10:40
MS41: Efficient simulation of random fields and applications (Part 1)
Chair: Laura Scarabosio
10:40
25 mins
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Fast random field generation with H-matrices
Michael Feischl, Frances Kuo, Ian Sloan
Abstract: We use the H-matrix technology to compute the approximate square root of a
covariance matrix in linear complexity. This allows us to generate normal and
log-normal random fields
on general point sets with optimal complexity. We derive rigorous error
estimates which show convergence of the method. Our approach requires only mild assumptions on the covariance function and on the point set.
Therefore, it might be also a nice alternative to the circulant embedding approach which is well-defined only for regular
grids and stationary covariance functions.
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11:05
25 mins
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The rational SPDE approach for Gaussian random fields with general smoothness
David Bolin, Kristin Kirchner
Abstract: A popular approach for modeling and inference in spatial statistics is to represent Gaussian random fields as solutions to stochastic partial differential equations (SPDEs) of the form $L^{\beta}u = \mathcal{W}$,
where $\mathcal{W}$ is Gaussian white noise, $L$ is a second-order differential operator, and $\beta>0$ is a parameter that determines the smoothness of $u$. However, this approach has been limited to the case $2\beta\in\mathbb{N}$, which excludes several important models
and makes it necessary to keep $\beta$ fixed during inference.
We propose a new method, the rational SPDE approach, which in spatial dimension $d\in\mathbb{N}$ is applicable for any $\beta>d/4$, and
thus remedies the mentioned limitation. The presented scheme combines a finite element discretization with a rational approximation of the function $x^{-\beta}$ to approximate $u$. For the resulting approximation,
an explicit rate of convergence to $u$ in mean-square sense is derived.
Furthermore, we show that our method has the same computational benefits
as in the restricted case $2\beta\in\mathbb{N}$ when used for simulation and inference. Numerical experiments and a statistical application
are used to illustrate the accuracy of the method, and to show that it facilitates likelihood-based inference for all model parameters including $\beta$.
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11:30
25 mins
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Boundary effects in PDE-based sampling of Gaussian Mat\'ern random fields
Ustim Khristenko, Laura Scarabosio, Piotr Swierczynski, Elisabeth Ullmann, Barbara Wohlmuth
Abstract: Samples of a Gaussian random field with Mat\'ern covariance can be generated efficiently by solving a differential equation with Gaussian white noise forcing. However, such equation is originally posed on the whole $\mathbb{R}^d$, and truncation to a bounded computational domain with artificial boundary conditions introduces unwanted boundary effects. To mitigate these, a common practice in spatial statistics is to embed the computational domain into a larger domain, and postulate convenient boundary conditions on the latter. In this talk, we provide a rigorous analysis for the error in the covariance of the sampled field introduced by the domain truncation, for homogeneous Dirichlet, homogeneous Neumann,
and periodic boundary conditions. We show that the error decays exponentially in the window size, independently of the type of boundary condition. Moreover, numerical experiments are presented to illustrate the theory and compare the local behavior of the error for the three choices of boundary conditions.
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