Title: The inexact Matrix-Newton framework for solving NEPv

Abstract:

The eigenvector-dependent nonlinear eigenvalue problem (NEPv), also known as nonlinear eigenvector problem, is a special type of eigenvalue problem where we seek to find k eigenpairs of a Hermitian matrix function H:ℂn,k→ℂn,n that depends nonlinearly on the eigenvectors itself. That is, we have to find V∈ℂn,k with orthonormal columns and Λ=ΛH∈ℂk,k such that H(V)V=VΛ

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NEPv arise in a variety of applications, most notably in quantum chemistry applications, such as discrete Kohn-Sham- or Gross-Pitaevskii-equations, and data science applications, such as robust linear discriminant analysis or trace ratio optimization.

This talk is concerned with solving NEPv by viewing it as a set of nonlinear matrix equations and using an inexact Newton method on a matrix level. In this setting, Newton's method is applied using the Fréchet derivative and exploiting the structure of the problem by using a global GMRES-approach to solve the Newton-update equation efficiently.

Preprint available from arXiv: An inexact Matrix-Newton method for solving NEPv, 2023, arxiv.org/abs/2311.09670.