Application of multilevel structured matrices for the solution of direct and inverse electromagnetic problems



электромагнитные волны, интегральные уравнения, метод Галеркина, блочные матрицы, параллельные программы, обратные задачи, многоуровневые матрицы


We consider the problem of electromagnetic wave scattering in the heterogeneous 3D half-space bounded by a perfectly conducting plane. Using a local heterogeneity model, we reduce this problem to a volume integral equation. Applying the Galerkin discretization on uniform Cartesian grids with special basis functions, we obtain a linear system with a three-level block matrix structured as TTT+THT. Taking into account this special structure of the matrix, we propose a parallel algorithm for the solution of the problem under consideration. The employment of this algorithm makes it possible to perform a numerical simulation of measurements with an accuracy sufficient for the solution of the inverse problem, i.e., for the study of heterogeneity structure. The results of solving the inverse problem with the use of Born approximation show a high accuracy of the method proposed. The work is partially supported by the Russian Foundation for Basic Research (04-07-90336, 05-01-00721) according to the programme of high-priority fundamental research of the Department of Mathematical Sciences of RAS «Computational and Information Technologies for the Solution of Large-Scale Problems».

Author Biographies

D.V. Savostjanov

E.E. Tyrtyshnikov


  1. Самохин А.Б. Исследование задач дифракции электромагнитных волн в локально-неоднородных средах // ЖВМ и МФ. 1990. 30, № 1. 107-121.
  2. Самохин А.Б. Интегральные уравнения и итерационные методы в электромагнитном рассеянии. М.: Радио и связь, 1998.
  3. Smirnov Yu.G., Tsupack A.A. Volume singular integral equations for solving diffraction problem of electromagnetic waves in microwave oven // Proc. of European Symp. on Numer. Meth. in Electromagnetics. 2002. 172-176.
  4. Ivakhnenko V.I., Tyrtyshnikov E.E. Block-Toeplitz-Structure-based solution strategies for CEM problems // 11th Annual Review of Progress in Applied Comp. Electromagnetics. Conf. Proceedings. Monterey, CA, 1995. 181-188.
  5. Еремин Ю.А., Ивахненко В.И. Строгие и приближенные модели царапины на основе метода интегральных уравнений // Дифф. уравнения. 2001. 37, № 10. 1386-1394.
  6. Воеводин В.В., Тыртышников Е.Е. Вычислительные процессы с теплицевыми матрицами. М.: Наука, 1987.
  7. Zwamborn A.P. M., Van der Berg. The three-dimensional weak form of the conjugate gradient FFT method for solving scattering problems // IEEE Trans. Microwave Theory Tech. 1992. MTT-40, 9. 1757-1765.
  8. Saad Y., Schultz M.H. GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems // SIAM F. Scientific and Stat. Comp. 1986. 7. 856-869.
  9. Gao G., Fang S., Torres-Verdin C. A new approximation for 3D electromagnetic scattering in the presence of anisotropic conductive media // 3DEMIII Workshop. Adelaide, 2003.



How to Cite

Савостьянов Д., Тыртышников Е. Application of Multilevel Structured Matrices for the Solution of Direct and Inverse Electromagnetic Problems // Numerical Methods and Programming (Vychislitel’nye Metody i Programmirovanie). 2005. 7. 1-16



Section 1. Numerical methods and applications

Most read articles by the same author(s)