On the Convergence of the Self-Consistent Field Iteration for a Class of Nonlinear Eigenvalue Problems

Yang, Chao; Gao, Weiguo; Meza, Juan C.

doi:10.1137/080716293

Cited by 51 publications

(69 citation statements)

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“…In that example, the iterate x (k) oscillates and the sequence {f (k) } is not monotonically increasing. This convergence behavior of the SCF iteration has been previously observed in electronic structure calculations (see, e.g., [11,[13][14][15] [14,15]) have made efforts to analyze and explain this phenomenon. In particular, because finding a dominant eigenvector x (k+1) of E (k) = E(x (k) ) is equivalent to solving the following maximizing problem:…”

Section: A Trust-region Scf Iterationmentioning

confidence: 72%

“…It is shown that, for this problem, the SCF-like iteration converges globally to the global maximizer and the convergence rate is quadratic under a generic condition. However, in [14], for a class of nonlinear eigenvalue problems, oscillation would take place and will generate a sequence of approximate solutions that contain two convergent subsequences, neither of which converges to the solution for the related nonlinear eigenvalue problem. For our discussed problem (1.1), we will first show that for the special case D = µW (µ > 0), good global convergence properties hold, but in general, the same phenomenon as was discussed in [14] occurs.…”

Section: Global Convergence Of the Scf Iterationmentioning

confidence: 98%

“…However, in [14], for a class of nonlinear eigenvalue problems, oscillation would take place and will generate a sequence of approximate solutions that contain two convergent subsequences, neither of which converges to the solution for the related nonlinear eigenvalue problem. For our discussed problem (1.1), we will first show that for the special case D = µW (µ > 0), good global convergence properties hold, but in general, the same phenomenon as was discussed in [14] occurs. We will further introduce, in Section 4, as a remedy for the failure of convergence, a trust-region SCF iteration, which has been proven to be effective for minimizing the Kohn-Sham total energy function [11,13,15].…”

Section: Global Convergence Of the Scf Iterationmentioning

confidence: 98%

“…This technique has been used to analyze the convergence of the SCF iteration, for instance, in [14,16]. For (1.1), in particular, we will analyze the distance between span(x (k) ) and span(x * ), where span(u) denotes the one-dimensional subspace spanned by u, and x * is the limit point of {x (k) }.…”

Section: Local Convergence Of the Scf Iterationmentioning

confidence: 99%

“…In this paper, by relying closely upon the global optimality condition that states that (1.1) is equivalent to a nonlinear extreme eigenvalue problem, we apply the so-called self-consistent-field (SCF) iteration to the resulting nonlinear eigenvalue problem. The SCF iteration is currently one of the most widely used algorithms for solving the Hartree-Fock and Kohn-Sham equations in electronic structure calculations (see, e.g., [10][11][12][13][14][15]); moreover, in dimension reduction and feature extraction, the effective algorithm discussed in [16] is also an SCF-like iteration for the trace quotient (or trace ratio) optimization problem. In every step, our proposed SCF-like iteration finds a dominant eigenpair of a relevant matrix, which can be achieved by some sophisticated eigensolvers, and thereby, is also suitable for large-scale problems.…”

Section: Introductionmentioning

confidence: 99%

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On a self-consistent-field-like iteration for maximizing the sum of the Rayleigh quotients

Zhang

2014

Journal of Computational and Applied Mathematics

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h i g h l i g h t s• A self-consistent-field method for optimizing the sum of the Rayleigh quotients is proposed.• Local convergence and global convergence are established under some conditions. • Quadratic convergence is proved for some situations.• Numerical testing shows efficiency, in comparison with other optimization-based methods. MSC:Keywords: Rayleigh quotient The nonlinear eigenvalue problem The self-consistent-field iteration Trust region a b s t r a c tIn this paper, we consider efficient methods for maximizing x ⊤ Bxwhere B, D are symmetric matrices, and W is symmetric and positive definite. This problem can arise in the downlink of a multi-user MIMO system and in the sparse Fisher discriminant analysis in pattern recognition. It is already known that the problem of finding a global maximizer is closely associated with a nonlinear extreme eigenvalue problem. Rather than resorting to some general optimization methods, we introduce a self-consistent-field-like (SCF-like) iteration for directly solving the resulting nonlinear eigenvalue problem. The SCF iteration is widely used for solving the nonlinear eigenvalue problems arising in electronic structure calculations. One attractive feature of the SCF for our problem is that once it converges, the limit point not only satisfies the necessary local optimality conditions automatically, but also, and most importantly, satisfies a global optimality condition, which generally is not achievable in some optimization-based methods. The global convergence and local quadratic convergence rate are proved for certain situations. For the general case, we then discuss a trust-region SCF iteration for stabilizing the SCF iteration, which is of good global convergence behavior. Our preliminary numerical experiments show that these algorithms are more efficient than some optimization-based methods.

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Section: A Trust-region Scf Iterationmentioning

confidence: 72%

Section: Global Convergence Of the Scf Iterationmentioning

confidence: 98%

Section: Global Convergence Of the Scf Iterationmentioning

confidence: 98%

Section: Local Convergence Of the Scf Iterationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On a self-consistent-field-like iteration for maximizing the sum of the Rayleigh quotients

Zhang

2014

Journal of Computational and Applied Mathematics

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An analysis for the DIIS acceleration method used in quantum chemistry calculations

2011

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This work features an analysis for the acceleration technique DIIS that is standardly used in most of the important quantum chemistry codes, e.g. in DFT and Hartree-Fock calculations and in the Coupled Cluster method. Taking up results from [23], we show that for the general nonlinear case, DIIS corresponds to a projected quasi-Newton/ secant method. For linear systems, we establish connections to the wellknown GMRES solver and transfer according (positive as well as negative) convergence results to DIIS. In particular, we discuss the circumstances under which DIIS exhibits superlinear convergence behaviour. For the general nonlinear case, we then use these results to show that a DIIS step can be interpreted as step of a quasi-Newton method in which the Jacobian used in the Newton step is approximated by finite differences and in which the according linear system is solved by a GMRES procedure, and give according convergence estimates.

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Numerical Solution of the Kohn-Sham Equation by Finite Element Methods with an Adaptive Mesh Redistribution Technique

Bao

Liu

2012

J Sci Comput

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On the Convergence of the Self-Consistent Field Iteration for a Class of Nonlinear Eigenvalue Problems

Cited by 51 publications

References 5 publications

On a self-consistent-field-like iteration for maximizing the sum of the Rayleigh quotients

On a self-consistent-field-like iteration for maximizing the sum of the Rayleigh quotients

An analysis for the DIIS acceleration method used in quantum chemistry calculations

Numerical Solution of the Kohn-Sham Equation by Finite Element Methods with an Adaptive Mesh Redistribution Technique

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