Research reports

Years: 2025 2024 2023 2022 2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991

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}
}

Disclaimer
© Copyright for documents on this server remains with the authors. Copies of these documents made by electronic or mechanical means including information storage and retrieval systems, may only be employed for personal use. The administrators respectfully request that authors inform them when any paper is published to avoid copyright infringement. Note that unauthorised copying of copyright material is illegal and may lead to prosecution. Neither the administrators nor the Seminar for Applied Mathematics (SAM) accept any liability in this respect. The most recent version of a SAM report may differ in formatting and style from published journal version. Do reference the published version if possible (see SAM Publications).