臺灣博碩士論文加值系統

在本論文中，我們使用量身定做的有限點法與多項式渾沌展開法求解偏微分方程式，應用在流體動力、影像處理以及財金相關等問題。

在第一部份中，考慮微小黏滯係數隨時間變化的擬線性伯格斯方程式。在量身定做的有限點法中，我們採用兼具時間和空間特性的基底，進行局部逼近，為伯格斯方程式實作兩種方法。第一種，先利用霍普—柯爾轉換，實作量身定做的有限點法。第二種，以原始伯格斯方程式區域解，作為量身定做的有限點法基底，利用疊代法逼近，取得下一個時間近似解方式施作。第二種量身定做的有限點法，特別適合使用在不連續及陡坡方程式。兩種量身定做的有限點法求解伯格斯方程式，在粗造網格中都可以獲得合理精確的數值解。

於第二部份中，我們將量身定做的有限點法應用至影像處理。採用異向性擴散對流偏微分方程式，來進行影像去雜訊過濾程序。在影像去雜訊和壓縮的過程中，引用四元樹資料結構，以做為多層式不規則網格資料儲存格式。此異向性擴散對流過濾程序，顯示此方法比起其他偏微分方程過濾器，可得到更精確的近似解。

而在第三部份裡，我們將量身定做的有限點法應用於歐式選擇權（布雷克—休斯方程式）。當我們將數值解和其他已知方法做比較後，發覺量身定做的有限點法其計算效能和精確度表現可說是十分優異。

最後，我們以多項式渾沌展開法求解隨機偏微分方程式。提出廣義和任意兩種多項式渾沌展開法的理論定義，並以這兩種多項式渾沌展開法求解隨機偏微分方程，包含簡單測試方程以及隨機波動率的布雷克—休斯方程式。首先，以對數常態分佈的波動率，檢驗廣義和任意兩種多項式渾沌展開法，計算布雷克—休斯方程式數值解的精確度。再者，以短期隨機波動率，引用任意多項式渾沌展開法，計算歐式選擇權利金的平均值和標準差。以此多項式渾沌展開法求解隨機波動率選擇權方式，目前對於金融數學界而言，尚屬新嘗試。

In this dissertation, we study the tailored finite point method (TFPM) and polynomial chaos expansion (PCE) scheme for solving partial differential equations (PDEs). These PDEs are related to fluid dynamics, imaging processing and finance problems.

In the first part, we concern on quasilinear time-dependent Burgers'' equations with small coefficients of viscosity. The selected basis functions for the TFPM method automatically fit the properties of the local solution in time and space simultaneously. We apply the Hopf-Cole transformation to derive the first TFPM-I scheme. For the second scheme, we approximate the solution by using local exact solutions and consider iterated processes to attain numerical solutions to the original form of the Burgers'' equation. The TFPM-II is particularly suitable for a solution with steep gradients or discontinuities. More importantly, the TFPM obtained numerical solutions with reasonable accuracy even on relatively coarse meshes for Burgers'' equations.

In the second part, we employ the application of the TFPM in an anisotropic convection-diffusion (ACD) filter for image denoising. A quadtree structure is implemented in order to allow multi-level storage during the denoising and compression process. The ACD filter exhibits the potential to get a more accurate approximated solution to the PDEs.

In the third part, we regard the TFPM for Black-Scholes equations, European option pricing. We compare the performance of our algorithm with other popular numerical schemes. The numerical experiments using the TFPM is more efficient and accurate compared to other well-known methods.

In the last part, we present the polynomial chaos expansion (PCE) for stochastic PDEs. We provide a review of the theory of generalized polynomial chaos expansion (gPCE) and arbitrary polynomial chaos expansion (aPCE) including the case analysis of test problems. We demonstrate the accuracy of the gPCE and aPCE for the Black-Scholes model with the log-normal random volatilities. Furthermore, we employ the aPCE scheme for arbitrary distributions of uncertainty volatilities with short term price data. This is the forefront of adopting the polynomial chaos expansion in the randomness of volatilities in financial mathematics.

