Vol. 110

Front:[PDF file] Back:[PDF file]
Latest Volume
All Volumes
All Issues
2010-11-11

A Novel Lanczos-Type Procedure for Computing Eigenelements of Maxwell and Helmholtz Problems

By Bruno Carpentieri, Yan-Fei Jing, and Ting-Zhu Huang
Progress In Electromagnetics Research, Vol. 110, 81-101, 2010
doi:10.2528/PIER10072204

Abstract

We introduce a novel variant of the Lanczos method for computing a few eigenvalues of sparse and/or dense non-Hermitian systems arising from the discretization of Maxwell- or Helmholtz-type operators in Electromagnetics. We develop a Krylov subspace projection technique built upon short-term vector recurrences that does not require full reorthogonalization and can approximate simultaneously both left and rigth eigenvectors. We report on experiments for solving eigenproblems arising in the analysis of dielectric waveguides and scattering applications from PEC structures. The theoretical and numerical results reported in this study will contribute to highlight the potential and enrich the database of this technology for solving generalized eigenvalue problems in Computational Electromagnetics.

Citation


Bruno Carpentieri, Yan-Fei Jing, and Ting-Zhu Huang, "A Novel Lanczos-Type Procedure for Computing Eigenelements of Maxwell and Helmholtz Problems," Progress In Electromagnetics Research, Vol. 110, 81-101, 2010.
doi:10.2528/PIER10072204
http://jpier.org/PIER/pier.php?paper=10072204

References


    1. Bai, A., D. Day, J. Demmel, and J. Dongarra, "A test matrix collection for non-hermitian eigenvalue problems,", Technical Report, Knoxville, TN, USA, 1997.
    doi:10.1137/1.9780898719581

    2. Bai, Z., T. Ericsson, and T. Kowalski, Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide, Chapter Symmetric Indefinite Lanczos Method, Society for Industrial and Applied Mathematics, Philadelphia, 2000.
    doi:10.5194/ars-7-23-2009

    3. Bandlow, B. and R. Schuhmann, "3D eigenmode calculation of metallic nano-structures," Advances in Radio Science (ARS), Vol. 7, 23-27, 2009.
    doi:10.1007/s00607-006-0161-7

    4. Carpentieri, B., "A matrix-free two-grid preconditioner for boundary integral equations in electromagnetism," Computing, Vol. 77, No. 3, 275-296, 2006.

    5. Carpentieri, B., "Fast iterative solution methods in electromagnetic scattering," Progress In Electromagnetic Research, Vol. 79, 151-178, 2007.
    doi:10.1137/S1064827502408591

    6. Carpentieri, B., I. S. Duff, and L. Giraud, "A class of spectral two-level preconditioners," SIAM J. Scientific Computing, Vol. 25, No. 2, 749-765, 2003.
    doi:10.1137/060654906

    7. Carpentieri, B., L. Giraud, and S. Gratton, "Additive and multiplicative two-level spectral preconditioning for general linear systems," SIAM J. Scientific Computing, Vol. 29, No. 4, 1593-1612, 2007.
    doi:10.2528/PIER09091402

    8. Chang, H.-W., Y.-H. Wu, S.-M. Lu, W.-C. Cheng, and M.-H. Sheng, "Field analysis of dielectric waveguide devices based on coupled transverse-mode integral equation-numerical investigation," Progress In Electromagnetics Research, Vol. 97, 159-176, 2009.
    doi:10.1137/1.9780898719192

    9. Cullum, J. and R. A. Willoughby, Lanczos Algorithms for Large Symmetric Eigenvalue Computations, Birkhaüser, Basel, 1985. Vol. 1. Theory, Vol. 2. Programs. Vol. 1 reprinted by SIAM, Philadelphia, PA, 2002.

