Erik Jansson
WASP and KAW postdoctoral fellow in the Cambridge Image Analysis Group and CRA at Wolfson College.
Generating non-stationary Gaussian random fields on hypersurfaces using surface finite element methods
Erik Jansson
Slides available at slides.com/erikjansson
Joint work with:
- Annika Lang (Chalmers/University of Gothenburg)
- Mike Pereira (Mines Paris - PSL University)
Made possible by:
-
A. Bonito, D. Guignard, and W. Lei, Numerical approximation of Gaussian random fields on
closed surfaces, Computational and Applied Mathematics. In press (2024) -
Dziuk, G., and Elliott, C. M., Finite element methods for surface PDEs. Acta Num., 22:289–396, 2013.
-
H. Fujita and T. Suzuki, Evolution problems, Handbook of Numerical Analysis, pp. 789–928,
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Whittle, P., Stochastic processes in several dimensions. Bull. Inst. Int. Stat., 40:974–994, 1963.
Talk based on:
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https://arxiv.org//2406.08185
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D. Boffi, Finite element approximation of eigenvalue problems, Acta Numerica, pp. 1-120 (2010).
-
A. Yagi, Abstract Parabolic Evolution Equations and Their Applications, Springer Monographs in Mathematics, Springer, Berlin, 2010.
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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
Bonus references
GRFs on Manifolds
The SPDE view: Consider GRFs that are solutions to elliptic stochastic partial differential equations on manifolds:
\mathcal{L}\mathcal Z = \mathcal{W}
In this talk: Manifolds = compact, boundary–less oriented embedded surfaces or curves in \(\mathbb{R}^3\) or \(\mathbb{R}^2\)
Elliptic differential operator
White noise
Two questions:
Sampling
Statistics
Elliptic differential operator
White noise
Two questions:
Sampling
Statistics
GRFs on Manifolds
In this talk: Manifolds = compact, boundary–less oriented embedded surfaces or curves in \(\mathbb{R}^3\) or \(\mathbb{R}^2\)
The SPDE view: Consider GRFs that are solutions to elliptic stochastic partial differential equations on manifolds:
\mathcal{L}\mathcal Z = \mathcal{W}
Elliptic differential operator
White noise
Two questions:
Computation
Statistics
GRFs on Manifolds
In this talk: Manifolds = compact, boundary–less oriented embedded surfaces or curves in \(\mathbb{R}^3\) or \(\mathbb{R}^2\)
The SPDE view: Consider GRFs that are solutions to elliptic stochastic partial differential equations on manifolds:
\mathcal{L}\mathcal{Z} = \mathcal{W}
The differential operator
D \colon T_x \mathcal M \to T_x \mathcal M \\
V \colon \mathcal M \to \mathbb R^+
Eigenpairs of \( \mathcal{L} \): \((\lambda_i,e_i)\)
\mathsf A_{\mathcal M}(u,v) = \int_{\mathcal M} D \nabla_{\mathcal M} u {\cdot} \nabla_{\mathcal M} v \mathrm d x
{+} \int_{\mathcal{M}} V uv \mathrm d x
+ Conditions
\(\mathsf A_\mathcal{M}\) defines elliptic operator \(\mathcal L\)
The Model
~\mathcal{Z} = \gamma(\mathcal L) \mathcal{W}~
\(\gamma\colon\mathbb{R}_+ \to \mathbb{R}\) is a function satisfying sufficient decay properties (plus more)
~\mathcal{Z} = \sum_{i=1}^\infty \gamma(\lambda_i) W_i e_i~
\gamma(\mathcal L)^{-1}\mathcal{Z} = \mathcal{W}
formally
\mathcal{W} = \sum_{i=1}^\infty W_i e_i, \\ W_i \sim \mathsf N(0,1)
The Model
GRFs on Manifolds
Some examples
5
The Model: Example 1
\mathcal M = \mathbb{S}^2, \gamma(\lambda) = \lambda^{-\alpha}, \alpha > 1/2
D_{ij} = \delta_{ij}
V(x) = 10
\alpha = 1.5
\alpha = 0.55
(Whittle-) Matérn random fields given by SPDE \((\kappa^2-\Delta_{\mathbb{S}^2})^\beta \mathcal Z = \mathcal{W}\)!
