Multigrid Method for Nonlinear Eigenvalue Problems Based on Newton Iteration

In this paper, a novel multigrid method based on Newton iteration is proposed to solve nonlinear eigenvalue problems. Instead of handling the eigenvalue λ and eigenfunction u separately, we treat the eigenpair ( λ , u ) as one element in a product space R × H 0 1 ( Ω ) . Then in the presented multig...

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Published in:	Journal of scientific computing Vol. 94; no. 2; p. 42
Main Authors:	Xu, Fei, Xie, Manting, Yue, Meiling
Format:	Journal Article
Language:	English
Published:	New York Springer US 01.02.2023 Springer Nature B.V
Subjects:	Algorithms Approximation Boundary value problems Computational mathematics Computational Mathematics and Numerical Analysis Convergence Efficiency Eigenvalues Eigenvectors Estimates Finite element analysis Iterative methods Lagrange multiplier Mathematical and Computational Engineering Mathematical and Computational Physics Mathematics Mathematics and Statistics Nonlinear equations Theoretical Newton iteration Nonlinear eigenvalue problems Multigrid method
ISSN:	0885-7474, 1573-7691
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Abstract	In this paper, a novel multigrid method based on Newton iteration is proposed to solve nonlinear eigenvalue problems. Instead of handling the eigenvalue λ and eigenfunction u separately, we treat the eigenpair ( λ , u ) as one element in a product space R × H 0 1 ( Ω ) . Then in the presented multigrid method, only one discrete linear boundary value problem needs to be solved for each level of the multigrid sequence. Because we avoid solving large-scale nonlinear eigenvalue problems directly, the overall efficiency is significantly improved. The optimal error estimate and linear computational complexity can be derived simultaneously. In addition, we also provide an improved multigrid method coupled with a mixing scheme to further guarantee the convergence and stability of the iteration scheme. More importantly, we prove convergence for the residuals after each iteration step. For nonlinear eigenvalue problems, such theoretical analysis is missing from the existing literatures on the mixing iteration scheme.
AbstractList	In this paper, a novel multigrid method based on Newton iteration is proposed to solve nonlinear eigenvalue problems. Instead of handling the eigenvalue λ and eigenfunction u separately, we treat the eigenpair (λ,u) as one element in a product space R×H01(Ω). Then in the presented multigrid method, only one discrete linear boundary value problem needs to be solved for each level of the multigrid sequence. Because we avoid solving large-scale nonlinear eigenvalue problems directly, the overall efficiency is significantly improved. The optimal error estimate and linear computational complexity can be derived simultaneously. In addition, we also provide an improved multigrid method coupled with a mixing scheme to further guarantee the convergence and stability of the iteration scheme. More importantly, we prove convergence for the residuals after each iteration step. For nonlinear eigenvalue problems, such theoretical analysis is missing from the existing literatures on the mixing iteration scheme. In this paper, a novel multigrid method based on Newton iteration is proposed to solve nonlinear eigenvalue problems. Instead of handling the eigenvalue λ and eigenfunction u separately, we treat the eigenpair ( λ , u ) as one element in a product space R × H 0 1 ( Ω ) . Then in the presented multigrid method, only one discrete linear boundary value problem needs to be solved for each level of the multigrid sequence. Because we avoid solving large-scale nonlinear eigenvalue problems directly, the overall efficiency is significantly improved. The optimal error estimate and linear computational complexity can be derived simultaneously. In addition, we also provide an improved multigrid method coupled with a mixing scheme to further guarantee the convergence and stability of the iteration scheme. More importantly, we prove convergence for the residuals after each iteration step. For nonlinear eigenvalue problems, such theoretical analysis is missing from the existing literatures on the mixing iteration scheme.
