The Stabilized Biconjugate Gradient Fast Fourier Transform Method for Electromagnetic Scattering
Keywords:
The Stabilized Biconjugate Gradient Fast Fourier Transform Method for Electromagnetic ScatteringAbstract
An iterative method, the stabilized biconjugate gradient (BiCGSTAB) method, combined with the fast Fourier transform (FFT) for solving electromagnetic scattering problems is developed for the 3-D volume electric field integral equation. It converges significantly faster than the conventional conjugate gradient (CG) and biconjugate gradient (BiCG) fast Fourier transform methods. With this BCGS-FFT method, we can solve a large-scale volume integral equation with $20$ million unknowns on a single CPU workstation.
Downloads
References
P. Zwamborn and P. M. van den Berg, “The three dimensional
weak form of the conjugate gradient FFT method for solving
scattering problems,” IEEE Trans. Microwave Theory and
Tech., vol. 9, no. 40, pp. 1757-1766, 1992.
H. Gan and W. C. Chew, “A discrete BCG-FFT algorithm
for solving 3D inhomogeneous scatter problems,” J. Electro-
magn. Waves and Appl., vol. 9, no. 10, pp. 1339–1357, 1995.
M. R. Hestenes and E. Stiefel, “Methods of conjugate gra-
dients for solving linear systems,” J. Res. Nat. Bur. Stand.,
vol. 49, pp. 409–435, 1952.
T. K. Sarkar, ed., Application of conjugate gradient method
to electromagnetics and signal analysis, PIER 5, Progress in
Electromagnetics Research, New York: Elsevier, 1991.
ACES JOURNAL, VOL. 17, NO. 1, MARCH 2002, SI: APPROACHES TO BETTER ACCURACY/RESOLUTION IN CEM102
T. K. Sarkar, “The application of conjugate gradient method
the solution of operator equations arising in electromagnetic
scattering from wire antennas,” Radio Science, vol. 19, Sept.-
Oct. 1984, pp. 1156-1172.
T. K. Sarkar, “On the application of the generalized bicon-
jugate gradient method,” J. Electromagn. Waves and Appl.,
vol. 1, no. 3, pp. 223–242, 1987.
C. Lanczos, “An iteration method for the solution of the
eigenvalue problem of linear differential and integral oper-
ators,” J. Res. Nat. Bur. Stand., vol. 45, pp. 255–282, 1950.
C. Lanczos, “Solutions of systems of linear equations by mini-
mized iterations,” J. Res. Nat. Bur. Stand., vol. 49, pp. 33-53,
P. Sonneveld, CGS: “A fast Lanczos-type solver for nonsym-
metric linear system,” SIAM J. Sci. Statist. Comput., 10
(1989), pp. 36-52.
H. A. van der Vorst, “Bi-CGSTAB: a fast and smoothly
converging variant of Bi-CG for the solution of nonsymmetric
linear systems,” SIAM J. Sci. Stat. Comput., vol. 13, no. 2,
pp. 631-644, March 1992.
T. F. Chan, E. Gallopoulos, V. Simoncini, T. Szeto and
C. H. Tong, “A quasi-minimal residual variant of the BI-
CGSTAB algorithm for nonsymmetric system,” SIAM J. Sci.
Comput., vol. 15, no. 2, pp. 338-347, 1994.
R. W. Freund, “A transpose-free quasi-minimal residual al-
gorithm for non-Hermitian linear systems,” SIAM J. Sci.
Comput., vol. 14, no. 2, pp. 470-482, March 1993.
R. W. Freund and N. M. Nachtigal, “QMR, A quasi-minimal
residual method for non-Hermitian linear systems,” Numer.
Math., 60 (1991), pp. 315-339.
R. W. Freund and N. M. Nachtigal, “An implementation of
the QMR method based on coupled two-term recurrences,”
SIAM J. Sci. Comput., 15 (1994), pp. 313-337.
M. Sauren and H. M. Bucker, “On deriving the quasi-
minimal residual method,” SIAM Rev., vol. 40, no. 4, pp.
-926, December 1998.
New York, Wiley, 1989.
M. F. Catedra, R. P. Torres, J. Basterrechea, and E.
Gago, The CG-FFT Method: Application of Signal Process-
ing Techniques to Electromagnetics, Boston: Artech House,
Z. Q. Zhang, and Q. H. Liu, “Three-dimensional weak-form
conjugate- and biconjugate-gradient FFT methods for vol-
ume integral equations,” Microwave Opt. Tech. Lett., vol. 29,
no. 5, pp. 350–356, 2001.
C. F. Smith, A. F. Peterson, and R. Mittra, “The biconju-
gate gradient method for electromagnetic scattering,” IEEE
Trans. Antenna Propagat., vol AP-38, no. 6, pp. 938–940,
June 1990.
W. C. Chew, Waves and Fields in Inhomogeneous Media,
IEEE Press, 1995.
B. Yang and J. S. Hesthaven, “A pseudospectral method
for time-domain computation of electromagnetic scattering
by bodies of revolution,” IEEE Trans. Antennas Propagat.,
vol. 47, pp. 132–141, 1999
