论文标题
卡尔曼滤波器技术的逆介质散射问题
Inverse medium scattering problems with Kalman filter techniques
论文作者
论文摘要
我们研究了从散射波的远场模式中重建未知的不均匀介质的逆介质散射问题。反向散射问题通常是不稳定的，也是非线性的，并且经常对迭代优化方法进行调整。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放入一个长矢量和映射中，并使用Tikhonov正则化方法迭代地求解线性化的大型系统方程，这称为Levenberg-Marquardt方案。但是，这在计算上很昂贵，因为当可用的测量数量增加时，我们必须构建较大的系统方程。在本文中，我们提出了基于卡尔曼过滤器的两种重建算法。一种是等同于Levenberg-Marquardt方案的算法，另一种是受到扩展的Kalman滤波器的启发。对于算法推导，我们迭代地将Kalman滤波器应用于我们的非线性方程的线性化方程。我们提出的算法顺序更新了状态空间的状态和标准的重量，这避免了大型系统方程的构建并保留了过去更新的信息。最后，我们提供数值示例来证明我们提出的算法。
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 the Levenberg-Marquardt scheme. However, this is computationally expensive because we must construct the larger system equations when the number of available measurements increases. 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. Our proposed algorithms sequentially update the state and the weight of the norm for the state space, which avoids the construction of a large system equation and retains the information of past updates. Finally, we provide numerical examples to demonstrate our proposed algorithms.
