Adaptive Galerkin approximation algorithms for Kolmogorov equations in infinite dimensions (Q373235)

The authors propose an adaptive spectral-Galerkin approximation for the finite-dimensional approximation to forward Kolmogorov (or Fokker-Planck) equations in infinite-dimensional spaces (which are connected to parabolic stochastic partial differential equations). To this end the authors develop (i) a space-time variational formulation and (ii) adaptive Wiener-Hermite polynomial chaos Galerkin discretizations of such equations. Then, also (iii) the well-posedness of these equations in the Hilbert space of square-integrable functions (w.r.t. some Gaussian measure) on an infinite-dimensional Hilbert space is established. The general results are then applied to obtain an approximation based on a wavelet polynomial chaos Riesz basis and the resulting approximations are (iv) proved to converge in a quasioptimal sense, i.e., they produce sequences of finite dimensional approximations which possess the optimal convergence rates afforded by best \(N\)-term approximations. Therefore, it is inferred that the performance of the algorithm is independent of the dimension.