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Optimality of Adaptive Galerkin methods for random parabolic partial differential equations

by C. Gittelson and R. Andreev and Ch. Schwab

(Report number 2013-09)

Abstract
Galerkin discretizations of a class of parametric and random parabolic partial differential equations (PDEs) are considered.The parabolic PDEs are assumed to depend on a vector \(y=(y_1,y_2,...)\) of possibly countably many parameters \(y_j\) which are assumed to take values in \([-1,1]\). Well-posedness of weak formulations of these parametric equation in suitable Bochner spaces is established. Adaptive Galerkin discretizations of the equation based on a tensor product of a generalized polynomial chaos in the parameter domain \(\Gamma = [-1,1]^\mathbb{N}\), and of suitable wavelet bases in the time interval \(I=[0,T]\) and the spatial domain \(D\subset \mathbb{R}^d\) are proposed and their optimality is established.

Keywords: partial differential equations with random coefficients, parabolic differential equations, uncertainty quantification, stochastic finite element methods, operator equations, adaptive methods, wavelets

@Techreport{GAS13_505,
  author = {C. Gittelson and R. Andreev and Ch. Schwab},
  title = {Optimality of Adaptive Galerkin methods for random parabolic partial differential equations},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2013-09},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2013/2013-09.pdf },
  year = {2013}
}

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