指導教授：王偉仲臺灣大學：數學研究所許奕中Hsu, Yi-ChungYi-ChungHsu2014-11-302018-06-282014-11-302018-06-282014http://ntur.lib.ntu.edu.tw//handle/246246/264029光子晶體(Photonic crystal)是奈米級的週期性晶體結構，可以被設計成電子波傳導(electric wave propagation)的通道，其原理是，一種光子晶體結構能完全反射某一段頻率的電子波，我們稱這段頻率為能隙帶(band gap)。我們可透過數值方法解馬克思威方程(Maxwell''s equation)，轉換成一個大型的一般特徵值問題(generalized eigenvalue problem)並找到光子晶體結構所對應的能隙帶。雖然現今的運算處理器進步神速，我們仍視光子晶體的相關計算為一大難事。為了快速有效地找到擁有最大能隙帶的光子晶體結構(optimal design)，我們嘗試使用共克利金(Co-Kriging)產生能隙帶的代理模型(Surrogate)，共克利金能容許我們使用不同精度(multi-fidelity)所解出的能隙帶資料，運用其並建造一個相對可信的代理模型。我們也結合其附屬的選點方式，期望進步法(Expected Improvement)，其功能可使代理模型找到極值點。預期能使用較少的計算時間，找到其最佳結構。另外，我們也嘗試使用一個小技巧，在過程中縮小搜尋的範圍(Zoom in)，加速了最佳化收斂的過程。Photonic crystal is a kind of nano-scale deletric periodic system, which is usually used to construct a tunnel for eletricwave propagation. .This idea is based on the absolute reflection of some frequencies of electricwave, these frequencies are continuous and called "photonic band gap" (band gap in simplicity). We use approximation method to convert Maxwell''s equations to a large-scale generalized eigenvalue problem, and solve it to archeive the band gap with respect to different photonic crystal design. Our propose is to find an optimal design and maximize the band gap, we try to use Co-Kriging, an alternative of Kriging, to construct surrogate for band gap, Co-Kriging allows us to use two different fidelity of experimental points to construct surrogate.中文摘要1 Abstract 2 Table of Contents 3 List of Figures 7 List of Tables 8 1 Introduction 9 2 Problem Definition 11 2.1 Photonic Crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Maximization Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 Eigenvalue Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4 Bandgap Extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.5 Algorithm: Bandgap Extraction via IPLM . . . . . . . . . . . . . . . 15 3 Methods and Procedure 17 3.1 Surrogate-based Auto Tuning Overview . . . . . . . . . . . . . . . . . 17 3.1.1 Derivative-based Optimization . . . . . . . . . . . . . . . . . 17 3.1.2 Concept of Surrogate-based Auto Tuning . . . . . . . . . . . . 17 3.1.3 Algorithm: Theoretical Surrogate-based Auto Tuning . . . . . 19 3.2 Initial Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.2.1 Irregular Space Filling . . . . . . . . . . . . . . . . . . . . . . 20 3.2.2 High-Low Fidelity samples Distribution . . . . . . . . . . . . 20 3.3 Observation Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.4 Surrogate Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.4.1 Kriging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.4.2 Co-Kriging . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.4.3 Why We Choose Co-Kriging . . . . . . . . . . . . . . . . . . . 29 3.5 Infill Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.5.1 Exploiting and Exploring . . . . . . . . . . . . . . . . . . . . 30 3.5.2 Fidelity Selection for new chosen sample . . . . . . . . . . . . 30 3.5.3 Low-fidelity Sample Duplicately Chosen . . . . . . . . . . . . 30 3.5.4 Algorithm: Sampling and Fidelity Choosing . . . . . . . . . . 32 3.6 Shrinking (Zoom in) strategy . . . . . . . . . . . . . . . . . . . . . . 33 3.6.1 Creating Sub-window . . . . . . . . . . . . . . . . . . . . . . 33 3.6.2 Condition to ”Zoom in” . . . . . . . . . . . . . . . . . . . . . 34 3.6.3 Future Work to ”Zoom out” . . . . . . . . . . . . . . . . . . . 35 3.7 Stopping Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4 Numerical Results and Disccusion 37 4.1 Environment Setting and Test Cases . . . . . . . . . . . . . . . . . . 37 4.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.2.1 Amount and Distribution of Initial Samples . . . . . . . . . . 38 4.2.2 ”Zoom In” Effect . . . . . . . . . . . . . . . . . . . . . . . . . 40 References 44 Appendices 46 Appendix A Kriging 46 A.1 Assumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 A.2 Parameters Estimation for Kriging . . . . . . . . . . . . . . . . . . . 48 A.2.1 Regression Coefficient beta . . . . . . . . . . . . . . . . . . . . . 49 A.2.2 Maximum Likelihood Estimate for Varience . . . . . . . . . . 49 A.2.3 Primary Parameters: theta, p . . . . . . . . . . . . . . . . . . . . 50 A.3 Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 Appendix B Co-Kriging 53 B.1 Autoregressive Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 B.1.1 Estimate sigma^2_c , beta_ c, theta_ c, p_c . . . . . . . . . . . . . . . . . . . . . . 55 B.1.2 Estimate sigma^2_d beta_d, theta_ d, p_d and . . . . . . . . . . . . . . . . . . 55 B.1.3 Estimate beta . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 B.2 Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Appendix C Exploring Criteria 59 C.1 Prediction Error Measurement . . . . . . . . . . . . . . . . . . . . . . 59 C.2 Expected Improvement . . . . . . . . . . . . . . . . . . . . . . . . . . 60585912 bytesapplication/pdf論文公開時間：2017/08/25論文使用權限：同意有償授權(權利金給回饋學校)光子晶體能隙帶電腦實驗統計最佳化thesishttp://ntur.lib.ntu.edu.tw/bitstream/246246/264029/1/ntu-103-R01221020-1.pdf