運用克利金微分代理模型的最佳化

Abstract

光子晶體 (photonic crystal) 是由不同折射率的介質以週期性排列形成之晶體結構，其物理特性可使某連續頻段之光波完全反射，此頻段稱為該光子晶體之能隙帶 (band gap)，想要找出特定光子晶體 結構的能隙帶需解其對應之眾多大型廣義特徵值問題 (generalized eigenvalue problem) 進而求出特徵曲線 (eigencurve) 的極值，其龐大計算量使得尋找光子晶體能隙帶相當耗費時間，因此我們考慮用代理模型 (surrogate)加速此過程。 代理模型是一種用來模擬黑箱函數(black-box function)的統計模型，可讓我們用較低的計算成本觀察目標函數的特性，藉由輸入黑箱 函數在觀測點的函數值並選取適當的基底函數 (basis function)，我們可以構造出適當的光滑近似函數來預測黑箱函數於其他各處的可能值，在比較一區域內的各預測值後，便可進一步找到此黑箱函數可能的區域極值發生處，這可應用在黑箱函數的最佳化問題上。 本論文嘗試使用克利金基底 (Kriging basis) 建構代理模型，為了改善其估計的精準度，再加入黑箱函數在觀測點的一次微分值而得到克利金微分代理模型，並以尋找光子晶體能隙帶問題為例，分別使用克利金代理模型和克利金微分代理模型做最佳化，從而比較兩者精度差異。

A photonic crystal is a kind of periodic structure composed of materials with different dielectric constants. It has a physical characteristic of total re- flecting photons with some frequencies. These groups of frequencies form a continuous interval called photonic band gap. To find the band gap of a pho- tonic crystal, one needs to solve numerous corresponding generalized eigen- value problems then search the extremes of the eigencurves. The process is time-consuming so we introduce the surrogate model to accelerate it. The surrogate model is a statistical model that describes the behavior of a black-box function (a function without explicit formulation). By inputting sample points and function values at there, we can choose proper basis func- tions then construct a smooth approximate function to predict function values at points we are interested in. After investigating the surrogate model, we can find the possible places in the domain that the extremes of the function may appear. In this thesis, the core technique we used in surrogate is the Kriging basis functions. To improve accuracy, we modify the model with first derivatives at sample points. We test the new model by some experiments such as find- ing the band gap of a photonic crystal. The results of comparing Kriging with first derivatives model to original Kriging model are also included.