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In this dissertation we obtain an e cient hybrid numerical method for the solution of stochastic di erential equations (SDEs). Speci cally, our method chooses between two numerical methods (Euler and Milstein) over a particular discretization interval depending on the value of the simulated Brownian increment driving the stochastic process. This is thus a new1 adaptive method in the numerical analysis of stochastic di erential equation. Mauthner (1998) and Hofmann et al (2000) have developed a general framework for adaptive schemes for the numerical solution to SDEs, [30, 21]. The former presents a Runge-Kutta-type method based on stepsize control while the latter considered a one-step adaptive scheme where the method is also adapted based on step size control. Lamba, Mattingly and Stuart, [28] considered an adaptive Euler scheme based on controlling the drift component of the time-step method. Here we seek to develop a hybrid algorithm that switches between euler and milstein schemes at each time step over the entire discretization interval, depending on the outcome of the simulated Brownian motion increment. The bias of the hybrid scheme as well as its order of convergence is studied. We also do a comparative analysis of the performance of the hybrid scheme relative to the basic numerical schemes of Euler and Milstein. / Mathematical Sciences / M.Sc. (Applied Mathematics)

In this thesis, we describe Schensted's algorithm for finding the length of a longest increasing subsequence of a finite sequence. Schensted's algorithm also constructs a bijection between permutations of the first N natural numbers and standard bitableaux of size N. We also describe the Hillman-Grassl algorithm which constructs a bijection between reverse plane partitions and the solutions in natural numbers of a linear equation involving hook lengths. Pascal programs and sample output for both algorithms appear in the appendix. In addition, we describe the operations on standard bitableaux corresponding to the operations of inverting and reversing permutations. Finally, we show that these operations generate the dihedral group D_4

A Dissertation submitted to the Faculty of Science, University of the Witwatersrand, in fulfillment of the requirements for the degree of Master of Science. Johannesburg, 2013 / Many of the problems in mathematics have very elegant solutions. As complex, real–world geometries come into play, however, this elegance is often lost. This is particularly the case with meshes of physical, real–world problems. Domain mapping helps to move problems from some geometrically complex domain to a regular, easy to use domain. Shape transformation, specifically, allows one to do this in 2D domains where mesh construction can be difficult. Numerical methods usually work over some mesh on the target domain. The structure and detail of these meshes affect the overall computation and accuracy immensely. Unfortunately, building a good mesh is not always a straight forward task. Finite Element Analysis, for example, typically requires 4–10 times the number of tetrahedral elements to achieve the same accuracy as the corresponding hexahedral mesh. Constructing this hexahedral mesh, however, is a difficult task; so in practice many people use tetrahedral meshes instead. By mapping the geometrically complex domain to a regular domain, one can easily construct elegant meshes that bear useful properties. Once a domain has been mapped to a regular domain, the mesh can be constructed and calculations can be performed in the new domain. Later, results from these calculations can be transferred back to the original domain. Using harmonic functions, source domains can be parametrised to spaces with many different desired properties. This allows one to perform calculations that would be otherwise expensive or inaccurate. This research implements and extends the methods developed in Voruganti et al. [2006 2008] for domain mapping using harmonic functions. The method was extended to handle cases where there are voids in the source domain, allowing the user to map domains that are not topologically equivalent to the equivalent dimension hypersphere. This is accomplished through the use of various boundary conditions as the void is mapped to the target domains which allow the user to reshape and shrink the void in the target domain. The voids can now be reduced to arcs, radial lines and even shrunk to single points. The algorithms were implemented in two and three dimensions and ultimately parallelised to run on the Centre for High Performance Computing clusters. The parallel code also allows for arbitrary dimension genus-0 source domains. Finally, applications, such as remeshing and robot path planning were investigated and illustrated.

