A replica exchange preconditioned Crank-Nicolson Langevin dynamic MCMC method for Bayesian inverse problems

TL;DR

This paper introduces repCNLD, a novel MCMC method combining replica exchange and preconditioned Langevin dynamics with Crank-Nicolson discretization, effectively sampling high-dimensional, multi-modal distributions in Bayesian inverse problems.

Contribution

The paper develops repCNLD with Crank-Nicolson discretization and multi-variance extension, improving convergence speed and mode capturing in complex high-dimensional Bayesian inverse problems.

Findings

Accelerates convergence of MCMC in high dimensions.

Effectively captures all modes in multi-modal distributions.

Reduces computational costs with multi-variance extension.

Abstract

This paper proposes a replica exchange preconditioned Langevin diffusion discretized by the Crank-Nicolson scheme (repCNLD) to handle high-dimensional and multi-modal distribution problems. Sampling from high-dimensional and multi-modal distributions is a challenging question. The performance of many standard MCMC chains deteriorates as the dimension of parameters increases, and many MCMC algorithms cannot capture all modes if the energy function is not convex. The proposed repCNLD can accelerate the convergence of the single-chain pCNLD, and can capture all modes of the multi-modal distributions. We proposed the Crank-Nicolson discretization, which is robust. Moreover, the discretization error grows linearly with respect to the time step size. We extend repCNLD to the multi-variance setting to further accelerate the convergence and save computation costs. Additionally, we derive an unbiased estimator of the swapping rate for the multi-variance repCNLD method, providing a guide for the choice of the low-fidelity model used in the second chain. We test our methods with high-dimensional Gaussian mixture models and high-dimensional nonlinear PDE inverse problems. Particularly, we employ the discrete adjoint method to efficiently calculate gradients for nonlinear PDE inverse problems.