Schur complement preconditioners for surface integral-equation formulations of dielectric problems solved with the multilevel fast multipole algorithm

SIAM Journal on Scientific Computing

Volumn 33, Issue 5, 2011, Pages 2440-2467

  Malas, Tahir ^a   Gürel, Levent ^a  

^a BILKENT UNIVERSITY   (Turkey)

Author keywords

Computational electromagnetics; Dielectric problems; Integral equation methods; Partitioned matrices; Preconditioning; Schur complement reduction method; Sparse approximate inverse preconditioners

Indexed keywords

COMPUTATIONAL ELECTROMAGNETICS; INTEGRAL EQUATIONS; INVERSE PROBLEMS; ITERATIVE METHODS; LINEAR SYSTEMS; PHOTONIC CRYSTALS; REAL TIME SYSTEMS;

APPROXIMATE INVERSE PRECONDITIONER; DIELECTRIC PROBLEM; INTEGRAL EQUATION METHODS; MATRIX; PARTITIONED MATRIX; PRECONDITIONING; SCHUR COMPLEMENT; SCHUR COMPLEMENT REDUCTION METHOD; SPARSE APPROXIMATE INVERSE; SPARSE-APPROXIMATE-INVERSE PRECONDITIONER;

MATRIX ALGEBRA;

EID: 79952611271     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/090780808     Document Type: Conference Paper

References (63)

