Regularity and convergence analysis in Sobolev and Hölder spaces for generalized Whittle–Matérn fields

Abstract

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.