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## Time-domain inverse scattering problem for two-dimensional frequency-dispersive lossy media

Liu Guang-Dong, Zhang Ye-Rong
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• #### 摘要

为了重建二维有耗色散介质的电参数分布,基于Debye模型,应用泛函分析和变分法,提出一种时域逆散射新方法.该方法首先以最小二乘准则构造目标函数,将逆问题表示为约束最小化问题,接着应用罚函数法转化为无约束最小化问题,然后基于变分计算导出闭式的Lagrange函数关于特征参数的Fréchet导数,最后借助梯度算法和时域有限差分法迭代反演Debye模型参数.为了对抗噪声污染和逆问题的病态特性,采用了一阶Tikhonov正则化方法.数值应用中,利用Polak-Ribière-Polyak非线性共轭梯度法,对二维乳

#### Abstract

A time-domain inverse scattering method for reconstruction of dispersive dielectric properties of two-dimensional (2D) lossy media based on the Debye model by using functional analysis and variational methods is developed. Firstly, the approach formulates a cost functional to turn the inverse problem into a constrained minimization problem according to least squares criterion, then the resulting constrained minimization problem is transformed into an unconstrained minimization problem by using a penalty function technique, and then the closed Fréchet derivatives of the Lagrange function with respect to the properties are derived based on the calculus of variations. Finally, one can solve the resulting problem by using any gradient-based algorithm and the finite-difference time-domain (FDTD) method. Also, the first-order Tikhonov’s regularization is adopted to cope with noise and the ill-posedness of the problem. In numerical example, the presented algorithm is applied to a 2D breast model with the help of the Polak-Ribière-Polyak (PRP) conjugate gradient (CG) method, and the results demonstrate its feasibility.

#### 作者及机构信息

###### 1. 南京邮电大学电子科学与工程学院,南京 210003
• 基金项目: 国家自然科学基金重点项目(批准号:60671065)资助的课题.

#### 参考文献

 [1] Zhang Y R, Nie Z P, Ruan Y Z 1997 Acta Elec. Sin. 25 100 (in Chinese) [张业荣、聂在平、阮颖铮 1997 电子学报 25 100] [2] Gustafsson M, He S 1999 Math. Comput. Simulat. 50 525 [3] Takenaka T, Jia H, Tanaka T 2000 J. Electromagn. Waves Appl. 14 1609 [4] Zhang H, He S L, Chen P, Sun W 2001 Acta Phys. Sin. 50 1481 (in Chinese) [张 航、何赛灵、陈 攀、孙 威 2001 物理学报 50 1481] [5] Liang Z C, Jin Y Q 2002 Acta Phys. Sin. 51 2239 (in Chinese) [梁子长、金亚秋 2002物理学报 51 2239] [6] Rekanos I T, Raisanen A 2003 IEEE Trans. Magn. 39 1381 [7] Abenius E, Strand B 2006 Int. J. Numer. Meth. Eng. 68 650 [8] Winters D W, Bond E J, van Veen B D, Hagness S C 2006 IEEE Trans. Antenn. Propag. 54 3517 [9] Liu D, Wang F, Huang Q X, Yan J H, Chi Y, Cen K F 2008 Acta Phys. Sin. 57 4812 (in Chinese) [刘 冬、王 飞、黄群星、严建华、池 涌、岑可法 2008 物理学报 57 4812] [10] Winters D W, Shea J D, Kosmas P, van Veen B D, Hagness S C 2009 IEEE Trans. Med. Imaging 28 969 [11] Wei B, Ge D B, Wang F 2008 Acta Phys. Sin. 57 6290 (in Chinese) [魏 兵、葛德彪、王 飞 2008 物理学报 57 6290] [12] Zhang Y Q, Ge D B 2009 Acta Phys. Sin. 58 4573 (in Chinese) [张玉强、葛德彪 2009 物理学报 58 4573] [13] Wei B, Li X Y, Wang F, Ge D B 2009 Acta Phys. Sin. 58 6174 (in Chinese) [魏 兵、李小勇、王 飞、葛德彪 2009 物理学报 58 6174] [14] Ge D B, Yan Y B 2005 Finite-Difference Time-Domain Method for Electromagnetic Waves (2nd Ed.) (Xi’an: Xidian University Press) p288 (in Chinese) [葛德彪、闫玉波 2005 电磁波时域有限差分法(第二版)(西安:西安电子科技大学出版社)第288页] [15] Taflove A, Hagness S C 2005 Computational Electrodynamics: The Finite-Difference Time-Domain Method (3rd Ed.) (Norwood: Artech House) p353 [16] Gabriel S, Lau R W, Gabriel C 1996 Phys. Med. Biol. 41 2271 [17] Tikhonov A N, Arsenin V Y 1979 SIAM Rev. 21 266 [18] Dai Y H 2003 Math. Comput. 17 1317 [19] Kuhn H W, Tucker A W 1951 Nonlinear Programming (California: University of California Press) p481

