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

by L. Herrmann and M. Keller and Ch. Schwab

(Report number 2019-41)

Abstract
We analyze rates of convergence for quasi-Monte Carlo (QMC) integration for Bayesian inversion of linear, elliptic PDEs 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 \emph{Besov priors} on the admissible uncertain inputs. We extend the QMC convergence theory from the Gaussian case, and in particular establish sufficient conditions on the uncertain inputs for achieving dimension-independent convergence rates \(>1/2\) of QMC integration with randomly shifted lattice rules. We apply the theory to a concrete class of linear, 2nd order elliptic boundary value problems with log-Besov uncertain diffusion coefficient.

Keywords: Quasi-Monte Carlo methods, Bayesian inverse problems, Besov priors, high-dimensional integration, elliptic partial differential equations

@Techreport{HKS19_845,
  author = {L. Herrmann and M. Keller and Ch. Schwab},
  title = {Quasi-Monte Carlo Bayesian estimation under Besov priors in elliptic inverse problems},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2019-41},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2019/2019-41.pdf },
  year = {2019}
}

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