The dependency of spectral gaps on the convergence of the inverse iteration for a nonlinear eigenvector problem
Patrick Henning
TLDR
In this paper , the generalized inverse iteration for computing ground states of the Gross-Pitaevskii eigenvector problem (GPE) is considered, and the authors prove explicit linear convergence rates that depend on the maximum eigenvalue in magnitude of a weighted linear eigen value problem.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.read more
Citations
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An efficient two level approach for simulating Bose-Einstein condensates
TL;DR: In this paper , the authors propose a solution to solve the problem of the problem: this paper ] of "uniformity" and "uncertainty" of the solution.
A two level approach for simulating Bose-Einstein condensates by localized orthogonal decomposition
Patrick Henning,Johan Warnegaard +1 more
TL;DR: In this paper , the authors considered the numerical computation of ground states and dynamics of single-component Bose-Einstein condensates (BECs) with a multiscale finite element approach known as Localized Orthogonal Decomposition.
Journal ArticleDOI
On the convergence of Sobolev gradient flow for the Gross-Pitaevskii eigenvalue problem
TL;DR: In this article , the convergence of three projected Sobolev gradient flows to the ground state of the Gross-Pitaevskii eigenvalue problem was studied, where the gradient flows were constructed as the gradient flow of the GPE with respect to the $H^1_0$-metric and two equivalent metrics on $H_0^1.
References
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Bose-Einstein condensation
TL;DR: The Bose-Einstein condensation (BEC) phenomenon was first introduced by Bose as discussed by the authors, who derived the Planck law for black-body radiation by treating the photons as a gas of identical particles.
Journal ArticleDOI
Structure of a quantized vortex in boson systems
TL;DR: In this paper, a theory of the elementary line vortex excitations is developed for a system of weakly repelling bosons, characterised by the presence of a finite fraction of the particles in a single particle state of integer angular momentum.
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Computing the Ground State Solution of Bose--Einstein Condensates by a Normalized Gradient Flow
Weizhu Bao,Qiang Du +1 more
TL;DR: A continuous normalized gradient flow (CNGF) is presented and its energy diminishing property is proved, which provides a mathematical justification of the imaginary time method used in the physics literature to compute the ground state solution of Bose--Einstein condensates (BEC).
Journal ArticleDOI
Mathematical theory and numerical methods for Bose-Einstein condensation
Weizhu Bao,Yongyong Cai +1 more
TL;DR: In this article, the authors mainly review recent results on mathematical theory and numerical methods for Bose-Einstein condensation (BEC), based on the Gross-Pitaevskii equation (GPE).
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