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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Vydáno v:	Journal of scientific computing Ročník 94; číslo 2; s. 42
Hlavní autoři:	Xu, Fei, Xie, Manting, Yue, Meiling
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	New York Springer US 01.02.2023 Springer Nature B.V
Témata:	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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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. 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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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