Multilevel MCMC Bayesian Inversion of Parabolic PDEs under Gaussian Prior

Abstract

We analyze the convergence of a multi-level Markov Chain Monte-Carlo (MLMCMC) algorithm for the Bayesian estimation of solution functionals for linear, parabolic partial differential equations subject to uncertain diffusion coefficient. The multilevel convergence analysis is performed 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 neither uniformly coercive nor uniformly bounded in terms of the realizations of the unknown gaussian random field. The path-wise, multi-level discretization in space and time considered is based on standard, first order, Lagrangean simplicial Finite Elements in the spatial domain and on 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 time-step. The MCMC algorithms covered by our analysis comprise the standard, indepence 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 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 [15] for the corresponding elliptic forward problems. This 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 sampler and pCN are analyzed in detail. Applicability of our analysis to other versions of MCMC is discussed.