Inverse medium scattering problems with Kalman filter techniques

TL;DR

This paper introduces two Kalman filter-based algorithms for inverse medium scattering problems, offering computational efficiency and improved handling of nonlinear, ill-posed inverse problems in reconstructing inhomogeneous media.

Contribution

The paper develops novel Kalman filter algorithms that avoid large system construction, enhancing computational efficiency in nonlinear inverse scattering reconstructions.

Findings

Algorithms effectively reconstruct inhomogeneous media from scattering data.

Proposed methods outperform traditional iterative schemes in computational efficiency.

Numerical examples validate the accuracy and stability of the algorithms.

Abstract

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.