#### 施引文献

•  [1] Zhang Y R, Nie Z P, Ruan Y Z 1997 Acta Elec. Sin. 25 100 (in Chinese) [张业荣、聂在平、阮颖铮 1997 电子学报 25 100] [2] Gustafsson M, He S 1999 Math. Comput. Simulat. 50 525 [3] Takenaka T, Jia H, Tanaka T 2000 J. Electromagn. Waves Appl. 14 1609 [4] Zhang H, He S L, Chen P, Sun W 2001 Acta Phys. Sin. 50 1481 (in Chinese) [张 航、何赛灵、陈 攀、孙 威 2001 物理学报 50 1481] [5] Liang Z C, Jin Y Q 2002 Acta Phys. Sin. 51 2239 (in Chinese) [梁子长、金亚秋 2002物理学报 51 2239] [6] Rekanos I T, Raisanen A 2003 IEEE Trans. Magn. 39 1381 [7] Abenius E, Strand B 2006 Int. J. Numer. Meth. Eng. 68 650 [8] Winters D W, Bond E J, van Veen B D, Hagness S C 2006 IEEE Trans. Antenn. Propag. 54 3517 [9] Liu D, Wang F, Huang Q X, Yan J H, Chi Y, Cen K F 2008 Acta Phys. Sin. 57 4812 (in Chinese) [刘 冬、王 飞、黄群星、严建华、池 涌、岑可法 2008 物理学报 57 4812] [10] Winters D W, Shea J D, Kosmas P, van Veen B D, Hagness S C 2009 IEEE Trans. Med. Imaging 28 969 [11] Wei B, Ge D B, Wang F 2008 Acta Phys. Sin. 57 6290 (in Chinese) [魏 兵、葛德彪、王 飞 2008 物理学报 57 6290] [12] Zhang Y Q, Ge D B 2009 Acta Phys. Sin. 58 4573 (in Chinese) [张玉强、葛德彪 2009 物理学报 58 4573] [13] Wei B, Li X Y, Wang F, Ge D B 2009 Acta Phys. Sin. 58 6174 (in Chinese) [魏 兵、李小勇、王 飞、葛德彪 2009 物理学报 58 6174] [14] Ge D B, Yan Y B 2005 Finite-Difference Time-Domain Method for Electromagnetic Waves (2nd Ed.) (Xi’an: Xidian University Press) p288 (in Chinese) [葛德彪、闫玉波 2005 电磁波时域有限差分法(第二版)(西安:西安电子科技大学出版社)第288页] [15] Taflove A, Hagness S C 2005 Computational Electrodynamics: The Finite-Difference Time-Domain Method (3rd Ed.) (Norwood: Artech House) p353 [16] Gabriel S, Lau R W, Gabriel C 1996 Phys. Med. Biol. 41 2271 [17] Tikhonov A N, Arsenin V Y 1979 SIAM Rev. 21 266 [18] Dai Y H 2003 Math. Comput. 17 1317 [19] Kuhn H W, Tucker A W 1951 Nonlinear Programming (California: University of California Press) p481
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##### 出版历程
• 收稿日期:  2009-11-26
• 修回日期:  2010-01-07
• 刊出日期:  2010-05-05

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