Solving Linear Bayesian Inverse Problems Using A Fractional Total Variation-Gaussian Prior And Transport Map

The Bayesian inference is widely used in many scientific and engineering problems,especially in the linear inverse problems in infinite-dimensional setting where the unknowns are functions.In such problems,choosing an appropriate prior distribution is an important task.In particular,the function to infer has much detail information,such as many sharp jumps,corners,and the discontinuous and nonsmooth oscillation.Hence,in this paper,we present a fractional order total variation-Gaussian(FTG)hybrid prior to deal with such problems.For the numerical implementations of linear inverse problems in function spaces,we also propose an efficient independence sampler based on a transport map.Firstly,this paper briefly introduce the fundamentals of Bayesian inference and related theories for the linear inverse problems in infinite-dimensional setting.The basic form of hybrid prior and the definitions of fractional derivatives are given in order to present the FTG hybrid prior.Next the fractional Sobolev space is chosen as the space of unknown function.Then the FTG hybrid prior is proposed,which the fractional order total variation(FTV)term is used to capture the detail information of the unknowns and simultaneously uses the Gaussian measure to ensure that it results in a well-defined posterior measure.Secondly,the the hierarchical Bayesian framework is applied to flexibly determine the regularization parameter.Next the transport maps in the context of Bayesian inverse problems is introduced,and this paper also explains how the diagonal transport maps can be constructed from reference measure and discusses a numerical solution for the corresponding optimization problem.Then the independence sampler using a proposal distribution derived from a diagonal map is described.Finally,this paper presents a range of numerical examples for the linear inverse problems from the one-dimensional deconvolution problem and inverse source identification problem to the two-dimensional limited computed tomography(CT)reconstruction in medical imaging and the image denoising.From those numerical experiments,the FTG prior can effectively reduces blocky effects and have good performance for the detail information in the unknowns,especially for recovering textures in images.And those results also illustrate the high efficiency of the diagonal map-based independence sampler algorithm.