Wavelet-based priors accelerate maximum-a-posteriori optimization in Bayesian inverse problems

TL;DR

This paper introduces wavelet-based priors for Bayesian inverse problems, demonstrating that they enable faster maximum-a-posteriori optimization for reconstructing fields with sharp interfaces from noisy data.

Contribution

It develops an adjoint method for optimizing Besov-norm-regularized functionals using Haar wavelets, showing improved convergence over traditional priors.

Findings

Wavelet priors facilitate localized reconstruction of permeability fields.

The proposed method accelerates MAP optimization compared to trigonometrically-based priors.

Effective reconstruction of sharp interfaces from noisy measurements.

Abstract

Wavelet (Besov) priors are a promising way of reconstructing indirectly measured fields in a regularized manner. We demonstrate how wavelets can be used as a localized basis for reconstructing permeability fields with sharp interfaces from noisy pointwise pressure field measurements in the context of the elliptic inverse problem. For this we derive the adjoint method of minimizing the Besov-norm-regularized misfit functional (this corresponds to determining the maximum a posteriori point in the Bayesian point of view) in the Haar wavelet setting. As it turns out, choosing a wavelet--based prior allows for accelerated optimization compared to established trigonometrically--based priors.