# Operator decompositions for inverse problems

** Speaker:** Joel Rosenfeld, University of South Florida

** Date:** May 9, 2024

** Abstract:** Dynamic Mode Decomposition is a tool to extract a reduced order model of time series data that has seen a lot of success in the analysis of fluid dynamical data. Originally, DMD began as a matrix decomposition approach, introduced by Schmidt in 2008. Over the past decade, DMD has evolved to involve the Koopman operator as a theoretical underpinning. The Koopman operator exchanges a finite dimensional nonlinear dynamical system for a linear operator over an infinite dimensional function space.

We will examine Koopman based DMD from the perspective of kernel functions and as a methodology for resolving a certain class of inverse problems. We will then give a general framework for handling inverse problems through function theoretic operators and kernel spaces. The talk will conclude with a variety of examples, including scattered data approximation and other problems.