Uncertainty-Aware Numerical Solutions of ODEs by Bayesian Filtering

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dc.contributor.advisor Hennig, Philipp (Prof. Dr.)
dc.contributor.author Kersting, Hans
dc.date.accessioned 2021-03-11T10:01:23Z
dc.date.available 2021-03-11T10:01:23Z
dc.date.issued 2021-03-11
dc.identifier.other 1751189252
dc.identifier.uri http://hdl.handle.net/10900/113263
dc.identifier.uri http://nbn-resolving.de/urn:nbn:de:bsz:21-dspace-1132632 de_DE
dc.identifier.uri http://dx.doi.org/10.15496/publikation-54639
dc.description.abstract Numerical analysis is the branch of mathematics that studies algorithms that compute approximations of well-defined, but analytically-unknown mathematical quantities. Statistical inference, on the other hand, studies which judgments can be made on unknown parameters in a statistical model. By interpreting the unknown quantity of interest as a parameter and providing a statistical model that relates it to the available numerical information (the `data'), we can thus recast any problem of numerical approximation as statistical inference. In this way, the field of probabilistic numerics introduces new 'uncertainty-aware' numerical algorithms that capture all relevant sources of uncertainty (including all numerical approximation errors) by probability distributions. While such recasts have been a decades-long success story for global optimization and quadrature (under the names of Bayesian optimization and Bayesian quadrature), the equally important numerical task of solving ordinary differential equations (ODEs) has been, until recently, largely ignored. With this dissertation, we aim to further shed light on this area of previous ignorance in three ways: Firstly, we present a first rigorous Bayesian model for initial value problems (IVPs) as statistical inference, namely as a stochastic filtering problem, which unlocks the employment of all Bayesian filters (and smoothers) to IVPs. Secondly, we theoretically analyze the properties of these new ODE filters, with a special emphasis on the convergence rates of Gaussian (Kalman) ODE filters with integrated Brownian motion prior, and explore their potential for (active) uncertainty quantification. And, thirdly, we demonstrate how employing these ODE filters as a forward simulator engenders new ODE inverse problem solvers that outperform classical 'uncertainty-unaware' ('likelihood-free') approaches. This core content is presented in Chapter 2. It is preceded by a concise introduction in Chapter 1 which conveys the necessary concepts and locates our work in the research environment of probabilistic numerics. The final Chapter 3 concludes with an in-depth discussion of our results and their implications. en
dc.language.iso en de_DE
dc.publisher Universität Tübingen de_DE
dc.rights ubt-podok de_DE
dc.rights.uri http://tobias-lib.uni-tuebingen.de/doku/lic_mit_pod.php?la=de de_DE
dc.rights.uri http://tobias-lib.uni-tuebingen.de/doku/lic_mit_pod.php?la=en en
dc.subject.classification Maschinelles Lernen , Gewöhnliche Differentialgleichung de_DE
dc.subject.ddc 004 de_DE
dc.subject.ddc 510 de_DE
dc.subject.other Probabilistische Numerik de_DE
dc.subject.other Probabilistic Numerics en
dc.subject.other Machine Learning en
dc.subject.other Ordinary Differential Equations en
dc.title Uncertainty-Aware Numerical Solutions of ODEs by Bayesian Filtering en
dc.type Dissertation de_DE
dcterms.dateAccepted 2020-12-17
utue.publikation.fachbereich Informatik de_DE
utue.publikation.fakultaet 7 Mathematisch-Naturwissenschaftliche Fakultät de_DE

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