The main topic of this diploma thesis is the modeling of nonlinear effects in ultrasonic fields. The work deals with application of finite difference method (FDTD) on the Westervelt equation and the subsequent creation of the model of ultrasonic fields in MATLAB. This thesis also includes theoretical analysis of ultra-acoustic and technical aspects of diagnostic ultrasonography. In addition, this document includes verification of theoretical assumptions by using created model.

Two-stage stochastic programming problem with PDE constraint, specially elliptic equation is formulated. The computational scheme is proposed, whereas the emphasis is put on approximation techniques. We introduce method of approximation of random variables of stochastic problem and utilize suitable numerical methods, finite difference method first, then finite element method. There is also formulated a mathematical programming problem describing a membrane deflection with random load. It is followed by determination of the acceptableness of using stochastic optimization rather than deterministic problem and assess the quality of approximations based on Monte Carlo simulation method and the theory of interval estimates. The resulting mathematical models are implemented and solved in the general algebraic modeling system GAMS. Graphical and numerical results are presented.

<div>Accurate, fast, large-scale, and circuit-intuitive electromagnetic analysis is of critical importance to the design of integrated circuits (IC) and systems. Existing methods for the analysis of integrated circuits and systems have not satisfactorily achieved these performance goals. In this work, rapid modeling and simulation methods are developed for large-scale and circuit-intuitive electromagnetic analysis of integrated circuits and systems. The derived model is correct from zero to high frequencies where Maxwell's equations are valid. In addition, in the proposed model, we are able to analytically decompose the layout response into static and full-wave components with neither numerical computation nor approximation. This decomposed yet rigorous model greatly helps circuit diagnoses since now designers are able to analyze each component one by one, and identify which component is the root cause for the design failure. Such a decomposition also facilitates efficient layout modeling and simulation, since if an IC is dominated by RC effects, then we do not have to compute the full-wave component; and vice versa. Meanwhile, it makes parallelization straightforward. In addition, we develop fast algorithms to obtain each component of the inverse rapidly. These algorithms are also applicable for solving general partial differential equations for fast electromagnetic analysis.</div><div><br></div><div>The fast algorithms developed in this work are as follows. First, an analytical method is developed for finding the nullspace of the curl-curl operator in an arbitrary mesh for an arbitrary order of curl-conforming vector basis function. This method has been applied successfully to both a finite-difference and a finite-element based analysis of general 3-D structures. It can be used to obtain the static component of the inverse efficiently. An analytical method for finding the complementary space of the nullspace is also developed. Second, using the analytically found nullspace and its complementary space, a rigorous method is developed to overcome the low-frequency breakdown problem in the full-wave analysis of general lossy problems, where both dielectrics and conductors can be lossy and arbitrarily inhomogeneous. The method is equally valid at high frequencies without any need for changing the formulation. Third, with the static component part solved, the full-wave component is also ready to obtain. There are two ways. In the first way, the full-wave component is efficiently represented by a small number of high-frequency modes, and a fast method is created to find these modes. These modes constitute a significantly reduced order model of the complementary space of the nullspace. The second way is to utilize the relationship between the curl-curl matrix and the Laplacian matrix. An analytical method to decompose the curl-curl operator to a gradient-divergence operator and a Laplacian operator is developed. The derived Laplacian matrix is nothing but the curl-curl matrix's Laplacian counterpart. They share the same set of non-zero eigenvalues and eigenvectors. Therefore, this Laplacian matrix can be used to replace the original curl-curl matrix when operating on the full-wave component without any computational cost, and an iterative solution can converge this modified problem much faster irrespective of the matrix size. The proposed work has been applied to large-scale layout extraction and analysis. Its performance in accuracy, efficiency, and capacity has been demonstrated.</div>

