The Kernel Trick - Data-Driven Dynamics | Lecture 7

While EDMD is a powerful method for approximating the Koopman operator from data, it has limitations. A major drawback is that to implement EDMD on high-dimensional data one typically needs a large and robust dictionary, which in turn makes computations prohibitively slow or even impossible. In this lecture we provide a work-around for such a situation by introducing the kernel trick. We show that one can take advantage of kernel functions to perform computations that scale with the number of data points instead of the size of the dictionary. The result is a simple, yet powerful, extension of EDMD for high-dimensional data and a connection with Reproducing Kernel Hilbert Spaces.

Coding demonstration in MATLAB comes from KernelDMD.m here: https://github.com/jbramburger/DataDrivenDynSyst

Koopman analysis of the Burgers equation: https://arxiv.org/abs/1712.06369

Get the book here: https://epubs.siam.org/doi/10.1137/1.9781611978162

Scripts and notebooks to reproduce all examples: https://github.com/jbramburger/DataDrivenDynSyst

This book provides readers with:

- methods not found in other texts as well as novel ones developed just for this book;

- an example-driven presentation that provides background material and descriptions of methods without getting bogged down in technicalities;

- examples that demonstrate the applicability of a method and introduce the features and drawbacks of their application; and

- a code repository in the online supplementary material that can be used to reproduce every example and that can be repurposed to fit a variety of applications not found in the book.

More information on the instructor: https://hybrid.concordia.ca/jbrambur/

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