We analyze several types of Galerkin approximations of a Gaussian random field Z: D× Ω→ R indexed by a Euclidean domain D⊂ R^d whose covariance structure is determined by a negative fractional power L^-2β of a second-order elliptic differential operator L: = - ∇ · (A∇) + κ². Under minimal assumptions on the domain D, the coefficients A: D→ R^d×d, κ: D→ R, and the fractional exponent β> 0 , we prove convergence in L_q(Ω; H^σ(D)) and in L_q(Ω; C^δ(D¯)) at (essentially) optimal rates for (1) spectral Galerkin methods and (2) finite element approximations. Specifically, our analysis is solely based on H^1+α(D) -regularity of the differential operator L, where 0 < α≤ 1. For this setting, we furthermore provide rigorous estimates for the error in the covariance function of these approximations in L_∞(D× D) and in the mixed Sobolev space H^σ,σ(D× D) , showing convergence which is more than twice as fast compared to the corresponding L_q(Ω; H^σ(D)) -rate. We perform several numerical experiments which validate our theoretical results for (a) the original Whittle–Matérn class, where A≡IdRd and κ≡ const. , and (b) an example of anisotropic, non-stationary Gaussian random fields in d= 2 dimensions, where A: D→ R^{2 × 2} and κ: D→ R are spatially varying.

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We show that if a sequence of piecewise affine linear processes converges in the strong sense with a positive rate to a stochastic process that is strongly Hölder continuous in time, then this sequence converges in the strong sense even with respect to much stronger Hölder norms and the convergence rate is essentially reduced by the Hölder exponent. Our first application hereof establishes pathwise convergence rates for spectral Galerkin approximations of stochastic partial differential equations. Our second application derives strong convergence rates of multilevel Monte Carlo approximations of expectations of Banach-space-valued stochastic processes.@en