ArticleNumber	42
Author	Yue, Meiling Xie, Manting Xu, Fei
Author_xml	– sequence: 1 givenname: Fei surname: Xu fullname: Xu, Fei email: xufei@lsec.cc.ac.cn organization: Institute of Computational Mathematics, Department of Mathematics, Faculty of Science, Beijing University of Technology – sequence: 2 givenname: Manting surname: Xie fullname: Xie, Manting organization: Center for Applied Mathematics, Tianjin University – sequence: 3 givenname: Meiling surname: Yue fullname: Yue, Meiling organization: School of Mathematics and Statistics, Beijing Technology and Business University
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Cites_doi	10.1137/130919398 10.1007/978-1-4757-4338-8 10.1137/1034116 10.1051/m2an/2011038 10.1145/321296.321305 10.4208/jcm.2009.27.4.018 10.1007/s00211-003-0498-1 10.1016/j.jcp.2017.11.024 10.1103/RevModPhys.53.603 10.1016/j.jcp.2012.04.036 10.1002/mma.793 10.1007/s11425-015-0234-x 10.1137/17M115935X 10.1007/s10915-010-9358-1 10.1016/j.jcp.2012.04.002 10.1103/PhysRev.140.A1133 10.1137/120880604 10.1017/S0962492904000212 10.1016/j.advwatres.2011.12.013 10.1016/0041-5553(62)90031-9 10.1007/978-94-015-8527-9 10.1016/j.jcp.2016.07.009 10.1002/nla.617 10.1090/S0025-5718-99-01180-1 10.1016/j.cma.2011.02.008 10.1007/978-3-662-02427-0 10.1088/0951-7715/17/2/010 10.1090/S0025-5718-1977-0431719-X 10.1007/978-1-4612-3172-1 10.1137/S1064827503422956 10.1088/0951-7715/16/1/307 10.1137/18M1206151
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Comput.2013355S277S29810.1137/1208806041284.82009 BrandtAMulti-level adaptive solutions to boundary-value problemsMath. Comput.19773133339010.1090/S0025-5718-1977-0431719-X0373.65054 StasiakPMatsenMWEfficiency of pseudo-spectral algorithms with anderson mixing for the SCFT of periodic block-copolymer phasesEur. Phys. J. E20113411019 BrennerSScottLThe Mathematical Theory of Finite Element Methods1994New YorkSpringer10.1007/978-1-4757-4338-80804.65101 CiarletPGLionsJLFinite Element Methods, Handbook of Numerical Analysis1991AmsterdamNorth Holland Anderson, D.G.: Iterative procedures for nonlinear integral equations. J. Assoc. Comput. Mach. 12, 547–560 (1965) HackbuschWMulti-grid Methods and Applications1985BerlinSpringer10.1007/978-3-662-02427-00595.65106 BaoGHuGLiuDAn h-adaptive finite element solver for the calculations of the electronic structuresJ. Comput. 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References_xml	– reference: YserentantHOn the regularity of the electronic Schrödinger equation in Hilbert spaces of mixed derivativesNumer. Math.200498473175910.1007/s00211-003-0498-11062.35100 – reference: BaoWDuQComputing the ground state solution of Bose–Einstein condensates by a normalized gradient flowSIAM J. Sci. Comput.2004251674169710.1137/S10648275034229561061.82025 – reference: BrandtAMulti-level adaptive solutions to boundary-value problemsMath. Comput.19773133339010.1090/S0025-5718-1977-0431719-X0373.65054 – reference: PollockSRebholzLXiaoMAnderson-accelerated convergence of picard iterations for incompressible Navier-Stokes equationsSIAM J. Numer. Anal.201957261563710.1137/18M12061511412.65198 – reference: ChenHXieHXuFA full multigrid method for eigenvalue problemsJ. Comput. Phys.201632274775910.1016/j.jcp.2016.07.0091352.65459 – reference: FedorenkoRPA relaxation method for solving elliptic difference equationsUSSR Comput. Math. Math. Phys.1961141092109610.1016/0041-5553(62)90031-90163.39303 – reference: ZhouAFinite dimensional approximations for the electronic ground state solution of a molecular systemMath. Methods Appl. Sci.20073042944710.1002/mma.7931119.35095 – reference: CancèsEChakirRMadayYNumerical