\mathcal M = \mathbb{S}^2, \gamma(\lambda) = \lambda^{-0.75}
D_{ij} = \delta_{ij}
V(x) = \begin{cases} 10^5 \quad \text{if } x_2^6+x_1^3-x_3^2 \in (0.1,0.5)\\
10 \quad \text{else}
\end{cases}
The Model: Example 2
D(x)\nabla_{ \mathbb S^2} u = {\color{red} ρ_1}(\nabla_{ \mathbb S^2} f (x) \cdot \nabla_{ \mathbb S^2} u)\nabla_{ \mathbb S^2} f (x) \\
\qquad+ {\color{blue} \rho_2 }(x \times \nabla_{ \mathbb S^2} f (x) \cdot \nabla_{ \mathbb S^2}u )(x \times \nabla_{ \mathbb S^2} f (x) )
\(\rho_1\) small, \(\rho_2\) large: field is elongated tangentially along level sets of \(f\)
\(\rho_1\) large, \(\rho_2\) small: field is elongated orthogonally along level sets of \(f\)
The Model: Example 3
\mathcal M = \mathbb{S}^2, \gamma(\lambda) = \lambda^{-0.75}
V(x) = 10 \\
f(x) = x_2, \rho_1 = 1, \rho_2 = 25
The Model: Example 3
\mathcal M = \mathbb{S}^2, \gamma(\lambda) = \lambda^{-0.75}
The Model: Example
- Preferred directions
- Local activations
- Local correlation length
- Different regularity by \(\gamma\)
Sampling the fields
Problem: \((\lambda_i,e_i)\) is not available!
Sampling \(\iff\) Evaluating \(\mathcal Z = \sum_{i=1}^\infty \gamma(\lambda_i) W_i e_i\)
Can \((\lambda_i,e_i)\) be approximated?
Maybe? Solving an SPDE, try with FEM!
Main computational tool: FEM
On Euclidean spaces
Just triangulate the domain!
\mathcal{D}
\mathcal{D}
Put the pieces back together, get the original domain!
Main computational tool: FEM
On manifolds!
Step 1: Triangulate the domain
\mathcal{M}
\mathcal{M}_h
Issue: Approximate solutions live on \(\mathcal{M}_h\), not \(\mathcal{M}\)!
Put the pieces back together, don't get the same domain!
Step 2: FEM space \(S_h \subset H^1(\mathcal{M}_h)\) of p.w., continuous, linear functions
Given \(\eta: \mathcal{M}_h \to \mathbb{R}\), \(\eta^\ell= \eta \circ a\) 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
Main computational tool: FEM
Discretization of operators
\mathcal M, \, H^{1}(\mathcal M)
Original
\mathcal{L}
Eigenpairs
(\lambda_i,e_i)
Discrete 1
\mathcal M, \, S_h^\ell
\mathcal{L}_h
Eigenpairs
(\lambda_{i,h},e_{i,h})
Discrete 2
\mathcal M_h, \, S_h
\mathsf L_h
Eigenpairs
(\Lambda_{i,h},E_{i,h})
How to sample?
\mathcal{Z}=\sum_{i\in\N}\gamma(\lambda_{i})W_ie_i
Use eigenpairs \((\Lambda_{i,h},E_{i,h})\) of \(\mathsf{L}_h\)
First idea: Use these to approximate
\mathcal{Z} \approx \sum_{i=1}^{N_h} \gamma(\Lambda_{i,h})\overline{W}_i E_{i,h}
How to sample?
In practice:
\sum_{i=1}^{N_h} \gamma(\Lambda_{i,h})\overline{W}_i E_{i,h} = \sum_{i=1}^{N_h} Z_i \psi_i
Works with known mesh, "unknown" manifold:
Find a mesh and you can simulate GRFs on it!
