Title: Challenges for Eigenvalue Computations in Breakthrough Applications

Abstract: In this talk, we will present our recent work on solving large-scale eigenvalue problems in which the coefficient matrices depend nonlinearly on the eigenvalues. Nonlinear eigenvalue problems (NLEVPs) arise in a variety of applications in science and engineering, e.g., dynamic analysis of structures or computational nanoelectronics, to mention just a few, and are both mathematically and practically far more challenging than their well-known linear counterparts. First, we will introduce two new Cauchy integral-based approaches for solving large-scale nonlinear eigenvalue problems: the nonlinear FEAST algorithm that allows calculating large number of interior eigenvalues in parallel, and the rational approximation method that allows solving a large-scale problems efficiently by exploiting the special structure of the newly constructed linearization. We will illustrate presented methods with several physically-motivated numerical examples to show that they are highly efficient and flexible when solving large-scale problems arising in computational physics, chemistry, material sciences etc.