Dimension-Independent MCMC Sampling for Inverse Problems with Non-Gaussian Priors

The computational complexity of Markov chain Monte Carlo (MCMC) methods for the exploration of complex probability measures is a challenging and important problem in both statistics and the applied sciences. A challenge of particular importance arises in Bayesian inverse problems where the target distribution may be supported on an infinite-dimensional state space. In practice this involves the approximation of the infinite-dimensional target measure defined on sequences of spaces of increasing dimension bearing the risk of an increase of the computational error. Previous results have established dimension-independent bounds on the Monte Carlo error of MCMC sampling for Gaussian prior measures. We extend these results by providing a simple recipe for also obtaining these bounds in the case of non-Gaussian prior measures and by studying the design of proposal chains for the Metropolis-Hastings algorithm with dimension-independent performance. This study is motivated by an elliptic inverse problem with non-Gaussian prior that arises in groundwater flow. We explicitly construct an efficient Metropolis-Hastings proposal based on local proposals in this case, and we provide numerical evidence supporting the theory.