Regularity and convergence analysis in Sobolev and Hölder spaces for generalized Whittle-Matérn fields
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We analyze several Galerkin approximations of a Gaussian random field ZD×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∇) + κ^2. Under minimal assumptions on the domain D, the coefficients AD→R^d× d, κD→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 (i) spectral Galerkin methods and (ii) 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. For the well-known example of such Gaussian random fields, the original Whittle-Matérn class, where L=-Δ + κ^2 and κ≡const., we perform several numerical experiments which validate our theoretical results.
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