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Solving linear Bayesian inverse problems using a fractional total variation-Gaussian (FTG) prior and transport map

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Abstract

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. Especially when the function to infer has much detail information, such as many sharp jumps, corners, and the discontinuous and nonsmooth oscillation, the so-called total variation-Gaussian (TG) prior is proposed in function space to address it. However, the TG prior is easy to lead the blocky (staircase) effect in numerical results. In this work, we present a fractional order-TG (FTG) hybrid prior to deal with such problems, where 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. For the numerical implementations of linear inverse problems in function spaces, we also propose an efficient independence sampler based on a transport map, which uses a proposal distribution derived from a diagonal map, and the acceptance probability associated to the proposal is independent of discretization dimensionality. And in order to take full advantage of the transport map, the hierarchical Bayesian framework is applied to flexibly determine the regularization parameter. Finally we provide some numerical examples to demonstrate the performance of the FTG prior and the efficiency and robustness of the proposed independence sampler method.

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Acknowledgements

The work described in this paper was supported by the NSF of China (12271151) and NSF of Hunan (2020JJ4166).

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Authors and Affiliations

  1. School of Mathematics, Hunan University, Changsha, 410082, China

    Zejun Sun & Guang-Hui Zheng

  2. School of Mathematics, Hunan Provincial Key Laboratory of Intelligent Information Processing and Applied Mathematics, Hunan University, Changsha, 410082, China

    Zejun Sun & Guang-Hui Zheng

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  1. Zejun Sun
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  2. Guang-Hui Zheng
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Correspondence to Guang-Hui Zheng.

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Sun, Z., Zheng, GH. Solving linear Bayesian inverse problems using a fractional total variation-Gaussian (FTG) prior and transport map. Comput Stat 38, 1811–1849 (2023). https://doi.org/10.1007/s00180-023-01332-w

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