Linear response eigenvalue problem solved by extended locally optimal preconditioned conjugate gradient methods
The locally optimal block preconditioned 4-d conjugate gradient method （LOBP4dCG） for the linear response eigenvalue problem was proposed by Bai and Li （2013） and later was extended to the generalized linear response eigenvalue problem by Bai and Li （2014）. We put forward two improvements to the met...
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| Published in | Science China. Mathematics Vol. 59; no. 8; pp. 1443 - 1460 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Beijing
Science China Press
01.08.2016
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| Subjects | |
| Online Access | Get full text |
| ISSN | 1674-7283 1869-1862 |
| DOI | 10.1007/s11425-016-0297-1 |
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| Abstract | The locally optimal block preconditioned 4-d conjugate gradient method （LOBP4dCG） for the linear response eigenvalue problem was proposed by Bai and Li （2013） and later was extended to the generalized linear response eigenvalue problem by Bai and Li （2014）. We put forward two improvements to the method： A shifting deflation technique and an idea of extending the search subspace. The deflation technique is able to deflate away converged eigenpairs from future computation, and the idea of extending the search subspace increases convergence rate per iterative step. The resulting algorithm is called the extended LOBP4dCG （ELOBP4dCG）. Numerical results of the ELOBP4dCG strongly demonstrate the capability of deflation technique and effec- tiveness the search space extension for solving linear response eigenvalue problems arising from linear response analysis of two molecule systems. |
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| AbstractList | The locally optimal block preconditioned 4-d conjugate gradient method (LOBP4dCG) for the linear response eigenvalue problem was proposed by Bai and Li (2013) and later was extended to the generalized linear response eigenvalue problem by Bai and Li (2014). We put forward two improvements to the method: A shifting deflation technique and an idea of extending the search subspace. The deflation technique is able to deflate away converged eigenpairs from future computation, and the idea of extending the search subspace increases convergence rate per iterative step. The resulting algorithm is called the extended LOBP4dCG (ELOBP4dCG). Numerical results of the ELOBP4dCG strongly demonstrate the capability of deflation technique and effectiveness the search space extension for solving linear response eigenvalue problems arising from linear response analysis of two molecule systems. The locally optimal block preconditioned 4-d conjugate gradient method （LOBP4dCG） for the linear response eigenvalue problem was proposed by Bai and Li （2013） and later was extended to the generalized linear response eigenvalue problem by Bai and Li （2014）. We put forward two improvements to the method： A shifting deflation technique and an idea of extending the search subspace. The deflation technique is able to deflate away converged eigenpairs from future computation, and the idea of extending the search subspace increases convergence rate per iterative step. The resulting algorithm is called the extended LOBP4dCG （ELOBP4dCG）. Numerical results of the ELOBP4dCG strongly demonstrate the capability of deflation technique and effec- tiveness the search space extension for solving linear response eigenvalue problems arising from linear response analysis of two molecule systems. |
| Author | BAI ZhaoJun LI RenCang LIN WenWeia |
| AuthorAffiliation | Department of Computer Science and Department of Mathematics, University of California at Davis, Davis, CA 95616, USA Department of Mathematics, University of Texas at Arlington, Arlington, TX 76019, USA Department of Applied Mathematics, Taiwan Chiao Tung University, Hsinehu 300, China |
| Author_xml | – sequence: 1 givenname: ZhaoJun surname: Bai fullname: Bai, ZhaoJun email: bai@cs.ucdavis.edu organization: Department of Computer Science and Department of Mathematics, University of California at Davis – sequence: 2 givenname: RenCang surname: Li fullname: Li, RenCang organization: Department of Mathematics, University of Texas at Arlington – sequence: 3 givenname: WenWei surname: Lin fullname: Lin, WenWei organization: Department of Applied Mathematics, Taiwan Chiao Tung University |
