Quasi-Monte Carlo Bayesian estimation under Besov priors in elliptic inverse problems

2020/11/30 by Lukas Herrmann, Magdalena Keller, Christoph Schwab · 15 citations
Decision Sciences · Mathematics · #Applied mathematics #Bayesian probability #Computer science #Hybrid Monte Carlo #Markov chain Monte Carlo #Mathematical Approximation and Integration #Mathematical analysis #Mathematical optimization #Mathematics #Monte Carlo method #Prior probability #Probabilistic and Robust Engineering Design #Quasi-Monte Carlo method #Rate of convergence #Statistical Methods and Inference #Statistics

paper · doi:10.1090/mcom/3615

published in Mathematics of Computation 90(330), 1831-1860 (American Mathematical Society)

openalex publication_date 2020/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06

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 <italic>Besov priors</italic> 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 slash 2"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">1/2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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.

Citations

• A Multivariate Faa di Bruno Formula with Applications

Cited by

• Parabolic PDE-constrained optimal control under uncertainty with entropic risk measure using quasi-Monte Carlo integration
• Generalized Dimension Truncation Error Analysis for High-Dimensional Numerical Integration: Lognormal Setting and Beyond
• On the optimality of dimension truncation error rates for a class of parametric partial differential equations
• Quasi-Monte Carlo for Bayesian design of experiment problems governed by parametric PDEs
• A median QMC method for unbounded integrands over \mathbb R^s in weighted unanchored Sobolev spaces
• Approximation of differential entropy in Bayesian optimal experimental design
• A localized consensus-based sampling algorithm

Related

• Quasi-Monte Carlo and Multilevel Monte Carlo Methods for Computing Posterior Expectations in Elliptic Inverse Problems
• Multilevel higher order Quasi-Monte Carlo Bayesian Estimation
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• Besov priors for Bayesian inverse problems