    10. Cullum, J. and R. A. Willoughby, "A practical procedure for computing eigenvalues of large sparse nonsymmetric matrices," Large Scale Eigenvalue Problems, Proceedings of the IBM Europe Institute Workshop on Large Scale Eigenvalue Problems, Jane Cullum and Ralph A. Willoughby, Editors, North-Holland Mathematics Studies, Vol. 127, 193-240, North-Holland, 1986.
    doi:10.2528/PIERB08062601

    11. Danesfahani , R., S. Hatamzadeh-Varmazyar, E. Babolian, and Z. Masouri, "Applying Shannon wavelet basis functions to the method of moments for evaluating the radar cross section of the conducting and resistive surfaces," Progress In Electromagnetics Research B, Vol. 8, 257-292, 2008.
    doi:10.1137/S0895479895292503

    12. Day, D., "An efficient implementation of the nonsymmetric lanczos algorithms," SIAM J. Matrix Anal. Appl., Vol. 18, 566-589, 1997.

    13. Dong, H., A. Chronopoulos, J. Zou, and A. Gopinath, "Vectorial integrated finite-difference analysis of dielectric waveguides," Integrated Photonics Research, Vol. 10, OSA Technical Digest Series (Optical Society of America), 1991.
    doi:10.2528/PIER09062604

    14. Dong, J., "Surface wave modes in chiral negative refraction grounded slab waveguides," Progress In Electromagnetics Research, Vol. 95, 153-166, 2009.
    doi:10.2528/PIER09092501

    15. Du, Y. and B. Liu, "A numerical method for electromagnetic scattering from dielectric rough surfaces based on the stochastic second degree method," Progress In Electromagnetics Research, Vol. 97, 327-342, 2009.
    doi:10.2528/PIERL08112706

    16. Essid, C., M. B. B. Salah, K. Kochlef, A. Samet, and A. B. Kouki, "Spatial-spectral formulation of method of moment for rigorous analysis of microstrip structures," Progress In Electromagnetics Research Letters, Vol. 6, 17-26, 2009.
    doi:10.2528/PIERB08091606

    17. Fan, Z., D.-Z. Ding, and R.-S. Chen, "The efficient analysis of electromagnetic scattering from composite structures using hybrid Cfie-Iefie," Progress In Electromagnetics Research B, Vol. 10, 131-143, 2008.
    doi:10.1049/el:19901367

    18. Fernandez, F. A. and Y. Lu, "Variational finite element analysis of dielectric waveguide with non spurious solutions," Electron. Letter, Vol. 26, No. 25, 2125-2126, 1990.
    doi:10.1137/0914009

    19. Freund, R., M. H. Gutknecht, and N. Nachtigal, "An implementation of the look-ahead Lanczos algorithm for non-Hermitian matrices," SIAM J. Scientific Computing, Vol. 14, 137-158, 1993.
    doi:10.1145/225545.225551

    20. Freund, R. W. and N. M. Nachtigal, "QMRPACK: A package of QMR algorithms," ACM Transactions on Mathematical Software, Vol. 22, 46-77, 1996.

    21. Galick, A. and T. Kerkhoven, "Iterative solution of the eigenvalue problem for a dielectric waveguide," IEEE Trans. MTT, Vol. 40, 699-705, 1991.
    doi:10.1017/S0962492900002737

    22. Gutknecht, M. H., "Lanczos-type solvers for nonsymmetric linear systems of equations," Acta Numerica, Vol. 6, 271-397, 1997.
    doi:10.2528/PIER09071409

    23. Hasar, U. C., "Permittivity determination of fresh cement-based materials by an open-ended waveguide probe using amplitudeonly measurements," Progress In Electromagnetics Research, Vol. 97, 27-43, 2009.

    24. Hernández, V., J. E. Román, A. Tomás, and V. Vidal, "A survey of software for sparse eigenvalue problems,", SLEPc Technical Report STR-6, Universidad Politecnica de Valencia, 2007.
    doi:10.2528/PIER09101204

    25. Hu, L., L.-W. Li, and T. S. Yeo, "Analysis of scattering by large inhomogeneous bi-anisotropic objects using AIM," Progress In Electromagnetics Research, Vol. 99, 21-36, 2009.

