Fast iterative solvers for numerical simulations of scattering and radiation on thin wires

Xian-Ming Gu, Ting-Zhu Huang, Xi-Le Zhao, Hou-Biao Li and Liang Li

Journal of Electromagnetic Waves and Applications, 2015, vol. 29, issue 10, 1281-1296

Abstract: The method of moments for the simplified scattering and radiation on thin wire models can lead to the large-scale linear systems with non-Hermitian Toeplitz-like coefficient matrices. Since such reduced linear systems are typically dense and ill conditioned, iterative solvers are rarely considered to deal with the discretized linear systems in these electromagnetic model simulations. However, utilizing the Toeplitz-like structure of coefficient matrices, we can use the fast Toeplitz matrix-vector product (MVP) to refresh iterative solvers for the discretized linear systems. Compared with the traditional solvers, the fast numerical method can reduce the computational cost from O(m3)$ \mathcal O (m^3) $ to O(mlogm)$ \mathcal O (m\log m) $ and the storage from O(m2)$ \mathcal O (m^2) $ to O(m)$ \mathcal O (m) $ without using any lossy compression. Due to the Toeplitz-like structure of coefficient matrices, circulant preconditioners are considered to further accelerate the convergence of iterative solvers. Since the circulant matrices can be diagonalized via fast Fourier transforms (FFTs), the fast Toeplitz MVP is utilized to retain the total computational complexity of the proposed methods at O(mlogm)$ \mathcal O (m\log m) $, where m$ m $ is the number of spatial grid nodes. Theoretical analyses of the spectra of the preconditioned linear systems are also preliminarily investigated. Finally, extensive numerical experiments involving both scattering and radiation issues are employed to demonstrate the effectiveness of the proposed methods in aspects of the number of iterations and CPU time elapsed.

Date: 2015
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DOI: 10.1080/09205071.2015.1042559

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