Speaker
Description
Sparse priors, such as the Laplace prior, are of considerable interest in Bayesian inverse problems because they promote sparsity and preserve edges in the solution, which are often more appropriate than the smooth reconstructions obtained with Gaussian priors. However, sampling from the resulting non-Gaussian posteriors is challenging, particularly in high-dimensional settings. To address this, we build on the randomize-then-optimize (RTO) framework and its extension to $\ell_1$-type priors via variable transformation. We propose a method for posterior sampling in Bayesian inverse problems with Laplace priors that converts the prior into a standard Gaussian in the transformed space. Within the RTO framework, this results in a nonlinear optimization problem at each iteration, which we solve using the Levenberg–Marquardt algorithm. To accelerate the linear system solves at each Levenberg–Marquardt iteration, we employ randomized preconditioners and investigate the effect of using single and half precision arithmetic in these solves. Performance is evaluated through numerical experiments on image deconvolution and computed tomography problems.