DrCaroline Wormell

Project Opportunities:

1.Project title: Quantifying the error of pattern-finding algorithms in dynamical systems

Project summary:

There are many interesting dynamical systems whose laws of evolution we have limited information about, from economies, climate systems, power grids, and fluid flow. We would like to predict these systems’ future and identify unexpected patterns in their behaviour, but we may only have a limited amount of observations to work with. Various tools exist to facilitate this: most of these involve approximating these systems' Koopman operator. The Koopman operator is a linear operator on functions that encodes composition by the dynamics: the key to studying many dynamical systems is in its eigenvalues and eigenfunctions. In many chaotic systems, such as turbulent fluids, the stability of the approximated Koopman operator can be highly variable depending on what is being measured. This makes error quantification very important in applications.

This project will develop mathematically rigorous tools to estimate the error of Koopman operator eigenvalues and eigenvectors obtained from data, by generalising a notion known as the sampling pseudospectrum. We will begin by studying a kernel-based algorithm known as kernel Extended Dynamical Mode Decomposition. This will involve applying, and where needed developing, rigorous approximation theory and probability theory in the context of dynamical systems. A strong mathematical background with at least one of functional analysis or probability theory would be desirable.

This Scholarship is funded by the Australian Research Council (ARC) and aims to support a PhD candidate undertaking research within the School of Mathematics and Statistics, Faculty of Science, at the University of Sydney.

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