1. G. Alléon, M. Benzi, and L. Giraud, Sparse approximate inverse preconditioning for dense linear systems arising in computational electromagnetics, Numer. Algorithms, 16 (1997), pp. 1-15. (Pubitemid 127429642)
2. Z.-Z. Bai, Structured preconditioners for nonsingular matrices of block two-by-two structures, Math. Comp., 75 (2006), pp. 791-815. (Pubitemid 43681472)
3. S. Balay, W. D. Gropp, L. C. McInnes, and B. F. Smith, PETSc Users Manual, Tech. Report ANL-95/11, Revision 2.1.5, Argonne National Laboratory, 2004.
4. M. Benzi, Preconditioning techniques for large linear systems: A survey, J. Comput. Phys., 182 (2002), pp. 418-477. (Pubitemid 36010064)
5. M. Benzi and G. H. Golub, A preconditioner for generalized saddle point problems, SIAM J. Matrix Anal. Appl., 26 (2004), pp. 20-41.
6. M. Benzi, G. H. Golub, and J. Liesen, Numerical solution of saddle point problems, Acta Numer., 14 (2005), pp. 1-137. (Pubitemid 40771317)
7. M. Benzi and M. Tůma, A sparse approximate inverse preconditioner for nonsymmetric linear systems, SIAM J. Sci. Comput., 19 (1998), pp. 968-994. (Pubitemid 128622541)
8. M. Benzi and M. Tůma, A comparative study of sparse approximate inverse preconditioners, Appl. Numer. Math., 30 (1999), pp. 305-340.
9. A. Brandt, Multilevel computations of integral transforms and particle interactions with oscillatory kernels, Comput. Phys. Comm., 65 (1991), pp. 24-38. (Pubitemid 21665553)
10. B. Carpentieri, I. S. Duff, L. Giraud, and G. Sylvand, Combining fast multipole techniques and an approximate inverse preconditioner for large electromagnetism calculations, SIAM J. Sci. Comput., 27 (2005), pp. 774-792.
11. Y. Chang and R. F. Harrington, A surface formulation for characteristic modes of material bodies, IEEE Trans. Antennas and Propagation, 25 (1977), pp. 789-795. (Pubitemid 8580447)
12. W. C. Chew, J.-M. Jin, E. Michielssen, and J. Song, eds., Fast and Efficient Algorithms in Computational Electromagnetics, Artech House, Norwood, MA, 2001.
13. E. Chow and Y. Saad, Approximate inverse techniques for block-partitioned matrices, SIAM J. Sci. Comput., 18 (1997), pp. 1657-1675. (Pubitemid 127639219)
14. E. Chow and Y. Saad, Experimental study of ILU preconditioners for indefinite matrices, J. Comput. Appl. Math., 86 (1997), pp. 387-414. (Pubitemid 127423443)
15. E. Chow and Y. Saad, Approximate inverse preconditioners via sparse-sparse iterations, SIAM J. Sci. Comput., 19 (1998), pp. 995-1023. (Pubitemid 128622542)
16. E. de Sturler and J. Liesen, Block-diagonal and constraint preconditioners for nonsymmetric indefinite linear systems. Part I: Theory, SIAM J. Sci. Comput., 26 (2005), pp. 1598-1619. (Pubitemid 41360306)
17. J. J. Dongarra, I. S. Duff, D. C. Sorensen, and H. A. van der Vorst, Numerical Linear Algebra for High-Performance Computers, SIAM, Philadelphia, 1998.
18. D. A. Dunavant, High degree efficient symmetrical Gaussian quadrature rules for the triangle, Internat. J. Numer. Methods Engrg., 21 (1985), pp. 1129-1148. (Pubitemid 15530458)
19. H. C. Elman, D. J. Silvester, and A. J. Wathen, Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics, Oxford University Press, New York, 2005.
20. Ö. Ergül and L. Gürel, Enhancing the accuracy of the interpolations and anterpolations in MLFMA, IEEE Trans. Antennas and Propagation Lett., 5 (2006), pp. 467-470.
21. Ö. Ergül and L. Gürel, Optimal interpolation of translation operator in multilevel fast multipole algorithm, IEEE Trans. Antennas and Propagation, 54 (2006), pp. 3822-3826. (Pubitemid 46357907)
22. Ö. Ergül and L. Gürel, Comparison of integral-equation formulations for the fast and accurate solution of scattering problems involving dielectric objects with multilevel fast multipole algorithm, IEEE Trans. Antennas and Propagation, 57 (2009), pp. 176-187.
23. J. M. Ford, K. Chen, and D. Evans, On a recursive Schur preconditioner for iterative solution of a class of dense matrix problems, Int. J. Comput. Math., 80 (2003), pp. 105-122. (Pubitemid 41710572)
24. G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed., Johns Hopkins University Press, Baltimore, MD, 1996.
25. R. D. Graglia, On the numerical integration of the linear shape functions times the 3-D Green's function or its gradient on a plane triangle, IEEE Trans. Antennas and Propagation, 41 (1993), pp. 1448-1455.
26. M. J. Grote and T. Huckle, Parallel preconditioning with sparse approximate inverses, SIAM J. Sci. Comput., 18 (1997), pp. 838-853. (Pubitemid 127640885)
27. L. Gürel and W. C. Chew, Recursive algorithms for calculating the scattering from N strips or patches, IEEE Trans. Antennas and Propagation, 38 (1990), pp. 507-515. (Pubitemid 20702109)
28. L. Gürel and W. C. Chew, A recursive T-matrix algorithm for strips and patches, Radio Sci., 27 (1992), pp. 387-401.
29. L. Gürel and Ö Ergül, Comparisons of FMM implementations employing different formulations and iterative solvers, in Proceedings of the IEEE Antennas and Propagation Society International Symposium, Vol. 1, 2003, pp. 19-22.
30. L. Gürel and Ö. Ergül, Singularity of the magnetic-field integral equation and its extraction, IEEE Antennas Wireless Propagat. Lett., 4 (2005), pp. 229-232. (Pubitemid 43131774)
31. L. Gürel and T. Malas, Iterative near-field preconditioner for the multilevel fast multipole algorithm, SIAM J. Sci. Comput., 32 (2010), pp. 1929-1949.
32. W. W. Hager, Updating the inverse of a matrix, SIAM Rev., 31 (1989), pp. 221-239.
