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

Analytic regularity and polynomial approximation of stochastic, parametric elliptic multiscale PDEs

by V. H. Hoang and Ch. Schwab

(Report number 2011-07)

Abstract
A class of second order, elliptic PDEs in divergence form with stochastic and anisotropic conductivity coefficients and $n$ known, separated microscopic length scales $\epsilon_i$, $i=1,...,n$ in a bounded domain $D\subset R^d$ is considered. Neither stationarity nor ergodicity of these coefficients is assumed. Sufficient conditions are given for the random solution to converge $P$-a.s, as $\epsilon_i\rightarrow 0$, to a stochastic, elliptic one-scale limit problem in a tensorized domain of dimension $(n+1)d$. It is shown that this stochastic limit problem admits best $N$-term "polynomial chaos" type approximations which converge at a rate $\sigma>0$ that is determined by the summability of the random inputs' Karhúnen-Loève expansion. The convergence of the polynomial chaos expansion is shown to hold $P$-a.s. and uniformly with respect to the scale parameters $\epsilon_i$. Regularity results for the stochastic, one-scale limiting problem are established. An error bound for the approximation of the random solution at finite, positive values of the scale parameters $\epsilon_i$ is established in the case of two scales, and in the case of $n>2$ scales convergence is shown, albeit without giving a convergence rate in this case.

Keywords:

@Techreport{HS11_102,
  author = {V. H. Hoang and Ch. Schwab},
  title = {Analytic regularity and polynomial approximation of stochastic, parametric elliptic multiscale PDEs},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2011-07},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2011/2011-07.pdf },
  year = {2011}
}

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).