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

Abstract

We analyze several Galerkin approximations of a Gaussian random field $\mathcal{Z}:\mathcal{D}\times \Omega \to \mathbb{R}$ indexed by a Euclidean domain $\mathcal{D}\subset {\mathbb{R}}^d$ whose covariance structure is determined by a negative fractional power $L^{-2\beta }$ of a second-order elliptic differential operator $L:=-\nabla \cdot (A\nabla )+{\kappa }^2$. Under minimal assumptions on the domain $\mathcal{D}$, the coefficients $A:\mathcal{D}\to {\mathbb{R}}^{d\times d}$, $\kappa :\mathcal{D}\to \mathbb{R}$, and the fractional exponent $\beta >0$, we prove convergence in $L_q(\Omega ;H^{\sigma }(\mathcal{D}))$ and in $L_q(\Omega ;C^{\delta }(\overline{\mathcal{D}}))$ at (essentially) optimal rates for (i) spectral Galerkin methods and (ii) finite element approximations. Specifically, our analysis is solely based on $H^{1+\alpha }(\mathcal{D})$-regularity of the differential operator $L$, where $0<\alpha \le 1$. For this setting, we furthermore provide rigorous estimates for the error in the covariance function of these approximations in $L_{\infty }(\mathcal{D}\times \mathcal{D})$ and in the mixed Sobolev space $H^{\sigma ,\sigma }(\mathcal{D}\times \mathcal{D})$, showing convergence which is more than twice as fast compared to the corresponding $L_q(\Omega ;H^{\sigma }(\mathcal{D}))$-rate. For the well-known example of such Gaussian random fields, the original Whittle-Matérn class, where $L=-\Delta +{\kappa }^2$ and $\kappa \equiv const.$, we perform several numerical experiments which validate our theoretical results.

41 pages, 2 figures

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