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 $$\mathscr {Z}:\mathscr {D}\times \varOmega \rightarrow \mathbb {R}$$ Z : D × Ω → R indexed by a Euclidean domain $$\mathscr {D}\subset \mathbb {R}^d$$ D ⊂ R d whose covariance structure is determined by a negative fractional power $$L^{-2\beta }$$ L - 2 β of a second-order elliptic differential operator $$L:= -\nabla \cdot (A\nabla ) + \kappa ^2$$ L : = - ∇ · ( A ∇ ) + κ 2 . Under minimal assumptions on the domain $$\mathscr {D}$$ D , the coefficients $$A:\mathscr {D}\rightarrow \mathbb {R}^{d\times d}$$ A : D → R d × d , $$\kappa :\mathscr {D}\rightarrow \mathbb {R}$$ κ : D → R , and the fractional exponent $$\beta >0$$ β > 0 , we prove convergence in $$L_q(\varOmega ; H^\sigma (\mathscr {D}))$$ L q ( Ω ; H σ ( D ) ) and in $$L_q(\varOmega ; C^\delta (\overline{\mathscr {D}}))$$ 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 $$H^{1+\alpha }(\mathscr {D})$$ H 1 + α ( D ) -regularity of the differential operator L, where $$0<\alpha \le 1$$ 0 < α ≤ 1 . For this setting, we furthermore provide rigorous estimates for the error in the covariance function of these approximations in $$L_{\infty }(\mathscr {D}\times \mathscr {D})$$ L ∞ ( D × D ) and in the mixed Sobolev space $$H^{\sigma ,\sigma }(\mathscr {D}\times \mathscr {D})$$ H σ , σ ( D × D ) , showing convergence which is more than twice as fast compared to the corresponding $$L_q(\varOmega ; H^\sigma (\mathscr {D}))$$ 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 $$A\equiv \mathrm {Id}_{\mathbb {R}^d}$$ A ≡ Id R d and $$\kappa \equiv {\text {const.}}$$ κ ≡ const. , and (b) an example of anisotropic, non-stationary Gaussian random fields in $$d=2$$ d = 2 dimensions, where $$A:\mathscr {D}\rightarrow \mathbb {R}^{2\times 2}$$ A : D → R 2 × 2 and $$\kappa :\mathscr {D}\rightarrow \mathbb {R}$$ κ : D → R are spatially varying.

Finite Element Representations of Gaussian Processes: Balancing Numerical and Statistical Accuracy

“…∞ for a particular FE discretization when 𝒟 is a hyperrectangle. The result holds for general dimension 𝐷 while being less sharp than the one-dimensional result in [7], which also allows for more general domains. However, our main objective is to illustrate that the plug-in character of the framework [39] allows to seamlessly translate 𝐿 ∞ and 𝐿 2 error bounds for the approximation 𝑢 ℎ ≈ 𝑢 into sufficient choices of 𝑛 ℎ in terms of 𝑁 in regression and classification settings.…”

“…We have used this framework to analyze the Matérn covariance approach and FE approximations on a hyperrectangle. Similar analysis can be carried out for more general Matérn-type GPs that replace ℒ = 𝜅 2 − Δ with a nonstationary or anisotropic elliptic operator on more general domains, where the prior approximation error bounds using FE are starting to emerge [3,1,7]. The criterion (2.11) and (2.12) can still be used for setting 𝑛 ℎ in these cases although there would not be an easily computable covariance function approach to compare with.…”

“…The Bayesian nonparametrics framework in [39] guarantees that if the rate of convergence of the GP prior approximation 𝑢 ℎ ≈ 𝑢 is fast enough, then the corresponding posteriors contract at the same rate. Establishing convergence rates for approximations 𝑢 ℎ ≈ 𝑢 is an active research area on numerical analysis of FE solution of fractional SPDEs [3,1,7]. As part of our analysis, we derive a crude estimate of the approximation error…”

Multilevel approximation of Gaussian random fields: Covariance compression, estimation and spatial prediction

“…For a linear, second-order (so that r = 2) elliptic (surface) differential operator L on M in divergence form, models of this type have been developed, e.g., in [10,45]. Moreover, computationally efficient methods to sample from such random fields or to employ the models in statistical applications, involving for instance inference or spatial predictions, have been discussed recently, e.g., in [8,9,16,36]. The following proposition extends and unifies these approaches, admitting rather general operators L (which, in the classic Matérn case, see [46,63], is the Laplace-Beltrami operator L = −∆ M ∈ OP S 2 1,0 (M), with r = 2 and constant correlation length parameter κ > 0).…”

Surface finite element approximation of spherical Whittle--Matérn Gaussian random fields

“…Another possibility is to use finite element techniques in order to approximate solutions to SPDE (1). Finite element approaches have been recently studied in the case of Euclidean domains, see for instance [4,3,6].…”