The J-method for the Gross–Pitaevskii eigenvalue problem
TLDR
In this article, a modification of the J-method mimics an energy-decreasing discrete gradient flow is proposed, which allows both the selective approximation of excited states as well as the amplification of convergence beyond linear rates.Abstract:
This paper studies the J-method of [E. Jarlebring, S. Kvaal, W. Michiels. SIAM J. Sci. Comput. 36-4:A1978-A2001, 2014] for nonlinear eigenvector problems in a general Hilbert space framework. This is the basis for variational discretization techniques and a mesh-independent numerical analysis. A simple modification of the method mimics an energy-decreasing discrete gradient flow. In the case of the Gross–Pitaevskii eigenvalue problem, we prove global convergence towards an eigenfunction for a damped version of the J-method. More importantly, when the iterations are sufficiently close to an eigenfunction, the damping can be switched off and we recover a local linear convergence rate previously known from the discrete setting. This quantitative convergence analysis is closely connected to the J-method’s unique feature of sensitivity with respect to spectral shifts. Contrary to classical gradient flows, this allows both the selective approximation of excited states as well as the amplification of convergence beyond linear rates in the spirit of the Rayleigh quotient iteration for linear eigenvalue problems. These advantageous convergence properties are demonstrated in a series of numerical experiments involving exponentially localized states under disorder potentials and vortex lattices in rotating traps.read more
Citations
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Localization and delocalization of ground states of Bose-Einstein condensates under disorder.
TL;DR: This paper studies the localization behaviour of Bose-Einstein condensates in disorder potentials, modeled by a Gross-Pitaevskii eigenvalue problem on a bounded interval and quantifies exponential localization of the ground state.
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Sobolev gradient flow for the Gross-Pitaevskii eigenvalue problem: global convergence and computational efficiency
Patrick Henning,Daniel Peterseim +1 more
TL;DR: In this paper, a new normalized Sobolev gradient flow for the Gross-Pitaevskii eigenvalue problem is proposed, which is based on an energy inner product that depends on time through the density of the flow itself.
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Localized computation of eigenstates of random Schr\"odinger operators
Robert Altmann,Daniel Peterseim +1 more
TL;DR: A reliable numerical scheme which provides localized approximations of low-energy eigenstates of the linear random Schrodinger operator and is based on a preconditioned inverse iteration including an optimal multigrid solver which spreads information only locally.
Journal ArticleDOI
The dependency of spectral gaps on the convergence of the inverse iteration for a nonlinear eigenvector problem
TL;DR: 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.
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Localization and Delocalization of Ground States of Bose--Einstein Condensates Under Disorder
TL;DR: In this article , the authors studied the localization behavior of Bose-Einstein condensates in disorder potentials, modeled by a Gross-Pitaevskii eigenvalue problem on a bounded interval.
References
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Book
Iterative Solution of Nonlinear Equations in Several Variables
J.M. Ortega,Werner C. Rheinboldt +1 more
TL;DR: In this article, the authors present a list of basic reference books for convergence of Minimization Methods in linear algebra and linear algebra with a focus on convergence under partial ordering.
Journal ArticleDOI
Theory of Bose-Einstein condensation in trapped gases
TL;DR: In this article, the authors reviewed the Bose-Einstein condensation of dilute gases in traps from a theoretical perspective and provided a framework to understand the main features of the condensation and role of interactions between particles.
Journal ArticleDOI
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.
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