The CBiCG class of algorithms for complex symmetric linear systems with applications in several electromagnetic model problems. (English) Zbl 1344.65041
Summary: Frequency domain formulations of computational electromagnetic problems often require the solutions of complex-valued non-Hermitian systems of equations, which are still symmetric. For this kind of problems a whole class of sub-variant solver methods derived from the complex-valued Bi-Conjugate Gradient method is available. This class of methods contains established iterative methods as the Conjugate Orthogonal Conjugate Gradient (COCG) method, Bi-Conjugate Gradient Conjugate Residual (BiCGCR) method and Conjugate \(A\)-Orthogonal Conjugate Residual (COCR) method. The mathematical equivalence of the BiCGCR method and COCR method is shown and preconditioned variants of the various solvers are derived. An efficient kind of two-step preconditioning technique is also proposed. Numerical experiments involving e.g. electro-quasistatic frequency domain simulation are employed to show the difference in the convergence behaviors of these iterative methods and effectiveness of the two-step preconditioning techniques.
MSC:
| 65F10 | Iterative numerical methods for linear systems |
| 78M10 | Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory |
| 78M12 | Finite volume methods, finite integration techniques applied to problems in optics and electromagnetic theory |
Keywords:
conjugate gradient; BiCG method; complex symmetric matrix; Krylov subspace method; computational electromagneticsReferences:
| [1] | van Rienen, U.; Clemens, M.; Weiland, T., Simulation of low-frequency fields on high-tension insulators with light contaminations, IEEE Trans. Magn., 32, 3, 816-819 (1996) |
| [2] | Weiland, T., On the unique numerical solution of Maxwellian eigenvalue problems in three dimensions, Part. Accel., 7, 227-242 (1985) |
| [3] | Clemens, M.; Weiland, T., Discrete electromagnetism with the finite integration technique, Prog. Electromagn. Res. (PIER), 32, 65-87 (2001) |
| [4] | Saad, Y., Iterative Methods for Sparse Linear Systems (2003), SIAM: SIAM Philadephia, USA · Zbl 1002.65042 |
| [5] | Clemens, M.; Schuhmann, R.; van Rienen, U.; Weiland, T., Modern Krylov subspace methods in electromagnetic field computation using the finite integration theory, Appl. Comput. Electromagn. Soc. J., 11, 1, 70-84 (1996), Special Issue on Applied Mathematics: Meeting the challenges presented by Computational Electromagnetics |
| [6] | Lanczos, C., Solution of systems of linear equations by minimized itertions, J. Res. Natl. Bur. Stand., 49, 33-53 (1952) |
| [7] | Jacobs, D. A.H., The exploitation of sparsity by iterative methods, (Duff, I. S., Sparse Matrices and their Uses (1981), Springer-Verlag: Springer-Verlag Berlin, Germany), 191-222 · Zbl 0469.65016 |
| [8] | Jacobs, D. A.H., A generalization of the conjugate gradient method to solve complex systems, IMA J. Numer. Anal., 6, 447-452 (1986) · Zbl 0614.65028 |
| [9] | Sonneveld, P., CGS, a fast Lanczos-type solver for nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 10, 36-52 (1989) · Zbl 0666.65029 |
| [10] | van der Vorst, H. A., Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 13, 631-644 (1992) · Zbl 0761.65023 |
| [11] | Freund, R. W., A transpose-free quasi-minimal residual algorithm for non-Hermitian linear systems, SIAM J. Sci. Comput., 14, 470-482 (1993) · Zbl 0781.65022 |
| [12] | Chan, T. F.; Gallopoulos, E.; Simoncini, V.; Szeto, T.; Tong, C. H., A quasi-minimal residual variant of the Bi-CGSTAB algorithm for nonsymmetric systems, SIAM J. Sci. Comput., 15, 338-347 (1994) · Zbl 0803.65038 |
| [14] | Clemens, M.; Weiland, T.; van Rienen, U., Comparison of Krylov-type methods for complex linear systems applied to high-voltage problems, IEEE Trans. Magn., 34, 5, 3335-3338 (1998) |
