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

TL;DR

This paper introduces a novel Hadamard fractional total variation-Gaussian prior for infinite-dimensional Bayesian inverse problems, providing theoretical analysis and demonstrating its effectiveness through numerical experiments.

Contribution

The paper develops a new hybrid prior in fractional Sobolev spaces for Bayesian inverse problems, with comprehensive theoretical analysis and practical sampling methods.

Findings

The HFTG prior captures texture details and avoids step effects.

The Bayesian method with HFTG prior is well-posed and accurate.

Numerical results confirm the effectiveness of the proposed approach.

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 step 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 pCN algorithm. Finally, numerical results under different models indicate that the Bayesian inference method with HFTG prior is effective and accurate.