A density matrix approach to the convergence of the self-consistent field iteration

Upadhyaya, Parikshit; Jarlebring, Elias; Rubensson, Emanuel H.

doi:10.3934/naco.2020018

Cited by 18 publications

(16 citation statements)

References 22 publications

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“…While the convergence of several SCF and direct minimization algorithms has been analyzed from a mathematical point of view (see e.g. [16,38,42,53,60,64] and references therein), the two approaches have not been compared in a systematic way to our knowledge. The purpose of this paper is to contribute to fill this gap, by focusing on very simple representatives of each class, namely the damped SCF iteration and the gradient descent.…”

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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“…In general, SCF exhibits linear local convergence when it converges. Convergence can be characterized in terms of gaps [35] (see also [34,Theorem 3.1]). Rather than reviewing the details of the convergence results, we refer the reader to the general characterizations, e.g., in [3,19,20,35] and the references therein.…”

Section: Local Convergence Of Algorithmmentioning

confidence: 99%

“…SCF is an iterative method that involves solving a linear eigenvalue problem in each step until convergence or self-consistency. The convergence of SCF and its variants has been studied in a number of works which can be classified into two broad categories: the optimization-based approach of looking at (1) as the optimality conditions of a minimization problem [8,[18][19][20] or different matrix analysis-based approaches [34,35]. For a discussion of similarities and differences among the two approaches, see [9].…”

Section: Introductionmentioning

confidence: 99%

Implicit algorithms for eigenvector nonlinearities

Jarlebring

Upadhyaya

2021

Numer Algor

Self Cite

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We study and derive algorithms for nonlinear eigenvalue problems, where the system matrix depends on the eigenvector, or several eigenvectors (or their corresponding invariant subspace). The algorithms are derived from an implicit viewpoint. More precisely, we change the Newton update equation in a way that the next iterate does not only appear linearly in the update equation. Although the modifications of the update equation make the methods implicit, we show how corresponding iterates can be computed explicitly. Therefore, we can carry out steps of the implicit method using explicit procedures. In several cases, these procedures involve a solution of standard eigenvalue problems. We propose two modifications, one of the modifications leads directly to a well-established method (the self-consistent field iteration) whereas the other method is to our knowledge new and has several attractive properties. Convergence theory is provided along with several simulations which illustrate the properties of the algorithms.

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“…Popular methods for solving the nonlinear eigenvalue problem are for example the Self Consistent Field Iteration (SCF), cf. [19,23,24,29,52], which requires to solve a linearized eigenvalue problem in each iteration. Algorithms that belong to the SCF class are for instance the Roothaan algorithm [48] or the optimal damping algorithm proposed in [23].…”

Section: Introductionmentioning

confidence: 99%

The J-method for the Gross–Pitaevskii eigenvalue problem

2021

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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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A density matrix approach to the convergence of the self-consistent field iteration

Cited by 18 publications

References 22 publications

Convergence Analysis of Direct Minimization and Self-Consistent Iterations

Convergence Analysis of Direct Minimization and Self-Consistent Iterations

Implicit algorithms for eigenvector nonlinearities

The J-method for the Gross–Pitaevskii eigenvalue problem

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