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Exponential convergence of Sobolev gradient descent for a class of nonlinear eigenproblems.
TLDR
Using the Łojasiewicz inequality, it is shown that a Sobolev gradient descent method with adaptive inner product converges exponentially fast to the ground state for the Gross-Pitaevskii eigenproblem.Abstract:
We propose to use the Łojasiewicz inequality as a general tool for analyzing the convergence rate of gradient descent on a Hilbert manifold, without resorting to the continuous gradient flow. Using this tool, we show that a Sobolev gradient descent method with adaptive inner product converges exponentially fast to the ground state for the Gross-Pitaevskii eigenproblem. This method can be extended to a class of general high-degree optimizations or nonlinear eigenproblems under certain conditions. We demonstrate this generalization by several examples, in particular a nonlinear Schrodinger eigenproblem with an extra high-order interaction term. Numerical experiments are presented for these problems.read more
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Convergence analysis of direct minimization and self-consistent iterations
TL;DR: This article compares from a numerical analysis perspective two simple representatives, the damped self-consistent field iterations and the gradient descent algorithm, of the two classes of methods competing in the field: SCF and direct minimization methods.
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Fast Global Convergence for Low-rank Matrix Recovery via Riemannian Gradient Descent with Random Initialization.
TL;DR: A new global analysis framework for a class of low-rank matrix recovery problems on the Riemannian manifold is proposed and the convergence guarantee is nearly optimal and almost dimension-free, which fully explains the numerical observations.
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The $J$-method for the Gross-Pitaevskii eigenvalue problem
TL;DR: A local linear rate of convergence is established and global convergence towards an eigenfunction is proved in the case of the Gross-Pitaevskii eigenvalue problem and this quantitative convergence analysis is closely connected to the J-method's unique feature of sensitivity with respect to spectral shifts.
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The J-method for the Gross–Pitaevskii eigenvalue problem
TL;DR: 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.
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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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A new Sobolev gradient method for direct minimization of the Gross-Pitaevskii energy with rotation
Ionut Danaila,Parimah Kazemi +1 more
TL;DR: This paper improves traditional steepest descent methods for the direct minimization of the Gross-Pitaevskii (GP) energy with rotation at two levels by defining a new inner product to equip the Sobolev space and derive the corresponding gradient.
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Computation of Ground States of the Gross--Pitaevskii Functional via Riemannian Optimization
Ionut Danaila,Bartosz Protas +1 more
TL;DR: In this article, the authors combine concepts from Riemannian optimization and the theory of Sobolev gradients to derive a new conjugate gradient method for direct minimization of the Gross-Pitaevskii energy functional with rotation.
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Two-Level discretization techniques for ground state computations of Bose-Einstein condensates
TL;DR: In this paper, a new methodology for computing ground states of Bose-Einstein condensates based on finite element discretizations on two different scales of numerical resolution is presented.
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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: A new normalized Sobolev gradient flow for the Gross-Pitaevskii eigenvalue problem based on an energy inner product that depends on time through the density of the flow itself and converges to an eigenfunction is proposed.
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Convergence analysis of direct minimization and self-consistent iterations
TL;DR: In this article, the numerical solution of subspace optimization problems, consisting of minimizing a smooth functional over the set of orthogonal projectors of fixed rank, is studied. But this paper is not concerned with the optimization of subspaces.