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Dimension Truncation in QMC for Affine-Parametric Operator Equations

by R. N. Gantner

(Report number 2017-03)

Abstract
An application of quasi-Monte Carlo methods of significant recent interest in the MCQMC community is the quantification of uncertainties in partial differential equation models. Uncertainty quantification for both forward problems and Bayesian inverse problems leads to high-dimensional integrals that are well-suited for QMC approximation. One of the approximations required in a general formulation as an affine-parametric operator equation is the truncation of the formally infinite-parametric operator to a finite number of dimensions. To date, a numerical study of the available theoretical convergence rates for this error have to the author's knowledge not been published. We present novel results for a selection of model problems, the computation of which has been enabled by recently developed, higher-order QMC methods based on interlaced polynomial lattice rules. Surprisingly, the observed rates are one order better in the case of integration over the parameters than the commonly cited theory suggests; a proof of this higher rate is included, resulting in a theoretical statement consistent with the observed numerics.

Keywords: uncertainty quantification, affine-parametric operator equations, Bayesian Inversion, QMC, interlaced polynomial lattice rules

@Techreport{G17_699,
  author = {R. N. Gantner},
  title = {Dimension Truncation in QMC for Affine-Parametric Operator Equations},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2017-03},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2017/2017-03.pdf },
  year = {2017}
}

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