Abstract
In this paper we address the problem of the prohibitively large computational cost of existing Markov chain Monte Carlo methods for large-scale applications with high-dimensional parameter spaces, e.g., in uncertainty quantification in porous media flow. We propose a new multilevel Metropolis-Hastings algorithm and give an abstract, problem-dependent theorem on the cost of the new multilevel estimator based on a set of simple, verifiable assumptions. For a typical model problem in subsurface flow, we then provide a detailed analysis of these assumptions and show significant gains over the standard Metropolis-Hastings estimator. Numerical experiments confirm the analysis and demonstrate the effectiveness of the method with consistent reductions of more than an order of magnitude in the cost of the multilevel estimator over the standard Metropolis-Hastings algorithm for tolerances ε < 10 - 2.
| Original language | English |
|---|---|
| Pages (from-to) | 509-545 |
| Number of pages | 37 |
| Journal | Siam Review |
| Volume | 61 |
| Issue number | 3 |
| Early online date | 7 Aug 2019 |
| DOIs | |
| Publication status | Published - 2019 |
Funding
\ast Published electronically August 7, 2019. This paper originally appeared in SIAM/ASA Journal on Uncertainty Quantification, Volume 3, 2015, pages 1075--1108, under the title ``A Hierarchical Multilevel Markov Chain Monte Carlo Algorithm with Applications to Uncertainty Quantification in Subsurface Flow."" https://doi.org/10.1137/19M126966X Funding: Part of this work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract DE-AC52-07A27344 and released under LLNL-JRNL-630212. \dagger Institute of Data Science and Artificial Intelligence, College of Engineering Mathematics and Physical Sciences, University of Exeter, Exeter EX4 4QF, UK, and The Alan Turing Institute, London NW1 2DB, UK ([email protected]). \ddagger Maxar Technologies, Westminster, CO 80234 ([email protected]). \S Institut for Applied Mathematics and Interdisciplinary Center for Scientific Computing, Heidelberg University, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany ([email protected]). \P University of Edinburgh, School of Mathematics, James Clerk Maxwell Building, Edinburgh EH9 3FD, UK ([email protected]).
Keywords
- Bayesian approach
- Elliptic PDEs with random coefficients
- Finite element analysis
- Log-normal coefficients
- Metropolis-Hastings algorithm
- Multilevel Monte Carlo
ASJC Scopus subject areas
- Theoretical Computer Science
- Computational Mathematics
- Applied Mathematics