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ANOVA-GP Modeling for High-Dimensional Bayesian Inverse Problems

Author

Listed:
  • Xiaoyu Shi

    (School of Information Science and Technology, ShanghaiTech University, Shanghai 201210, China
    These authors contributed equally to this work.)

  • Hanyu Zhang

    (School of Information Science and Technology, ShanghaiTech University, Shanghai 201210, China
    These authors contributed equally to this work.)

  • Guanjie Wang

    (School of Statistics and Mathematics, Shanghai Lixin University of Accounting and Finance, Shanghai 201209, China)

Abstract

Markov chain Monte Carlo (MCMC) stands out as an effective method for tackling Bayesian inverse problems. However, when dealing with computationally expensive forward models and high-dimensional parameter spaces, the challenge of repeated sampling becomes pronounced. A common strategy to address this challenge is to construct an inexpensive surrogate of the forward model, which cuts the computational cost of individual samples. While the Gaussian process (GP) is widely used as a surrogate modeling strategy, its applicability can be limited when dealing with high-dimensional input or output spaces. This paper presents a novel approach that combines the analysis of variance (ANOVA) decomposition method with Gaussian process regression to handle high-dimensional Bayesian inverse problems. Initially, the ANOVA method is employed to reduce the dimension of the parameter space, which decomposes the original high-dimensional problem into several low-dimensional sub-problems. Subsequently, principal component analysis (PCA) is utilized to reduce the dimension of the output space on each sub-problem. Finally, a Gaussian process model with a low-dimensional input and output is constructed for each sub-problem. In addition to this methodology, an adaptive ANOVA-GP-MCMC algorithm is proposed, which further enhances the adaptability and efficiency of the method in the Bayesian inversion setting. The accuracy and computational efficiency of the proposed approach are validated through numerical experiments. This innovative integration of ANOVA and Gaussian processes provides a promising solution to address challenges associated with high-dimensional parameter spaces and computationally expensive forward models in Bayesian inference.

Suggested Citation

  • Xiaoyu Shi & Hanyu Zhang & Guanjie Wang, 2024. "ANOVA-GP Modeling for High-Dimensional Bayesian Inverse Problems," Mathematics, MDPI, vol. 12(2), pages 1-18, January.
  • Handle: RePEc:gam:jmathe:v:12:y:2024:i:2:p:301-:d:1320816
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    References listed on IDEAS

    as
    1. Li, Jinglai, 2015. "A note on the Karhunen–Loève expansions for infinite-dimensional Bayesian inverse problems," Statistics & Probability Letters, Elsevier, vol. 106(C), pages 1-4.
    2. Luo, Chunling & Shen, Lijuan & Xu, Ancha, 2022. "Modelling and estimation of system reliability under dynamic operating environments and lifetime ordering constraints," Reliability Engineering and System Safety, Elsevier, vol. 218(PA).
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