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Numerical Approximation of Gaussian random fields on Closed Surfaces

Andrea Bonito, Diane Guignard, Wenyu Lei

TL;DR

This paper addresses the efficient numerical simulation of Gaussian Matérn random fields on closed surfaces by recasting the problem as a fractional SPDE $(\kappa^2 I-\Delta_\gamma)^s u = w$ and using a Balakrishnan integral representation $u=L^{-s}w$ with $L=\kappa^2 I-\Delta_\gamma$. It develops a two-step method combining a sinc quadrature for the integral with a surface finite element discretization on an approximate surface, while avoiding eigenpair calculations by projecting the white noise into a finite element space. A comprehensive error analysis is provided: the sinc quadrature exhibits exponential convergence independent of the FE mesh size, and strong as well as mean-square error estimates account for geometric errors from surface approximation and eigenvalue discrepancies. Numerical experiments on the unit sphere and a torus demonstrate the method’s accuracy and show how the parameters $s$ (smoothness) and $\kappa$ (correlation length) influence the resulting GRFs, confirming the theoretical rates and the practicality of the approach for surface-based uncertainty quantification.

Abstract

We consider the numerical approximation of Gaussian random fields on closed surfaces defined as the solution to a fractional stochastic partial differential equation (SPDE) with additive white noise. The SPDE involves two parameters controlling the smoothness and the correlation length of the Gaussian random field. The proposed numerical method relies on the Balakrishnan integral representation of the solution and does not require the approximation of eigenpairs. Rather, it consists of a sinc quadrature coupled with a standard surface finite element method. We provide a complete error analysis of the method and illustrate its performances by several numerical experiments.

Numerical Approximation of Gaussian random fields on Closed Surfaces

TL;DR

This paper addresses the efficient numerical simulation of Gaussian Matérn random fields on closed surfaces by recasting the problem as a fractional SPDE and using a Balakrishnan integral representation with . It develops a two-step method combining a sinc quadrature for the integral with a surface finite element discretization on an approximate surface, while avoiding eigenpair calculations by projecting the white noise into a finite element space. A comprehensive error analysis is provided: the sinc quadrature exhibits exponential convergence independent of the FE mesh size, and strong as well as mean-square error estimates account for geometric errors from surface approximation and eigenvalue discrepancies. Numerical experiments on the unit sphere and a torus demonstrate the method’s accuracy and show how the parameters (smoothness) and (correlation length) influence the resulting GRFs, confirming the theoretical rates and the practicality of the approach for surface-based uncertainty quantification.

Abstract

We consider the numerical approximation of Gaussian random fields on closed surfaces defined as the solution to a fractional stochastic partial differential equation (SPDE) with additive white noise. The SPDE involves two parameters controlling the smoothness and the correlation length of the Gaussian random field. The proposed numerical method relies on the Balakrishnan integral representation of the solution and does not require the approximation of eigenpairs. Rather, it consists of a sinc quadrature coupled with a standard surface finite element method. We provide a complete error analysis of the method and illustrate its performances by several numerical experiments.
Paper Structure (29 sections, 9 theorems, 180 equations, 5 figures, 5 tables)

This paper contains 29 sections, 9 theorems, 180 equations, 5 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Random Fields
  3. Random Fields in Euclidean Domains
  4. Properties of the KL Expansion
  5. Gaussian Random Fields
  6. Random Fields on Surfaces
  7. Dunford--Taylor Formulation of
  8. Differential Operators on Surfaces
  9. Solution Operator
  10. Fractional Powers of Elliptic Operators
  11. Regularity of the White Noise and the Solution
  12. Dunford--Taylor Integral Representation
  13. Numerical Scheme
  14. Finite Element Methods on Discrete Surfaces
  15. Lifted Finite Element Method on
  16. ...and 14 more sections

Key Result

Proposition 3.1

Let $\widetilde{w}$ be a Gaussian white noise. For $\tfrac{n-1}{4}<s<1$, the expression is in $L^2(\Omega; L^2(\gamma))$ and coincides with $L^{-s} \widetilde{w}$, i.e.

Figures (5)

  • Figure 1: Log-log plot of sinc approximation error $|e(\widetilde{\lambda})|$ for $\widetilde{\lambda}\in [2,10^7]$.
  • Figure 2: Errors $e_{\rm strong}$ for $\kappa=0.5$ and different values of the truncation parameter $\mathtt{L}$ using $K=10000$ Monte Carlo samples.
  • Figure 3: Errors $e_{\rm weak}$ for $\kappa=2$ (left) and $\kappa=8$ (right) using $K=1000$ Monte Carlo samples. The dashed lines indicate the behavior predicted by Theorem \ref{['t:weak-convergence']}.
  • Figure 4: Numerical solution $U_k(\cdot,\omega_m)$ to Problem \ref{['e:spde']} on the unit sphere when $\kappa=0.5$ and $s=0.55$ (left), $s=0.75$ (middle), and $=0.95$ (right).
  • Figure 5: Numerical solution to Problem \ref{['e:spde']} on the torus defined in \ref{['def:torus']} when $\kappa=0.5$ and $s=0.55$ (left), $s=0.75$ (middle), and $s=0.95$ (right).

Theorems & Definitions (21)

  • Remark 2.1
  • Proposition 3.1
  • proof
  • Remark 4.1
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • Lemma 4.4
  • ...and 11 more