Erik Jansson
WASP and KAW postdoctoral fellow in the Cambridge Image Analysis Group and CRA at Wolfson College.
Generating Gaussian random fields using surface finite element methods
Erik Jansson
Slides available at slides.com/erikjansson
Joint work with:
- Mihály Kovács (Pázmány Péter Catholic University/Chalmers)
- Annika Lang (Chalmers/University of Gothenburg)
- Mike Pereira (Mines Paris - PSL University)
Made possible by:
-
Bolin, D., Kirchner, K., Kovács, M., Numerical solution of fractional elliptic stochastic PDEs with spatial white noise, IMA J. Numer. Anal., 40(2):1051–1073, 2020
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Dziuk, G., and Elliott, C. M., Finite element methods for surface PDEs. Acta Num., 22:289–396, 2013.
-
Lang A., Pereira, M., Galerkin–Chebyshev approximation of Gaussian random fields on compact Riemannian manifolds (preprint)
-
Whittle, P., Stochastic processes in several dimensions. Bull. Inst. Int. Stat., 40:974–994, 1963.
GRFs on manifolds
The SPDE view: GRFs are solutions to elliptic stochastic partial differential equations on manifolds:
\mathcal{L}u = \mathcal{W}
In this talk: Manifolds = compact, boundary–less 2D embedded surfaces in \(\mathbb{R}^3\)
Elliptic differential operator
White noise
Two questions:
Computation
Statistics
GRFs on manifolds
The SPDE view: GRFs are solutions to elliptic stochastic partial differential equations on manifolds:
\mathcal{L}u = \mathcal{W}
In this talk: Manifolds = compact, boundary–less 2D embedded surfaces in \(\mathbb{R}^3\)
Elliptic differential operator
White noise
Two questions:
Computation
Statistics
GRFs on manifolds
The SPDE view: GRFs are solutions to elliptic stochastic partial differential equations on manifolds:
\mathcal{L}u = \mathcal{W}
In this talk: Manifolds = compact, boundary–less 2D embedded surfaces in \(\mathbb{R}^3\)
Elliptic differential operator
White noise
Two questions:
Sampling
Statistics
Main computational tool: FEM
On surfaces!
Step 1: Triangulate the surface
Step 2: FEM space \(S_h \subset H^1(\mathcal{M}_h)\) of p.w., continuous, linear functions
\mathcal{M}
\mathcal{M}_h
Problem 1: Approximate solutions live on \(\mathcal{M}_h\), not \(\mathcal{M}\)!
Given \(\eta: \mathcal{M}_h \to \mathbb{R}\), \(\eta^\ell= \eta \circ p^{-1}\) is on \(\mathcal{M}\)!
Step 3: Key tool in surface finite elements: the lift
Takeway: FEM error similar to flat case, up to a "geometry error" term
Main computational tool: FEM
{\mathcal{T_h}}^\ell
{\mathcal{T_h}}
\nu_{\mathcal{T_h}}
p(x) = x-d_s(x) \nu
x
\nu
Whittle–Matérn fields on the sphere
\(\left(\kappa^2-\Delta_{\mathbb{S}^2}\right)^\beta u=\mathcal{W}\), \(\beta>1/2, \kappa \neq 0\)
Question: What to do with fractional operator?
Dunford–Taylor integral representation:
(\kappa^2- \Delta_{\mathbb{S}^2})^{-\beta} =\frac{\sin( \pi \beta)}{\pi}\int_{-\infty}^{\infty} e^{2 \beta y} \left( I+e^{2y}(\kappa^2 - \Delta_{\mathbb{S}^2})\right)^{-1} \, \mathrm{d} y\\
Whittle–Matérn fields on the sphere
\(\left(\kappa^2-\Delta_{\mathbb{S}^2}\right)^\beta u=\mathcal{W}\), \(\beta>1/2, \kappa \neq 0\)
Question: What to do with noise?
\mathcal{W}\sim \sum_{l=1}^\infty\sum_{m=-l}^l a_{l,m}Y_{l,m}
\mathcal{W}_L= \sum_{l=1}^L\sum_{m=-l}^l a_{l,m}Y_{l,m}
Computable by standard SFEM!
(\kappa^2 -\Delta_{\mathbb{S}^2})^{-\beta} \approx Q^{\beta}_k \propto \sum_{l=-K^{-}}^{K^+} e^{2\beta y_{l}}{\left(I+e^{2 y_l}\left(\kappa^2- \Delta_{\mathbb{S}^2}\right)\right)}^{-1} \\
Question: What to do with fractional operator?
Dunford–Taylor integral sinc quadrature:
Whittle–Matérn fields on the sphere
In E.J, Kovács, M & Lang, A, (2022) strong error bounds are proved.
\|u-u_{L,h}^{\ell}\|_* \sim \mathcal{O}(h^2)
Problem: How to approximate noise?
Solution: Different approach...
\mathcal{Z}=\sum_{i\in\N}\gamma(\lambda_{i})W_ie_i
\((\lambda_i,e_i)\) are eigenpairs of \(\mathcal{L}\)
Use power spectral density \(\gamma : \mathbb{R}_+\rightarrow \mathbb{R} \)
Random weights \((W_i : i\in\mathbb{N})\) are Gaussian
Problem: Eigenfunctions are not known.
General surfaces: an outlook
General surfaces: an outlook
\mathcal{Z}=\sum_{i\in\N}\gamma(\lambda_{i})W_ie_i
Solution (?): Approximation with SFEM basis functions
\mathcal{Z} \approx \sum_{i=1}^{N_h} \gamma(\lambda_{i,h})\overline{W}_i \psi_i
General surfaces: an outlook
\mathcal{L}(\cdot) = c(x)+\nabla \cdot (A(x)\nabla(\cdot))
c(x):
small
large
A(x):
Pullback of \(\R^3\) metric
\begin{bmatrix}
0.1 & 0 & 0\\
0 & 0.1 & 0 \\
0&0 & 1
\end{bmatrix}
Inverse correlation length: small values, bigger spots.
Smaller correlation length along \(z\)-axis, elongation in that direction!
General surfaces: an outlook
Nordstat23
By Erik Jansson
