臺灣博碩士論文加值系統

本論文主要是提出一個當反應函數為未知或難以定義時，依然能有效率地尋找有效點的演算法，其中有效點為其所對應之反應值屬於特定的目標區域內。在此目標區域可以是反應
函數之極值，有界區間，正值，等等。不同於一般對於反應函數的假設，在此我們視反應值的取得為一演化過程。因此，我們考慮用自回歸過程來配適這個演化過程。為了尋找有效點，我們利用當時的反應值以及來自自回歸過程所得到的預測值分別構成的兩個代用曲面，並分別從這兩個曲面上選取可能的有效點。最後我們還利用演化過程的收斂條件來提高新演算法的效率。在本論文中我們展示數個模擬實驗的結果和兩個從動態系統中尋找正Lyapunov指數的實際例子。從這些結果顯示出我們的演算法是有效率的且實用的。

We develop an algorithm to find so-called effective points
$\mathbf{x}\in\mathcal{R}^n$ such that the corresponding responses
$f(\mathbf{x})\in\mathcal{R}$ belong to a specific region of
interest. Examples of a region of interest include extreme values,
bounded intervals, positivity, and others. Here the responses are
obtained iteratively with respect to the evolution $t$, and these
evolution processes are fitted by auto-regressive processes. To find
the effective points, the true yet unknown response surface is
approximated by a surrogate model. Then possible effective points
are selected from two surrogate surfaces, which are constructed
based on the predictions of the AR processes and the current values
of the evolution processes respectively. The convergency criteria
for evolution processes are also used here for improving the
efficiency of our novel algorithm. Several simulations and two real
examples for finding positive Lyapunov exponents of a dynamical
system are demonstrated. Computational results show that the novel
algorithms is efficient and practical.

1 Introduction 1
2 The New Algorithm 2
2.1 The Basis-Based Response Surface Method . . . . . . . . . . . . . . . . 2
2.2 New Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3 Computing Experiments 8
3.1 The Lyapunov Exponents . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.2 The simulation cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4 Experimental Results 13
4.1 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.2 The results for the two experiments of L.E. . . . . . . . . . . . . . . . . 16
4.3 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5 Conclusion 30
A Appendix: The ‾gure of evolution processes for Lyaounov Exponents 32
References 38

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