Uniqueness and modified Newton method for cracks from the far field patterns with a fixed incident direction

TL;DR

This paper proves the unique determination of cracks from far field patterns with fixed incident direction, even in unknown media, and introduces a modified Newton method for effective shape reconstruction using multi-frequency data.

Contribution

It establishes the uniqueness of crack identification from far field data with fixed incident direction and develops a modified Newton method for improved numerical reconstruction.

Findings

Uniqueness of crack determination with fixed incident direction.

Modified Newton method enhances shape reconstruction accuracy.

Multi-frequency data improves reconstruction quality.

Abstract

We consider the inverse cracks scattering problems from the far field patterns with a fixed incident direction. We firstly show that the sound-soft cracks can be uniquely determined by the multi-frequency far field patterns with a fixed incident direction. The proof is based on a low frequency asymptotic analysis of the scattered field. One important feature of the uniqueness result is that the background can even be an unknown inhomogeneous medium. A modified Newton method is then proposed for the numerical reconstruction of the shapes and locations of the cracks. Compared to the classical Newton method, the modified Newton method relaxes the dependence of a good initial guess and can be applied for multiple cracks. Numerical examples in two dimensions are presented to demonstrate the feasibility and effectiveness of the modified Newton method. In particular, the quality of the reconstructions can be greatly improved if we use the measurements properly with two frequencies or two incident directions.