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Many scientific and engineering problems require to perform Bayesianinferences in function spaces, in which the unknowns are of infinite dimension.In such problems, choosing an appropriate prior distribution is an importanttask. In particular we consider problems where the function to infer is subjectto sharp jumps which render the commonly used Gaussian measures unsuitable. Onthe other hand, the so-called total variation (TV) prior can only be defined ina finite dimensional setting, and does not lead to a well-defined posteriormeasure in function spaces. In this work we present a TV-Gaussian (TG) prior toaddress such problems, where the TV term is used to detect sharp jumps of thefunction, and the Gaussian distribution is used as a reference measure so thatit results in a well-defined posterior measure in the function space. We alsopresent an efficient Markov Chain Monte Carlo (MCMC) algorithm to draw samplesfrom the posterior distribution of the TG prior. With numerical examples wedemonstrate the performance of the TG prior and the efficiency of the proposedMCMC algorithm.

A TV-Gaussian prior for infinite-dimensional Bayesian inverse problems and its numerical implementations