15:45
MS41: Efficient simulation of random fields and applications (Part 2)
Chair: Laura Scarabosio
15:45
25 mins
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Empirically driven orthonormal bases for functional data analysis
Krzysztof Podgorski, Hiba Nassar
Abstract: In implementations of the functional data (FD) methods, the effect of the initial choice of an orthonormal basis has not been properly studied. Typically, several standard bases such as Fourier, wavelets, splines, etc. are considered to transform observed FD and a choice is made without any formal criteria indicating which of the bases is preferable for the initial transformation of the data. In an attempt to address this issue, we propose a strictly data-driven method of orthonormal basis selection. The method uses $B$-splines and utilizes recently introduced splinets efficiently orthonormalizing the $B$-splines. The algorithm learns from the data in the machine learning style of FD mining to efficiently place knots. The optimality criterion is based on the total mean square error and is utilized both in the learning algorithms and in comparison studies. The latter indicate efficiency that could be used to analyze FD obtained from a complex physical system.
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16:10
25 mins
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Heavy-tailed SPDE-random Field Priors for Bayesian Inversion
Lassi Roininen, Sari Lasanen, Karla Monterrubio-Gómez, Sara Wade
Abstract: We propose to study hierarchical non-Gaussian Markov random fields via their stochastic partial differential equation (SPDE) presentations. Instead of the often-used Gaussian driving noise, we use symmetric alpha-stable distributions in the SPDE formulation. Thus, we obtain heavy-tailed SPDE-random field priors. They are appealing in modelling material interfaces e.g. in subsurface imaging or medical tomography, as well as modelling rare events. We consider the interplay between finite-dimensional approximations of these models, and their infinite-dimensional counterparts. With these models and any practical Bayesian statistical inverse problem, MCMC sampling will pose significant challenges, so we finally present the current status of our computational solutions.
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16:35
25 mins
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Stable random sheets and their approximations in Bayesian inverse problems
Sari Lasanen Lasanen
Abstract: We consider stable analogs of the Brownian sheet as heavy-tailed priors in Bayesian inverse problems.
Especially, we verify that the correct sample path properties are captured by the approximations and that the corresponding approximated posterior distributions are consistent with respect to increasing of the dimensionality of the approximations.
The emphasis is on building the bridge from theory to practice.
We utilise Lévy-LePage series representations of stable random variables and random walk-type approximations.
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17:00
25 mins
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Computation for very large multiscale spatio-temporal conditional distributions
Finn Lindgren
Abstract: As part of the H2020 EUSTACE project, a multiscale spatio-temporal model for past historical daily temperatures was developed. The building blocks are stochastic partial differential equations at different spatial and temporal scales, discretised into Gaussian Markov random fields. These are then linked to several types of observations, both linearly and with non-linear transformations. To compute the posterior conditional mean and to compute posterior samples, iterative linear solver methods were developed, based on overlapping block preconditioning. The resulting methods made it possible to reconstruct daily temperatures across 165 years on a global 20km resolution, including uncertainty propagated from the complex observation model variability and biases.
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