本論文首度將田口最佳化方法(Taguchi’s Method)應用於二維介質柱體的逆散射問題。在正散射的分析部分，本研究以時域有限差分法為基礎，至於逆散射則被轉換為最佳化問題以進行求解，並和使用差異型演化法(Differential Evolution, DE)、自我適應之動態差異型演化法(Self-Adaptive Dynamic Differential Evolution, SADDE)求解的結果進行比較。
　　為了描述與重建柱體的形狀，在正散射部分，本研究採用傅立葉函數展開(Fourier series expansion)，但在逆散射部分則使用仿樣函數展開(cubic spline)，如此可確保柱體形狀建構的合理性。最後將時域有限差分法結合田口最佳化法以重建均勻介質柱體，並探討田口最佳化法之特性參數對重建結果的影響。
　　田口最佳化法和目前常用的演算法(Algorithm)有很大的不同，後者透過機率(Probability)的概念來完成全域的搜尋；相對的，田口最佳化法則排除機率，而以直交表(Orthogonal Arrays)展現的均勻分布特性，並結合將搜尋範圍不斷地遞減(Range Reduction) 的迭代方式，來執行全域的搜尋以找出最佳解。
　　本論文首先將田口最佳化法、差異型演化法與自我適應之動態差異型演化法用九種具不同特性之測試函數與以測試，進行50維、100維、250維等高維度函數的測試，發現田口最佳化法與SADDE分別對不同特性之測試函則展現出優越的特性。之後，當應用在逆散射問題時，田口最佳化法依然展現其優越收斂的特性，然而SADDE則沒能展現其在測試函數上的優越特性。為此，我們可看到針對二維介質柱體的逆散射問題，田口最佳化法所展現的強健性與一致性。

This thesis is the first to apply Taguchi optimization method to inverse scattering problem, for which a two-dimensional dielectric object is considered. The analysis of forward scattering part is based on the finite difference time domain (FDTD) method, while the inverse scattering part is tackled by transforming the problem into an optimization one, of which the Taguchi optimization method is chosen. The reconstructed results are compared with those obtained by Differential Evolution (DE), and Self-Adaptive Dynamic Differential Evolution (SADDE).

  To described and reconstructed the shape of the dielectric cylinder,  Fourier series expansion is used for the forward scattering part, while the cubic splines are employed for the inverse scattering part. In this way, the objective of shape reconstruction of this study is maintained reasonable. At the end, the FDTD method combined with Taguchi optimization method is applied to reconstruct the cylinder of the homogeneous medium. In addition, the parameter effects of Taguchi optimization method upon the reconstruction results are studied.

  Taguchi optimization method is quite different from those statistic methods commonly used for global optimization. The latter utilize the random characteristic to achieve the global searching, while the former, on the contrast, exclude the concept of probability. Taguchi optimization method actually utilize the uniform characteristic of the OA table, and combine with the mechanism of range reduction to achieve the iterative searching in a global way.

  In this thesis, the Taguchi optimization method, DE and SADDE are applied to test nine different benchmarked functions, at first. The performances are examined for those with dimensions of 50, 100 and 250, respectively. It is found that Taguchi optimization method, and SADDE exhibit superior performance in the test. Then, when applied to the inverse scattering problem of dielectric cylinders. Taguchi optimization method still exhibit good reconstruction results, while SADDE doesn’t. It is thus conclude that Taguchi optimization method is especially suited for the proposed inverse scattering problem.

目錄
中文摘要	III
英文摘要	V
第一章 簡介	1
1.1 研究動機與相關文獻	1
1.2 本研究之貢獻	8
1.3 各章內容簡述	9
第二章	時域有限差分法	10
2.1 馬克斯威爾方程式	10
2.2 馬克斯威爾方程式於FDTD方法中差分離散實現	12
2.2.1 Yee單胞(Yee cell)的空間解析方法與蛙跳式(Leap-frog)時間　步進計算方法 	13
2.2.2 FDTD更新方程式	14
2.3 數值色散現象與COURANT穩定準則	15
2.4 吸收邊界條件(ABSORBING BOUNDARY CONDITIONS)	17
第三章 全域最佳化演算法	19
3.1 田口最佳化法(TAGUCHI’S OPTIMIZATION METHOD)	19
3.2 差異型演化法(DIFFERENTIAL EVOLUTION) 	24
3.3 自我適應之動態差異型演化法(SELF-ADAPTIVE DYNAMIC DIFFERENTIAL EVOLUTION)	29
3.4 最佳化方法測試 	31
第四章 自由空間中二維介質柱體影像重建 	47
4.1模擬環境與相關參數設定	47
4.1.1模擬環境配置與參數設定	47
4.1.2 目標函數與最佳化方法搜尋參數	48
4.2最佳化方法重建自由空間中二為非均勻介質柱體影像	49
第五章 結論	65
參考文獻	67















