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

Sparse MCMC gpc Finite Element Methods for Bayesian Inverse Problems

by V. H. Hoang and Ch. Schwab and A. M. Stuart

(Report number 2012-23)

Abstract
Several classes of MCMC methods for the numerical solution of Bayesian Inverse Problems for partial differential equations (PDEs) with unknown random field coefficients are considered. A general framework for their numerical analysis is presented. The complexity of MCMC sampling for the unknown fields from the posterior density, as well as the convergence of the discretization error of the PDE of interest in the forward response map, is analyzed. Particular attention is given to bounds on the overall work required by the MCMC algorithms for achieving a prescribed error level E. We show that the computational complexity of straightforward combinations of MCMC sampling strategies with standard PDE solution methods is generally excessive. Two computational strategies for substantially reducing the computational complexity of MCMC methods for Bayesian inverse problems prising in PDEs are studied: a parametric, deterministic gpc-type (generalized polynomial chaos) representation of the forward solution map of the PDE with uncertain coefficients, which has been proposed and implemented in the engineering literature (e.g. [17, 15, 16]); and a new Multi-Level Monte Carlo sampling strategy of the Markov Chain (MLMCMC) with sampling from a multilevel discretization of the posterior and a multilevel discretization of the forward PDE. We compare the computational complexity of these gpc-MCMC and MLMCMC methods to that of the plain MCMC method, and provide sufficient conditions on the regularity of the unknown coefficient for both, the gpc-MCMC and MLMCMC method, to afford substantial complexity reductions over the plain MCMC approach.

Keywords:

@Techreport{HSS12_466,
  author = {V. H. Hoang and Ch. Schwab and A. M. Stuart},
  title = {Sparse MCMC gpc Finite Element Methods for Bayesian Inverse Problems},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2012-23},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2012/2012-23.pdf },
  year = {2012}
}

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).