Abstract
We analyze several types of Galerkin approximations of a Gaussian random field mathscr {Z}:mathscr {D}times varOmega rightarrow mathbb {R} indexed by a Euclidean domain mathscr {D}subset mathbb {R}^d whose covariance structure is determined by a negative fractional power L^{-2beta } of a second-order elliptic differential operator L:= -nabla cdot (Anabla ) + kappa ^2. Under minimal assumptions on the domain mathscr {D}, the coefficients A:mathscr {D}rightarrow mathbb {R}^{dtimes d}, kappa :mathscr {D}rightarrow mathbb {R}, and the fractional exponent beta >0, we prove convergence in L_q(varOmega ; H^sigma (mathscr {D})) and in L_q(varOmega ; C^delta (overline{mathscr {D}})) at (essentially) optimal rates for (1) spectral Galerkin methods and (2) finite element approximations. Specifically, our analysis is solely based on H^{1+alpha }(mathscr {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 }(mathscr {D}times mathscr {D}) and in the mixed Sobolev space H^{sigma ,sigma }(mathscr {D}times mathscr {D}), showing convergence which is more than twice as fast compared to the corresponding L_q(varOmega ; H^sigma (mathscr {D}))-rate. We perform several numerical experiments which validate our theoretical results for (a) the original Whittle–Matérn class, where Aequiv mathrm {Id}_{mathbb {R}^d} and kappa equiv {text {const.}}, and (b) an example of anisotropic, non-stationary Gaussian random fields in d=2 dimensions, where A:mathscr {D}rightarrow mathbb {R}^{2times 2} and kappa :mathscr {D}rightarrow mathbb {R} are spatially varying.
Highlights
1.1 Motivation and backgroundBy virtue of their practicality owing to the full characterization by their mean and covariance structure, Gaussian random fields (GRFs for short) are popular models for many applications in spatial statistics and uncertainty quantification, e.g., [4,7,19,32,39,41]
Several methodologies in these disciplines require the efficient simulation of GRFs at unstructured locations in various possibly non-convex Euclidean domains, and this topic has been intensively discussed in both areas, spatial statistics and computational mathematics, see, e.g., [3,8,14,18,21,28,31,36]
Note that the covariance structure of a GRF is uniquely determined by its covariance operator, in this case given by the negative fractional-order differential operator L−2β
Summary
By virtue of their practicality owing to the full characterization by their mean and covariance structure, Gaussian random fields (GRFs for short) are popular models for many applications in spatial statistics and uncertainty quantification, e.g., [4,7,19,32,39,41]. A computational approach which allows for arbitrary fractional exponents β > d/4 has been suggested in [2,3] To this end, a sinc quadrature combined with a Galerkin discretization of the differential operator L is applied to the Balakrishnan integral representation of the fractional-order inverse L−β. We remark that strong convergence of the sinc-Galerkin approximation with respect to the L2(Ω; L2(D))-norm, i.e., (3) for σ = 0, at the rate 2β − d/2 has already been proven in [3, Theorem 2.10]. Our results do generalize the analysis of [3] for the strong error to different norms, and to less regular differential operators This is of relevance for several practical applications, since the spatial domain, where the GRF is simulated, may be non-convex or the coefficient A may have jumps.
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