Multilevel Markov Chain Monte Carlo for Bayesian Inversion of Parabolic Partial Differential Equations under Gaussian Prior

We analyze the convergence of a multilevel Markov chain Monte Carlo (MLMCMC) algorithm for the Bayesian estimation of solution functionals for linear, parabolic partial differential equations subject to a log-Gaussian uncertain diffusion coefficient. Precisely, our multilevel convergence analysis is for a time-independent, log-Gaussian diffusion coefficient and for observations which are assumed to be corrupted by additive, centered, Gaussian observation noise. The elliptic spatial part of the parabolic PDE is assumed to be neither uniformly coercive nor uniformly bounded in terms of the realizations of the unknown Gaussian random field. The pathwise, multilevel discretization in space and time is a standard, first order, Lagrangian simplicial finite element method in the spatial domain and a first order, implicit timestepping of backward Euler type ensuring good dissipation and unconditional stability, and resulting in first order convergence in terms of the spatial meshwidth and the timestep. The Markov chain Monte Carlo (MCMC) algorithms covered by our analysis comprise the standard independence sampler as well as various variants, such as pCN. We prove that the proposed MLMCMC algorithm delivers approximate Bayesian estimates of quantities of interest consistent to first order in the discretization parameter on the finest spatial/temporal discretization meshwidth and stepsize in overall work which scales essentially (i.e., up to terms which depend logarithmically on the discretization parameters) as that of one deterministic solve on the finest mesh. Our convergence analysis is based on the discretization-level dependent truncation of the increments, introduced first in [V. H. Hoang, J. H. Quek, and Ch. Schwab, Inverse Problems, 36 (2020), 035021] for the corresponding elliptic forward problems, which is an essential modification of the MLMCMC method developed for elliptic problems under uniform prior in [V. H. Hoang, Ch. Schwab, and A. M. Stuart, Inverse Problems, 29 (2013), 085010]. This modification is required to address measurability and integrability issues encountered in the Bayesian posterior density evaluated at consecutive discretization levels with respect to the Gaussian prior. Both independence and pCN samplers are analyzed in detail. Applicability of our analysis to other versions of MCMC is discussed.