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Analysis of multilevel MCMC-FEM for Bayesian inversion of log-normal diffusions

by V.H. Hoang and J.H. Quek and Ch. Schwab

(Report number 2019-05)

Abstract
We develop the Multilevel Markov Chain Monte Carlo Finite Element Method (MLMCMC-FEM for short) 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 \({\mathbb R}^d\). We provide a convergence analysis with absolute mean convergence rate estimates for the proposed modified MLMCMC method, showing in particular error vs. work bounds which are explicit in the discretization parameters. This work generalizes the MLMCMC algorithm and the error vs. work analysis for uniform prior measure from [21] which we also review here, to Gaussian priors. In comparison to [21], we show by mathematical proofs and numerical examples that the unboundedness of the parameter range under gaussian prior and the nonuniform 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 with log-normal 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 confirm our error vs. work analysis.

Keywords:

@Techreport{HQS19_809,
  author = {V.H. Hoang and J.H. Quek and Ch. Schwab},
  title = {Analysis of multilevel MCMC-FEM for Bayesian inversion of log-normal diffusions},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2019-05},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2019/2019-05.pdf },
  year = {2019}
}

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