A Numerical Method to Solve a Phaseless Coefficient Inverse Problem from a Single Measurement of Experimental Data
A Numerical Method to Solve a Phaseless Coefficient Inverse Problem from a Single Measurement of Experimental Data
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DOI：
10.1137/18m1179560
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发表时间：
2018-03
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通讯作者：
M. Klibanov;N. Koshev;Dinh-Liem Nguyen;L. Nguyen;A. Brettin;V. Astratov
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作者：
M. Klibanov;N. Koshev;Dinh-Liem Nguyen;L. Nguyen;A. Brettin;V. Astratov
本文提出了一种全局数值方法来解决无相位系数反问题：如何从位于测量板上的光探测器阵列处的全复值波场的强度（模平方）重建散射体的空间分布折射率。波场的传播由3D亥姆霍兹方程控制。我们的方法包括两个阶段。在第一阶段，我们使用渐近分析得到的散射波场的模的上估计。这种估计允许我们使用反演公式近似地重建测量板处的波场。这将无相位逆散射问题简化为相位逆散射问题。在第二阶段，我们应用最近开发的全局收敛的数值方法来重建所需的折射率从在第一阶段获得的全波。与优化方法不同，上述两阶段方法是全局的，因为它不需要对真实解进行良好的初始猜测。我们测试我们的数值计算模拟和实验数据的方法。虽然实验数据是嘈杂的，我们的方法产生相当准确的数值结果。
We propose in this paper a globally numerical method to solve a phaseless coefficient inverse problem: how to reconstruct the spatially distributed refractive index of scatterers from the intensity (modulus square) of the full complex valued wave field at an array of light detectors located on a measurement board. The propagation of the wave field is governed by the 3D Helmholtz equation. Our method consists of two stages. On the first stage, we use asymptotic analysis to obtain an upper estimate for the modulus of the scattered wave field. This estimate allows us to approximately reconstruct the wave field at the measurement board using an inversion formula. This reduces the phaseless inverse scattering problem to the phased one. At the second stage, we apply a recently developed globally convergent numerical method to reconstruct the desired refractive index from the total wave obtained at the first stage. Unlike the optimization approach, the two-stage method described above is global in the sense that it does not require a good initial guess of the true solution. We test our numerical method on both computationally simulated and experimental data. Although experimental data are noisy, our method produces quite accurate numerical results.