The inverse wave scattering problem seeks to estimate a heterogeneous, inaccessible medium, modeled by unknown variable coefficients in wave equations, from transient recordings of waves generated by probing signals. It is a widely studied inverse problem with important applications, that is typically formulated as a nonlinear least squares data fit optimization. For typical measurement setups and band-limited probing signals, the least squares objective function has spurious local minima far and near the true solution, so Newton-type optimization methods fail. We introduce a different approach, for electromagnetic inverse wave scattering in lossless, anisotropic media. Our reduced order model (ROM) is an algebraic, discrete time dynamical system derived from Maxwell's equations with four important properties: (1) It is data driven, without knowledge of the medium. (2) The data to ROM mapping is nonlinear and yet the ROM can be obtained in a non-iterative fashion. (3) It has a special algebraic structure that captures the causal Wave propagation. (4) The ROM interpolates the data on a uniform time grid. We show how to obtain from the ROM an estimate of the wave field at inaccessible points inside the unknown medium. The use of this wave is twofold: First, it defines a computationally inexpensive imaging function designed to estimate the support of reflective structures in the medium, modeled by jump discontinuities of the matrix valued dielectric permittivity. Second, it gives an objective function for quantitative estimation of the dielectric permittivity, that has better behavior than the least squares data fitting objective function. The methodology introduced in this paper applies to Maxwell's equations in three dimensions. To avoid high computational costs, we limit the study to a cylindrical domain filled with an orthotropic medium, so the problem becomes two dimensional.

翻译：逆波散射问题旨在通过探测信号产生的瞬态波记录，估计由波动方程中未知变系数建模的、无法直接探测的非均匀介质。这是一个具有重要应用且被广泛研究的逆问题，通常表述为非线性最小二乘数据拟合优化。对于典型的测量设置和带限探测信号，最小二乘目标函数在真实解的远近处均存在虚假局部极小值，导致牛顿型优化方法失效。我们提出了一种针对无耗散各向异性介质中电磁逆波散射的新方法。我们的降阶模型是基于麦克斯韦方程组推导的代数离散时间动力系统，具有四个重要特性：(1) 它是数据驱动的，无需介质先验知识；(2) 数据到ROM的映射是非线性的，但ROM可通过非迭代方式获得；(3) 具有捕捉因果波传播的特殊代数结构；(4) ROM在均匀时间网格上插值数据。我们展示了如何从ROM中获取未知介质内部不可达点处的波场估计值。该波场具有双重用途：首先，它定义了一种计算成本低廉的成像函数，用于估计介质中由矩阵值介电常数跳跃不连续性建模的反射结构支撑区域；其次，它为介电常数的定量估计提供了比最小二乘数据拟合目标函数更优的目标函数。本文方法适用于三维麦克斯韦方程组。为避免高昂计算成本，我们将研究范围限定在填充正交各向异性介质的柱形域内，从而使问题退化为二维情形。