Analysis of a multilevel Markov chain Monte Carlo finite element method for Bayesian inversion of log-normal diffusions

Abstract

We develop the multilevel Markov chain Monte Carlo finite element method (MLMCMC-FEM) to sample from the posterior density of the Bayesian inverse problems. The unknown is the diffusion coefficient of a linear, second-order divergence form, elliptic equation in a bounded, polytopal subdomain of R d . We provide a convergence analysis with absolute mean convergence rate estimates for the proposed modified MLMCMC-FEM showing in particular error versus work bounds, which are explicit in the discretization parameters. This work generalizes the MLMCMC-FEM algorithm and the error versus work analysis for the uniform prior measure from Hoang et al (2013 Inverse Problems 29), which we also review here, to linear, elliptic, divergence-form PDEs with a log-Gaussian uncertain coefficient and Gaussian prior measure. In comparison to Hoang et al (2013 Inverse Problems 29), we show by mathematical proofs and numerical examples that the unboundedness of the parameter range under Gaussian prior and the non-uniform ellipticity of the forward model require essential modifications in the MCMC sampling algorithm and in the error analysis. The proposed novel multilevel MCMC sampler applies to general Bayesian inverse problems for linear, second order elliptic divergence-form PDEs with log-Gaussian coefficients. It only requires a numerical forward solver with essentially optimal complexity for producing an approximation of the posterior expectation of a quantity of interest within a prescribed accuracy. Numerical examples using independence and pCN samplers are in agreement with our error versus work analysis.

Analyticity and sparsity in uncertainty quantification for PDEs with Gaussian random field inputs

“…Much attention has been devoted in recent years to the analysis of efficient simulation methods for the solution u. We mention only MC methods and their variants (see, e.g., [88,65,26,27] and the references there) and Quasi-Monte Carlo (QMC for short) integration (see, e.g., [54]), and Wiener-Hermite polynomial chaos approximation (see, e.g., [67,11] and the references there) and sparse-grid (a.k.a. "stochastic collocation") (see, e.g., [47] and the references there).…”

“…However, we can not, in general, expect uniform w.r. to y ∈ R J a-priori estimates, also of the higher derivatives, for smoother functions x → g(x, y) and x → f . Therefore, the parametric problem (1.2) is nonuniformly elliptic, [26,65]. In particular, therefore, also a-priori error bounds for various discretization schemes will contain this uniformity w.r.t.…”

Estimating posterior quantity of interest expectations in a multilevel scalable framework

“…Such methods include utilizing gradient and Hessian information to modify the MCMC proposal (requiring additional solvers beyond the forward PDE), as in References 4,5. Other methods include multilevel approaches that utilize coarse grid solvers to accelerate mixing, as in References 6,7 or—in addition to accelerating MCMC mixing—utilize a telescoping sum to perform variance reduction (similar to multilevel Monte Carlo 8‐11 ) resulting in fewer fine grid samples 12‐14 …”

Multilevel Hierarchical Decomposition of Finite Element White Noise with Application to Multilevel Markov Chain Monte Carlo

“…In particular, [27] developed an approach to both accelerate the mixing of the MCMC chain by using multiple levels, each with coarser spatial discretizations, and accelerate the sampling by performing variance reduction via multilevel Monte Carlo following the ideas of [38,34,8,19,56]. Analysis of a multilevel MCMC was completed in [41]. While promising speed up results have been shown, numerical testing has been limited to 2D spatial domains.…”