Multilevel Hierarchical Decomposition of Finite Element White Noise with Application to Multilevel Markov Chain Monte Carlo
Abstract
In this work we develop a new hierarchical multilevel approach to generate Gaussian random field realizations in an algorithmically scalable manner that is well suited to incorporating into multilevel Markov chain Monte Carlo (MCMC) algorithms. This approach builds off of other partial differential equation (PDE) approaches for generating Gaussian random field realizations; in particular, a single field realization may be formed by solving a reaction-diffusion PDE with a spatial white noise source function as the right-hand side. While these approaches have been explored to accelerate forward uncertainty quantification tasks, e.g., multilevel Monte Carlo, the previous constructions are not directly applicable to multilevel MCMC frameworks which build fine-scale random fields in a hierarchical fashion from coarse-scale random fields. Our new hierarchical multilevel method relies on a hierarchical decomposition of the white noise source function in $L^2$ which allows us to form Gaussian random field realizations across multiple levels of discretization in a way that fits into multilevel MCMC algorithmic frameworks. After presenting our main theoretical results and numerical scaling results to showcase the utility of this new hierarchical PDE method for generating Gaussian random field realizations, this method is tested on a four-level MCMC algorithm to explore its feasibility.
- Authors:
-
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
- Washington Univ., St. Louis, MO (United States)
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States); Portland State Univ., OR (United States)
- Publication Date:
- Research Org.:
- Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)
- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA)
- OSTI Identifier:
- 1843111
- Report Number(s):
- LLNL-JRNL-820098
Journal ID: ISSN 1064-8275; 1031374
- Grant/Contract Number:
- AC52-07NA27344
- Resource Type:
- Accepted Manuscript
- Journal Name:
- SIAM Journal on Scientific Computing
- Additional Journal Information:
- Journal Volume: 43; Journal Issue: 5; Journal ID: ISSN 1064-8275
- Publisher:
- Society for Industrial and Applied Mathematics (SIAM)
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 97 MATHEMATICS AND COMPUTING; Gaussian random field; nonlinear Bayesian inference; Markov chain Monte Carlo; multilevel Markov chain Monte Carlo; high-dimensional uncertainty quantification; algebraic multigrid
Citation Formats
Fairbanks, Hillary R., Villa, Umberto, and Vassilevski, Panayot S. Multilevel Hierarchical Decomposition of Finite Element White Noise with Application to Multilevel Markov Chain Monte Carlo. United States: N. p., 2021.
Web. doi:10.1137/20m1349606.
Fairbanks, Hillary R., Villa, Umberto, & Vassilevski, Panayot S. Multilevel Hierarchical Decomposition of Finite Element White Noise with Application to Multilevel Markov Chain Monte Carlo. United States. https://doi.org/10.1137/20m1349606
Fairbanks, Hillary R., Villa, Umberto, and Vassilevski, Panayot S. Tue .
"Multilevel Hierarchical Decomposition of Finite Element White Noise with Application to Multilevel Markov Chain Monte Carlo". United States. https://doi.org/10.1137/20m1349606. https://www.osti.gov/servlets/purl/1843111.
@article{osti_1843111,
title = {Multilevel Hierarchical Decomposition of Finite Element White Noise with Application to Multilevel Markov Chain Monte Carlo},
author = {Fairbanks, Hillary R. and Villa, Umberto and Vassilevski, Panayot S.},
abstractNote = {In this work we develop a new hierarchical multilevel approach to generate Gaussian random field realizations in an algorithmically scalable manner that is well suited to incorporating into multilevel Markov chain Monte Carlo (MCMC) algorithms. This approach builds off of other partial differential equation (PDE) approaches for generating Gaussian random field realizations; in particular, a single field realization may be formed by solving a reaction-diffusion PDE with a spatial white noise source function as the right-hand side. While these approaches have been explored to accelerate forward uncertainty quantification tasks, e.g., multilevel Monte Carlo, the previous constructions are not directly applicable to multilevel MCMC frameworks which build fine-scale random fields in a hierarchical fashion from coarse-scale random fields. Our new hierarchical multilevel method relies on a hierarchical decomposition of the white noise source function in $L^2$ which allows us to form Gaussian random field realizations across multiple levels of discretization in a way that fits into multilevel MCMC algorithmic frameworks. After presenting our main theoretical results and numerical scaling results to showcase the utility of this new hierarchical PDE method for generating Gaussian random field realizations, this method is tested on a four-level MCMC algorithm to explore its feasibility.},
doi = {10.1137/20m1349606},
journal = {SIAM Journal on Scientific Computing},
number = 5,
volume = 43,
place = {United States},
year = {Tue Jun 08 00:00:00 EDT 2021},
month = {Tue Jun 08 00:00:00 EDT 2021}
}
Works referenced in this record:
Dimension-independent likelihood-informed MCMC
journal, January 2016
- Cui, Tiangang; Law, Kody J. H.; Marzouk, Youssef M.
