Theory and Algorithms for Eigenvector-Dependent Nonlinear Eigenvalue Problems

Eigenvector-Dependent Nonlinear Eigenvalue Problems (NEPv) naturally arise in various areas of applied science and engineering. This project aims to develop new theories and algorithms for a special class of NEPv where the coefficient matrix function presents an affine-linear structure. Such a structure is shared by many practical NEPvs, but has largely be ignored in previous studies. By the affine-linear structure, we greatly simplies the anlysis of NEPv and establish reliable and efficient numerical algorithms to find the solution. Our planned contributions are three fold: Firstly, for theory we will develop two new techniques for analyzing the solution of al-NEP, including a novel geometric description and a generally applied variational characterization. Secondly, for numerical techniques we will focus on improving the popular SCF iteration, and will contribute a new geometric interpretation of SCF to deepen understanding of the algorithm, as well as two novel acceleration schemes to speed up SCF. Finally, for supporting future research we will collect representative NEPv from real applications with various backgrounds, and build the first benchmark NEPv repository. file:///ad.uky.edu/as/Admin/A&S%20Grants/PRE-AWARD/Mathematics/Lu_Ding/NSF%2012-1-2020/Abstract.txt[11/23/2020 4:02:54 PM]