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Tailored finite point method (TFPM) for solving one-dimensional
Burgers’ equations . . . . . . . . . . . . . . . . . . . . . . 5
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 The TFPM–I . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.1 The Hopf–Cole transformation . . . . . . . . . . . . . . . . 7
2.2.2 The TFPM–I . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.3 Stability analysis of the TFPM–I . . . . . . . . . . . . . . 11
2.2.4 Truncation error for the TFPM–I . . . . . . . . . . . . . . 12
2.3 The TFPM–II . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.1 Stability analysis for the TFPM–II . . . . . . . . . . . . . 16
2.3.2 Error analysis for the TFPM–II . . . . . . . . . . . . . . . 17
2.4 Numerical experiments . . . . . . . . . . . . . . . . . . . . 18
2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 22
3 TFPM for image denoising and compression . . . . . . . . . . . 30
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2 Nonlinear filters . . . . . . . . . . . . . . . . . . . . . . 31
3.2.1 The PM diffusion filter . . . . . . . . . . . . . . . . . . 31
3.2.2 The CD filter and ACD filter . . . . . . . . . . . . . . . . 32
3.3 TFPM for the ACD filter . . . . . . . . . . . . . . . . . . . 34
3.3.1 The TFPM for the discretization . . . . . . . . . . . . . . 34
3.3.2 Stability analysis . . . . . . . . . . . . . . . . . . . . . 36
3.4 Implementing the TFPM in adaptive grids . . . . . . . . . . . 37
3.4.1 The quadtree representation of images . . . . . . . . . . . 37
3.4.2 The TFPM in adaptive grids . . . . . . . . . . . . . . . . . 41
3.5 Numerical experiments . . . . . . . . . . . . . . . . .. . . . 42
3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . .. . . . 43
4 TFPM for the Black–Scholes equation . . . . . . . . . . . . . 46
4.1 Introduction of the Black–Scholes equation . . . . . . . . . . 46
4.2 The explicit and implicit TFPM for heat equations . . .. . . . 50
4.3 The sparse stiffness matrix with Dirichlet boundaries . . . . 51
4.4 Numerical experiments . . . . . . . . . . . . . . . . . . . . 52
4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 54
5 Stochastic Black–Scholes equations . . . . . . . . . . . . . . 58
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.2 Uncertainty volatilities . . . . . . . . . . . . . . . . . . . 59
5.3 Probability theory and random variables . . . . . . . . . . . 63
5.3.1 Measure on a Hilbert space . . . . . . . . . . . . . . . . . 64
5.3.2 Generalized polynomials chaos expansion (gPCE) . . . . . . . 65
5.3.3 Distribution fitting historical volatilities . . . . . . . . 67
5.4 Stochastic spectral methods . . . . . .. . . . . . . . . . . . 71
5.4.1 Monte Carlo sampling (MCS) . . . . . . . . . . . . . . . . . 71
5.4.2 Stochastic collocation method (SCM) . . . . . . . . . . . . 72
5.4.3 Stochastic Galerkin method (SGM) . . . . . . . . . . . . . . 74
5.5 Arbitrary polynomial chaos expansion (aPCE) . . . . . . . .. . 77
5.5.1 Construction of the aPCE . . . . . . . . . . . . . . . . . . 77
5.5.2 Orthonormal polynomial basis of the aPCE . . . . . . . . . . 78
5.6 PCE for test problems . . . . . . . . . . . . . . . . . . . . 79
5.6.1 Stochastic ODEs . . . . . . . . . . . . . . . . . . . . . . 79
5.6.2 Stochastic diffusion equations . . . . . . . . . . . . . . . 88
5.7 PCE for Black–Scholes equations . . . . . . . . . . . . . . . 94
5.8 Numerical experiments . . . . . . . . . . . . . . . . . . . . 97
5.9 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . 99
6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 110
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . 112

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