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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

@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}
}

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