SHARP ESTIMATION of CONVERGENCE RATE for SELF-CONSISTENT FIELD ITERATION to SOLVEEIGENVECTOR-DEPENDENT NONLINEAR EIGENVALUE PROBLEMS

Zhaojun Bai; Ren Cang Li; Ding Lu

doi:10.1137/20M136606X

SHARP ESTIMATION of CONVERGENCE RATE for SELF-CONSISTENT FIELD ITERATION to SOLVEEIGENVECTOR-DEPENDENT NONLINEAR EIGENVALUE PROBLEMS

Zhaojun Bai, Ren Cang Li, Ding Lu

Mathematics

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

Abstract

We present a comprehensive convergence analysis for the self-consistent field (SCF) iteration to solve a class of nonlinear eigenvalue problems with eigenvector dependency (NEPvs). Using the tangent-Angle matrix as an intermediate measure for approximation error, we establish new formulas for two fundamental quantities that characterize the local convergence behavior of the plain SCF: The local contraction factor and the local asymptotic average contraction factor. In comparison with previously established results, new convergence rate estimates provide much sharper bounds on the convergence speed. As an application, we extend the convergence analysis to a popular SCF variant-The level-shifted SCF. The effectiveness of the convergence rate estimates is demonstrated numerically for NEPvs arising from solving the Kohn-Sham equation in electronic structure calculation and the Gross-Pitaevskii equation for modeling of the Bose-Einstein condensation.

Original language	English
Pages (from-to)	301-327
Number of pages	27
Journal	SIAM Journal on Matrix Analysis and Applications
Volume	43
Issue number	1
DOIs	https://doi.org/10.1137/20M136606X
State	Published - 2022

Bibliographical note

Funding Information:
˚Received by the editors September 10, 2020; accepted for publication (in revised form) by K. Meerbergen October 25, 2021; published electronically February 28, 2022. https://doi.org/10.1137/20M136606X Funding: The first author was supported by NSF grant DMS-1913364. The second author was supported in part by NSF grants DMS-1719620 and DMS-2009689. The third author was supported by NSF grant DMS-2110731. :Department of Computer Science, University of California, Davis, Davis, CA 95616 USA (bai@cs.ucdavis.edu). ;Department of Mathematics, University of Texas at Arlington, Arlington, TX 76019 USA (rcli@uta.edu). \S Department of Mathematics, University of Kentucky, Lexington, KY 40506 USA (ding. lu@uky.edu).

Publisher Copyright:
© 2022 Society for Industrial and Applied Mathematics Publications. All rights reserved.

Keywords

convergence factor
level-shifted SCF
nonlinear eigenvalue problem
self-consistent field iteration

ASJC Scopus subject areas

Analysis

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10.1137/20M136606X

Cite this

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title = "SHARP ESTIMATION of CONVERGENCE RATE for SELF-CONSISTENT FIELD ITERATION to SOLVEEIGENVECTOR-DEPENDENT NONLINEAR EIGENVALUE PROBLEMS",

abstract = "We present a comprehensive convergence analysis for the self-consistent field (SCF) iteration to solve a class of nonlinear eigenvalue problems with eigenvector dependency (NEPvs). Using the tangent-Angle matrix as an intermediate measure for approximation error, we establish new formulas for two fundamental quantities that characterize the local convergence behavior of the plain SCF: The local contraction factor and the local asymptotic average contraction factor. In comparison with previously established results, new convergence rate estimates provide much sharper bounds on the convergence speed. As an application, we extend the convergence analysis to a popular SCF variant-The level-shifted SCF. The effectiveness of the convergence rate estimates is demonstrated numerically for NEPvs arising from solving the Kohn-Sham equation in electronic structure calculation and the Gross-Pitaevskii equation for modeling of the Bose-Einstein condensation.",

keywords = "convergence factor, level-shifted SCF, nonlinear eigenvalue problem, self-consistent field iteration",

