(3.238.96.184) 您好!臺灣時間:2021/05/08 03:21
字體大小: 字級放大   字級縮小   預設字形  
回查詢結果

詳目顯示:::

: 
twitterline
研究生:林玉端
研究生(外文):Yu--Tuan Lin
論文名稱:使用量身定做的有限點法與多項式渾沌展開法解流體動力、影像處理以及財金問題
論文名稱(外文):Implementations of Tailored Finite Point Method and Polynomial Chaos Expansion for Solving Problems Related to Fluid Dynamics, Image Processing and Finance
指導教授:王輝清施因澤
指導教授(外文):Hui-Ching WangYin--Tzer Shih
口試委員:陳焜燦楊肅煜黃杰森
口試日期:2016-07-07
學位類別:博士
校院名稱:國立中興大學
系所名稱:應用數學系所
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:2016
畢業學年度:104
語文別:英文
論文頁數:119
中文關鍵詞:量身定做的有限點法多項式渾沌展開法伯格斯方程式布雷克—休斯方程式
外文關鍵詞:Tailored Finite Point MethodPolynomial Chaos ExpansionBurgers’ EquationsBlack-Scholes Equations
相關次數:
  • 被引用被引用:0
  • 點閱點閱:107
  • 評分評分:系統版面圖檔系統版面圖檔系統版面圖檔系統版面圖檔系統版面圖檔
  • 下載下載:6
  • 收藏至我的研究室書目清單書目收藏:0
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

[1] E.N. Aksan. Quadratic B-spline finite element method for numerical solution of the Burgers’ equation. Appl. Math. Comput., 174:884–896, 2006.

[2] A. Asaithambi. Numerical solution of the Burgers’ equation by automatic differentiation. Appl. Math. Comput., 216:2700–2708, 2010.

[3] A.R. Bahadir and M. Sağlam. A mixed finite fifference and boundary element approach to one-dimensional Burgers’ equation. Appl. Math. Comput., 160:663–673, 2005.

[4] A.N. Brooks and T.J.R. Hughes. Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput. Meth. Appl. Mech. Eng., 32:199–259, 1982.

[5] J. Canny. A computational approach to edge detection. IEEE Trans. Pattern Anal. Machine Intell., 8:679–698, 1986.

[6] F. Catté, P. L. Lions, J. M. Morel, and T. Coll. Image selective smoothing
and edge detection by nonlinear diffusion. SIAM J. Numer. Anal., 29:182–193, 1992.

[7] J.D. Cole. On a quasi-linear parabolic equation occurring in aerodynamics. Quart. Appl. Math., 9(3):225–236, 1951.

[8] ROHAN Academic Computing. Image analysis. http://www-rohan.sdsu.edu/doc/matlab/toolbox/images.

[9] A. Dogan. A Galerkin finite element approach to Burgers’ equation. Appl. Math. Comput., 157:331–346, 2004.

[10] Albert Einstein. In Germ.: Über die von der molekularkinetischen theorieder Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten teilchen; In Eng: investigations on the theory of Brownian movement. Annalen
der Physik, 17(8):549–560, 1905.

[11] M. S. Eldred and J. Burkardt, editors. Comparison of Non-Intrusive Polynomial Chaos and Stochastic Collocation Methods for Uncertainty Quantification. American Institute of Aeronautics and Astronautics, 2009.

[12] Howard Elman. Lecture notes: Numerical methods for mathematical models posed with uncertainty.

[13] Oliver G. Ernst, Antje Mugler, Hans-Jörg Starkloff, and Elisabeth Ullmann. On the convergence of generalized polynomial chaos expansions. Mathematical Modelling and Numerical Analysis, M2AN 46:317–339, 2012.

[14] R. Finkel and J. L. Bentley. Quad trees: a data structure for retrieval on composite keys. Acta Inform., 4:1–9, 1974.

[15] F. Gao and C. Chi. Numerical solution of nonlinear Burgers’ equation using high accuracy multi-quadric quasi-interpolation. Appl. Math. Comput., 229:414–421, 2014.

[16] R.J. Gelinas and S. K. Doss. The moving finite element method: applications to general partial differential equations with multiple large gradients. J. Comput. Phys., 40:202–249, 1981.

[17] Marc Gerritsma, Jan-Bart van der Steen, Peter Vos, and George Karniadakis. Time-dependent generalized polynomial chaos. Journal of Computational Physics, 229:8333–8363, 2010.

[18] Roger Ghanem. The nonlinear Gaussian spectrum of log-normal stochastic process and variables. J. Appl. Mech, 66:964–973, 1999.

[19] Roger G. Ghanem and Pol D. Spanos. Stochastic Finite Elements: A Spectral Approach. Springer New York, 1991.

[20] Daniel Hackmann. Solving the Black Scholes equation using a finite difference method. http://math.yorku.ca/~dhackman/.


[21] Chuan-Hsiang Han and Yu-Tuan Lin. Accelerated variance reduction methods on GPU. In Proceedings of the 20th IEEE International Conference on Parallel and Distributed Systems, 2014.

