Research reports

Years: 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

Convergence rates of high dimensional Smolyak quadrature

by J. Zech and Ch. Schwab

(Report number 2017-27)

Abstract
We analyse convergence rates of Smolyak integration for parametric maps $u:U\to X$ taking values in a Banach space $X$, defined on the parameter domain $U=[-1,1]^{\mathbb{N}}$. For parametric maps which are sparse, as quantified by summability of their Taylor polynomial chaos coefficients, dimension-independent convergence rates superior to $N$-term approximation rates under the same sparsity are achievable. We propose a concrete Smolyak algorithm to apriori identify integrand-adapted sets of active multiindices (and thereby unisolvent sparse grids of quadrature points) via upper bounds for the integrands' Taylor gpc coefficients. For so-called ``$(\mathbf{b},\varepsilon)$-holomorphic'' integrands $u$ with $\mathbf{b}\in\ell^p(\mathbb{N})$ for some $p\in(0,1)$, we prove the dimension-independent convergence rate $2/p-1$ in terms of the number of quadrature points. The proposed Smolyak algorithm is proved to yield (essentially) the same rate in terms of the total computational cost for both nested and non-nested univariate quadrature points. Numerical experiments and a mathematical sparsity analysis accounting for cancellations in quadratures and in the combination formula demonstrate that the asymptotic rate $2/p-1$ is realized computationally for a moderate number of quadrature points under certain circumstances. By a refined analysis of model integrand classes we show that a generally large preasymptotic range otherwise precludes reaching the asymptotic rate $2/p-1$ for practically relevant numbers of quadrature points.

Keywords: generalized polynomial chaos, Smolyak Quadrature, sparsity, holomorphy

@Techreport{ZS17_723,
  author = {J. Zech and Ch. Schwab},
  title = {Convergence rates of high dimensional Smolyak quadrature},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2017-27},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2017/2017-27.pdf },
  year = {2017}
}

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).