最佳化對於當前電腦實驗以及物理方面是非常重要的一個問題，在本研究中探討最佳化問題在於昂貴函數計算假設下的問題。在過去做此方法是透過重複以下兩個步驟來解決此問題，首先以現有資訊建立代理模型，其次透過代理模型選取下一個點並加入現有資訊中。在本研究中代理模型採用Bayesian adaptive Radial Basis Function (BaRBF)來建模並依據Sample Expected Improvement (SEI)選擇標準來進行選點。BaRBF是採用馬可夫鏈蒙地卡羅對徑向基函數進行參數估計，而SEI是根據EI改進的選擇標準，是為了可以應用在BaRBF模型上。在本文中將提出局部貝式演算法(LBaRBF)，其起始概念為快速探索局部最佳值，當演算法探索到局部最佳值時，就重啟此算法，來探索下一個局部最佳值，如果演算法重啟足夠多次，達到探索全部的局部最佳值，也就代表了此演算法找到了全域最佳值。另外由於原先所採用的重啟架構所耗費成本過大，所以額外提出了逃脫局部貝氏演算法(E-LBaRBF)，透過加入一些新的探索點去改變現有的代理模型平面使其在不放棄所有資訊下，也可以達到相同效果。最後在模擬測試部分展示新的逃脫架構比起原先的重啟效益來的更好。

Optimization is a very issue problem for current computer experiments and physics. In this study, the expensive computational cost is considered. In the past, this optimization problem can be solved by repeating the following two steps. First, we build a surrogate model based on the current explored point set, and then add the next point through the surrogate model to the explored point set. In this study, the Bayesian adaptive Radial Basis Function (BaRBF) is adopted for the surrogate construction, and the Sample Expected Improvement (SEI) criterion is used to choose the next explored point. In this thesis, the local Bayesian RBF algorithm (LBaRBF) is proposed. Its key concept is to quickly explore the local optimal value. When the LBaRBF identifies a locally optimal point, it restarts the algorithm to identify another one. Thus once the LBaRBF can find all optimal points, it means that the global best one is also identified. In addition, due to the expensive computational cost assumption, we propose another algorithm, called escaped LBaRBF(E-LBaRBF). Instead of the restart step in LBaRBF, E-LBaRBF will update the surrogate model by adding extract explored points. In this thesis, several numerical experiments are used to demonstrate the performances of the proposed two methods.

Andrieu, C., Freitas, N. d., & Doucet, A. (2001). Robust full bayesian learning for radial basis networks. Neural computation, 13(10), 2359–2407.
Box, G. E., Draper, N. R., et al. (1987). Empirical model-building and response surfaces (Vol. 424). Wiley New York.
Chen, R.-B., Wang, W., & Wu, C. J. (2017). Sequential designs based on bayesian uncertainty quantification in sparse representation surrogate modeling. Technometrics, 59(2), 139–152.
Chen, R.-B., Wang, Y., & Wu, C. J. (2020). Finding optimal points for expensive functions using adaptive rbf-based surrogate model via uncertainty quantification. Journal of Global Optimization, 77, 919–948.
Chipman, H., Ranjan, P., & Wang, W. (2012). Sequential design for computer experiments with a flexible bayesian additive model. Canadian Journal of Statistics, 40(4), 663–678.
Eriksson, D., Bindel, D., & Shoemaker, C. A. (2019). pysot and poap: An event-driven asynchronous framework for surrogate optimization. arXiv preprint arXiv:1908.00420.
Fasshauer, G. E., & Zhang, J. G. (2007). On choosing “optimal＂shape parameters for rbf approximation. Numerical Algorithms, 45(1-4), 345–368.
George, E. I., & McCulloch, R. E. (1993). Variable selection via gibbs sampling. Journal of the American Statistical Association, 88(423), 881–889.
Gutmann, H.-M. (2001). A radial basis function method for global optimization. Journal of global optimization, 19(3), 201–227.
Jones, D. R. (2001). A taxonomy of global optimization methods based on response surfaces. Journal of global optimization, 21(4), 345–383.
Jones, D. R., Schonlau, M., & Welch, W. J. (1998). Efficient global optimization of expensive black-box functions. Journal of Global optimization, 13(4), 455–492.
Koutsourelakis, P.-S. (2009). Accurate uncertainty quantification using inaccurate computational models. SIAM Journal on Scientific Computing, 31(5), 3274–3300.
Mockus, J., Tiesis, V., & Zilinskas, A. (1978). The application of bayesian methods for
seeking the extremum. Towards global optimization, 2(117-129), 2.
Morris, M. D., & Mitchell, T. J. (1995). Exploratory designs for computational experiments. Journal of statistical planning and inference, 43(3), 381–402.
Regis, R. G., & Shoemaker, C. A. (2007). A stochastic radial basis function method for the global optimization of expensive functions. INFORMS Journal on Computing, 19(4), 497–509.
Zellner, A. (1986). On assessing prior distributions and bayesian regression analysis with g-prior distributions. Bayesian inference and decision techniques.