What about strong error?
\((Z_i)\) are Gaussian with covariance matrix determined by \(\gamma\) and FEM matrices
How to sample?
Z \sim N(0,\Sigma_Z)
\Sigma_Z = \sqrt{C}^{-T} \gamma^2(S) \sqrt{C}
C_{kl} = (\psi_k,\psi_l)_{L^2(\mathcal M_h)}
R_{kl} = \mathsf{A}_{\mathcal M_h}(\psi_k,\psi_l)
S = \sqrt{C}^{-1} R \sqrt{C}^{-T}
Chebyshev quadrature approximation of \(\gamma\)
Error analysis: first try
First try:
Problem:
\sum_{i\in\N}\gamma(\lambda_{i})W_ie_i - \sum_{i=1}^{N_h} \gamma(\Lambda_{i,h})\overline{W}_i E_{i,h}
\|e_i-E_{i,h}\|_{L^2} ??
Generally: Approximating eigenfunctions are hard!
Eigenvalues are fine, however!
Problematic discrete eigenfunctions
Generally: Approximating eigenfunctions are hard! Multiplicity!
From D. Boffi, Acta Num. (2010)
Noise and Field Approximations
Various white noise approximations, various approximations of \(\mathcal Z\)
\widetilde{\mathcal W} = P_h \mathcal W \, (\in S_h^\ell)
\widetilde{\mathcal Z}_h = \gamma(\mathcal L_h )P_h \mathcal W
\hat{\mathsf{W}} = \mathsf P_h (\sigma \widetilde{\mathcal{W}}^{-\ell})
\mathsf W = \mathsf P_h (\sigma^{1/2} \widetilde{\mathcal{W}}^{-\ell})
\hat{\mathsf{Z}}_h = \gamma(\mathsf L_h )\hat{\mathsf W}
\mathsf{Z}_h = \gamma(\mathsf L_h )\mathsf W
sd
sd
sd
\mathsf{Z}_h = \sum_{i=1}^{N_h} \gamma(\Lambda_{i,h})\overline{W}_i E_{i,h}
Strong error roadmap
\widetilde{\mathcal Z} = \gamma(\mathcal L ) \mathcal W
\hat{\mathsf{Z}}_h = \gamma(\mathsf L_h )\hat{\mathsf W}
\mathsf{Z}_h = \gamma(\mathsf L_h )\mathsf W
\widetilde{\mathcal Z}_h = \gamma(\mathcal L_h )P_h \mathcal W
??
Important result: errors on the form
\(\|\gamma(\mathcal L)f -\gamma(\mathcal L_h) f\|_{L^2}\)
Brief note on proof:
\gamma(\mathcal L) f = \frac{1}{2\pi i } \int_{\Gamma} \gamma(z)(z-\mathcal{L})^{-1}f \\
\gamma(\mathcal L_h) f = \frac{1}{2\pi i } \int_{\Gamma} \gamma(z)(z-\mathcal{L}_h)^{-1}f
Deterministic Error Result
Brief note on proof:
\gamma(\mathcal L) f = \frac{1}{2\pi i } \int_{\Gamma} \gamma(z)(z-\mathcal{L})^{-1}f \\
\gamma(\mathcal L_h) f = \frac{1}{2\pi i } \int_{\Gamma} \gamma(z)(z-\mathcal{L}_h)^{-1}f
Deterministic Error Result
Strong error bound
In the end:
\|\mathcal Z - \mathsf Z_h^\ell\|_{L^2} \lesssim K_\alpha(h) h^{\min\{\alpha -d/4; 2\}} \\
K_\alpha(h) = \begin{cases} 1 ~\text{ if } \alpha - d/4 < 2 \\ |\log h| \text{ if } \alpha - d/4 = 2 \\ |\log h|^{d/2-1/2} \text{ else }\end{cases}
In conclusion...
- We can sample non-stationary random fields on curves and surfaces using mesh only
- We can prove that the strong error is bounded
- Future: Time-dependent fields? Inference?
Numdiff-17
By Erik Jansson
