請用此 Handle URI 來引用此文件:
http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/55640
完整後設資料紀錄
DC 欄位 | 值 | 語言 |
---|---|---|
dc.contributor.advisor | 王偉仲(Wei-Chung Wang) | |
dc.contributor.author | Yi-Chung Hsu | en |
dc.contributor.author | 許奕中 | zh_TW |
dc.date.accessioned | 2021-06-16T04:14:21Z | - |
dc.date.available | 2017-08-25 | |
dc.date.copyright | 2014-08-25 | |
dc.date.issued | 2014 | |
dc.date.submitted | 2014-08-20 | |
dc.identifier.citation | [1] R Biswas, MM Sigalas, K-M Ho, and S-Y Lin. Three-dimensional photonic
band gaps in modified simple cubic lattices. Physical Review B, 65(20):205121, 2002. [2] Alexander Forrester, Andras Sobester, and Andy Keane. Engineering design via surrogate modelling: a practical guide. John Wiley & Sons, 2008. [3] Tsung-Ming Huang, Han-En Hsieh, Wen-Wei Lin, and Weichung Wang. Eigendecomposition of the discrete double-curl operator with application to fast eigensolver for three-dimensional photonic crystals. SIAM Journal on Matrix Analysis and Applications, 34(2):369–391, 2013. [4] Sajeev John. Strong localization of photons in certain disordered dielectric superlattices. Physical review letters, 58(23):2486, 1987. [5] Marc C Kennedy and Anthony O’Hagan. Predicting the output from a complex computer code when fast approximations are available. Biometrika, 87(1):1–13, 2000. [6] Danie G. Krige. A statistical approach to some basic mine valuation problems on the witwatersrand. Journal of Chemical, Metallurgical, and Mining Society of South Africa, 1951. [7] Lord Rayleigh. Xxvi. on the remarkable phenomenon of crystalline reflexion described by prof. stokes. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 26(160):256–265, 1888. [8] Jerome Sacks, William J Welch, Toby J Mitchell, and Henry P Wynn. Design and analysis of computer experiments. Statistical science, pages 409–423, 1989. [9] Thomas J Santner, Brian J Williams, and William Notz. The design and analysis of computer experiments. Springer, 2003. [10] Eli Yablonovitch. Inhibited spontaneous emission in solid-state physics and electronics. Physical review letters, 58(20):2059, 1987. [11] 李其澔. 使用離散粒子群演算法尋找最佳無摺疊均勻實驗設計. Master’s thesis, 臺灣大學, Jan 2014. | |
dc.identifier.uri | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/55640 | - |
dc.description.abstract | 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。 | zh_TW |
dc.description.abstract | 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. | en |
dc.description.provenance | Made available in DSpace on 2021-06-16T04:14:21Z (GMT). No. of bitstreams: 1 ntu-103-R01221020-1.pdf: 585912 bytes, checksum: 024efad6c9abf9c967ccd49bc8eef80a (MD5) Previous issue date: 2014 | en |
dc.description.tableofcontents | 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 . . . . . . . . . . . . . . . . . . . . . . . . . . 60 | |
dc.language.iso | zh-TW | |
dc.title | 結合高低精準度模型尋找最大能隙帶的光子晶體結構 | zh_TW |
dc.title | Two Different Fidelity Surrogates Assisted Optimization for Maximizing Photonic Crystal Bandgap | en |
dc.type | Thesis | |
dc.date.schoolyear | 102-2 | |
dc.description.degree | 碩士 | |
dc.contributor.oralexamcommittee | 陳瑞彬(Ray-Bing Chen),郭岳承(Yueh-Cheng Kuo) | |
dc.subject.keyword | 光子晶體能隙帶,電腦實驗,統計最佳化, | zh_TW |
dc.subject.keyword | Photonic crystal band gap,Computer experiment,Stochastic optimization,Kriging, | en |
dc.relation.page | 61 | |
dc.rights.note | 有償授權 | |
dc.date.accepted | 2014-08-20 | |
dc.contributor.author-college | 理學院 | zh_TW |
dc.contributor.author-dept | 數學研究所 | zh_TW |
顯示於系所單位: | 數學系 |
文件中的檔案:
檔案 | 大小 | 格式 | |
---|---|---|---|
ntu-103-1.pdf 目前未授權公開取用 | 572.18 kB | Adobe PDF |
系統中的文件,除了特別指名其著作權條款之外,均受到著作權保護,並且保留所有的權利。