- Journal of Computational Physics, Vol. 304
Markov chain Monte Carlo Using an Approximation
journal, December 2005
- Christen, J. Andrés; Fox, Colin
- Journal of Computational and Graphical Statistics, Vol. 14, Issue 4
Analysis of Boundary Effects on PDE-Based Sampling of Whittle--Matérn Random Fields
journal, January 2019
- Khristenko, U.; Scarabosio, L.; Swierczynski, P.
- SIAM/ASA Journal on Uncertainty Quantification, Vol. 7, Issue 3
Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients
journal, January 2011
- Cliffe, K. A.; Giles, M. B.; Scheichl, R.
- Computing and Visualization in Science, Vol. 14, Issue 1
Analysis of Circulant Embedding Methods for Sampling Stationary Random Fields
journal, January 2018
- Graham, I. G.; Kuo, F. Y.; Nuyens, D.
- SIAM Journal on Numerical Analysis, Vol. 56, Issue 3
MFEM: A modular finite element methods library
journal, January 2021
- Anderson, Robert; Andrej, Julian; Barker, Andrew
- Computers & Mathematics with Applications, Vol. 81
A Multilevel, Hierarchical Sampling Technique for Spatially Correlated Random Fields
journal, January 2017
- Osborn, Sarah; Vassilevski, Panayot S.; Villa, Umberto
- SIAM Journal on Scientific Computing, Vol. 39, Issue 5
Mitigating the influence of the boundary on PDE-based covariance operators
journal, January 2018
- Daon, Yair; Stadler, Georg
- Inverse Problems & Imaging, Vol. 12, Issue 5
A Computational Framework for Infinite-Dimensional Bayesian Inverse Problems Part I: The Linearized Case, with Application to Global Seismic Inversion
journal, January 2013
- Bui-Thanh, Tan; Ghattas, Omar; Martin, James
- SIAM Journal on Scientific Computing, Vol. 35, Issue 6
The egg model - a geological ensemble for reservoir simulation
journal, November 2014
- Jansen, J. D.; Fonseca, R. M.; Kahrobaei, S.
- Geoscience Data Journal, Vol. 1, Issue 2
A Computational Framework for Infinite-Dimensional Bayesian Inverse Problems, Part II: Stochastic Newton MCMC with Application to Ice Sheet Flow Inverse Problems
journal, January 2014
- Petra, Noemi; Martin, James; Stadler, Georg
- SIAM Journal on Scientific Computing, Vol. 36, Issue 4
A Stochastic Newton MCMC Method for Large-Scale Statistical Inverse Problems with Application to Seismic Inversion
journal, January 2012
- Martin, James; Wilcox, Lucas C.; Burstedde, Carsten
- SIAM Journal on Scientific Computing, Vol. 34, Issue 3
Fast Algorithms for Bayesian Uncertainty Quantification in Large-Scale Linear Inverse Problems Based on Low-Rank Partial Hessian Approximations
journal, January 2011
- Flath, H. P.; Wilcox, L. C.; Akçelik, V.
- SIAM Journal on Scientific Computing, Vol. 33, Issue 1
MCMC Methods for Functions: Modifying Old Algorithms to Make Them Faster
journal, August 2013
- Cotter, S. L.; Roberts, G. O.; Stuart, A. M.
- Statistical Science, Vol. 28, Issue 3
Algebraic Hybridization and Static Condensation with Application to Scalable $H$(div) Preconditioning
journal, January 2019
- Dobrev, V.; Kolev, T.; Lee, C. S.
- SIAM Journal on Scientific Computing, Vol. 41, Issue 3
A Hierarchical Multilevel Markov Chain Monte Carlo Algorithm with Applications to Uncertainty Quantification in Subsurface Flow
journal, January 2015
- Dodwell, T. J.; Ketelsen, C.; Scheichl, R.
