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

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 - 2 β of a second-order elliptic differential operator L : = - ∇ · (A ∇) + κ 2 . 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 ≡ Id R d 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. [ABSTRACT FROM AUTHOR]