Gaussian Random Fields (GRFs) are ubiquitous models of random functions in computational UQ. Their efficient numerical treatment as input data for PDEs, for modeling in spatial statistics and as key building block within larger UQ simulation loops continues to receive attention in numerical analysis, spatial statistics, and scientific computing.
This mini-symposium will present contributions at the forefront of research, addressing among others the fast and compressive multilevel simulation of GRFs, the impact of formatted numerical linear algebra (H- and H2-Matrix formats, Quantized Tensor Trains) on GRF simulation and identification, efficient covariance estimation algorithms for GRFs, the interplay of massively parallel PDE solvers and GRFs, the multilevel Monte Carlo and Quasi-Monte Carlo integration for GRF PDE inputs, as well as statistical applications.
10:30
Fast simulation of non-stationary Gaussian random fields
Kristin Kirchner | Seminar for Applied Mathematics, ETH Zurich | Switzerland
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Kristin Kirchner | Seminar for Applied Mathematics, ETH Zurich | Switzerland
We propose and analyze multilevel algorithms for the fast simulation of possibly non-stationary Gaussian random fields (GRFs for short) indexed, e.g., by a bounded Euclidean domain or by a compact manifold. A colored GRF, admissible for our algorithms, solves a stochastic fractional-order PDE driven by Gaussian white noise, where the underlying differential operator is assumed to be linear, elliptic, of second-order and in divergence form.
The proposed algorithms numerically approximate samples of the GRF on nested sequences of regular, simplicial partitions of Euclidean domains and manifolds, respectively. Work and memory to compute one approximate realization of the GRF scale essentially linear in the degrees of freedom, independent of the possibly low regularity of the GRF and at a certified accuracy.
Based on joint works with David Bolin, Sonja G. Cox, Lukas Herrmann, Mihály Kovács, and Christoph Schwab.
11:00
MLQMC Methods for Elliptic PDEs Driven by White Noise
Matteo Croci | University of Oxford | United Kingdom
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Matteo Croci | University of Oxford | United Kingdom
When solving partial differential equations driven by additive spatial white noise, the efficient sampling of white noise realizations can be challenging. In this talk we focus on the efficient sampling of white noise using quasi-random points in a finite element method and multilevel Quasi Monte Carlo (MLQMC) setting. This work is an extension of previous research on white noise sampling for MLMC. We express white noise as a wavelet series expansion that we divide in two parts. The first part is sampled using a randomized low-discrepancy sequence and contains a finite number of terms in order of decaying importance to ensure good QMC convergence. The second part is a correction term which is sampled using standard pseudo-random numbers. We show how the sampling of both terms can be performed in linear time and memory complexity in the number of mesh cells via a supermesh construction. Furthermore, our technique can be used to enforce the MLQMC coupling even in the case of non-nested mesh hierarchies. We demonstrate the efficacy of our method with numerical experiments.
11:30
Boundary effects in PDE-based sampling of Gaussian Matérn random fields
Laura Scarabosio | Technical University of Munich | Germany
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Laura Scarabosio | Technical University of Munich | Germany
Samples of a Gaussian random field with Matérn covariance can be generated efficiently by solving a differential equation with Gaussian white noise forcing. However, such an equation is originally posed on the whole 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.
12:00
- CANCELED - Spatial modeling of significant wave height using deformed SPDE models
David Bolin | King Abdullah University of Science and Technology (KAUST) | Saudi Arabia
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David Bolin | King Abdullah University of Science and Technology (KAUST) | Saudi Arabia
A non-stationary Gaussian random field model is developed based on a combination of the SPDE approach and the classical deformation method. With the deformation method, a stationary field is defined on a domain which is deformed so that the field is non-stationary on the new domain. We show that if the stationary field is a Matérn field defined as a solution to a fractional SPDE, the resulting non-stationary model can be represented as the solution to another fractional SPDE on the deformed domain. By defining the model in this way, the computational advantages of the SPDE approach can be combined with the deformation method's more intuitive parameterisation of non-stationarity. In particular it allows for essentially independent control over the non-stationary practical correlation range on one hand and the variance on the other hand. This has not been possible with previously proposed non-stationary SPDE models.
The model is tested on spatial data of significant wave height, a characteristic of ocean surface conditions which is important when estimating the wear and risks associated with a planned journey of a ship. The model is fitted to data from the north Atlantic and is used to compute wave height exceedance probabilities and the distribution of accumulated fatigue damage for ships traveling a popular shipping route. The model results agree well with the data, indicatinng that the model could be used for route optimization in naval logistics.