- SIAM/ASA Journal on Uncertainty Quantification, Vol. 3, Issue 1
Efficient White Noise Sampling and Coupling for Multilevel Monte Carlo with Nonnested Meshes
journal, January 2018
- Croci, M.; Giles, M. B.; Rognes, M. E.
- SIAM/ASA Journal on Uncertainty Quantification, Vol. 6, Issue 4
Whittle-Matérn priors for Bayesian statistical inversion with applications in electrical impedance tomography
journal, January 2014
- Roininen, Lassi; M. J. Huttunen, Janne; Lasanen, Sari
- Inverse Problems & Imaging, Vol. 8, Issue 2
Analysis of a multilevel Markov chain Monte Carlo finite element method for Bayesian inversion of log-normal diffusions
journal, February 2020
- Hoang, Viet Ha; Quek, Jia Hao; Schwab, Christoph
- Inverse Problems, Vol. 36, Issue 3
Multilevel Monte Carlo Path Simulation
journal, June 2008
- Giles, Michael B.
- Operations Research, Vol. 56, Issue 3
Scalable hierarchical PDE sampler for generating spatially correlated random fields using nonmatching meshes: Scalable hierarchical PDE sampler using nonmatching meshes
journal, January 2018
- Osborn, Sarah; Zulian, Patrick; Benson, Thomas
- Numerical Linear Algebra with Applications, Vol. 25, Issue 3
Further analysis of multilevel Monte Carlo methods for elliptic PDEs with random coefficients
journal, March 2013
- Teckentrup, A. L.; Scheichl, R.; Giles, M. B.
- Numerische Mathematik, Vol. 125, Issue 3
Geometric MCMC for infinite-dimensional inverse problems
journal, April 2017
- Beskos, Alexandros; Girolami, Mark; Lan, Shiwei
- Journal of Computational Physics, Vol. 335
Solving large-scale PDE-constrained Bayesian inverse problems with Riemann manifold Hamiltonian Monte Carlo
journal, October 2014
- Bui-Thanh, T.; Girolami, M.
- Inverse Problems, Vol. 30, Issue 11
Likelihood-informed dimension reduction for nonlinear inverse problems
journal, October 2014
- Cui, T.; Martin, J.; Marzouk, Y. M.
- Inverse Problems, Vol. 30, Issue 11
Preconditioning Markov Chain Monte Carlo Simulations Using Coarse-Scale Models
journal, January 2006
- Efendiev, Y.; Hou, T.; Luo, W.
- SIAM Journal on Scientific Computing, Vol. 28, Issue 2
An explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach: Link between Gaussian Fields and Gaussian Markov Random Fields
journal, August 2011
- Lindgren, Finn; Rue, Håvard; Lindström, Johan
- Journal of the Royal Statistical Society: Series B (Statistical Methodology), Vol. 73, Issue 4
Multi-level Monte Carlo Finite Element method for elliptic PDEs with stochastic coefficients
journal, April 2011
- Barth, Andrea; Schwab, Christoph; Zollinger, Nathaniel
- Numerische Mathematik, Vol. 119, Issue 1
Equation of State Calculations by Fast Computing Machines
journal, June 1953
- Metropolis, Nicholas; Rosenbluth, Arianna W.; Rosenbluth, Marshall N.
- The Journal of Chemical Physics, Vol. 21, Issue 6
Monte Carlo sampling methods using Markov chains and their applications
journal, April 1970
- Hastings, W. K.
- Biometrika, Vol. 57, Issue 1
The multi-level Monte Carlo finite element method for a stochastic Brinkman Problem
journal, March 2013
- Gittelson, Claude J.; Könnö, Juho; Schwab, Christoph
- Numerische Mathematik, Vol. 125, Issue 2
Scheduling Massively Parallel Multigrid for Multilevel Monte Carlo Methods
journal, January 2017
- Drzisga, D.; Gmeiner, B.; Rüde, U.
- SIAM Journal on Scientific Computing, Vol. 39, Issue 5
Taylor approximation and variance reduction for PDE-constrained optimal control under uncertainty
text, January 2018
- Chen, Peng; Villa, Umberto; Ghattas, Omar
- arXiv