A data-driven and model-based accelerated Hamiltonian Monte Carlo method for Bayesian elliptic inverse problems

TL;DR

This paper introduces a hybrid data-driven and model-based accelerated Hamiltonian Monte Carlo method to efficiently solve large-scale Bayesian inverse problems involving elliptic PDEs, reducing computational costs significantly.

Contribution

The paper proposes a novel approach combining data-driven basis construction with model-based PDE solving to accelerate HMC in Bayesian inverse problems.

Findings

Significant dimension reduction achieved in solution space.

Enhanced computational efficiency demonstrated in numerical examples.

Maintained accuracy while reducing computational cost.

Abstract

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) the 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 achieve 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. We present numerical examples to demonstrate the accuracy and efficiency of the proposed method.