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Multilevel quasi-Monte Carlo uncertainty quantification for advection-diffusion-reaction

by L. Herrmann and Ch. Schwab

(Report number 2019-06)

Abstract
We survey the numerical analysis of a class of deterministic, higher-order QMC integration methods in forward and inverse uncertainty quantification algorithms for advection-reaction-diffusion (ARD) equations in polygonal domains $D\subset \mathbb{R}^2$ with distributed uncertain inputs. We admit spatially heterogeneous material properties. For the parametrization of the uncertainty, we assume at hand systems of functions which are locally supported in $D$. Distributed uncertain inputs are written in countably parametric, deterministic form with locally supported representation systems. Parametric regularity and sparsity of solution families and of response functions in scales of weighted Kontrat'ev spaces in $D$ is quantified using analytic continuation.

Keywords: Higher order quasi-Monte Carlo, parametric operator equations, Bayesian inverse problems, uncertainty quantification

@Techreport{HS19_810,
  author = {L. Herrmann and Ch. Schwab},
  title = {Multilevel quasi-Monte Carlo uncertainty quantification for advection-diffusion-reaction},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2019-06},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2019/2019-06.pdf },
  year = {2019}
}

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