小波在解電磁積分方程式之研究

Abstract

本論文首先探討一些小波應用在電磁積分方程式的基本理論和結果，使用以小波為基底的方法，來解電磁積分方程式，能有效地將傳統動差解法(method of moments)之滿矩陣稀疏化，進而節省巨額計算量和時間，而且不會犧牲解的精確度。在電磁散射問題之數值結果顯示，對於動電積分方程式，以離散小波轉換(discrete wavelet transform)之阻抗矩陣之元素將可以稀疏至O(N2) (0<□<1)；而以離散小波封包(discrete wavelet packet)的方法，將可進一步稀疏至O(N4/3)。 接著，我們成功地將此方法運用在3度空間的生物電磁工程上。利用小波-基底(wavelet-based)方法來解一個張量積分方程式(tensor integral equation)，可以有效率地求出生物組織或大腦對電磁波吸收程度。我們發現使用這些方法比起傳統的動差解法能夠減少運算數目超過一個位數以上，當然可以減少計算時間和記憶體之需求。 其次，我們開發一個新的小波觀念－“可見能量(visible energy)”，定義為頻域內之單一母小波(mother wavelet)，在可見區內(kx≦k0)，所有不同解析尺度(scale)小波中，任一個位移(translation)小波之能量和。藉此定量分析之方式，能夠事先找尋最適當的小波來解電磁散射之問題。利用小波可見能量有兩點好處(1)可以節省大量選擇適當小波的時間，以達到最稀疏化之電磁阻抗矩陣的目的(2)可以幫助研究人員在開發新的小波種類時一個相當有用的標準，來判斷所發展的小波是否設計良好。因此，我們利用小波可見能量所開發的定量分析方式，對於有興趣研究小波的人員將會有很大的幫助。 除此之外，我們更進一步探討非正交小波之條件數(condition number)與可見能量在解電磁散射問題時，對於計算時間所造成的影響。我們發現較小可見能量之非正交小波會產生較大的條件數，既然較大的條件數，會導致病態式(ill-conditioning)小波基底轉換矩陣之產生，該矩陣將產生較多的迭代數(iterations)，反而可能需要花費較多的計算時間來求解，抵銷了即使有較好的可見能量所形成較好的矩陣稀疏程度所帶來的好處。 論文最後，我們指出未來數個可以更深入探討的研究課題和相關的小波應用。

In this thesis, we first study some preliminary theory and results on the application of wavelets to electromagnetic integral equation. The dense matrix resulting from an integral operator can be made less dense using wavelet-based methods with thresholding techniques to attain an arbitrary degree of solution accuracy. Numerical results to EM scattering problems has been shown that for electrodynamic integral equations with oscillatory kernel, the transformed matrices have about O(□N2) (0<□<1) nonzero elements or O(N4/3) nonzero elements when it is applied the discrete wavelet transform (DWT) method or the discrete wavelet packet method (DWP), respectively. A new application to 3-dimentional biological problems is also presented. A tensor integral equation in conjugation with wavelet-based method can effectively solve electromagnetic absorption in human brains. It is found that using these approaches can reduce operation numbers more than one order to compare with that of traditional MoM. It saves lots of computation time and memory requirements. Next, we exploit a novel concept of “visible energy” for a criterion to choose the suitable wavelets in solving the electrodynamic scattering problems. The visible energy is defined as the energy of all dilations of a single mother wavelet for an arbitrary translation in the spectral domain over the entire visible region in which the spatial frequency is smaller than the free-space spatial-frequency (wavenumber). There are two main advantages in using the visible energy of wavelets: (1) It can save a lot of computing time in choosing proper wavelets to solve EM integral problems; (2) It can be a useful criterion for judging the wavelet whether is well-designed or not when researchers develop a new wavelet. This quantitative concept will be very helpful to those who interested in the study of wavelets. Additionally, we further investigate the effects of condition number and visible energy on computation time in solving electromagnetic scattering problems by using nonorthogonal wavelets. It is found that the smaller visible energy of nonorthogonal wavelet produces higher condition number. Since the large condition numbers result in ill-conditioning wavelet basis transformed matrix, the matrix will lead to more iterations and may cost much computation time. This effect may cancel any benefits from higher sparsity obtained by smaller visible energy of nonorthogonal wavelets. Finally, we pinpointed several possible extensions and applications for further studies.