Adaptive Galerkin approximation algorithms for Kolmogorov equations in infinite dimensions

Space-time variational formulations and adaptive Wiener–Hermite polynomial chaos Galerkin discretizations of Kolmogorov equations in infinite dimensions, such as Fokker–Planck andOrnstein–Uhlenbeck equations for functions defined on an infinite-dimensional separable Hilbert space H , are developed. The wellposedness of these equations in the Hilbert space L2(H, μ) of functions on the infinite-dimensional domain H , which are square-integrablewith respect to aGaussian measure μ with trace class covariance operator Q on H , is proved. Specifically, for the infinite-dimensional Fokker–Planck equation, adaptive space-time Galerkin discretizations, based on a wavelet polynomial chaos Riesz basis obtained by tensorization of biorthogonal piecewise polynomial wavelet bases in time with a spatial Wiener–Hermite polynomial chaos arising from the Wiener–Itô decomposition of L2(H, μ), are introduced. The resulting space-time adaptiveWiener–Hermite polynomial Galerkin discretization algorithms of the infinite-dimensional PDE are proved to converge quasioptimally in the sense that they produce sequences of finite-dimensional approximations that attain the best possible convergence rates afforded by best N -term approximations of the solution from tensor-products of multiresolution (wavelet) time-discretizations and theWiener–Hermite polynomial chaos inL2(H, μ). 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 polynomial degree and regularity C. Schwab (B) Seminar for Applied Mathematics (SAM), ETH Zurich, HG G57.1, CH 8092 Zurich, Switzerland e-mail: [email protected] E. Süli Mathematical Institute, University of Oxford, 24–29 St Giles, Oxford OX1 3LB, UK e-mail: [email protected]