Algorithm For Inverse Acoustic Scattering Problem Based On Decomposition And Kaczmarz Method

As an important research field in mathematical physics,acoustic and electromagnetic wave scattering have broad applicaton prospects in many fields of science,such as non-destructive test-ing,medical imaging,exploration of oil and gas,space remote sensing,detection of modern radar,target stealth,seismic surveying and exploration of seabed resources.In this paper,we concern about the acoustic inverse scattering problem.The main purpose of this paper is to reconstruct the boundaries of the sound-soft obstacles by using the far-field patterns of scattered waves.In practice,the main difficulty lies in the ill-posedness and nonlinearity of the problem.Based on the original decomposition method,the content of this paper is divided into three parts:First,we study the direct problem of acoustic scattering and focus on the problem comes from the impermeable obstacles under two-dimensional Dirichlet boundary condition.Also we disperse the acoustic double-layer potentials with the help of the of Nystrom method.Meanwhile the numerical simulation of the direct problem is given.Second,we reconstruct the obstacle boundary under decomposition method and Newton method.The basic idea is using decomposition method to convert the problem into a nonlinear and ill-posed operator equation which maps the boundary into the total wave field on the boundary.In the linear ill-posed step,the near-field is reconstructed from the far-field data using the Tikhonov regular-ization method,whereas the non-linear step uses the Newton iteration method to find the boundary that meets the condition.Finally,taking into account that there exists instability for noise data when using the decom-position-Newton algorithm,this paper proposes the decomposition-Kaczmarz algorithm based on the decomposition-Newton algorithm.The algorithm uses the idea of cyclic iteration in the process of solving equations and the numerical examples show that this algorithm effectively improves the reconstruction for noise data.This is where the innovation of this article lies.Numerical simulations are given using Matlab in the above three parts.