A data-driven and model-based accelerated Hamiltonian Monte Carlo method for Bayesian elliptic inverse problems
A data-driven and model-based accelerated Hamiltonian Monte Carlo method for Bayesian elliptic inverse problems
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DOI：
10.1007/s11222-023-10262-y
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发表时间：
2021-04
影响因子：
2.2
通讯作者：
Sijing Li;Cheng Zhang;Zhiwen Zhang;Hongkai Zhao
中科院分区：
文献类型：
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作者：
Sijing Li;Cheng Zhang;Zhiwen Zhang;Hongkai Zhao
本文研究了一类椭圆型偏微分方程的贝叶斯反问题。具体来说，我们提出了一个数据驱动和基于模型的方法来加速汉密尔顿蒙特卡罗（HMC）方法在解决大规模贝叶斯逆问题。其关键思想是利用（基于模型）和构建（基于数据）的内在近似低维结构的基本问题，它由两个组件组成的训练组件，计算一组数据驱动的基础，以实现显着的降维的解决方案空间，和快速求解组件，计算的解决方案和其衍生物的新采样的椭圆PDE与构建的数据驱动的基础。因此，我们开发了一个有效的数据和基于模型的方法贝叶斯逆问题，并克服了典型的计算瓶颈HMC重复评估的哈密顿量涉及的解决方案（及其衍生物）建模的复杂系统，在我们的情况下，多尺度椭圆PDE。最后，我们提出的数值例子来证明所提出的方法的准确性和效率。
In this paper, we consider a Bayesian inverse problem modeled by elliptic partial differential equations (PDEs). Specifically, we propose a data-driven and model-based approach to accelerate the Hamiltonian Monte Carlo (HMC) method in solving large-scale Bayesian inverse problems. The key idea is to exploit (model-based) and construct (data-based) intrinsic approximate low-dimensional structure of the underlying problem which consists of two components—a training component that computes a set of data-driven basis to achieve significant dimension reduction in the solution space, and a fast solving component that computes the solution and its derivatives for a newly sampled elliptic PDE with the constructed data-driven basis. Hence we develop an effective data and model-based approach for the Bayesian inverse problem and overcome the typical computational bottleneck of HMC—repeated evaluation of the Hamiltonian involving the solution (and its derivatives) modeled by a complex system, a multiscale elliptic PDE in our case. Finally, we present numerical examples to demonstrate the accuracy and efficiency of the proposed method.