Title: Homotopy Methods for Solving Nonlinear Systems and Beyond

Abstract: This talk will cover some recent progress on homotopy
methods to solve nonlinear systems. I will start with homotopy methods for solving nonlinear PDEs with multiple solutions and bifurcations by coupling with domain decomposition and reduced basis methods. Examples from hyperbolic systems and free boundary problems will be used to demonstrate the ideas. Then a homotopy method will be introduced for solving maximum entropy problem to reconstruct probability density functions based on empirical data. I will also talk about a homotopy training algorithm for nonlinear optimization arising from machine learning.