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Multilevel approximation of Gaussian random fields: fast simulation

by L. Herrmann and K. Kirchner and Ch. Schwab

(Report number 2019-32)

Abstract
We propose and analyze several multilevel algorithms for the fast simulation of possibly non-stationary Gaussian random fields (GRFs for short) indexed, e.g., by a bounded domain \(\mathcal{D} \subset \mathbb{R}^n\) or, more generally, by a compact metric space \(\mathcal{X}\) such as a compact \(n\)-manifold \(\mathcal{M}\). A colored GRF \(\mathcal{Z}\), admissible for our algorithms, solves the stochastic fractional-order equation \(\mathcal{A}^\beta \mathcal{Z} = \mathcal{W}\) for some \(\beta>n/4\), where \(\mathcal{A}\) is a linear, local, second-order elliptic differential operator in divergence form and \(\mathcal{W}\) is white noise on \(\mathcal{X}\). We thus consider GRFs on \(\mathcal{X}\) with covariance operators of the form \(\mathcal{C}=\mathcal{A}^{-2\beta}\). The proposed algorithms numerically approximate samples of \(\mathcal{Z}\) on nested sequences \(\{\mathcal{T}_\ell\}_{\ell \geq 0}\) of regular, simplicial partitions \(\mathcal{T}_\ell\) of \(\mathcal{D}\) and \(\mathcal{M}\), respectively. Work and memory to compute one approximate realization of the GRF \(\mathcal{Z}\) on the triangulation \(\mathcal{T}_\ell\) with consistency \(\mathcal{O}(N_\ell^{-\rho})\), for some consistency order \(\rho>0\), scale essentially linear in \(N_\ell = \#(\mathcal{T}_\ell)\), independent of the possibly low regularity of the GRF. The algorithms are based on a sinc quadrature for an integral representation of (the application of) the negative fractional-order elliptic operator \(\mathcal{A}^{-\beta}\). For the proposed numerical approximation, we prove bounds of the computational cost and the consistency error in various norms.

Keywords: Gaussian random fields, Matérn covariances, spatial statistics, fractional operators, multilevel methods

@Techreport{HKS19_836,
  author = {L. Herrmann and K. Kirchner and Ch. Schwab},
  title = {Multilevel approximation of Gaussian random fields: fast simulation},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2019-32},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2019/2019-32.pdf },
  year = {2019}
}

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