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

by S. G. Cox and K. Kirchner

(Report number 2019-23)

Abstract
We analyze several Galerkin approximations of a Gaussian random field \(\mathcal{Z}\colon\mathcal{D}\times\Omega\to\mathbb{R}\) 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 \(A\colon\mathcal{D}\to\mathbb{R}^{d\times d}\), \(\kappa\colon\mathcal{D}\to\mathbb{R}\), 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 \(0<\alpha\leq 1\). 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 \(\kappa\equiv\operatorname{const.}\), we perform several numerical experiments which validate our theoretical results.

Keywords: Gaussian random fields, Matérn covariance, fractional operators, Hölder continuity, Galerkin approximations, finite element method

@Techreport{CK19_827,
  author = {S. G. Cox and K. Kirchner},
  title = {Regularity and convergence analysis in Sobolev and H\"older spaces for generalized Whittle-Matérn fields},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2019-23},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2019/2019-23.pdf },
  year = {2019}
}

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