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Adaptive Galerkin approximation algorithms for partial differential equations in infinite dimensions

by Ch. Schwab and E. Süli

(Report number 2011-69)

Abstract
Space-time variational formulations of infinite-dimensional Fokker-Planck (FP) and Ornstein-Uhlenbeck (OU) equations for functions on a separable Hilbert space H are developed. The well-posedness of these equations in the Hilbert space L2 (H,u) of functions on H, which are square-integrable with respect to a Gaussian measure u on H, is proved. Specifically, for the infinite-dimensional FP equation, adaptive space-time Galerkin discretizations, based on a tensorized Riesz basis, built from biorthogonal piecewise polynomial wavelet bases in time and the Wiener-Hermite polynomial chaos in the Wiener-It^o decomposition of L2 (H,u), are introduced. The resulting space-time adaptive Wiener-Hermite polynomial Galerkin discretizations of the infinite-dimensional PDE are proved to converge quasioptimally in the sense that they produce sequences of finite-dimensional approximations which attain the best possible algebraic rates afforded by the tensor-products of multiresolution (wavelet) time-discretizations and of systems of tensorized Wiener–Hermite polynomial chaos expansions in L2 (H,u). Here, quasioptimality is understood with respect to the nonlinear, best N-term approximation benchmark of the solution. As a consequence, the proposed adaptive Galerkin solution algorithms exhibit dimension-independent performance, which is optimal with respect to the algebraic best N-term rate afforded by the solution and the regularity of the multiresolution (wavelet) time-discretizations in the finite-dimensional case, in particular. All constants in our error and complexity bounds are shown to be independent of the number of “active” coordinates identified by the proposed adaptive Galerkin approximation algorithms. The computational work and memory required by the proposed algorithms scale linearly with the support size of the coefficient vectors in the approximations, with dimension-independent constants.

Keywords:

@Techreport{SS11_148,
  author = {Ch. Schwab and E. S\"uli},
  title = {Adaptive Galerkin approximation algorithms for partial differential equations in infinite dimensions},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2011-69},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2011/2011-69.pdf },
  year = {2011}
}

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