numerical examples.