Linearizing a nonlinear eigenvalue problem with quadratic rational eigenvector nonlinearities
Victor Janssens, Karl Meerbergen, Wim Michiels
TL;DR
The paper addresses nonlinear eigenvalue problems with eigenvector nonlinearities by focusing on a scalar quadratic rational nonlinearity and deriving a compact linearization that embeds NEPv eigenvalues into a linear eigenproblem via a two-parameter (MEP) framework. It introduces operator determinants $\\Delta_0, \\\Delta_1, \\\Delta_2$ and proves an equivalence to a GEP $\\Delta_1 z = \lambda \\\Delta_0 z$ (with $z = v \\otimes w$) that yields all NEPv eigenvalues while exposing spurious eigenpairs. To handle them, the authors develop Arnoldi-based filtering, including a one-sided (Filtering) and a two-sided projection approach, leveraging invariant subspaces to suppress roughly half of the spurious solutions and accelerate convergence. They also analyze singular-pencil scenarios ($\mathrm{rank}(C) < n-1$) and provide an upper bound on the number of NEPv solutions via a BKK-based argument, supplemented by numerical experiments on random dense, PDE-inspired, and low-rank scenarios. The results offer a scalable, structure-exploiting pathway to solving a broad class of NEPv arising in physics and data applications, with clear avenues for generalization to multiple nonlinearities and larger-scale problems.
Abstract
Nonlinear eigenvalue problems with eigenvector nonlinearities (NEPv) are algebraic eigenvalue problems whose matrix depends on the eigenvector. Applications range from computational quantum mechanics to machine learning. Due to its nonlinear behavior, existing methods almost exclusively rely on fixed-point iterations, the global convergence properties of which are only understood in specific cases. Recently, a certain class of NEPv with linear rational eigenvector nonlinearities has been linearized, i.e., the spectrum of the linear eigenvalue problem contains the eigenvalues of the NEPv. This linear problem is solved using structure exploiting algorithms to improve both convergence and reliability. We propose a linearization for a different class of NEPv with quadratic rational nonlinearities, inspired by the discretized Gross-Pitaevskii equation. The eigenvalues of this NEPv form a subset of the spectrum of a linear multiparameter eigenvalue problem which is equivalent to a system of generalized eigenvalue problems expressed in terms of operator determinants. A structure exploiting Arnoldi algorithm is used to filter a large portion of spurious solutions and to accelerate convergence.