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 gpc approximation for parametric and random 2nd order hyperbolic PDEs
by Ch. Schwab and V. H. Hoang
(Report number 2010-19)
Abstract
Initial boundary value problems of linear second order hyperbolic partial differential equations whose coefficients depend on countably many random parameters are reduced to a parametric family of deterministic initial boundary value problems on an infinite dimensional parameter space. This parametric family is approximated by Galerkin projection onto finitely supported polynomial systems in the parameter space. We establish uniform stability with respect to the support of the resulting coupled hyperbolic systems, and {provide sufficient smoothness and compatibility conditions on the data for the solution to exhibit analytic respectively Gevrey regularity with respect to the countably many parameters. Sufficient conditions for the $p$-summability of the generalized polynomial chaos expansion of the parametric solution in terms of the countably many input parameters are obtained and rates of convergence of best $N$-term polynomial chaos type approximations of the parametric solution are given. In addition, regularity both in space and time for the parametric family of solutions is proved for data satisfying certain compatibility conditions. The results allow obtaining convergence rates and stability of sparse space-time tensor product Galerkin discretizations in the parameter space.
Keywords: Wave Equation, generalized polynomial chaos, random media, best N-term approximation
BibTeX@Techreport{SH10_75, author = {Ch. Schwab and V. H. Hoang}, title = {Analytic regularity and gpc approximation for parametric and random 2nd order hyperbolic PDEs}, institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich}, number = {2010-19}, address = {Switzerland}, url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2010/2010-19.pdf }, year = {2010} }
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).