  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.