author = "Zhaojun Bai and Li, {Ren Cang} and Ding Lu",

note = "Funding Information: ˚Received by the editors September 10, 2020; accepted for publication (in revised form) by K. Meerbergen October 25, 2021; published electronically February 28, 2022. https://doi.org/10.1137/20M136606X Funding: The first author was supported by NSF grant DMS-1913364. The second author was supported in part by NSF grants DMS-1719620 and DMS-2009689. The third author was supported by NSF grant DMS-2110731. :Department of Computer Science, University of California, Davis, Davis, CA 95616 USA (bai@cs.ucdavis.edu). ;Department of Mathematics, University of Texas at Arlington, Arlington, TX 76019 USA (rcli@uta.edu). \S Department of Mathematics, University of Kentucky, Lexington, KY 40506 USA (ding. lu@uky.edu). Publisher Copyright: {\textcopyright} 2022 Society for Industrial and Applied Mathematics Publications. All rights reserved.",

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AU - Bai, Zhaojun

AU - Li, Ren Cang

AU - Lu, Ding

N1 - Funding Information: ˚Received by the editors September 10, 2020; accepted for publication (in revised form) by K. Meerbergen October 25, 2021; published electronically February 28, 2022. https://doi.org/10.1137/20M136606X Funding: The first author was supported by NSF grant DMS-1913364. The second author was supported in part by NSF grants DMS-1719620 and DMS-2009689. The third author was supported by NSF grant DMS-2110731. :Department of Computer Science, University of California, Davis, Davis, CA 95616 USA (bai@cs.ucdavis.edu). ;Department of Mathematics, University of Texas at Arlington, Arlington, TX 76019 USA (rcli@uta.edu). \S Department of Mathematics, University of Kentucky, Lexington, KY 40506 USA (ding. lu@uky.edu). Publisher Copyright: © 2022 Society for Industrial and Applied Mathematics Publications. All rights reserved.

PY - 2022

Y1 - 2022

N2 - We present a comprehensive convergence analysis for the self-consistent field (SCF) iteration to solve a class of nonlinear eigenvalue problems with eigenvector dependency (NEPvs). Using the tangent-Angle matrix as an intermediate measure for approximation error, we establish new formulas for two fundamental quantities that characterize the local convergence behavior of the plain SCF: The local contraction factor and the local asymptotic average contraction factor. In comparison with previously established results, new convergence rate estimates provide much sharper bounds on the convergence speed. As an application, we extend the convergence analysis to a popular SCF variant-The level-shifted SCF. The effectiveness of the convergence rate estimates is demonstrated numerically for NEPvs arising from solving the Kohn-Sham equation in electronic structure calculation and the Gross-Pitaevskii equation for modeling of the Bose-Einstein condensation.

AB - We present a comprehensive convergence analysis for the self-consistent field (SCF) iteration to solve a class of nonlinear eigenvalue problems with eigenvector dependency (NEPvs). Using the tangent-Angle matrix as an intermediate measure for approximation error, we establish new formulas for two fundamental quantities that characterize the local convergence behavior of the plain SCF: The local contraction factor and the local asymptotic average contraction factor. In comparison with previously established results, new convergence rate estimates provide much sharper bounds on the convergence speed. As an application, we extend the convergence analysis to a popular SCF variant-The level-shifted SCF. The effectiveness of the convergence rate estimates is demonstrated numerically for NEPvs arising from solving the Kohn-Sham equation in electronic structure calculation and the Gross-Pitaevskii equation for modeling of the Bose-Einstein condensation.

KW - convergence factor

KW - level-shifted SCF

KW - nonlinear eigenvalue problem

KW - self-consistent field iteration

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SHARP ESTIMATION of CONVERGENCE RATE for SELF-CONSISTENT FIELD ITERATION to SOLVEEIGENVECTOR-DEPENDENT NONLINEAR EIGENVALUE PROBLEMS

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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