The Mathematical Institute, University of Oxford, Eprints Archive

Adaptive Galerkin approximation algorithms for partial differential equations in infinite dimensions

Schwab, Christoph and Suli, Endre (2011) Adaptive Galerkin approximation algorithms for partial differential equations in infinite dimensions. Technical Report. Unspecified. (Submitted)

[img]
Preview
PDF
498Kb

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 $L^{2}(H,\mu)$ of functions on $H$, which are square-integrable with respect to a Gaussian measure $\mu$ 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 Hermite polynomial chaos in the Wiener-Itô decomposition of $L^{2}(H,\mu)$, are introduced and are shown to converge quasioptimally with respect to the nonlinear, best $N$-term approximation benchmark. As a consequence, the proposed adaptive Galerkin solution algorithms perform quasioptimally with respect to the best $N$-term approximation 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.

Item Type:Technical Report (Technical Report)
Subjects:H - N > Numerical analysis
Research Groups:Numerical Analysis Group
ID Code:1452
Deposited By:Lotti Ekert
Deposited On:02 Dec 2011 08:59
Last Modified:02 Dec 2011 08:59

Repository Staff Only: item control page