33. S. S. Haykin, Kalman Filtering and Neural Networks, John Wiley & Sons, New York, 2001.
34. G. C. Hsiao and R. E. Kleinman, Mathematical foundations for error estimation in numerical solutions of integral equations in electromagnetics, IEEE Trans. Antennas and Propagation, 45 (1997), pp. 316-328. (Pubitemid 127770783)
35. I. C. F. Ipsen, A note on preconditioning nonsymmetric matrices, SIAM J. Sci. Comput., 23 (2001), pp. 1050-1051. (Pubitemid 34721052)
36. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light, Princeton University Press, Princeton, NJ, 2008.
37. S. G. Johnson, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, Guided modes in photonic crystal slabs, Phys. Rev. B, 60 (1999), pp. 5751-5758.
38. L. Y. Kolotilina and A. Y. Yeremin, Factorized sparse approximate inverse preconditioning I. Theory, SIAM J. Matrix Anal. Appl., 14 (1993), pp. 45-58.
39. J. Lee, J. Zhang, and C.-C. Lu, Sparse inverse preconditioning of multilevel fast multipole algorithm for hybrid integral equations in electromagnetics, IEEE Trans. Antennas and Propagation, 52 (2004), pp. 158-175.
40. L. Little, Y. Saad, and L. Smoch, Block LU preconditioners for symmetric and nonsymmetric saddle point problems, SIAM J. Sci. Comput., 25 (2003), pp. 729-748.
41. T. W. Lloyd, J. M. Song, and M. Yang, Numerical study of surface integral formulations for low-contrast objects, IEEE Antennas Wireless Propagat. Lett., 4 (2005), pp. 482-485. (Pubitemid 43131841)
42. T. Malas and L. Gürel, Incomplete LU preconditioning with the multilevel fast multipole algorithm for electromagnetic scattering, SIAM J. Sci. Comput., 29 (2007), pp. 1476-1494.
43. T. Malas and L. Gürel, Accelerating the multilevel fast multipole algorithm with the sparseapproximate-inverse (SAI) preconditioning, SIAM J. Sci. Comput., 31 (2009), pp. 1968-1984.
44. T. Malas and L. Gürel, Iterative Schur Complement Preconditioners for the Simulation of Dielectric Problems in Computational Electromagnetics, Tech. Report, Bilkent University, 2009.
45. C. Müller, Foundations of the Mathematical Theory of Electromagnetic Waves, Springer-Verlag, New York, 1969.
46. M. A. Olshanskii and Y. V. Vassilevski, Pressure Schur complement preconditioners for the discrete Oseen problem, SIAM J. Sci. Comput., 29 (2007), pp. 2686-2704.
47. A. P. Pavacic, D. L. del Ro, J. R. Mosig, and G. V. Eleftheriades, Three-dimensional ray-tracing to model internal reflections in off-axis lens antennas, IEEE Trans. Antennas and Propagation, 54 (2006), pp. 604-612.
48. A. J. Poggio and E. K. Miller, Integral equation solutions of three-dimensional scattering problems, in Computer Techniques for Electromagnetics, R. Mittra, ed., Pergamon Press, Oxford, 1973, Chap. 4, pp. 159-264.
49. S. M. Rao, D. R. Wilton, and A. W. Glisson, Electromagnetic scattering by surfaces of arbitrary shape, IEEE Trans. Antennas and Propagation, 30 (1982), pp. 409-418.
50. S. M. Rao and D. R. Wilton, E-field, H-field, and combined-field solution for arbitrarily shaped three-dimensional dielectric bodies, Electromag., 10 (1990), pp. 407-421.
51. Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd ed., SIAM, Philadelphia, 2003.
52. G. Schmidlin, U. Fischer, Z. Andjelǐc, and C. Schwab, Preconditioning of the second-kind boundary integral equations for 3D eddy current problems, Internat. J. Numer. Methods Engrg., 51 (2001), pp. 1009-1031. (Pubitemid 32675962)
53. X. Q. Sheng, J.-M. Jin, J. Song, W. C. Chew, and C.-C. Lu, Solution of combined-field integral equation using multilevel fast multipole algorithm for scattering by homogeneous bodies, IEEE Trans. Antennas and Propagation, 46 (1998), pp. 1718-1726. (Pubitemid 128748333)
54. C. Siefert and E. de Sturler, Preconditioners for generalized saddle-point problems, SIAM J. Numer. Anal., 44 (2006), pp. 1275-1296. (Pubitemid 47135923)
55. V. Simoncini and D. B. Szyld, The effect of non-optimal bases on the convergence of Krylov subspace methods, Numer. Math., 100 (2005), pp. 711-733. (Pubitemid 40924291)
56. L. N. Trefethen and D. Bau, III, Numerical Linear Algebra, SIAM, Philadelphia, 1997.
57. T.-K. Wu and L. L. Tsai, Scattering from arbitrarily-shaped lossy dielectric bodies of revolution, Radio Sci., 12 (1977), pp. 709-718.
58. P. Ylä-Oijala and M. Taskinen, Calculation of CFIE impedance matrix elements with RWG and ̌n×RWG functions, IEEE Trans. Antennas and Propagation, 51 (2003), pp. 1837-1846.
59. P. Ylä-Oijala and M. Taskinen, Application of combined field integral equation for electromagnetic scattering by composite metallic and dielectric objects, IEEE Trans. Antennas and Propagation, 53 (2005), pp. 1168-1173. (Pubitemid 40466863)
60. P. Ylä-Oijala and M. Taskinen, Well-conditioned Muller formulation for electromagnetic scattering by dielectric objects, IEEE Trans. Antennas and Propagation, 53 (2005), pp. 3316-3323. (Pubitemid 41555334)
61. P. Ylä-Oijala, M. Taskinen, and S. Järvenpää, Surface integral equation formulations for solving electromagnetic scattering problems with iterative methods, Radio Sci., 40 (2005), article RS6002.
62. P. Ylä-Oijala, M. Taskinen, and S. Järvenpää, Analysis of surface integral equations in electromagnetic scattering and radiation problems, Eng. Anal. Boundary Elem., 32 (2008), pp. 196-209.
63. J. Zhang, On preconditioning Schur complement and Schur complement preconditioning, Electron. Trans. Numer. Anal., 10 (2000), pp. 115-130. (Pubitemid 38816969)