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Variational Bayesian inversion for the reaction coefficient in space-time nonlocal diffusion equations

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Abstract

In the paper, a variational Bayesian method is used to identify the reaction coefficient for space-time nonlocal diffusion equations using nonlocal averaged flux data. To show the posterior measure to be well-defined, we rigorously prove that the forward operator is continuous with respect to the unknown reaction field. Then, gradient-based prior information is proposed to explore oscillation features in the reaction coefficient. Moreover, the Bayesian inverse problem is shown to be well-posed in Hellinger distance. To accurately characterize the posterior density using uncorrelated samples, an efficient variational Bayesian method is used to estimate the reaction coefficient in the nonlocal models. A few numerical results are presented to illustrate the efficacy of the proposed approach and confirm some theoretic discoveries.

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Funding

L. Jiang received support of NSFC 11871378 and the Science Challenge Project (No. TZ2018001), and G. Zheng received support of NSFC 11301168.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Hunan University of Technology and Business, Changsha, 410205, China

    Xiaoyan Song

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

    Guang-Hui Zheng

  3. School of Mathematical Sciences, Tongji University, Shanghai, 200092, China

    Lijian Jiang

Authors
  1. Xiaoyan Song
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  2. Guang-Hui Zheng
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  3. Lijian Jiang
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Correspondence to Lijian Jiang.

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Communicated by: Bangti Jin

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Song, X., Zheng, GH. & Jiang, L. Variational Bayesian inversion for the reaction coefficient in space-time nonlocal diffusion equations. Adv Comput Math 47, 31 (2021). https://doi.org/10.1007/s10444-021-09850-1

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  • DOI: https://doi.org/10.1007/s10444-021-09850-1

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