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Uncertainty Quantification for Spectral Fractional Diffusion: Sparsity Analysis of Parametric Solutions

by L. Herrmann and Ch. Schwab and J. Zech

(Report number 2018-11)

Abstract
In bounded, polygonal domains \({\rm D} \subset \mathbb{R}^d\), we analyze solution regularity and sparsity for computational uncertainty quantification for spectral fractional diffusion. Two types of uncertainty are considered: i) uncertain, parametric diffusion coefficients, and ii) uncertain physical domains \({\rm D}\). For either of these problem classes, we analyze sparsity of countably-parametric solution families. Principal novel technical contribution of the present paper is a sparsity analysis for operator equations with distributed uncertain inputs which, in particular, may be given as a general gpc representation, generalizing earlier results which required an affine-parametric representation. The summability results established here imply best \(N\)-term approximation rate bounds as well as dimension-independent convergence rates of numerical approximation methods such as stochastic collocation, Smolyak and Quasi-Monte Carlo integration methods and compressed sensing or least-squares approximations.

Keywords: Fractional diffusion, nonlocal operators, uncertainty quantification, sparsity, generalized polynomial chaos

@Techreport{HSZ18_765,
  author = {L. Herrmann and Ch. Schwab and J. Zech},
  title = {Uncertainty Quantification for Spectral Fractional Diffusion: Sparsity Analysis of Parametric Solutions},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2018-11},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2018/2018-11.pdf },
  year = {2018}
}

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