A Rayleigh quotient minimization algorithm based on algebraic multigrid

Numerical Linear Algebra with Applications

Volumn 14, Issue 7, 2007, Pages 563-580

  Hetmaniuk, U ^a  

^a SANDIA NATIONAL LABORATORIES   (United States)

Author keywords

Algebraic multigrid; Preconditioned eigensolver; Symmetric generalized eigenvalue problem

Indexed keywords

EID: 34548476243     PISSN: 10705325     EISSN: None     Source Type: Journal    
DOI: 10.1002/nla.545     Document Type: Article

References (30)

1. Neymeyr K. Solving mesh eigenproblems with multigrid efficiency. In Numerical Methods for Scientific Computing, Variational Problems and Applications, Kuznetsoz Y, Neittaanmäki P, Pironneau O (eds). CIMNE: Barcelona, 2003.
2. Arbenz P, Hetmaniuk U, Lehoucq R, Tuminaro R. A comparison of eigensolvers for large-scale 3D modal analysis using AMG-preconditioned iterative methods. International Journal for Numerical Methods in Engineering 2005; 64:204-236.
3. Borzi A, Borzi G. Algebraic multigrid methods for solving generalized eigenvalue problems. International Journal for Numerical Methods in Engineering 2006; 65:1186-1196.
4. Lashuk I, Argentati M, Ovtchinnikov E, Knyazev A. Preconditioned eigensolver LOBPCG in Hypre and PETSc. In Domain Decomposition Methods in Science and Engineering. Lecture Notes in Computational Science and Engineering, vol. 55. Springer: Berlin, 2007; 635-642.
5. Brandt A, McCormick S, Ruge J. Multigrid methods for differential eigenproblems. SIAM Journal on Scientific and Statistical Computing 1983; 4:244-260.
6. Hackbusch W. On the computation of approximate eigenvalues and eigenfunctions of elliptic operators by means of a multi-grid method. SIAM Journal on Numerical Analysis 1979; 16:201-215.
7. Heuveline V, Bertsch C. On multigrid methods for the eigenvalue computation of nonselfadjoint elliptic operators. East-West Journal of Numerical Mathematics 2000; 8(4):275-297.
8. Deuflhard P, Friese T, Schmidt F. A nonlinear multigrid eigenproblem solver for the complex Helmholtz equation. Technical Report SC 97-55, Konrad-Zuse-Zentrum, Berlin, 1997.
9. Heiskanen M, Torsti T, Puska M, Nieminen R. Multigrid method for electronic structure calculations. Physical Review B 2001; 63(245106).
10. Mandel J, McCormick S. A multilevel variational method for Au = λBu on composite grids. Journal of Computational Physics 1989; 80:442-452.
11. Schmidt F. Simultaneous computation of the lowest eigenvalues and eigenvectors of the Helmholtz equation. Technical Report TR 95-13, Konrad-Zuse-Zentrum, Berlin, 1995.
12. Hetmaniuk U, Lehoucq R. Multilevel methods for eigenspace computations in structural dynamics. In Domain Decomposition Methods in Science and Engineering. Lecture Notes in Computational Science and Engineering, vol. 55. Springer: Berlin, 2007; 103-114.
13. Brandt A, McCormick S, Ruge J. Algebraic multigrid (AMG) for sparse matrix equations. In Sparsity and its Applications, Evans DJ (ed.). Cambridge University Press: Cambridge, MA, 1984.
14. Cai Z, Mandel J, McCormick S. Multigrid methods for nearly singular linear equations and eigenvalue problems. SIAM Journal on Numerical Analysis 1997; 34:178-200.
15. Stuben K. A review of algebraic multigrid. Journal of Computational and Applied Mathematics 2001; 128: 281-309.
16. Vaněk P, Mandel J, Brezina M. Algebraic multigrid based on smoothed aggregation for second and fourth order problems. Computing 1996; 56:179-196.
17. Döhler B. Ein neues Gradientenverfahren zur simultanen Berechnung der kleinsten oder grössten Eigenwerte des allgemeinen Eigenwertproblems. Numerische Mathematik 1982; 40:79-91.
18. Gee M, Siefert C, Hu J, Tuminaro R, Sala M. ML 5.0 smoothed aggregation user's guide. Technical Report SAND2006-2649, Sandia National Laboratories, 2006.
19. Stathopoulos A, Wu K. A block orthogonalization procedure with constant synchronization requirements. SIAM Journal on Scientific Computing 2002; 23:2165-2182.
20. Dongarra J, Du Croz J, Duff I, Hammarling S. A set of level 3 basic linear algebra subprograms. ACM Transactions on Mathematical Software 1990; 16:1-17.
21. Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J, Du Croz J, Greenbaum A, Hammarling S, McKenney A, Sorensen D. LAPACK Users' Guide. SIAM: Philadelphia, PA, 1999.
22. Bai Z, Demmel J, Dongarra J, Ruhe A, van der Vorst H (eds). Templates for the Solution of the Algebraic Eigenvalue Problems: A Practical Guide. SIAM: Philadelphia, PA, 2000.
23. Feng Y. An integrated Davidson and multigrid solution approach for very large scale symmetric eigenvalue problems. Computer Methods in Applied Mechanics and Engineering 2001; 190(28):3543-3563.
24. van der Vorst H. Computational methods for large eigenvalue problems. Handbook of Numerical Analysis, vol. III. Elsevier: Amsterdam, 2002; 3-179.
25. Wu K, Saad Y, Stathopoulos A. Inexact Newton preconditioning techniques for large symmetric eigenvalue problems. Electronic Transactions on Numerical Analysis 1998; 7:202-214.
26. Zhou Y. Studies on Jacobi-Davidson, Rayleigh quotient iteration, inverse iteration, generalized Davidson, and Newton updates. Numerical Linear Algebra with Applications 2006; 13(8):621-642.
27. Parlett BN. The Symmetric Eigenvalue Problem. SIAM: Philadelphia, PA, 1998.
28. Heroux M, Bartlett R, Howie V, Hoekstra R, Hu J, Kolda T, Lehoucq R, Long K, Pawlowski R, Phipps E, Salinger A, Thornquist H, Tuminaro R, Willenbring J, Williams A, Stanley K. An overview of the Trilinos project. ACM Transactions on Mathematical Software 2005; 31(3):397-423.
29. Adams M, Brezina M, Hu J, Tuminaro R. Parallel multigrid smoothing: polynomial versus Gauss-Seidel. Journal of Computational Physics 2003; 188(2):593-610.
30. Lehoucq R, Sorensen D, Yang C. ARPACK Users' Guide: Solution of Large-scale Eigenvalue Problems by Implicitly Restarted Arnoldi Methods. SIAM: Philadelphia, PA, 1998.