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Prior distributions for Bayesian inference that rely on the $l_1$-norm of theparameters are of considerable interest, in part because they promote parameterfields with less regularity than Gaussian priors (e.g., discontinuities andblockiness). These $l_1$-type priors include the total variation (TV) prior andthe Besov $B^s_{1,1}$ space prior, and in general yield non-Gaussian posteriordistributions. Sampling from these posteriors is challenging, particularly inthe inverse problem setting where the parameter space is high-dimensional andthe forward problem may be nonlinear. This paper extends therandomize-then-optimize (RTO) method, an optimization-based sampling algorithmdeveloped for Bayesian inverse problems with Gaussian priors, to inverseproblems 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 posteriordistribution of the transformed parameters is amenable to Metropolized samplingvia RTO. We demonstrate this approach on several deconvolution problems and anelliptic PDE inverse problem, using TV or Besov $B^s_{1,1}$ space priors. Ourresults show that the transformed RTO algorithm characterizes the correctposterior distribution and can be more efficient than other samplingalgorithms. The variable transformation can also be extended to othernon-Gaussian priors.

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