[22] H. Han. Private communication with Prof. Han.

[23] H. Han and Z. Huang. A tailored finite point method for the Helmholtz equation with high wave numbers in heterogeneous medium. J. Comp. Math., 26:728–739, 2008.

[24] H. Han and Z. Huang. Tailored finite point method for a singular perturbation problem with variable coefficients in two dimensions. J. Sci. Comp., 41:200–220, 2009.

[25] H Han, Z Huang, and RB Kellogg. A tailored finite point method for a singular perturbation problem on an unbounded domain. J Sci Comp, 36:243–261, 2008.

[26] H. Han, J.J.H. Miller, and M. Tang. A parameter-uniform tailored finite point method for singularly perturbed linear ODE systems. J. Comp. Math, 31:422–438, 2013.

[27] H. Han, Y.-T. Shih, and C.-C. Tsai. Tailored finite point method for numerical solutions of singular perturbed eigenvalue problems. Adv. Appl. Math. Mech., 6(3):376–402, 2014.

[28] A. Hashemian and H.M. Shodja. A meshless approach for solution of Burgers’ equation. J. Comput. Appl. Math., 220:226–239, 2008.

[29] I.A. Hassanien, A.A. Salama, and H.A. Hosham. Fourth-order finite difference method for solving Burgers equation. Appl. Math. Comput., 170:781–800, 2005.

[30] Steven L. Heston. A closed-form solution for options with stochastic volatility with applications to bond and currency options. The Review of Financial
Studies, 6(2):327–343, 1993.

[31] E. Hopf. The partial differential equation ut + uux = νuxx. Commun. Pure Appl. Math, 3:201–230, 1950.

[32] P. W. Hsieh, Y. T. Shih, and S. Y. Yang. A tailored finite point method for solving steady MHD duct flow problems with boundary layers. Commun. Comput. Phys., 10:161–182, 2011.

[33] Z. Huang and X. Yang. Tailored finite point method for first order wave equation. J. Sci. Comput., 49:351–366, 2011.

[34] John C. Hull. Options, Futures, and Other Derivatives. Pearson Education, 2009.

[35] Investopedia. http://www.investopedia.com/.

[36] C. Johson and K. Szepessy. On the convergence of a finite element method for a nonlinear hyperbolic conservation law. Math. Comput., 49:427–444, 1987.

[37] Davar Khoshnevisan. Probability. AMS, 2007.

[38] Emil Barandt Kærgaard. Spectral methods for uncertainty quantification. Master’s thesis, Technical University of Denmark, 2013.

[39] S. Kutulay, A.R. Bahadir, and A. Özdeş. Numerical solution of the onedimensional Burgers’ equation: Explicit and exact-explicit finite difference methods. J. Comput. Appl. Math., 103:251–261, 1999.

[40] S. Kutulay, A. Esen, and I. Dag. Numerical solutions of the Burgers’ equation by the least-squares quadratic B-spline finite element method. J. Comput. Appl. Math., 167:21–33, 2004.

[41] W. Liao. An implicit fourth-order compact finite difference scheme for onedimensional Burgers’ equation. Appl. Math. Comput., 206(2):755–764, 2008.

[42] W. Liao and J. Zhu. Efficient and accurate finite difference schemes for solving one-dimensional Burgers’ equation. Int. J. Comput. Math., 88(12):2575–2590, 2011.

[43] Yen-Ru Lin. Discrete nonlinear diffusion model for image denoising by using the tailored finite point method. Master’s thesis, Department of Applied Mathematics, National Chung Hsing University, 2014.

[44] Yu-Tuan Lin, Yin-Tzer Shih, and Chih-Ching Tsai. An anisotropic convectiondiffusion model using tailored finite point method for image denoising and compression. Communications in Computational Physics, 2016.

[45] Yu-Tuan Lin, Yin-Tzer Shih, and Hui-Ching Wang. An explicit and implicit tailored finite point method for option pricing simulation. In Proceedings of the 6th International Asia Conference on Industrial Engineering and Management
Innovation, pages 189–199. Springer, 2015.

[46] F. Liu and J. Liu. Anisotropic diffusion for image denoising based on diffusion tensors. J. Vis. Commun. Image R., 23:516–521, 2012.

[47] X. Ma and N. Zabaras. High-dimensional stochastic model representation technique for the solution of stochastic pdes. J. Comput. Phys., 229:3884–3915, 2010.

[48] O. P. Le Maître and Omar M. Knio. Spectral Methods for Uncertainty Quantification with Application to Computational Fluid Dynamics. Springer Netherlands, 2010.

[49] Robert C. Merton. Theory of rational option pricing. Bell Journal of Economics and Management Science (The RAND Corporation), 4(1):141–183, 1973.

[50] Motoi J. Namihira. Probabilistic Uncertainty Analysis and Its Applications in Option Models. PhD thesis, Florida State University, 2013.

