Multilevel preconditioned iterative eigensolvers for Maxwell eigenvalue problems. (English) Zbl 1127.65021
The authors investigate different eigensolvers to compute a small number (5-10) of the smallest eigenvalues of the homogeneous Helmholtz equation discretized by the finite element method. The practical background of this problem originates in accelerator physics where the lowest eigenmodes of resonant rf (radio frequency) accelerating structures are of central interest in the design phase.
Discretization by finite elements leads to a real symmetric generalized matrix eigenvalue problem. Different eigenvalue solvers and preconditioners are carefully described and compared for a realistic, practical application problem. These are especially the Jacobi-Davidson algorithm, the locally optimal block preconditioned conjugate gradient (LOBPCG) algorithm and a multilevel preconditioner which combines a hierarchical basis and a smoothed aggregation algebraic multigrid (AMG) preconditioner. As a result of the extensive study the Jacobi-Davidson algorithms turns out to be superior to LOBPCG for the given Maxwell eigenvalue problem.
The well-compiled study gives important information for further research not only in the field of Maxwell’s eigenvalue problem.
Discretization by finite elements leads to a real symmetric generalized matrix eigenvalue problem. Different eigenvalue solvers and preconditioners are carefully described and compared for a realistic, practical application problem. These are especially the Jacobi-Davidson algorithm, the locally optimal block preconditioned conjugate gradient (LOBPCG) algorithm and a multilevel preconditioner which combines a hierarchical basis and a smoothed aggregation algebraic multigrid (AMG) preconditioner. As a result of the extensive study the Jacobi-Davidson algorithms turns out to be superior to LOBPCG for the given Maxwell eigenvalue problem.
The well-compiled study gives important information for further research not only in the field of Maxwell’s eigenvalue problem.
Reviewer: Ursula van Rienen (Rostock)
MSC:
| 65F15 | Numerical computation of eigenvalues and eigenvectors of matrices |
| 65N55 | Multigrid methods; domain decomposition for boundary value problems involving PDEs |
| 78M10 | Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory |
Keywords:
Maxwell equation; Helmholtz equation; generalized eigenvalue problem; Jacobi-Davidson; LOBPCG; smoothed aggregation AMG preconditionerReferences:
| [1] | Arbenz, P.; Geus, R., A comparison of solvers for large eigenvalue problems originating from Maxwell’s equations, Numer. Linear Algebra Appl., 6, 1, 3-16 (1999) · Zbl 0982.65039 |
| [2] | Arbenz, P.; Geus, R.; Adam, S., Solving Maxwell eigenvalue problems for accelerating cavities, Phys. Rev. ST Accel. Beams, 4, 022001 (2001), Electronic journal available from |
| [3] | Bai, Z.; Demmel, J.; Dongarra, J.; Ruhe, A.; van der Vorst, H., Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide (2000), SIAM: SIAM Philadelphia, PA · Zbl 0965.65058 |
| [4] | Bank, R. E., Hierarchical bases and the finite element method, Acta Numerica, 5, 1-43 (1996) · Zbl 0865.65078 |
| [5] | Bochev, P. B.; Garasi, C. J.; Hu, J. J.; Robinson, A. C.; Tuminaro, R. S., An improved algebraic multigrid method for solving Maxwell’s equations, SIAM J. Sci. Comput., 25, 2, 623-642 (2003) · Zbl 1163.76337 |
| [6] | Ciarlet, P. G., The Finite Element Method for Elliptic Problems, Studies in Mathematics and Its Applications, vol. 4 (1978), North-Holland: North-Holland Amsterdam · Zbl 0445.73043 |
| [7] | Demmel, J. W.; Eisenstat, S. C.; Gilbert, J. R.; Li, X. S.; Liu, J. W.H., A supernodal approach to sparse partial pivoting, SIAM J. Matrix Anal. Appl., 20, 3, 720-755 (1999) · Zbl 0931.65022 |
| [8] | Demmel, J. W.; Gilbert, J. R.; Li, X. S., SuperLU Users’ Guide (December 2002), available from |
| [9] | Dongarra, J. J.; Duff, I. S.; Sorensen, D. C.; van der Vorst, H. A., Numerical Linear Algebra for High-Performance Computers (1998), SIAM: SIAM Philadelphia, PA · Zbl 0914.65014 |
| [10] | Feng, Y. T.; Owen, D. R.J., Conjugate gradient methods for solving the smallest eigenpair of large symmetric eigenvalue problems, Internat. J. Numer. Methods Engrg., 39, 13, 2209-2229 (1996) · Zbl 0880.73074 |
| [11] | Fokkema, D. R.; Sleijpen, G. L.G.; van der Vorst, H. A., Jacobi-Davidson style QR and QZ algorithms for the partial reduction of matrix pencils, SIAM J. Sci. Comput., 20, 1, 94-125 (1998) · Zbl 0924.65027 |
