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Stabilizing block diagonal preconditioners for complex dense matrices in electromagnetics. (English) Zbl 1202.65040
Summary: Preconditioning techniques are widely used to speed up the convergence of iterative methods for solving large linear systems with sparse or dense coefficient matrices. For certain application problems, however, the standard block diagonal preconditioner makes the Krylov iterative methods converge more slowly or even diverge. To handle this problem, we apply diagonal shifting and stabilized singular value decomposition to each diagonal block, which is generated from the multilevel fast multiple algorithm (MLFMA), to improve the stability and efficiency of the block diagonal preconditioner. Our experimental results show that the improved block diagonal preconditioner maintains the computational complexity of MLFMA, converges faster and also reduces the CPU cost.
MSC:
65F08 Preconditioners for iterative methods
65F10 Iterative numerical methods for linear systems
65F50 Computational methods for sparse matrices
65Y20 Complexity and performance of numerical algorithms
65F20 Numerical solutions to overdetermined systems, pseudoinverses
78A55 Technical applications of optics and electromagnetic theory
78M15 Boundary element methods applied to problems in optics and electromagnetic theory
Software:
BILUTM; Scaleme
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[1] Benzi, M.; Bertaccini, D., Approximate inverse preconditioning for shifted linear systems, Bit, 43, 231-244, (2003) · Zbl 1037.65043
[2] Carpentieri, B.; Duff, I.S.; Giraud, L.; Magolu mongu Made, M., Sparse symmetric preconditioners for dense linear systems in electromagnetics, Numer. linear alg. appl., 11, 753-771, (2004) · Zbl 1164.65340
[3] Chew, W.C.; Jin, J.M.; Midielssen, E.; Song, J.M., Fast and efficient algorithms in computational electromagnetics, (2001), Artech House Boston
[4] Coifman, R.; Rokhlin, V.; Wandzura, S., The fast multipole method for the wave equation: a Pedestrian prescription, IEEE antennas propaga. mag., 35, 3, 7-12, (1993)
[5] M. Degiorgi, G.G. Tiberi, A. Monorchio, G. Manara, R. Mittra An SVD-based method for analyzing electromagnetic scattering from plates and faceted bodies using physical optics bases. In Proceeding of IEEE Antennas and Propagation Society International Symposium. 1(A), 147-150 (2005).
[6] Kershaw, D.S., On the problem of unstable pivots in incomplete LU-conjugate gradient method, J. comput. phys., 38, 114-123, (1980) · Zbl 0442.65022
[7] Kolundzija, B.M., Electromagnetic modeling of composite metallic and dielectric structures, IEEE trans. micro. theory tech., 47, 7, 1021-1032, (1995)
[8] Landesa, L.; Taboada, J.M.; Obelleiro, F.; Rodrigues, J.L.; Mourino, C.; Gomez, A., Solution of a very large integral-equation problems with single-level FMM, Microwave opt. technol. lett., 51, 10, 2451-2453, (2009)
[9] Lee, J.; Zhang, J.; Lu, C.C., Incomplete LU preconditioning for large scale dense complex linear systems from electromagnetic wave scattering problems, J. comput. phys., 185, 158-175, (2003) · Zbl 1017.65027
[10] Lee, J.; Zhang, J.; Lu, C.C., Sparse inverse preconditioning of multilevel fast multipole algorithm for hybrid integral equations in electromagnetics, IEEE trans. antennas propaga., 52, 9, 2277-2287, (2004) · Zbl 1368.78050
[11] Lee, J.; Zhang, J.; Lu, C.C., Performance of preconditioned Krylov iterative methods for solving hybrid integral equations in electromagnetics, Appl. comput. electromagn. soc. J., 18, 4, 54-61, (2003)
[12] Li, L.; Liu, Z.J.; Dong, X.L.; Thompson, J.A.; Carin, L., Scalable multilevel fast multipole method for multiple targets in the vicinity of a half space, IEEE trans. geosci. remote sens., 41, 4, 791-802, (2003)
[13] Lu, C.C.; Chew, W.C., A multilevel algorithm for solving a boundary integral equation of wave scattering, IEEE trans. micro. opt. tech. lett., 7, 10, 466-470, (1994)
[14] Lu, C.C.; Chew, W.C., A coupled surface-volume integral equation approach for the calculation of electromagnetic scattering from composite metallic and material targets, IEEE trans. antennas propaga., 48, 12, 1866-1868, (2000)
[15] Manteuffel, T.A., An incomplete factorization techniques for positive definite linear systems, Math. comput., 34, 473-494, (1980) · Zbl 0422.65018
[16] Pan, X.-M.; Sheng, X.-Q., A sophisticated parallel MLFMA for scattering by extremely large targets, IEEE antennas propaga. mag., 50, 3, 129-138, (2008)
[17] Rao, S.M.; Wilton, D.R.; Glisson, A.W., Electromagnetic scattering by surface of arbitrary shape, IEEE trans. antennas propaga., AP-30, 3, 409-418, (1982)
[18] Rokhlin, V., Rapid solution of integral equations of scattering theory in two dimensions, J. comput. phys., 86, 2, 414-439, (1990) · Zbl 0686.65079
[19] Saad, Y., Iterative methods for sparse linear systems, (1996), PWS Publishing New York · Zbl 1002.65042
[20] Y. Saad, M. Sosonkina, Enhanced preconditioners for large sparse least squares problems. Report UMSI-2001-1, Minnesota Supercomputer Institute, University of Minnesota, Minneapolis (2001).
[21] Saad, Y.; Zhang, J., BILUTM: A domain-based multilevel block ILUT preconditioner for general sparse matrices, SIAM J. matrix anal. appl., 21, 1, 279-299, (1999) · Zbl 0942.65045
[22] Saad, Y.; Zhang, J., Enhanced multi-level block ILU preconditioning strategies for general sparse linear systems, J. comput. appl. math., 130, 99-118, (2001) · Zbl 1010.65014
[23] Shark, T.K.; Rao, S.M.; Djordievic, A.R., Electromagnetic scattering and radiation from finite microstrip structures, IEEE trans. micro. opt. tech., 38, 11, 1568-1575, (1990)
[24] Song, J.M.; Lu, C.C.; Chew, W.C., Multilevel fast multipole algorithm for electromagnetic scattering by large complex objects, IEEE trans. antennas propaga., AP-45, 10, 1488-1493, (1997)
[25] Vaupel, T.; Hansen, V., Electrodynamic analysis of combined microstrip and coplanar/slotline structure with 3-D components based on a surface/volume integral equation approach, IEEE trans. micro. theory tech., 47, 9, 1150-1155, (1999)
[26] Velamparambil, S.V.; Song, J.; Chew, W.C.; Gallivan, K., Scaleme: a portable, scalable multipole engine for electromagnetic and acoustic integral equation solvers, IEEE antennas propaga. symp., 3, 1774-1777, (1998)
[27] Wang, Y.; Lee, J.; Zhang, J., A short survey on preconditioning techniques for large scale dense complex linear systems in electromagnetics, Int. J. comput. math., 8, 1211-1223, (2007) · Zbl 1123.65032
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