針對一Impedance Crack的逆散射問題

Abstract

本計畫的目的是要藉由部分已知的Far field partern 來尋找一散射問題的crack。我 們從已知的資料，也就是Far field partern U∞著手。為了找出crack，也就是此散射問題 的障礙物，我們必須解所謂的Far field 方程式 F(Γ)=U∞。因為此方程式具有非線性及 ill-posed 的性質，我們需要Newton 法(將此問題線性化)及Tikhonov regularization(使問題 可解) ， 而此方法則需計算F 的Frechet 導數。當我們成功的證明F 為Frechet 可微及 歸納出F』的性質後，我們則可以建立數值的F』模型。而在此計畫的第二階段，我們 將重建一些例子藉此說明理論部分的可行性。

The aim of this project is to find the crack of a scattering problem with impedance boundary conditions from the knowledge of the far field pattern of the scattered field at a set of discrete points. We start with the given far field pattern u∞ . To determine the crack, we have to solve the so-called far field equation, ( ) F u∞ Γ = . Because of the non-linearity and the ill-posedness of this equation, we will use both Newton's method and Tikhonov regularization. We must therefore compute the Frechet derivative of the far field operator F (w.r.t the boundary Γ). After sucessfully clasifying the derivative of the far field operator, we can then build our numerical model. At the second stage of this project, some real reconstructions of the crack will be computed to justify our theoretical part.