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Non-stationary Gaussian random fields on hypersurfaces: Sampling and strong error analysis

Erik Jansson, Annika Lang, Mike Pereira

TL;DR

This work extends the SPDE-based sampling framework to non-stationary Gaussian random fields on hypersurfaces by representing fields as $\mathcal{Z}=\gamma(\mathcal{L})\mathcal{W}$ with a variable-coefficient elliptic operator $\mathcal{L}$. It develops a robust SFEM-based sampling pipeline and a Galerkin–Chebyshev approach that avoids explicit eigenfunction computations, enabling efficient non-stationary field generation on curved domains. The authors derive sharp strong convergence rates that depend on the amplitude-spectral-density parameter $\alpha$ and surface dimension, using a deterministic SFEM error analysis plus functional-calculus tools, and validate them with numerical experiments on circles and spheres. The results provide a practical, theoretically grounded method for simulating localized, directionally biased Gaussian fields on manifolds, with explicit guidance on polynomial truncation and error control.

Abstract

A flexible model for non-stationary Gaussian random fields on hypersurfaces is introduced.The class of random fields on curves and surfaces is characterized by an amplitude spectral density of a second order elliptic differential operator.Sampling is done by a Galerkin--Chebyshev approximation based on the surface finite element method and Chebyshev polynomials. Strong error bounds are shown with convergence rates depending on the smoothness of the approximated random field. Numerical experiments that confirm the convergence rates are presented.

Non-stationary Gaussian random fields on hypersurfaces: Sampling and strong error analysis

TL;DR

This work extends the SPDE-based sampling framework to non-stationary Gaussian random fields on hypersurfaces by representing fields as with a variable-coefficient elliptic operator . It develops a robust SFEM-based sampling pipeline and a Galerkin–Chebyshev approach that avoids explicit eigenfunction computations, enabling efficient non-stationary field generation on curved domains. The authors derive sharp strong convergence rates that depend on the amplitude-spectral-density parameter and surface dimension, using a deterministic SFEM error analysis plus functional-calculus tools, and validate them with numerical experiments on circles and spheres. The results provide a practical, theoretically grounded method for simulating localized, directionally biased Gaussian fields on manifolds, with explicit guidance on polynomial truncation and error control.

Abstract

A flexible model for non-stationary Gaussian random fields on hypersurfaces is introduced.The class of random fields on curves and surfaces is characterized by an amplitude spectral density of a second order elliptic differential operator.Sampling is done by a Galerkin--Chebyshev approximation based on the surface finite element method and Chebyshev polynomials. Strong error bounds are shown with convergence rates depending on the smoothness of the approximated random field. Numerical experiments that confirm the convergence rates are presented.
Paper Structure (21 sections, 17 theorems, 201 equations, 6 figures)

This paper contains 21 sections, 17 theorems, 201 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Deterministic theory: geometry, functional analysis and finite elements
  3. Geometric and functional analytic setting
  4. SFEM--Galerkin approximation
  5. Deterministic error analysis
  6. Stochastic theory: random fields on surfaces and convergence of SFEM approximation
  7. Random fields on surfaces
  8. Approximation of random fields with surface finite elements
  9. Error analysis of the SFEM discretization
  10. Case $\alpha\le 1$
  11. Case $\alpha>1$
  12. Error analysis of the Galerkin--Chebyshev approximation
  13. Numerical experiments
  14. Deterministic proofs
  15. Proof of \ref{['prop:spectral']}
  16. ...and 6 more sections

Key Result

Proposition 2.1

Let $\delta>0$ be the coercivity constant defined in eq:coerc. There exists a set of eigenpairs $\lbrace (\lambda_i, e_i)\rbrace_{i\in\amsmathbb{N}}$ of $\mathcal{L}$ consisting of a sequence of increasing real-valued eigenvalues $0<\delta \le \lambda_1 \le \lambda_2 \le \cdots$ with $\lambda_i \rig

Figures (6)

  • Figure 1: Examples of random field samples generated with our method.
  • Figure 2: One-dimensional illustration of the lift. The lift is along the normal vector $\nu$ to the surface $\mathcal{M}$.
  • Figure 3: Contours used to define the Cauchy--Stieltjes integral representation of operators.
  • Figure 4: Some examples to highlight the influence of the different model parameters.
  • Figure 5: First numerical experiment (on the circle).
  • ...and 1 more figures

Theorems & Definitions (34)

  • Proposition 2.1
  • Remark 2.2
  • Lemma 2.3: Eigenvalue error bounds
  • Proposition 2.4
  • Lemma 2.5
  • proof : Proof of \ref{['prop:det']}
  • Proposition 2.6
  • Theorem 3.1
  • Remark 3.2
  • Lemma 3.3
  • ...and 24 more