不均匀固体的衍射逆问题
Inverse Problem of Diffraction by an Inhomogeneous Solid with a Piecewise Hoelder Refractive Index
AI总结：
研究通过两步法重构不均匀固体的折射率，利用电流和入射波确定折射率分布，验证了方法的有效性。
摘要：
本文考虑了重构不均匀固体P未知折射率k(x)的问题。假设折射率是分段Hölder函数。将原边界值问题转化为积分Lippman-Schwinger方程。入射波由位于P外的点源定义。逆问题的解分为两步：首先确定inhomogeneity区域的电流J=(k² -k₀²)u；其次用J(x)和入射波u₀表达所需函数k(x)。证明了在分段常数函数类中，第一类积分方程解的唯一性。通过求解给定折射率的测试问题验证了两步方法，近似解与精确解的比较证实了该方法的有效性。
英文摘要：
The problem of reconstruction of an unknown refractive index $k(x)$ of an inhomogeneous solid $P$ is considered. The refractive index is assumed to be a piecewise-Hölder function The original boundary value problem for the Helmholtz equation is reduced to the integral Lippman-Schwinger equation. The incident wave is defeined by a point source located outside $P.$ The solution of the inverse problem is obtained in two steps. First, the "current" $ J = (k^2 - k_0^2)u$ is determined in the inhomogeneity region. Second, the desired function $k (x) $ is expressed via the current $ J (x) $ and the incident wave $u_0.$ The uniqueness of the solution $J$ to the first-kind integral equation is proved in the class of piecewise-constant functions. The two-step method was verified by solving a test problem with a given refractive index. The comparison between the approximate solutions and the exact one approved the efficiency of the proposed method.
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