[51] F. Nobile, R. Tempone, and C. Webster. A sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J. Numer. Anal., 2008.

[52] Bernt Arne Odegaard. Financial numerical recipes in c++. http://finance.bi.no/~bernt/gcc_prog/recipes/recipes.pdf, 2014.

[53] University of Maryland. http://www.burgers.umd.edu/burgers.html.

[54] A. O’Hagan. Polynomial chaos: a tutorial and critique from a statisticians perspective. http://www.tonyohagan.co.uk/academic/pdf/Polynomial-chaos.pdf, 2013.

[55] S. Oladyshkin and W. Nowak. Data-driven uncertainty quantification using the arbitrary polynomial chaos expansion. Reliability Engineering & System Safety, 106:179–190, 2012.

[56] K. Pandey, L. Verma, and A.K. Verma. On a finite difference scheme for Burgers’ equation. Appl. Math. Comput., 215:2206–2214, 2009.

[57] P. Perona and J. Malik. Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Machine Intell., 12:629–639, 1990.

[58] Mass Per Pettersson, Gianluca Iaccarino, and Jan Nordström. Polynomial Chaos Methods for Hyperbolic Partial Differential Equations. Springer International Publishing, 2015.

[59] T. Preußer and M. Rumpf. An adaptive finite element method for large scale image processing. Journal of Visual Communication and Image Representation, 11(2):183–195, 2000.

[60] Roland Pulch and Cathrin van Emmerich. Polynomial chaos for simulating random volatilities. Mathematics and Computers in Simulation, 80(2):245–255, 2009.

[61] Mark Richardson. Numerical Methods for Option Pricing. University of Oxford, 2009.

[62] Y. Shih, C. Chien, and C. Chuang. An adaptive parameterized block-based singular value decomposition for image de-noising and compression. Appl. Math. Comput., 218:10370–10385, 2012.

[63] Y. Shih and H. C. Elman. Modified streamline diffusion schemes for convection diffusion problems. Meth. Appl. Mech. Eng., 147:137–151, 1999.

[64] Y. Shih, C. Rei, and H. Wang. A novel PDE based image restoration: convection-diffusion equation for image denoising. J. Comput. Appl. Math., 231:771–779, 2009.

[65] Y.-T. Shih, R. B. Kellogg, and Y. Chang. Characteristic tailored finite point method for convection dominated convection-diffusion-reaction problems. J. Sci. Comput., 47:189–215, 2011.

[66] Y.-T. Shih, R. B. Kellogg, and P. Tsai. A tailored finite point method for convection-diffusion-reaction problems. J. Sci. Comput., 43(2):239–260, 2010.

[67] YT Shih and HC Elman. Iterative methods for stabilized discrete convection diffusion problems. IMA, Numerical Analysis, 20(3):333–385, 2000.

[68] Albert N. Shiryaev. Problems in Probability. Springer, 2012.

[69] Steven Shreve. Stochastic Calculus for Finance II: Continuous Time Models. Springer-Verlag New York, 2004.

[70] A. N. Tikhonov. Solution of incorrectly formulated problems and the regularization method. Soviet Math. Dokl., 4:1035 – 1038, 1963.

[71] Chih-Ching Tsai, Yin-Tzer Shih, Yu-Tuan Lin, and Hui-Ching Wang. Tailored finite point method for solving one-dimensional Burgers’ equation. International Journal of Computer Mathematics, 2016.

[72] J. Weickert. Anisotropic diffusion in image processing. ECMI Series, Teubner-Verlag, Stuttgart, Germany, 1998.

[73] Joachim Weickert. Anisotropic Diffusion in Image Processing. PhD thesis, University of Copenhagen, Copenhagen, Denmark, 1998.

[74] N. Wiener. The homogeneous chaos. American Journal of Mathematics, 60:897–936, 1938.

[75] WikiPedia. https://en.wikipedia.org.

[76] W.L. Wood. An exact solution for Burgers’ equation. Comm. Numer. Meth. Eng., 22(7):797–798, 2006.

[77] Dongbin Xiu. Numerical Methods for Stochastic Computations: A Spectral Method Approach. Princeton University Press, 2010.

[78] Dongbin Xiu and G. E. Karniadakis. The Wiener–Askey polynomial chaos for stochastic differential equations. SIAM. J. Sci. Comput., 24:619–644, 2002.

[79] Dongbin Xiu and Jie Shen. Efficient stochastic galerkin methods for random diffusion equations. Journal of Computational Physics, 228:266 – 281, 2009.

[80] P.-G. Zhang and J.-P. Wang. A predictor-corrector compact finite difference scheme for Burgers’ equation. Appl. Math. Comput., 219:892–898, 2012.

QRCODE
 
 
 
 
 
                                                                                                                                                                                                                                                                                                                                                                                                               
第一頁 上一頁 下一頁 最後一頁 top
無相關論文
 
無相關期刊
 
無相關點閱論文
 
系統版面圖檔 系統版面圖檔