Quasi-Monte Carlo Bayesian estimation under Besov priors in elliptic inverse problems
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- by Lukas Herrmann, Magdalena Keller and Christoph Schwab;
- Math. Comp. 90 (2021), 1831-1860
- DOI: https://doi.org/10.1090/mcom/3615
- Published electronically: March 10, 2021
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Uncorrected version: Original version posted March 10, 2021
Corrected version: This paper was updated to correct typos introduced by the publisher.
Abstract:
We analyze rates of convergence for quasi-Monte Carlo (QMC) integration for Bayesian inversion of linear, elliptic partial differential equations with uncertain input from function spaces. Adopting a Riesz or Schauder basis representation of the uncertain inputs, function space priors are constructed as product measures on spaces of (sequences of) coefficients in the basis representations. The numerical approximation of the posterior expectation, given data, then amounts to a high- or infinite-dimensional numerical integration problem. We consider in particular so-called Besov priors on the admissible uncertain inputs. We extend the QMC convergence theory from the Gaussian case, and establish sufficient conditions on the uncertain inputs for achieving dimension-independent convergence rates greater than $1/2$ of QMC integration with randomly shifted lattice rules. We apply the theory to a concrete class of linear, second order elliptic boundary value problems with log-Besov uncertain diffusion coefficient.References
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Bibliographic Information
- Lukas Herrmann
- Affiliation: Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenbergerstrasse 69, 4040 Linz, Austria
- MR Author ID: 1247312
- ORCID: 0000-0003-3402-6420
- Email: lukas.herrmann@ricam.oeaw.ac.at
- Magdalena Keller
- Affiliation: Seminar for Applied Mathematics, ETH Zürich, Rämistrasse 101, CH-8092 Zürich, Switzerland
- Email: magdalena_keller@bluewin.ch
- Christoph Schwab
- Affiliation: Seminar for Applied Mathematics, ETH Zürich, Rämistrasse 101, CH-8092 Zürich, Switzerland
- MR Author ID: 305221
- Email: christoph.schwab@sam.math.ethz.ch
- Received by editor(s): August 5, 2019
- Received by editor(s) in revised form: August 7, 2020
- Published electronically: March 10, 2021
- Additional Notes: The authors acknowledge the computational resources provided by the EULER cluster of ETH Zürich (https://scicomp.ethz.ch/wiki/Euler). The first author acknowledges partial support by the Swiss National Science Foundation under grant SNF 159940.
- © Copyright 2021 by the authors
- Journal: Math. Comp. 90 (2021), 1831-1860
- MSC (2020): Primary 35R60, 62F15, 65M32; Secondary 65C05, 65N21, 65N30
- DOI: https://doi.org/10.1090/mcom/3615
- MathSciNet review: 4273117