    26. Jin, J., The Finite Element Method in Electromagnetics, John Wiley & Sons, Inc., New York, 1993.
    doi:10.2528/PIER09101901

    27. Jing, Y.-F., B. Carpentieri, and T.-Z. Huang, "Experiments with Lanczos biconjugate A-orthonormalization methods for MoM discretizations of Maxwell's equations," Progress In Electromagnetics Research, Vol. 99, 427-451, 2009.
    doi:10.1016/j.jcp.2009.05.022

    28. Jing, Y.-F., T.-Z. Huang, Y. Zhang, L. Li, G.-H. Cheng, Z.-G. Ren, Y. Duan, T. Sogabe, and B. Carpentieri, "Lanczos-type variants of the COCR method for complex nonsymmetric linear systems," Journal of Computational Physics, Vol. 228, No. 17, 6376-6394, 2009.
    doi:10.1137/0719030

    29. Kahan, W., B. N. Parlett, and E. Jiang, "Residual bounds on approximate eigensystems of nonnormal matrices," SIAM Journal on Numerical Analysis, Vol. 19, No. 3, 470-484, 1982.
    doi:10.2528/PIER09072804

    30. Kusiek, A. and J. Mazur, "Analysis of scattering from arbitrary configuration of cylindrical objects using hybrid finite-difference mode-matching method," Progress In Electromagnetics Research, Vol. 97, 105-127, 2009.
    doi:10.2528/PIER09071704

    31. Liu, Z.-L. and J. Yang, "Analysis of electromagnetic scattering with higher-order moment method and Nurbs model," Progress In Electromagnetics Research, Vol. 96, 83-100, 2009.
    doi:10.2528/PIER09092604

    32. Malek, F., "The analytical design of a folded waveguide traveling wave tube and small signal gain analysis using Madey's theorem," Progress In Electromagnetics Research, Vol. 98, 137-162, 2009.
    doi:10.1016/0024-3795(91)90381-6

    33. Morgan, R. B., "Computing interior eigenvalues of large matrices," Linear Algebra Appl., Vol. 154/156, 289-309, 1991.
    doi:10.1002/nla.1680020205

    34. Paige, C. C., B. N. Parlett, and H. A. van der Vorst, "Approximate solutions and eigenvalue bounds from Krylov subspaces," Numer. Linear Algebra Appl., Vol. 2, 115-133, 1995.
    doi:10.1137/0613036

    35. Parlett, B., "Reduction to tridiagonal form and minimal realizations," SIAM J. Matrix Analysis and Applications, Vol. 13, 567-593, 1992.

    36. Parlett, B., D. Taylor, and Z.-S. Liu, "A look-ahead lanczos algorithm for unsymmetric matrices," Math. Comp., Vol. 44, 105-124, 1985.
    doi:10.2528/PIER09071703

    37. Ping, X. W. and T.-J. Cui, "The factorized sparse approximate inverse preconditioned conjugate gradient algorithm for finite element analysis of scattering problems," Progress In Electromagnetics Research, Vol. 98, 15-31, 2009.
    doi:10.2528/PIER09090905

    38. Rothwell, E. J., A. K. Temme, and B. R. Crowgey, "Pulse reflection from a dielectric discontinuity in a rectangular waveguide," Progress In Electromagnetics Research, Vol. 97, 11-25, 2009.
    doi:10.1137/S1064827502406415

    39. Simoncini, V. and D. B. Szyld, "Theory of inexact Krylov subspace methods and applications to scientific computing," SIAM J. Sci. Comput., Vol. 25, No. 2, 454-477, 2003.

    40. Wilkinson, J. H., The Algebraic Eigenvalue Problem, Oxford University Press, Walton Street, Oxford OX2 6DP, UK, 1965.
    doi:10.2528/PIER09090901

    41. Zhang, H., S. Y. Tan, and H. S. Tan, "A flanged parallel-plate waveguide probe for microwave imaging of tumors," Progress In Electromagnetics Research, Vol. 97, 45-60, 2009.