Bayesian inverse problems with $l_1$ priors: a Randomize-then-Optimize approach

TL;DR

This paper extends the Randomize-then-Optimize (RTO) sampling method to Bayesian inverse problems with $l_1$-type priors by using a variable transformation, enabling efficient sampling of complex, non-Gaussian posteriors.

Contribution

The authors develop a variable transformation approach that allows RTO to handle $l_1$-type priors in inverse problems, broadening its applicability to non-Gaussian posteriors.

Findings

Transformed RTO accurately characterizes the posterior distribution.

The method outperforms other sampling algorithms in efficiency.

Applicable to deconvolution and PDE inverse problems.

Abstract

Prior distributions for Bayesian inference that rely on the $l_1$-norm of the parameters are of considerable interest, in part because they promote parameter fields with less regularity than Gaussian priors (e.g., discontinuities and blockiness). These $l_1$-type priors include the total variation (TV) prior and the Besov $B^s_{1,1}$ space prior, and in general yield non-Gaussian posterior distributions. Sampling from these posteriors is challenging, particularly in the inverse problem setting where the parameter space is high-dimensional and the forward problem may be nonlinear. This paper extends the randomize-then-optimize (RTO) method, an optimization-based sampling algorithm developed for Bayesian inverse problems with Gaussian priors, to inverse problems with $l_1$-type priors. We use a variable transformation to convert an $l_1$-type prior to a standard Gaussian prior, such that the posterior distribution of the transformed parameters is amenable to Metropolized sampling via RTO. We demonstrate this approach on several deconvolution problems and an elliptic PDE inverse problem, using TV or Besov $B^s_{1,1}$ space priors. Our results show that the transformed RTO algorithm characterizes the correct posterior distribution and can be more efficient than other sampling algorithms. The variable transformation can also be extended to other non-Gaussian priors.