本論文主要研究污染源追蹤和重構的反問題以及倒向隨機微分方程的數值求解。 / 論文的第一部份考慮污染源追蹤及重構的反問題。它的目的是重構反應對流擴散系統中的未知污染源的位置以及強度。污染源的追蹤和重構在工程、化學、生物以及環境等領域有廣泛的應用。我們將同時重構靜態單點污染源的位置以及強度。在本論文中，我們提出了一個基於對偶概率的算法，它將污染源追蹤重構的反問題轉化為Volterra積分反問題。對於污染源的位置和污染物釋放強度的可重構性，文中也進行了理論上的分析和討論。數值結果表明此方法是高效穩定的。隨後，我們將對偶概率方法推廣應用與追蹤和重構動態單點污染源隨時間的軌跡以及強度。數值結果顯示，我們的方法要比多數現有的方法為有效，計算成本也大大降低。 / 論文的第二部份討論倒向隨機微分方程的數值求解。倒向隨機微分方程在隨機控制、生物、化學反應，尤其是數理金融上有重要的應用。論文中所提出的數值方法，主要是基於倒向隨機微分方程的置換解的概念。置換解的適定性分析不涉及鞅表示論，從而更靈活，更容易推廣。利用置換解的理論，文中所涉及的誤差分析都不需要用到鞅表示論。對於一般的倒向隨機微分方程，我們提出了一種簡單的倒向算法，并證明了它是半階收斂的。但是，在算法的實際應用中只可能選取有限個基函數，從而帶入了截斷誤差。截斷誤差在簡單倒向算法中會隨時間累加，導致誤差是半階增長的。為了克服這個缺點，我們提出了一種新的算法。這種算法無需進行皮卡迭代，並且在理論上我們證明了，使用這個新的算法，截斷誤差是可控的，它不會隨時間增加。隨後，我們對馬爾科夫情況的倒向隨機微分方程提出了幾個高階的數值算法，並且給出了嚴格的誤差分析。我們的數值實驗結果表明，文中所提出的方法精度高，穩定性強，且計算成本小。 / In this thesis, we shall propose some numerical methods for solving two important classes of application problems, namely the inverse heat source problems and the backward stochastic differential equations. / The inverse heat source problems are to recover the source terms in a convection-diffusion- reaction system. These inverse problems have wide applications in many areas, such as engineering, chemistry, biology, pollutant tracking, and so on. We shall first investigate the simultaneous reconstruction of the location and strength of a static singular source. An adjoint probabilistic algorithm is proposed, which turns the inverse heat source problem into an inverse Volterra integral problem. The identifiability of the location and strength of a singular source is also discussed, and numerical results are presented to show the robustness and effectiveness of the method. Then we extend the adjoint probabilistic method to reconstruct the source trace and release history of a singular moving point source. Numerical examples show that the adjoint probabilistic method is more efficient and less expensive than most existing efficient numerical methods. / The second part of the thesis is devoted to numerical solutions of some nonlinear backward stochastic differential equations (BSDEs). BSDEs are widely used in various fields like stochastic control, biology, chemistry reaction, especially mathematical finance. Our numerical methods are based on a new framework about the transposition solution to BSDEs. The proof of the well-posedness of the transposition solution does not involve Martingale representation, neither does our error analysis for the numerical schemes proposed in this thesis. For general BSDEs, we first propose a simple backward scheme, which is proved to have an accuracy of half order. However, in the real application of the scheme, it is only possible to choose a finite subset of basis functions, which will generate truncation error. The truncation error accumulates backward in time, leading to the increment of the numerical error up to a half order. To overcome this drawback, we propose a new numerical scheme without Picard iterations and prove that the truncation error is bounded independent of time partitions. Afterwards, we propose some higher order schemes for Markovian BSDEs with rigorous error analysis. Finally, numerical simulations are presented to demonstrate that the proposed methods are accurate, stable and less expensive than most existing ones. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Wang, Shiping. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 126-133). / Abstract also in Chinese. / Abstract --- p.i / Acknowledgement --- p.v / Chapter 1 --- Introduction to inverse heat source problems and BSDEs --- p.1 / Chapter 1.1 --- Inverse heat source problems --- p.2 / Chapter 1.2 --- Backward stochastic differential equations --- p.7 / Chapter 1.3 --- Outline of the thesis --- p.11 / Chapter Part I: --- Numerical Method for Inverse Heat Source Problem --- p.13 / Chapter 2 --- Inverse heat source: static point source --- p.14 / Chapter 2.1 --- Reformulation of the forward problem --- p.15 / Chapter 2.2 --- Inverse source problem and its identifiability --- p.21 / Chapter 2.2.1 --- Identifiability of partial time in one dimensional cases --- p.21 / Chapter 2.2.2 --- Identifiability of two dimensional cases --- p.25 / Chapter 2.3 --- Algorithm to solve the inverse problem --- p.26 / Chapter 2.4 --- Numerical experiments --- p.29 / Chapter 3 --- Inverse heat source: moving point source --- p.41 / Chapter 3.1 --- Reformulation of the problem --- p.42 / Chapter 3.2 --- Algorithm and numerical examples --- p.43 / Chapter 3.2.1 --- Algorithm to recover source trace and strength --- p.44 / Chapter 3.2.2 --- Numerical examples --- p.45 / Chapter Part II: --- Numerical Methods to Backward Stochastic Differential Equations --- p.55 / Chapter 4 --- Preliminaries --- p.56 / Chapter 4.1 --- Notations and definitions --- p.56 / Chapter 4.2 --- Useful lemmas and theorems --- p.60 / Chapter 4.3 --- Existing schemes for forward SDEs --- p.66 / Chapter 5 --- Numerical algorithms to BSDEs and error estimates --- p.68 / Chapter 5.1 --- A simple backward algorithm for BSDEs and its error estimate --- p.69 / Chapter 5.1.1 --- A simple backward algorithm --- p.69 / Chapter 5.1.2 --- Error estimate for simple backward scheme --- p.71 / Chapter 5.2 --- A new explicit backward algorithm for BSDEs and its error estimates --- p.85 / Chapter 5.2.1 --- A new explicit backward algorithm --- p.85 / Chapter 5.2.2 --- Error estimate for explicit backward scheme --- p.86 / Chapter 6 --- Higher order schemes of Markovian cases and error estimates --- p.91 / Chapter 6.1 --- Error estimate of 1-order scheme for Markovian BSDEs --- p.92 / Chapter 6.2 --- 2-order scheme for Markovian BSDEs and its error estimate --- p.100 / Chapter 6.2.1 --- 2-order scheme for Markovian BSDEs --- p.100 / Chapter 6.2.2 --- Error estimate of 2-order scheme --- p.102 / Chapter 7 --- Simulation results for BSDEs --- p.106 / Chapter 7.1 --- Basis functions --- p.107 / Chapter 7.2 --- Numerical simulations --- p.108 / Chapter 7.2.1 --- Application on option pricing --- p.108 / Chapter 7.2.2 --- Numerical examples on Markovian BSDEs --- p.114 / Chapter 8 --- Conclusions and future work --- p.123 / Chapter 8.1 --- Conclusions --- p.123 / Chapter 8.2 --- Future work --- p.124 / Bibliography --- p.126
| Identifer | oai:union.ndltd.org:cuhk.edu.hk/oai:cuhk-dr:cuhk_328670 |
| Date | January 2013 |
| Contributors | Wang, Shiping, Chinese University of Hong Kong Graduate School. Division of Mathematics. |
| Source Sets | The Chinese University of Hong Kong |
| Language | English, Chinese |
| Detected Language | English |
| Type | Text, bibliography |
| Format | electronic resource, electronic resource, remote, 1 online resource (xiii, 133 leaves) : ill. (some col.) |
| Rights | Use of this resource is governed by the terms and conditions of the Creative Commons “Attribution-NonCommercial-NoDerivatives 4.0 International” License (http://creativecommons.org/licenses/by-nc-nd/4.0/) |
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