A Hadamard fractional total variation-Gaussian (HFTG) prior for Bayesian inverse problems

Abstract

This paper studies the infinite-dimensional Bayesian inference method with Hadamard fractional total variation-Gaussian (HFTG) prior for solving inverse problems. First, Hadamard fractional Sobolev space is established and proved to be a separable Banach space under some mild conditions. Afterwards, the HFTG prior is constructed in this separable fractional space, and the proposed novel hybrid prior not only captures the texture details of the region and avoids staircase effects, but also provides a complete theoretical analysis in the infinite dimensional Bayesian inversion. Based on the HFTG prior, the well-posedness and finite-dimensional approximation of the posterior measure of the Bayesian inverse problem are given, and samples are extracted from the posterior distribution using the standard preconditioned Crank-Nicolson (pCN) algorithm. Finally, numerical results under different models indicate that the Bayesian inference method with HFTG prior is effective and accurate.