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Higher order Quasi-Monte Carlo integration for Bayesian Estimation

by J. Dick and R. Gantner and Q. Le Gia and Ch. Schwab

(Report number 2016-13)

Abstract
We analyze combined Quasi-Monte Carlo quadrature and Finite Element approximations in Bayesian estimation of solutions to countably-parametric operator equations with holomorphic dependence on the parameters as considered in [Cl. Schillings and Ch. Schwab: Sparsity in Bayesian Inversion of Parametric Operator Equations. Inverse Problems, 30, (2014)]. Such problems arise in numerical uncertainty quantification and in Bayesian inversion of operator equations with distributed uncertain inputs, such as uncertain coefficients, uncertain domains or uncertain source terms and boundary data. We show that the parametric Bayesian posterior densities belong to a class of weighted Bochner spaces of functions of countably many variables, with a particular structure of the QMC quadrature weights: up to a (problem-dependent, and possibly large) finite dimension \(S\) product weights can be used, and beyond this dimension, weighted spaces with so-called SPOD weights, recently introduced in [J. Dick, F.Y. Kuo, Q.T. Le Gia, D. Nuyens and Ch. Schwab, Christoph Higher order QMC Petrov-Galerkin discretization for affine parametric operator equations with random field inputs. SIAM J. Numer. Anal. 52 (2014), 2676--2702.], are used to describe the solution regularity. We establish error bounds for higher order Quasi-Monte Carlo quadrature for the Bayesian estimation based on [J. Dick, Q.T. LeGia and Ch. Schwab, Higher order Quasi-Monte Carlo integration for holomorphic, parametric operator equations, Report 2014-23, SAM, ETH Zürich]. It implies, in particular, regularity of the parametric solution and of the countably-parametric Bayesian posterior density in SPOD weighted spaces. This, in turn, implies that the Quasi-Monte Carlo quadrature methods in [J. Dick, F.Y. Kuo, Q.T. Le Gia, D. Nuyens, Ch. Schwab, Higher order QMC Galerkin discretization for parametric operator equations, SINUM (2014)] are applicable to these problem classes, with dimension-independent convergence rates \(O(N^{-1/p})\) of \(N\)-point HoQMC approximated Bayesian estimates, where \(0 < p < 1\) depends only on the sparsity class of the uncertain input in the Bayesian estimation. Fast component-by-component (CBC for short) construction [R.N. Gantner and Ch. Schwab Computational Higher Order Quasi-Monte Carlo Integration, Report 2014-25, SAM, ETH Zürich] allow efficient Bayesian estimation with up to \(10^3\) parameters.

Keywords: Quasi-Monte Carlo, Lattice rules, digital nets, parametric operator equations, infinite-dimensional quadrature, Bayesian inverse problems, Uncertainty Quantification, CBC construction, SPOD weights.

@Techreport{DGLS16_650,
  author = {J. Dick and R. Gantner and Q. Le Gia and Ch. Schwab},
  title = {Higher order Quasi-Monte Carlo integration for Bayesian Estimation},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2016-13},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2016/2016-13.pdf },
  year = {2016}
}

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