| [12] | Freund, R.; Nachtigal, N. M., Software for simplified Lanczos and QMR algorithms, Appl. Numer. Math., 19, 319-341 (1995) · Zbl 0853.65041 |
| [13] | R. Geus, The Jacobi-Davidson Algorithm for Solving Large Sparse Symmetric Eigenvalue Problems, Ph.D. Thesis No. 14734, ETH Zurich, available at URL http://e-collection.ethbib.ethz.ch/show?type=diss&nr=14734, 2002; R. Geus, The Jacobi-Davidson Algorithm for Solving Large Sparse Symmetric Eigenvalue Problems, Ph.D. Thesis No. 14734, ETH Zurich, available at URL http://e-collection.ethbib.ethz.ch/show?type=diss&nr=14734, 2002 |
| [14] | Girault, V.; Raviart, P.-A., Finite Element Methods for the Navier-Stokes Equations, Springer Series in Computational Mathematics, vol. 5 (1986), Springer: Springer Berlin · Zbl 0413.65081 |
| [15] | Hestenes, M. R.; Karush, W., A method of gradients for the calculation of the characteristic roots and vectors of a real symmetric matrix, J. Res. National Bureau Standards, 47, 45-61 (1951) |
| [16] | Hiptmair, R.; Neymeyr, K., Multilevel method for mixed eigenproblems, SIAM J. Sci. Comput., 23, 6, 2141-2164 (2002) · Zbl 1013.65120 |
| [17] | J. Hu, C. Tong, R.S. Tuminaro, ML 2.0 smoothed aggregation user’s guide, Tech. Report SAND2001-8028, Sandia National Laboratories, December 2000; J. Hu, C. Tong, R.S. Tuminaro, ML 2.0 smoothed aggregation user’s guide, Tech. Report SAND2001-8028, Sandia National Laboratories, December 2000 |
| [18] | Kikuchi, F., Mixed and penalty formulations for finite element analysis of an eigenvalue problem in electromagnetism, Comput. Methods Appl. Mech. Engrg., 64, 509-521 (1987) · Zbl 0644.65087 |
| [19] | Knyazev, A. V., Toward the optimal preconditioned eigensolver: Locally optimal block preconditioned conjugate gradient method, SIAM J. Sci. Comput., 23, 2, 517-541 (2001) · Zbl 0992.65028 |
| [20] | Knyazev, A.; Neymeyr, K., Efficient solution of symmetric eigenvalue problems using multigrid preconditioners in the locally optimal block conjugate gradient method, Electron. Trans. Numer. Anal., 15, 38-55 (2003) · Zbl 1031.65126 |
| [21] | Knyazev, A.; Neymeyr, K., A geometric theory for preconditioned inverse iteration III: A short and sharp convergence estimate for generalized eigenvalue problems, Linear Algebra Appl., 358, 1-3, 95-114 (2003) · Zbl 1037.65039 |
| [22] | Lehoucq, R. B.; Sorensen, D. C.; Yang, C., ARPACK Users’ Guide: Solution of Large-Scale Eigenvalue Problems by Implicitely Restarted Arnoldi Methods (1998), SIAM: SIAM Philadelphia, PA, The software and this manual are available at · Zbl 0901.65021 |
| [23] | Longsine, D. E.; McCormick, S. F., Simultaneous Rayleigh-quotient minimization methods for \(A x = \lambda B x\), Linear Algebra Appl., 34, 195-234 (1980) · Zbl 0475.65021 |
| [24] | Parlett, B. N., The Symmetric Eigenvalue Problem (1980), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ, Republished by SIAM, Philadelphia, PA, 1998 · Zbl 0431.65016 |
| [25] | Perdon, A.; Gambolati, G., Extreme eigenvalues of large sparse matrices by Rayleigh quotient and modified conjugate gradients, Comput. Methods Appl. Mech. Engrg., 56, 125-156 (1986) · Zbl 0579.65028 |
| [26] | Reitzinger, S.; Schöberl, J., An algebraic multigrid method for finite element discretizations with edge elements, Numer. Linear Algebra Appl., 9, 3, 223-238 (2002) · Zbl 1071.65170 |
| [27] | Silvester, P. P.; Ferrari, R. L., Finite Elements for Electrical Engineers (1996), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0587.73157 |
| [28] | Sleijpen, G. L.G.; van der Vorst, H. A., A Jacobi-Davidson iteration method for linear eigenvalue problems, SIAM J. Matrix Anal. Appl., 17, 2, 401-425 (1996) · Zbl 0860.65023 |
| [29] | Stüben, K., An introduction to algebraic multigrid, (Trottenberg, U.; Oosterlee, C. W.; Schüller, A., Multigrid (2000), Academic Press: Academic Press New York), 413-532 |
| [30] | Vaněk, P.; Mandel, J.; Brezina, M., Algebraic multigrid based on smoothed aggregation for second and fourth order problems, Computing, 56, 3, 179-196 (1996) · Zbl 0851.65087 |
| [31] | Yserentant, H., Hierarchical bases, (O’Malley, R. E., ICIAM 91: Proceedings of the Second International Conference on Industrial and Applied Mathematics (1992), SIAM: SIAM Philadelphia, PA), 256-276 |
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