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Convergence rate analysis of MCMC-FEM for Bayesian inversion of log-normal diffusion problems

by V.H. Hoang and Ch. Schwab

(Report number 2016-19)

Abstract
Markov Chain Monte Carlo (MCMC) methods for the numerical solution of Bayesian Inverse Problems for linear second order, divergence form elliptic partial differential equations (PDEs) with lognormal random field coefficients are analyzed. The analysis of the MCMC Finite Element discretization for uniformly elliptic, random diffusion problems of [14] is extended. The complexity of MCMC sampling for the uncertain input fields from the posterior density, as well as the MCMC error due to discretization of the PDE of interest in the forward response map, are analyzed in the abstract framework of MCMC methods of Meyn and Tweedie [16]. Particular attention is given to bounds on the overall work required by the MCMC algorithms for achieving a prescribed error level \(\varepsilon > 0\). We prove convergence rate estimates and bound the computational complexity of straightforward combinations of MCMC sampling strategies with Finite Element approximation of solution of the forward PDE. Due to the non-uniform ellipticity, the computational complexity analysis of the MCMC-FEM is probabilistic.

Keywords:

@Techreport{HS16_656,
  author = {V.H. Hoang and Ch. Schwab},
  title = {Convergence rate analysis of MCMC-FEM for Bayesian inversion of log-normal diffusion problems},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2016-19},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2016/2016-19.pdf },
  year = {2016}
}

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