由散射問題產生的二維內部傳輸特徵值問題之數值研究

Abstract

逆散射問題中，我們的目標是由遠場圖(far field pattern)來決定散射物體的形狀。在數值模擬中，我們首先解決內部傳輸特徵值問題，二維內部傳輸特徵值問題可以由兩個拉普拉斯(Laplacian)特徵值問題在某些限制形成的。通過解決該問題的幾個最小傳輸特徵值，我們可以構造物體在邊界上的總場(total field)。利用邊界上的信息，我們可以透過求解亥姆霍茲方程(Helmholtz equation)在狄利克雷邊界條件（Dirichlet boundary condition）和羅賓邊界條件(Robin boundary condition)下找到散射場(scattered field)。我們使用有限差分法和有限體積法將內部傳輸特徵值問題等價地轉換成二次特徵值問題，藉由解二次特徵值問題的幾個最小特徵值來計算在邊界上的總場。最後，我們解散射問題來觀察不同障礙中的散射場，並在最後一節給出一些數值結果。

The goal of the inverse problem is to determine the shapes of scattering objects using knowledge of the far field pattern. In numerical simulations, we firstly solve the interior transmission eigenvalue problem, and then the 2-dimensional interior transmission eigenvalue problem can be formed by two Laplacian eigenvalue problems under some restrictions. By solving a few lowest transmission eigenvalues of this problem, we can construct the total field on the boundary of the scattering object. Utilizing the information on the boundary, we can find the scattered field by solving the Helmholtz equation under Dirichlet and Robin boundary conditions. To carry out the the procedure outlined above, we use the finite difference method and the finite volume method to equivalently transfer the interior transmission eigenvalue problem to the quadratic eigenvalue problem. We then we solve a few lower eigenvalues of the quadratic eigenvalue problem to compute the total field on the boundary, finally, we solve the scattering problem to observe the scattered field with different obstacles and present some numerical results in the final section.