The dependency of spectral gaps on the convergence of the inverse iteration for a nonlinear eigenvector problem

TL;DR

This paper analyzes the convergence behavior of inverse iteration methods for the nonlinear Gross-Pitaevskii eigenvector problem, establishing explicit rates linked to spectral gaps and extending results to related inverse iteration schemes.

Contribution

It provides the first local convergence analysis for inverse iteration on the GPE without damping, connecting spectral gaps to convergence rates and explaining spectral shift effects.

Findings

Explicit linear convergence rates depending on spectral gaps

Extension of results to gradient flow and damped inverse iterations

Numerical experiments confirming theoretical predictions

Abstract

In this paper we consider the generalized inverse iteration for computing ground states of the Gross-Pitaevskii eigenvector problem (GPE). For that we prove explicit linear convergence rates that depend on the maximum eigenvalue in magnitude of a weighted linear eigenvalue problem. Furthermore, we show that this eigenvalue can be bounded by the first spectral gap of a linearized Gross-Pitaevskii operator, recovering the same rates as for linear eigenvector problems. With this we establish the first local convergence result for the basic inverse iteration for the GPE without damping. We also show how our findings directly generalize to extended inverse iterations, such as the Gradient Flow Discrete Normalized (GFDN) proposed in [W. Bao, Q. Du, SIAM J. Sci. Comput., 25 (2004)] or the damped inverse iteration suggested in [P. Henning, D. Peterseim, SIAM J. Numer. Anal., 53 (2020)]. Our analysis also reveals why the inverse iteration for the GPE does not react favourably to spectral shifts. This empirical observation can now be explained with a blow-up of a weighting function that crucially contributes to the convergence rates. Our findings are illustrated by numerical experiments.