Stability for the inverse source problems in elastic and electromagnetic waves
Stability for the inverse source problems in elastic and electromagnetic waves
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DOI：
10.1016/j.matpur.2019.06.006
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发表时间：
2017-03
期刊：
影响因子：
--
通讯作者：
G. Bao;Peijun Li;Yue Zhao
中科院分区：
文献类型：
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作者：
G. Bao;Peijun Li;Yue Zhao
本文讨论了时谐弹性波方程和电磁波方程的逆源问题。目标是根据辐射波场的边界测量分别确定外力和电流密度。由于模型系统的不适定性和复杂性，这些问题具有挑战性。建立了这两个逆源问题的唯一性和稳定性。基于连续和离散多频数据，建立了统一的递增稳定性理论。稳定性估计由两部分组成：Lipschitz型数据偏差和源函数的高频尾部。随着频率上限的增加，后者减小，因此变得可以忽略不计。稳定性增强的结果表明，利用多频率数据可以克服反问题的不适定性。该方法以积分方程组和解析延拓为基础，只需要狄里克莱特数据。该分析采用格林张量的渐近展开和Dirichlet-to-Neumann映射的透明边界条件。此外，首次建立了Navier方程和Maxwell方程的逆源问题的稳定性。
This paper concerns the inverse source problems for the time-harmonic elastic and electromagnetic wave equations. The goal is to determine the external force and the electric current density from boundary measurements of the radiated wave field, respectively. The problems are challenging due to the ill-posedness and complex model systems. Uniqueness and stability are established for both of the inverse source problems. Based on either continuous or discrete multi-frequency data, a unified increasing stability theory is developed. The stability estimates consist of two parts: the Lipschitz type data discrepancy and the high frequency tail of the source functions. As the upper bound of frequencies increases, the latter decreases and thus becomes negligible. The increasing stability results reveal that ill-posedness of the inverse problems can be overcome by using multi-frequency data. The method is based on integral equations and analytical continuation, and requires the Dirichlet data only. The analysis employs asymptotic expansions of Green's tensors and the transparent boundary conditions by using the Dirichlet-to-Neumann maps. In addition, for the first time, the stability is established on the inverse source problems for both the Navier and Maxwell equations.