圖目錄
圖2.1 FDTD中二維Yee單胞於TMz模態(左)與TEz模態(右)表示圖	13
圖2.2 FDTD中電磁場計算時序圖	14
圖3.1田口最佳化法流程圖	22
圖3.2 差異型演化法流程圖	25
圖3.3 差異型進化法中突變方法一的示意圖。	27
圖3.4 差異型進化法中突變方法二的示意圖。	28
圖3.5 測試函數函數圖形	31
圖3.6 三種最佳化方法於50-D測試函數收斂特性比較	34
圖3.7 三種最佳化方法於100-D測試函數收斂特性比較	37
圖3.8 三種最佳化方法於250-D測試函數收斂特性比較	41
圖4.1  自由空間中任意形狀介質柱體模擬環境示意圖	47
圖4.2  入射電場波形與頻譜分佈。(a)入射電場時域波形，(b)入射電　
場頻譜分佈	48
圖4.3正確結構重建圖	52
圖4.4差異型演化法	52
圖4.5自我適應之動態差異型演化法	52
圖4.6 (a)田口最佳化法(rr=0.1)	52
圖4.6 (b)田口最佳化法(rr=0.2)	53
圖4.6 (c)田口最佳化法(rr=0.3)	53
圖4.6 (d)田口最佳化法(rr=0.4)	53
圖4.6 (e)田口最佳化法(rr=0.5)	53
圖4.6 (f)田口最佳化法(rr=0.6)	53
圖4.6 (g)田口最佳化法(rr=0.7)	53
圖4.6 (h)田口最佳化法(rr=0.8)	54
圖4.6 (i)田口最佳化法(rr=0.9)	54
圖4.6 (j)田口最佳化法(rr=0.99)	54
圖4.7 TM、DE、SADDE三種最佳化法之比較	54
圖4.8正確結構重建圖	57
圖4.9差異型演化法	57
圖4.10自我適應之動態差異型演化法	57
圖4.11 (a)田口最佳化法(rr=0.1)	57
圖4.11 (b)田口最佳化法(rr=0.2)	58
圖4.11 (c)田口最佳化法(rr=0.3)	58
圖4.11 (d)田口最佳化法(rr=0.4)	58
圖4.11 (e)田口最佳化法(rr=0.5)	58
圖4.11 (f)田口最佳化法(rr=0.6)	58
圖4.11 (g)田口最佳化法(rr=0.7)	58
圖4.11 (h)田口最佳化法(rr=0.8)	59
圖4.11 (i)田口最佳化法(rr=0.9)	59
圖4.11 (j)田口最佳化法(rr=0.99)	59
圖4.12 TM、DE、SADDE三種最佳化法之比較	59
圖4.13正確結構重建圖	62
圖4.14差異型演化法重建圖	62
圖4.15自我適應之動態差異型演化法重建圖	62
圖4.16 (a)田口最佳化法(rr=0.1)	62
圖4.16 (b)田口最佳化法(rr=0.2)	63
圖4.16 (c)田口最佳化法(rr=0.3)	63
圖4.16 (d)田口最佳化法(rr=0.4)	63
圖4.16 (e)田口最佳化法(rr=0.5)	63
圖4.16 (f)田口最佳化法(rr=0.6)	63
圖4.16 (g)田口最佳化法(rr=0.7)	63
圖4.16 (h)田口最佳化法(rr=0.8)	64
圖4.16 (i)田口最佳化法(rr=0.9)	64
圖4.16 (j)田口最佳化法(rr=0.99)	64
圖4.17 TM、DE、SADDE三種最佳化法之比較	64
























表目錄
表3.1 田口直交表OA(18,5,3,2)	23
表3.2 測試函數(benchmark functions)表	32
表3.3 田口最佳化法於測試函數收斂表	46
表3.4 測試函數最佳化方法效能比較表	46
表4.1 均勻介質物體田口最佳化法重建誤差率	52
表4.2多層介質物體田口最佳化法整體重建誤差率	56
表4.3多層介質物體田口最佳化法介電係數1.5的重建誤差率	56
表4.4多層介質物體田口最佳化法介電係數2.5的重建誤差率	56
表4.5多層介質物體田口最佳化法介電係數5.5的重建誤差率	57
表4.6雙層介質物體田口最佳化法整體重建誤差率	61
表4.7雙層介質物體田口最佳化法介電係數1.0的重建誤差率	62
表4.8雙層介質物體田口最佳化法介電係數2.1的重建誤差率	62

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