| [15] | van der Vorst, H. A.; Melissen, J., A Petrov-Galerkin type method for solving \(A x = b\), where \(A\) is symmetric complex, IEEE Trans. Magn., 26, 2, 706-708 (1990) |
| [16] | Sogabe, T.; Zhang, S.-L., A COCR method for solving complex symmetric linear systems, J. Comput. Appl. Math., 199, 297-303 (2007) · Zbl 1108.65028 |
| [17] | Stiefel, E., Relaxationsmethoden bester Strategie zur Lösung linearer Gleichungssysteme, Comment. Math. Helv., 29, 157-179 (1955) · Zbl 0066.36703 |
| [18] | Sleijpen, G. L.G.; van der Vorst, H. A.; Fokkema, D. R., BiCGstab \(( \ell )\) and other hybrid Bi-CG methods, Numer. Algorithms, 7, 75-109 (1994) · Zbl 0810.65027 |
| [19] | Hahne, P., Zur numerischen berechnung zeitharmonischer elektromagnetischer felder (1992), Technische Hochschule Darmstadt: Technische Hochschule Darmstadt Darmstadt, Germany, (Ph.D. thesis) |
| [20] | Deuflhard, P.; Bornemann, F., (Numerische Mathematik II, Integration Gewöhnlicher Differentialgleichungen. Numerische Mathematik II, Integration Gewöhnlicher Differentialgleichungen, De-Gruyter-Lehrbuch Edition (1994), De Gruyter Verlag: De Gruyter Verlag Berlin, New York) · Zbl 0856.65080 |
| [21] | Sogabe, T., Extensions of the conjugate residual method (2006), University of Tokyo: University of Tokyo Tokyo, Japan, Avaiable online at: http://www.ist.aichi-pu.ac.jp/person/sogabe/thesis.pdf |
| [22] | Jing, Y.-F.; Huang, T.-Z.; Zhang, Y.; Li, L.; Cheng, G.-H.; Ren, Z.-G.; Duan, Y.; Sogabe, T.; Carpentieri, B., Lanczos-type variants of the COCR method for complex nonsymmetric linear systems, J. Comput. Phys., 228, 6376-6394 (2009) · Zbl 1173.65316 |
| [23] | Freund, R. W.; Nachtigal, N. M., Software for simplified Lanczos and QMR algorithms, Appl. Numer. Math., 19, 319-341 (1995) · Zbl 0853.65041 |
| [24] | Elman, H. C.; O’Leary, D. P., Eigenanalysis of some preconditioned Helmholtz problems, Numer. Math., 83, 231-257 (1999) · Zbl 0934.65119 |
| [25] | Erlangga, Y. A.; Vuik, C.; Oosterlee, C. W., Comparison of multigrid and incomplete LU shifted-Laplace preconditioners for the inhomogeneous Helmholtz equation, Appl. Numer. Math., 56, 648-666 (2006) · Zbl 1094.65041 |
| [26] | Li, L.; Lanteri, S.; Perrussel, R., A hybridizable discontinuous Galerkin method combined to a Schwarz algorithm for the solution of 3D time-harmonic Maxwell’s equation, J. Comput. Phys., 256, 563-581 (2014) · Zbl 1349.78069 |
| [27] | Schmidthäußler, D.; Clemens, M., Reduction of linear subdomains for non-linear electro-quasistatic field simulations, IEEE Trans. Magn., 49, 5, 1669-1672 (2013) |
| [28] | van Rienen, U., (Numerical Methods in Computational Electrodynamics: Linear Systems in Practical Applications. Numerical Methods in Computational Electrodynamics: Linear Systems in Practical Applications, Lecture Notes in Computational Science and Engineering, vol. 12 (2001), Springer-Verlag: Springer-Verlag New York, USA) · Zbl 0977.78023 |
| [29] | Reitzinger, S.; Schreiber, U.; van Rienen, U., Electro-quasistatic calculation of electric field strength on high-voltage insulators with an algebraic multigrid algorithm, IEEE Trans. Magn., 39, 4, 2129-2132 (2003) |
| [30] | Arridge, S. R.; Egger, H.; Schlottbom, M., Preconditioning of complex symmetric linear systems with applications in optical tomography, Appl. Numer. Math., 74, 35-48 (2013) · Zbl 1302.65069 |
| [31] | Li, L.; Huang, T.-Z.; Jing, Y.-F.; Zhang, Y., Application of the incomplete Cholesky factorization preconditioned Krylov subspace method to the vector finite element method for 3-D electromagnetic scattering problems, Comput. Phys. Comm., 181, 271-276 (2010) · Zbl 1205.78059 |
| [33] | Gu, X.-M.; Huang, T.-Z.; Li, L.; Li, H.-B.; Sogabe, T.; Clemens, M., Quasi-minimal residual variants of the COCG and COCR methods for complex symmetric linear systems in electromagnetic simulations, IEEE Trans. Microw. Theory Techn., 62, 12, 2859-2867 (2014) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
