Parikshit Upadhyaya: The Eigenvector-Dependent Nonlinear Eigenvalue Problem

Abstract:

Nonlinear eigenproblems where the nonlinearity in the problem is present due to the dependence on the eigenvectors arise in several fields ranging from machine learning to quantum chemistry. In this talk, we will briefly look at these applications and then discuss the common numerical algorithms used to solve such problems and their convergence properties. In particular, we will talk about the Self-Consistent Field (SCF) iteration which is the most popular algorithm when one is interested in computing multiple eigenpairs. We will end by discussing Quasi-Newton algorithms for the case when one is interesting in only one eigenpair, and show that the SCF iteration is a Quasi-Newton algorithm in disguise.