Inhomogeneous Priors for Bayesian Inverse Problems

TL;DR

This paper introduces inhomogeneous priors for Bayesian inverse problems, enabling better modeling of spatially varying unknowns and improving reconstruction and uncertainty quantification in large-scale, data-limited scenarios.

Contribution

The authors develop a new class of nonstationary Whittle-Matern-type priors via convolution with white noise, compatible with Bayesian theory and efficient sampling methods.

Findings

Enhanced reconstruction quality in denoising and tomography.

Improved uncertainty quantification in data-limited cases.

Efficient numerical schemes with quantified approximation errors.

Abstract

Many inverse problems arising in engineering and applied sciences involve unknown quantities with pronounced spatial inhomogeneity, such as localized defects or spatially varying material properties, making reliable uncertainty quantification particularly challenging. While Bayesian inverse problem methodologies provide a principled framework for assessing reconstruction reliability, commonly used Gaussian priors, such as Whittle-Matern models, impose globally homogeneous assumptions that limit their ability to capture such structure in large-scale settings. We introduce a new class of inhomogeneous priors defined via convolution with white noise, yielding nonstationary Whittle-Matern-type random fields with a rigorous mathematical construction. These priors fit naturally within existing Bayesian well-posedness theory and enable efficient sampling by reducing prior realizations to the solution of a pseudo-differential equation, for which we develop numerical schemes with quantified approximation error. Numerical experiments in one-dimensional denoising and two-dimensional limited-angle X-ray tomography demonstrate significant improvements in reconstruction quality and uncertainty quantification, particularly in data-limited scenarios.