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

First order k-th moment finite element analysis of nonlinear operator equations with stochastic data

by A. Chernov and Ch. Schwab

(Report number 2011-51)

Abstract
We develop and analyze a class of efficient algorithms for uncertainty quan- tification of nonlinear operator equations. The algorithm are based on sparse Galerkin discretizations of tensorized linearizations at nominal parameters. Specifically, for a class of abstract nonlinear, parametric operator equations J(_,u) = 0 for random parameters _ with realizations in a neighborhood of a nominal parameter _0. Under some structural assumptions on the parame- ter dependence, by the implicit function theorem, J(_,u) = 0 admits locally unique solutions u=S(_) for all values _ in some neighborhood of _0. Random parameters_(!)=_0+r(!), are shown to imply a unique random solution u(!)=S(_(!)). We derive a multilinear, tensorized operator equation for the deterministic computation of k-th order statistical moments of the solution fluctuations u(!) - S(_0), provided that statistical moments of the random parameter perturbation r(!) are known. We present a sparse tensor Galerkin discretization for the tensorized first order perturbation equation. We prove a shift theorem for the k-point correlation equation in anisotropic smoothness scales and deduce that sparse tensor Galerkin discretizations of this equation converge in accuracy vs. complexity which equals, up to logarithmic terms, that of the Galerkin discretization of a single instance of the mean field problem. We illustrate the abstract theory for nonstationary parabolic diffusion problems in random domains. We verify Fréchet differentiability by means of shape calculus, and establish the Hadamard principle that the first order, k-th moment equation is completely specified in terms of data on the boundary of the nominal space-time cylinder. We perform boundary reduction of this parabolic evolution problem and propose a novel sparse tensor space-time Galerkin dis- cretization. In conjunction with the sparse tensor Galerkin approximation of the k-point correlation, it reduces the complexity of the Galerkin discretization to O(N(logN)k-1) where N denotes the number of degrees of freedom for a stationary problem on the boundary of the nominal domain (rather than on the space-time cylinder), thereby generalizing (25) to the boundary reduction of parabolic problems.

Keywords: Nonlinear operator equations, random parameters, deterministic methods, Fréchet derivative, sparse tensor approximation, random domain.

@Techreport{CS11_86,
  author = {A. Chernov and Ch. Schwab},
  title = {First order k-th moment finite element analysis of nonlinear operator equations with stochastic data },
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2011-51},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2011/2011-51.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).