Inverse medium scattering problems with Kalman filter techniques
Inverse medium scattering problems with Kalman filter techniques
复制标题
DOI：
10.1088/1361-6420/ac836f
复制
发表时间：
2022-07
期刊：
影响因子：
2.1
通讯作者：
T. Furuya;R. Potthast
中科院分区：
文献类型：
--
作者：
T. Furuya;R. Potthast
我们研究逆介质散射问题，从散射波的远场模式重建未知的非均匀介质。逆散射问题通常是不适定和非线性的，通常采用迭代优化方法。解决此问题的一种自然迭代方法是将所有可用的测量和映射分别放入一个长向量和映射中，并使用 Tikhonov 正则化方法（称为 Levenberg-Marquardt 方案）迭代求解线性化大系统方程。然而，这在计算上是昂贵的，因为当可用测量的数量增加时，我们必须构建更大的系统方程。在本文中，我们提出了两种基于卡尔曼滤波器的重建算法。一种是相当于 Levenberg-Marquardt 方案的算法，另一种是受到扩展卡尔曼滤波器的启发。对于算法推导，我们迭代地将卡尔曼滤波器应用于非线性方程的线性化方程。通过应用卡尔曼滤波器，我们提出的算法顺序更新状态和状态空间范数的权重，避免了构建大型系统方程，并保留了过去更新的信息。最后，我们提供数值示例来演示所提出的算法。
We study the inverse medium scattering problem to reconstruct the unknown inhomogeneous medium from the far field patterns of scattered waves. The inverse scattering problem is generally ill-posed and nonlinear, and the iterative optimization method is often adapted. A natural iterative approach to this problem is to place all available measurements and mappings into one long vector and mapping, respectively, and to iteratively solve the linearized large system equation using the Tikhonov regularization method, which is called Levenberg–Marquardt scheme. However, this is computationally expensive because we must construct the larger system equations when the number of available measurements is increasing. In this paper, we propose two reconstruction algorithms based on the Kalman filter. One is the algorithm equivalent to the Levenberg–Marquardt scheme, and the other is inspired by the extended Kalman filter. For the algorithm derivation, we iteratively apply the Kalman filter to the linearized equation for our nonlinear equation. By applying the Kalman filter, our proposed algorithms sequentially update the state and the weight of the norm for the state space, which avoids the construction of large system equation, and retains the information of past updates. Finally, we provide numerical examples to demonstrate the proposed algorithm.