New Preprint! Koopman Approximation & My Master's studies

 Finite-Data Error Bounds for Approximating the Koopman Operator: That is the name of my Masters thesis, but I have now a new preprint that shares its namesake. In this paper, me and my collaborators (Rachel Morris, Jason Bramburger and Simone Brugiapaglia) prove new error bounds for approximating the Koopman operator using Extended Dynamic Mode Decomposition (EDMD). In particular we build a bridge between least squares approximation to EDMD in order to conduct an in-depth study of the Monte Carlo rate of convergence, and we show how can improve upon it in settings with sufficient regularity to obtain an exponential convergence rate. We use the same ideas to study compressed sensing inspired methods for the recovery of the Koopman operator in sparse setting and do a numerical investigation of the methods proposed. If that sounds interesting to you, head over to the following links for the paper and the code. 

A retrospective of this work: Overall, I am very proud to have worked on that project. This was my first real research experience and with it, I feel that I grew tremendously as a mathematician. The numerical investigations in the early stages showed me how one can use empirical results to refine the results that we were pursuing and the importance of writing good code and experiments (still working on that skill). I then got to experience the feeling of writing genuinely new mathematical results that led to my Masters thesis, and now a research paper. I have presented this work at international conferences and got to have discussions with very interesting people over the course of the three years that I have been working on this.

I want to express my gratitude towards the people that made this work possible. First of all, my supervisors Jason Bramburger and Simone Brugiapaglia, who entrusted me with this idea and continue to do so for my PhD. Then to Rachel Morris, without whom this paper would be far from finished, and finally to Ben Adcock, whose insight turned out to be crucial for understanding some of the results in this work. 

With this now finished, I am excited to move on to different projects, I am sure you will be hearing more about it in a future blog post! If you want to hear more about the work done in my research group, you can check out Simone's blog, or Jason's and Rachel's websites.