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| Cites_doi | 10.1137/110838972 10.1016/0021-9991(88)90081-2 10.1137/110838960 10.1088/0953-8984/21/39/395502 10.1016/j.cam.2008.10.071 10.1016/0024-3795(92)90231-X 10.1134/1.568257 10.1016/0024-3795(93)00126-K 10.1080/03081087.2013.803242 10.1137/1.9780898719604 10.1007/978-3-642-61852-9 10.1137/140990735 10.1016/j.jcp.2006.02.007 10.1137/S1064827500366124 10.1137/1.9781611971446 10.1007/BF02141743 10.1016/0029-5582(61)90364-9 10.1145/1067967.1067973 10.1016/j.laa.2012.12.003 10.1007/s11075-015-9987-4 10.1016/j.laa.2010.06.034 10.1137/S1064827500382579 10.1063/1.1385368 10.1103/PhysRevB.40.12255 10.1137/0717059 10.1007/s10543-014-0472-6 10.1007/s00211-014-0681-6 10.1063/1.448223 |
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| Keywords | linear response 81Q15 conjugate-gradient 65L15 deflation eigenvalue problem 15A18 |
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| Notes | eigenvalue problem, linear response, deflation, conjugate-gradient, deflation The locally optimal block preconditioned 4-d conjugate gradient method （LOBP4dCG） for the linear response eigenvalue problem was proposed by Bai and Li （2013） and later was extended to the generalized linear response eigenvalue problem by Bai and Li （2014）. We put forward two improvements to the method： A shifting deflation technique and an idea of extending the search subspace. The deflation technique is able to deflate away converged eigenpairs from future computation, and the idea of extending the search subspace increases convergence rate per iterative step. The resulting algorithm is called the extended LOBP4dCG （ELOBP4dCG）. Numerical results of the ELOBP4dCG strongly demonstrate the capability of deflation technique and effec- tiveness the search space extension for solving linear response eigenvalue problems arising from linear response analysis of two molecule systems. 11-5837/O1 ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 23 |
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| References | BaiZLiR-CMinimization principle for the linear response eigenvalue problem, II: ComputationSIAM J Matrix Anal Appl201334392416304681010.1137/1108389721311.65102 ThoulessD JThe Quantum Mechanics of Many-Body Systems1972New YorkAcademic0103.23502 ThoulessD JVibrational states of nuclei in the random phase approximationNuclear Phys196122789512119310.1016/0029-5582(61)90364-90091.23103 MarquesM ACastroARubioAAssessment of exchange-correlation functionals for the calculation of dynamical properties of small clusters in time-dependent density functional theoryJ Chem Phys20011153006301410.1063/1.1385368 ZhouYLiR-CBounding the spectrum of large Hermitian matricesLinear Algebra Appl2011435480493279458710.1016/j.laa.2010.06.0341221.15022 ImakuraADuLSakuraiTError bounds of Rayleigh-Ritz type contour integral-based eigensolver for solving generalized eigenvalue problemsNumer Algor201671103120343934810.1007/s11075-015-9987-41333.65039 LiR-CBaiZGaoWSuY FRayleigh quotient based optimization methods for eigenvalue problemsMatrix Functions and Matrix Equations, vol. 19. Series in Contemporary Applied Mathematics2015SingaporeWorld Scientific7610810.1142/9789814675772_0004 MoneyJYeQEIGIFP: A MATLAB program for solving large symmetric generalized eigenvalue problemsACM Trans Math Software200531270279226679310.1145/1067967.10679731070.65531 YeQAn adaptive block Lanczos algorithmNumer Algor19961297110142355010.1007/BF021417430859.65032 KnyazevA VToward the optimal preconditioned eigensolver: Locally optimal block preconditioned conjugate gradient methodSIAM J Sci Comput200123517541186126310.1137/S10648275003661240992.65028 QuillenPYeQA block inverse-free preconditioned Krylov subspace method for symmetric generalized eigenvalue problemsJ Comput Appl Math201023312981313255936510.1016/j.cam.2008.10.0711186.65044 GolubGYeQAn inverse free preconditioned Krylov subspace methods for symmetric eigenvalue problemsSIAM J Sci Comput200224312334192442710.1137/S10648275003825791016.65017 LiR-CZhangL-HConvergence of block Lanczos method for eigenvalue clustersNumer Math201513183113338332910.1007/s00211-014-0681-61334.65073 OlsenJJørgensenPLinear and nonlinear response functions for an exact state and for an MCSCF stateJ Chem Phys1985823235326410.1063/1.448223 LiangXLiR-CExtensions of Wielandt’s min-max principles for positive semi-definite pencilsLinear Multilinear Algebra20146210321048324694210.1080/03081087.2013.8032421307.15015 HetmaniukULehoucqRBasis selection in LOBPCGJ Comput Phys2006218324332226795510.1016/j.jcp.2006.02.0071104.65031 SaadYOn the rates of convergence of the Lanczos