analysis of nonlinear eigenvalue problemsJ. Sci. Comput.2010451–39011710.1007/s10915-010-9358-11203.65237 – reference: CaiYZhangLBaiZLiROn an eigenvector-dependent nonlinear eigenvalue problemSIAM J. Matrix Anal. Appl.20183931360138210.1137/17M115935X1401.65036 – reference: Anderson, D.G.: Iterative procedures for nonlinear integral equations. J. Assoc. Comput. Mach. 12, 547–560 (1965) – reference: ChenHHeLZhouAFinite element approximations of nonlinear eigenvalue problems in quantum physicsComput. Methods Appl. Mech. Engrg.2011200211846186510.1016/j.cma.2011.02.0081228.81026 – reference: LinLYangCElliptic preconditioner for accelerating the self-consistent field iteration in Kohn-Sham density functional theorySIAM J. Sci. Comput.2013355S277S29810.1137/1208806041284.82009 – reference: LottPAWalkerHFWoodwardCSYangUMAn accelerated Picard method for nonlinear systems related to variably saturated flowAdv. Water Resour.2012389210110.1016/j.advwatres.2011.12.013 – reference: KohnWShamLSelf-consistent equations including exchange and correlation effectsPhys. Rev. A19651401133113810.1103/PhysRev.140.A1133 – reference: LiebEHThomas-Fermi and related theories of atoms and moleculesRev. Mod. Phys.19815360364110.1103/RevModPhys.53.6031114.81336 – reference: CancèsEChakirRMadayYNumerical analysis of the planewave discretization of some orbital-free and Kohn-Sham modelsESAIM Math. Model. Numer. Anal.20124634138810.1051/m2an/20110381278.82003 – reference: HuGXieHXuFA multilevel correction adaptive finite element method for Kohn–Sham equationJ. Comput. Phys.201835543644910.1016/j.jcp.2017.11.0241380.65371 – reference: ZhouAAn analysis of finite-dimensional approximations for the ground state solution of Bose-Einstein condensatesNonlinearity20041754155010.1088/0951-7715/17/2/0101051.35094 – reference: ShaidurovVVMultigrid Methods for Finite Element1995NetherlandsKluwer Academic Publics10.1007/978-94-015-8527-90837.65118 – reference: BaoGHuGLiuDAn h-adaptive finite element solver for the calculations of the electronic structuresJ. Comput. Phys.2012231144967497910.1016/j.jcp.2012.04.0021245.65125 – reference: TothAKelleyCTConvergence analysis for Anderson accelerationSIAM J. Numer. Anal.201553280581910.1137/1309193981312.65083 – reference: BrennerSScottLThe Mathematical Theory of Finite Element Methods1994New YorkSpringer10.1007/978-1-4757-4338-80804.65101 – reference: ChenHLiuFZhouAA two-scale higher-order finite element discretization for Schrödinger equationJ. Comput. Math.2009273153371212.65432 – reference: FangHSaadYTwo classes of multisecant methods for nonlinear accelerationNumer. 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Snippet	In this paper, a novel multigrid method based on Newton iteration is proposed to solve nonlinear eigenvalue problems. Instead of handling the eigenvalue λ and... In this paper, a novel multigrid method based on Newton iteration is proposed to solve nonlinear eigenvalue problems. Instead of handling the eigenvalue λ and...
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SubjectTerms	Algorithms Approximation Boundary value problems Computational mathematics Computational Mathematics and Numerical Analysis Convergence Efficiency Eigenvalues Eigenvectors Estimates Finite element analysis Iterative methods Lagrange multiplier Mathematical and Computational Engineering Mathematical and Computational Physics Mathematics Mathematics and Statistics Nonlinear equations Theoretical
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Title	Multigrid Method for Nonlinear Eigenvalue Problems Based on Newton Iteration
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