Sobolev Gradient Flow for the Gross--Pitaevskii Eigenvalue Problem: Global Convergence and Computational Efficiency

Henning, Patrick; Peterseim, Daniel

doi:10.1137/18m1230463

Cited by 44 publications

(97 citation statements)

References 36 publications

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“…This set is naturally equipped with the structure of a Riemannian manifold, and this allows the use of Riemann optimization algorithms [1,21]. Direct minimization algorithms are preferred for the Gross-Pitaevskii model with magnetic fields [3,18,27,29], for which the Aufbau principle is not satisfied in general. Gradient-type [2,17,49,61,66], Newton-type [5,15,68], and trust-region methods have also been designed to solve (1.1) for larger values of N .…”

Section: Introductionmentioning

confidence: 99%

Convergence Analysis of Direct Minimization and Self-Consistent Iterations

Cancès¹,

Kemlin²,

Levitt³

2021

SIAM J. Matrix Anal. Appl.

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This article is concerned with the numerical solution of subspace optimization problems, consisting of minimizing a smooth functional over the set of orthogonal projectors of fixed rank. Such problems are encountered in particular in electronic structure calculation (Hartree-Fock and Kohn-Sham Density Functional Theory-DFT-models). We compare from a numerical analysis perspective two simple representatives, the damped self-consistent field (SCF) iterations and the gradient descent algorithm, of the two classes of methods competing in the field: SCF and direct minimization methods. We derive asymptotic rates of convergence for these algorithms and analyze their dependence on the spectral gap and other properties of the problem. Our theoretical results are complemented by numerical simulations on a variety of examples, from toy models with tunable parameters to realistic Kohn-Sham computations. We also provide an example of chaotic behavior of the simple SCF iterations for a nonquadratic functional.

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Section: Introductionmentioning

confidence: 99%

Convergence Analysis of Direct Minimization and Self-Consistent Iterations

Cancès¹,

Kemlin²,

Levitt³

2021

SIAM J. Matrix Anal. Appl.

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“…It gives a more accurate description of the dynamics of Bosonic gases at ultra low temperature. With the presence of the nonlinear term \beta | u| 2 u, linear eigensolvers would fail, and the optimization of its variational form becomes the state-of-art solver; see, e.g., [22]. Apart from the PGD based on the L 2 metric, there can be other PGD algorithms based on other types of metrics and with different convergence theories, whose analysis is beyond the scope of this paper.…”

Section: Variational Eigenproblem On a Spherementioning

confidence: 99%

Analysis of Asymptotic Escape of Strict Saddle Sets in Manifold Optimization

Hou¹,

Li²,

Zhang³

2020

SIAM Journal on Mathematics of Data Science

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In this paper, we provide some analysis on the asymptotic escape of strict saddles in manifold optimization using the projected gradient descent algorithm (PGD). One of our main contributions is that we extend the current analysis to include nonisolated and possibly continuous saddle sets with complicated geometry. We prove that the PGD is able to escape strict critical submanifolds under certain conditions on the geometry and the distribution of the saddle point sets. We also show that the PGD may fail to escape strict saddles under weaker assumptions even if the saddle point set has zero measure. We apply this saddle analysis to the phase retrieval problem on the low-rank matrix manifold, prove that there are only a finite number of saddles, and that in a specific region, they are strict saddles with high probability. We also show the potential application of our analysis for a broader range of manifold optimization problems.

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“…Here we also mention the Projected Sobolev Gradient Flows (PSGFs), cf. [27,33,34,36,39,46,47,57], which form a subclass of the gradient flow methods. Sobolev gradients are the Riesz representants of the Fréchet derivative of the energy functional in a suitable Hilbert space.…”

Section: Introductionmentioning

confidence: 99%

“…Global convergence results for solving the GPEVP are very rare in the literature. To the best of our knowledge, global convergence had been so far only established for a damped PSGF suggested in [36], where corresponding analytical results can be found in [36,57].…”

Section: Introductionmentioning

confidence: 99%

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The J-method for the Gross–Pitaevskii eigenvalue problem

2021

Self Cite

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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.

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Sobolev Gradient Flow for the Gross--Pitaevskii Eigenvalue Problem: Global Convergence and Computational Efficiency

Cited by 44 publications

References 36 publications

Convergence Analysis of Direct Minimization and Self-Consistent Iterations

Convergence Analysis of Direct Minimization and Self-Consistent Iterations

Analysis of Asymptotic Escape of Strict Saddle Sets in Manifold Optimization

The J-method for the Gross–Pitaevskii eigenvalue problem

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