A replica exchange preconditioned Crank-Nicolson Langevin dynamic MCMC method with multi-variance strategy for Bayesian inverse problems

Abstract

The preconditioned Crank-Nicolson Markov Chain Monte Carlo (pCN-MCMC) method is an efficient approach for large-scale Bayesian inference problems. It is constructed by discretizing a stochastic partial differential equation (SPDE). When the gradient information is coupled in this SPDE, the corresponding SPDE is also referred to as the Langevin diffusion. In this work, we present the replica exchange preconditioned Crank-Nicolson Langevin dynamic (repCNLD) Monte Carlo method to speed up the convergence of the pCN-MCMC and tackle the challenge of multimodal distribution. We use the Crank-Nicolson scheme to discretize the repLD and set the preconditioned matrix as the covariance matrix of the Gaussian prior. The discretization error is analyzed. Moreover, we use the multi-variance strategy (m-repCNLD) to improve computational efficiency. Perturbations are introduced into the scheme due to the likelihood estimation error, leading to the biased swapping rate. To correct the bias, we provide an unbiased estimator derived from some assumptions on the forward solvers. The accuracy and efficiency of the proposed method are demonstrated by sampling Gaussian mixture distribution and solving PDE-based nonlinear inverse problems numerically.