Research reports

Years: 2025 2024 2023 2022 2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991

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}
}

Disclaimer
© Copyright for documents on this server remains with the authors. Copies of these documents made by electronic or mechanical means including information storage and retrieval systems, may only be employed for personal use. The administrators respectfully request that authors inform them when any paper is published to avoid copyright infringement. Note that unauthorised copying of copyright material is illegal and may lead to prosecution. Neither the administrators nor the Seminar for Applied Mathematics (SAM) accept any liability in this respect. The most recent version of a SAM report may differ in formatting and style from published journal version. Do reference the published version if possible (see SAM Publications).