and the block-Lanczos methodsSIAM J Numer Anal19801568770658875510.1137/07170590456.65016 CarsonECommunication-avoiding Krylov Subspace Methods in Theory and Practice2015BerkeleyUniversity of California FlaschkaULinW-WWuJ-LA KQZ algorithm for solving linear-response eigenvalue equationsLinear Algebra Appl199216593123114974810.1016/0024-3795(92)90231-X0759.65013 BaiZLiR-CMinimization principles and computation for the generalized linear response eigenvalue problemBIT Numer Math2014543154317795410.1007/s10543-014-0472-61293.65053 GiannozziPQUANTUM ESPRESSO: A modular and open-source software project for quantum simulations of materialsJ Phys Condensed Matter20092139550210.1088/0953-8984/21/39/395502 RoccaDTime-Dependent Density Functional Perturbation Theory: New algorithms with Applications to Molecular Spectra2007TriesteThe International School for Advanced Studies Kovač-StrikoJVeselićKTrace minimization and definiteness of symmetric pencilsLinear Algebra Appl1995216139158131998110.1016/0024-3795(93)00126-K0821.15008 LiangXLiR-CBaiZTrace minimization principles for positive semi-definite pencilsLinear Algebra Appl201343830853106301805910.1016/j.laa.2012.12.0031262.15010 WenZZhangYBlock algorithms with augmented Rayleigh-Ritz projections for large-scale eigenpair computation2015 OlsenJJensen AaH JJørgensenPSolution of the large matrix equations which occur in response theoryJ Comput Phys19887426528210.1016/0021-9991(88)90081-20636.65034 GolubG HVan LoanC FMatrix Computations1996Baltimore-MarylandJohns Hopkins University Press0865.65009 RoccaDIterative diagonalization of non-hermitian eigenproblems in time-dependent density functional and manybody perturbation theory2012BostonPresentation at Session B39, the APS Marching Meeting TsiperE VVariational procedure and generalized Lanczos recursion for small-amplitude classical oscillationsJETP Letters19997075175510.1134/1.568257 BaiZLiR-CMinimization principles for the linear response eigenvalue problem, I: TheorySIAM J Matrix Anal Appl20123310751100302346510.1137/1108389601263.65078 AndersonEBaiZBischofCLAPACK Users’ Guide1999PhiladelphiaSIAM10.1137/1.97808987196040755.65028 LancasterPYeQVariational properties and Rayleigh quotient algorithms for symmetric matrix pencilsOper Theory Adv Appl19894024727810383170676.15007 DemmelJApplied Numerical Linear Algebra1997PhiladelphiaSIAM10.1137/1.97816119714460879.65017 RingPSchuckPThe Nuclear Many-Body Problem1980New YorkSpringer-Verlag10.1007/978-3-642-61852-9 CarsonEDemmelJAccuracy of the s-step Lanczos method for the symmetric eigenproblem in finite precisionSIAM J Matrix Anal Appl201536793819335763110.1137/1409907351319.65024 TeterMPayneMAllanDSolution of Schr¨odinger equation for large systemsPhys Rev B198940122551226310.1103/PhysRevB.40.12255 A Imakura (297_CR13) 2016; 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| References_xml | – reference: OlsenJJørgensenPLinear and nonlinear response functions for an exact state and for an MCSCF stateJ Chem Phys1985823235326410.1063/1.448223 – reference: Kovač-StrikoJVeselićKTrace minimization and definiteness of symmetric pencilsLinear Algebra Appl1995216139158131998110.1016/0024-3795(93)00126-K0821.15008 – reference: LiR-CBaiZGaoWSuY FRayleigh quotient based optimization methods for eigenvalue problemsMatrix Functions and Matrix Equations, vol. 19. Series in Contemporary Applied Mathematics2015SingaporeWorld Scientific7610810.1142/9789814675772_0004 – reference: ThoulessD JVibrational states of nuclei in the random phase approximationNuclear Phys196122789512119310.1016/0029-5582(61)90364-90091.23103 – reference: CarsonEDemmelJAccuracy of the s-step Lanczos method for the symmetric eigenproblem in finite precisionSIAM J Matrix Anal Appl201536793819335763110.1137/1409907351319.65024 – reference: ThoulessD JThe Quantum Mechanics of Many-Body Systems1972New YorkAcademic0103.23502 – reference: YeQAn adaptive block Lanczos algorithmNumer Algor19961297110142355010.1007/BF021417430859.65032 – reference: KnyazevA VToward the optimal preconditioned eigensolver: Locally optimal block preconditioned conjugate gradient methodSIAM J Sci Comput200123517541186126310.1137/S10648275003661240992.65028 – reference: ImakuraADuLSakuraiTError bounds of Rayleigh-Ritz type contour integral-based eigensolver for solving generalized eigenvalue problemsNumer Algor201671103120343934810.1007/s11075-015-9987-41333.65039 – reference: CarsonECommunication-avoiding Krylov Subspace Methods in Theory and Practice2015BerkeleyUniversity of California – reference: LiangXLiR-CBaiZTrace