In this thesis we look at acoustic wave propagation under water. We look in particular at waves generated by a point source and what happens with the propagation when we model the bottom as flat or as curvilinear. We assume the source to be working at a certain frequency and therefore we model this problem by solving the Helmholtz equation. Since Helmholtz equation has some unwanted numerical properties we are interested in finding new numerical methods that could accelerate the solver. In this thesis we use the Waveholtz iteration, which solves Helmholtz equation by connecting it to the time-dependent wave equation. We use finite differences and the SBP-SAT method to approximate the spatial problem numerically and for modelling the sea bottom we use curvilinear coordinates.  To compare the Waveholtz iteration we also solve Helmholtz equation with a naive solver. The naive solver consists of approximating the equation with finite differences and then solving the linear system of equation by some iterative solver, which for our tests will be GMRES. The results show that the Waveholtz iteration converges in less iterations than our naive solver. It also shows that the number of iterations stays unchanged when changing our discretization, which otherwise is a big problem for our naive solver. This allows us to increase the accuracy of our numerical solution without changing the computation time too much.  We show that the number of iterations increases according to theory for an increasing frequency, and that for open problems we even see a smaller increase. For certain resonant frequencies in Helmholtz equation we do not expect the Waveholtz iteration to converge. In the neighbourhood of these frequencies the convergence becomes slow and we need many iterations for a solution of a certain accuracy. By reformulating the Waveholtz iteration as a Krylov solution we can see that resonances in Helmholtz equation have a smaller impact of the convergence. / I detta examensarbete undersöker vi akustisk vågutbredning i vatten. Vi kollar specifikt på vågor som genereras av en punktkälla och vad som sker när vi modellerar botten som plan eller som kurvlinjär. Då vi antar att punktkällan arbetar vid en bestämd frekvens, kommer vi modellera det fysikaliska problemet genom att lösa Helmholtz ekvation. Helmholtz ekvation har dock några numeriska egenskaper som är oönskade, och därför finns ett intresse av att hitta nya numeriska metoder som löser ekvationen. I detta examensarbete undersöker vi Waveholtz iteration, som löser Helmholtz ekvation genom att koppla den till den tidsberoende vågekvationen. Vi använder finita differenser och SBP-SAT metoden för att approximera det rumsliga problemet numeriskt. För att ge en detaljerad beskrivning av botten använder vi kurvlinjära koordinater. För att jämföra Waveholtz iterationen med något löser vi även Helmholtz med hjälp av en naiv lösare. Den naiva lösaren består av att approximera problemet med finita differenser och sedan lösa det linjära systemet rakt av med en iterativ lösare (vilket för våra fall kommer vara GMRES). Resultatet visar att Waveholtz iteration konvergerar på ett lägre antal iterationer än vår naiva lösare. Det visar även att antalet iterationer inte förändras när vi ändrar diskretisering, vilket annars är ett problem för vår naiva lösare. Detta innebär att vi kan få en högre noggrannhet utan att förlänga beräkningstiden alltför mycket.  Vi visar även att antalet iterationer ökar som förväntat med en ökad frekvens, samt att för öppna problem så ökar antalet iteration mindre än enligt teorin. Vid vissa resonanta frekvenser i Helmholtz ekvation förväntar vi oss att Waveholtz iteration inte kommer konvergerar. I närheten av dessa frekvenser blir konvergensen långsam och vi behöver många iterationer för att lösa problemet. Genom att formulera Waveholtz iteration som en Krylov lösning kommer resonanser i Helmholtz ekvation ge en mindre negativ effekt på konvergensen än om den är formulerad som en fixpunkts iteration.