O FBST (Full Bayesian Significance Test) é um procedimento para testar hipóteses precisas, apresentado por Pereira e Stern (1999), e baseado no cálculo da probabilidade posterior do conjunto tangente ao conjunto que define a hipótese nula. Este procedimento é uma alternativa Bayesiana aos testes de significância usuais. Neste trabalho, estudamos a relação entre os resultados do FBST e de um teste freqüentista, o TRVG (Teste da Razão de Verossimilhanças Generalizado), através de alguns problemas clássicos de testes de hipóteses. Apresentamos, também, todos os procedimentos computacionais utilizados para a resolução automática dos dois testes para grandes amostras, necessária ao estudo da relação entre os testes. / FBST (Full Bayesian Significance Test) is a procedure to test precise hypotheses, presented by Pereira and Stern (1999), which is based on the calculus of the posterior probability of the set tangent to the set that defines the null hypothesis. This procedure is a Bayesian alternative to the usual significance tests. In the present work we study the relation between the FBST\'s results and those of a frequentist test, GLRT (Generalised Likelihood Ratio Test) through some classical problems in hypotesis testing. We also present all computer procedures that compose the automatic solutions for applying FBST and GLRT on big samples what was necessary for studying the relation between both tests.

by Chi-Fai Chan. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references (leaves 53-54). / Abstract also in Chinese. / Chapter Chapter 1 --- INTRODUCTION --- p.1 / Chapter §1.1 --- Polynomial Interpolation --- p.1 / Chapter §1.2 --- Conjugate Gradient Type Methods --- p.6 / Chapter §1.3 --- Outline of the Thesis --- p.10 / Chapter Chapter 2 --- INTEGRAL EQUATIONS --- p.11 / Chapter §2.1 --- Integral Equations --- p.11 / Chapter §2.2 --- Numerical Treatments of Second Kind Integral Equations --- p.15 / Chapter Chapter 3 --- FAST ALGORITHM FOR SECOND KIND INTEGRAL EQUATIONS --- p.20 / Chapter §3.1 --- Introduction --- p.20 / Chapter §3.2 --- The Approximation --- p.24 / Chapter §3.3 --- Error Analysis --- p.35 / Chapter §3.4 --- Numerical Examples --- p.40 / Chapter §3.5 --- Concluding Remarks --- p.51 / References --- p.53

by Li Jingzhi. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2006. / Includes bibliographical references (leaves 56-57). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Parameter identification problems --- p.1 / Chapter 1.2 --- Overview of existing numerical methods --- p.2 / Chapter 1.3 --- Outline of the thesis --- p.4 / Chapter 2 --- General Framework --- p.6 / Chapter 2.1 --- Abstract inverse problem --- p.6 / Chapter 2.2 --- Abstract multilevel models --- p.7 / Chapter 2.3 --- Abstract MMC algorithm --- p.9 / Chapter 3 --- Dual Viewpoint and Convergence Condition --- p.15 / Chapter 3.1 --- Dual viewpoint of nonlinear multigrid method --- p.15 / Chapter 3.2 --- Convergence condition of MMC algorithm --- p.16 / Chapter 4 --- Applications of MMC Algorithm for Parameter Identification in Elliptic and Parabolic Systems --- p.20 / Chapter 4.1 --- Notations --- p.20 / Chapter 4.2 --- Parameter identification in elliptic systems I --- p.21 / Chapter 4.3 --- Parameter identification in elliptic systems II --- p.23 / Chapter 4.4 --- Parameter identification in parabolic systems I --- p.24 / Chapter 4.5 --- Parameter identification in parabolic systems II --- p.25 / Chapter 5 --- Numerical Experiments --- p.27 / Chapter 5.1 --- Test problems --- p.27 / Chapter 5.2 --- Smoothing property of gradient methods --- p.28 / Chapter 5.3 --- Numerical examples --- p.29 / Chapter 6 --- Conclusion Remarks --- p.55 / Bibliography --- p.56

Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Ao, Weiwei. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 178-186). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 1.1 --- Interior Spike Solutions for Lin-Ni-Takagi Problem --- p.7 / Chapter 1.1.1 --- Background and Main Results --- p.7 / Chapter 1.1.2 --- Sketch of the Proof of Theorem 1.1.1 --- p.12 / Chapter 1.2 --- The A2 and B2 Chern-Simons System --- p.14 / Chapter 1.2.1 --- Background --- p.14 / Chapter 1.2.2 --- Previous Results --- p.19 / Chapter 1.2.3 --- Main Results --- p.20 / Chapter 1.2.4 --- Sketch of the Proof for A₂ Case --- p.21 / Chapter 1.2.5 --- Sketch of the Proof for B₂ Case --- p.26 / Chapter 1.3 --- Organization of the Thesis --- p.27 / Chapter 2 --- The Lin-Ni-Takagi Problem --- p.29 / Chapter 2.1 --- Notation and Some Preliminary Analysis --- p.29 / Chapter 2.2 --- Linear Theory --- p.35 / Chapter 2.3 --- The Non Linear Projected Problem --- p.40 / Chapter 2.4 --- An Improved Estimate --- p.43 / Chapter 2.5 --- The Reduced Problem: A Maximization Procedure --- p.50 / Chapter 2.6 --- Proof of Theorem 1.1.1 --- p.58 / Chapter 2.7 --- More Applications and Some Open Problems --- p.60 / Chapter 3 --- The Chern-Simons System --- p.66 / Chapter 3.1 --- Proof of Theorem 1.2.1 in the A₂ Case --- p.66 / Chapter 3.1.1 --- Functional Formulation of the Problem --- p.66 / Chapter 3.1.2 --- First Approximate Solution --- p.68 / Chapter 3.1.3 --- Invertibility of Linearized Operator --- p.72 / Chapter 3.1.4 --- Improvements of the Approximate Solution: O(ε) Term --- p.76 / Chapter 3.1.5 --- Next Improvement of the Approximate Solution: O(ε²) Term --- p.78 / Chapter 3.1.6 --- A Nonlinear Projected Problem --- p.82 / Chapter 3.1.7 --- Proof of Theorem 1.2.1 for A₂ under Assumption (i) --- p.85 / Chapter 3.1.8 --- Proof of Theorem 1.2.1 for A₂ under Assumption (ii) --- p.94 / Chapter 3.1.9 --- Proof of Theorem 1.2.1 for A₂ under Assumption (iii) --- p.99 / Chapter 3.2 --- Proof of Theorem 1.2.1 in the B₂ Case --- p.100 / Chapter 3.2.1 --- Functional Formulation of the Problem for B₂ Case --- p.100 / Chapter 3.2.2 --- Classi cation and Non-degeneracy for B₂ Toda system --- p.101 / Chapter 3.2.3 --- Invertibility of Linearized Operator --- p.105 / Chapter 3.2.4 --- Improvements of the Approximate Solution --- p.106 / Chapter 3.2.5 --- Proof of Theorem 1.2.1 for B₂ under Assumption (i) --- p.112 / Chapter 3.2.6 --- Proof of Theorem 1.2.1 for B₂ under Assumption (ii) --- p.122 / Chapter 3.2.7 --- Proof of Theorem 1.2.1 for B₂ under Assumption (iii) --- p.127 / Chapter 3.3 --- Open Problems --- p.128 / Chapter 4 --- Appendix --- p.129 / Chapter 4.1 --- B₂ and G₂ Toda System with Singular Source --- p.129 / Chapter 4.1.1 --- Case 1: B₂ Toda system with singular source --- p.130 / Chapter 4.1.2 --- Case 2: G₂ Toda system with singular source --- p.136 / Chapter 4.2 --- The Calculations of the Matrix Q₁ --- p.148 / Chapter 4.3 --- The Calculations of the Matrix Q₁ --- p.169 / Bibliography --- p.178

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