minimization principles for positive semi-definite pencilsLinear Algebra Appl201343830853106301805910.1016/j.laa.2012.12.0031262.15010 – reference: MarquesM ACastroARubioAAssessment of exchange-correlation functionals for the calculation of dynamical properties of small clusters in time-dependent density functional theoryJ Chem Phys20011153006301410.1063/1.1385368 – reference: TeterMPayneMAllanDSolution of Schr¨odinger equation for large systemsPhys Rev B198940122551226310.1103/PhysRevB.40.12255 – reference: WenZZhangYBlock algorithms with augmented Rayleigh-Ritz projections for large-scale eigenpair computation2015 – reference: BaiZLiR-CMinimization principles for the linear response eigenvalue problem, I: TheorySIAM J Matrix Anal Appl20123310751100302346510.1137/1108389601263.65078 – reference: ZhouYLiR-CBounding the spectrum of large Hermitian matricesLinear Algebra Appl2011435480493279458710.1016/j.laa.2010.06.0341221.15022 – reference: AndersonEBaiZBischofCLAPACK Users’ Guide1999PhiladelphiaSIAM10.1137/1.97808987196040755.65028 – reference: RingPSchuckPThe Nuclear Many-Body Problem1980New YorkSpringer-Verlag10.1007/978-3-642-61852-9 – reference: LiR-CZhangL-HConvergence of block Lanczos method for eigenvalue clustersNumer Math201513183113338332910.1007/s00211-014-0681-61334.65073 – reference: DemmelJApplied Numerical Linear Algebra1997PhiladelphiaSIAM10.1137/1.97816119714460879.65017 – reference: OlsenJJensen AaH JJørgensenPSolution of the large matrix equations which occur in response theoryJ Comput Phys19887426528210.1016/0021-9991(88)90081-20636.65034 – reference: LiangXLiR-CExtensions of Wielandt’s min-max principles for positive semi-definite pencilsLinear Multilinear Algebra20146210321048324694210.1080/03081087.2013.8032421307.15015 – reference: QuillenPYeQA block inverse-free preconditioned Krylov subspace method for symmetric generalized eigenvalue problemsJ Comput Appl Math201023312981313255936510.1016/j.cam.2008.10.0711186.65044 – reference: LancasterPYeQVariational properties and Rayleigh quotient algorithms for symmetric matrix pencilsOper Theory Adv Appl19894024727810383170676.15007 – reference: SaadYOn the rates of convergence of the Lanczos and the block-Lanczos methodsSIAM J Numer Anal19801568770658875510.1137/07170590456.65016 – reference: BaiZLiR-CMinimization principle for the linear response eigenvalue problem, II: ComputationSIAM J Matrix Anal Appl201334392416304681010.1137/1108389721311.65102 – reference: MoneyJYeQEIGIFP: A MATLAB program for solving large symmetric generalized eigenvalue problemsACM Trans Math Software200531270279226679310.1145/1067967.10679731070.65531 – reference: RoccaDIterative diagonalization of non-hermitian eigenproblems in time-dependent density functional and manybody perturbation theory2012BostonPresentation at Session B39, the APS Marching Meeting – reference: GiannozziPQUANTUM ESPRESSO: A modular and open-source software project for quantum simulations of materialsJ Phys Condensed Matter20092139550210.1088/0953-8984/21/39/395502 – reference: HetmaniukULehoucqRBasis selection in LOBPCGJ Comput Phys2006218324332226795510.1016/j.jcp.2006.02.0071104.65031 – reference: RoccaDTime-Dependent Density Functional Perturbation Theory: New algorithms with Applications to Molecular Spectra2007TriesteThe International School for Advanced Studies – reference: GolubGYeQAn inverse free preconditioned Krylov subspace methods for symmetric eigenvalue problemsSIAM J Sci Comput200224312334192442710.1137/S10648275003825791016.65017 – reference: FlaschkaULinW-WWuJ-LA KQZ algorithm for solving linear-response eigenvalue equationsLinear Algebra Appl199216593123114974810.1016/0024-3795(92)90231-X0759.65013 – 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| Snippet | The locally optimal block preconditioned 4-d conjugate gradient method （LOBP4dCG） for the linear response eigenvalue problem was proposed by Bai and Li （2013）... The locally optimal block preconditioned 4-d conjugate gradient method (LOBP4dCG) for the linear response eigenvalue problem was proposed by Bai and Li (2013)... |
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| SubjectTerms | Algorithms Applications of Mathematics Conjugate gradient method Convergence Eigenvalues Mathematical models Mathematics Mathematics and Statistics Optimization Searching Subspaces 共轭梯度方法 局部优化 局部最优 搜索子空间 收敛速度 特征值问题 线性响应 预条件共轭梯度法 |
| Title | Linear response eigenvalue problem solved by extended locally optimal preconditioned conjugate gradient methods |
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