國內外對BOT實質選擇權之研究，大多集中於BOT主體本身所隱含之各種選擇權價值，鮮少論及BOT附屬事業之選擇權價值。惟因交通運輸BOT主體之自償率往往偏低，故須以保證最小運量或特許經營附屬事業等方式，來吸引潛在投資者。附屬事業對整個BOT計畫價值的影響頗大，如何針對其選擇權之價值加以分析為本研究的主要課題。個案將以台灣南北高速鐵路計畫為例，對其附屬事業之放棄選擇權加以探討。 本研究將主要探討下列課題，並提出研究結果： 一、有限差分法及蒙地卡羅模擬法計算選擇權價值，其間之差異？本 文利用有限差分法及蒙地卡羅模擬法來各別求算BOT 附屬事業的 放棄選擇權價值，一來了解 BOT 附屬事業的放棄選擇權價的大 小，二來比較有限差分法及蒙地卡羅模擬法兩者間的差異。 二、BOT附屬事業的放棄選擇權是否受主體事業的經營績效所影響？ 三、BOT附屬事業是否須考量履約保證金之設計？ 四、BOT主體決定經營或放棄時，是否會影響其附屬事業之放棄選擇 權價值？ 本研究係以蒙地卡羅模擬法及有限差分法單獨估計台灣南北高速鐵路附屬事業之放棄選擇權價值，並建立運輸主體與附屬事業間價值的關聯，再以蒙地卡羅模擬法作更精確的估算。另使用單因子變異數分析及Tukey's Multiple Comparison Method之統計方法，驗證BOT主體與其附屬事業選擇權間之相關性，期能有助於日後BOT計畫之參與者評估及決策使用。 / The study of the Real Option Analysis (ROA) thesis of BOT generally focus is on the principal parts of the project only, rarely is considered the option of ancillary business of BOT. Because the self-liquidation-ratio of the transportation of BOT is low, it needs the government financial support (minimum traffic guarantee or revenue enhancements) to attract the interest of intended investors. The influence of the ancillary business of BOT is huge, so how to evaluate the option is the big issue of thesis. The case focus on the Taiwan High Speed Rail BOT project, and will study the option value of it’s ancillary business. Thesis will discuss the following issues, and develop the result of study. 1.The calculation difference between Monte Carlo Simulation & Finite Difference Method to work out option value, Thesis will use the Monte Carlo Simulation & Finite Difference Method to work out the abandon option value of ancillary business of BOT. To get the abandon option value and compare the calculation difference between Monte Carol Simulation & Finite Difference Method. 2.Will the abandon option value of ancillary business of BOT be influenced by the principal parts of the project？ 3.Is there a need to consider the performance security deposit of ancillary business of BOT？ 4.Will the decision of BOT impact the abandon option value of ancillary business or not？ The thesis will use the Monte Carol Simulation & Finite Difference Method to calculate the abandon option value of ancillary business of Taiwan North-South High-Speed Railway Project (THR), and create the relation between the BOT & it’s ancillary business. The thesis will use the ANOVA & Tuley’s Multiple Comparison Method to validate the relationship, and hope it will let the participator to consider in the future.

Neste trabalho são apresentados os resultados do desenvolvimento e teste de esquemas upwind de alta resolução para o controle da difusão numérica em leis de conservação gerais e problemas em dinâmica dos fluidos. Em particular, são derivados dois novos esquemas: o ALUS (Adaptive Linear Upwind Scheme) e o TOPUS (Third-Order Polynomial Upwind Scheme). Esses esquemas são testados no transporte de escalares, em equações 1D tipo convecção-difusão, em sistemas hiperbólicos 1D, nas equações de Euler 2D da dinâmica dos gases e nas equações de Navier-Stokes incompressíveis 2D/3D. Os esquemas são então associados a uma modelagem algébrica não linear para a simulação de problemas de escoamentos incompressíveis turbulentos 2D com/sem superfícies livres / In this work, results of the development and testing of high-resolution upwind schemes for controlling of the numerical diffusion for general conservation laws and fluid dynamics problems are presented. In particular, two new high-resolution upwind schemes are derived, namely, the ALUS (Adaptive Linear Upwind Scheme) and the TOPUS (Third-Order Polynomial Upwind Scheme). These schemes are tested in scalar transport, 1D convection-diffusion equations, 1D hyperbolic systems, 2D Euler equations of the gas dynamics, and in 2D/3D incompressible Navier-Stokes equations. The schemes are then combined with a nonlinear Reynolds stress algebraic equation model for the simulation of 2D incompressible turbulent flows with/without free surfaces

Apesar de as EDPS que modelam leis de conservação e problemas em dinâmica dos fluídos serem bem estabelecidas, suas soluções numéricas continuam ainda desafiadoras. Em particular, há dois desafios associados à computação e ao entendimento desses problemas: um deles é a formação de descontinuidades (choques) e o outro é o fenômeno turbulência. Ambos os desafios podem ser atribuídos ao tratamento dos termos advectivos não lineares nessas equações de transporte. Dentro deste canário, esta tese apresenta o estudo do desenvolvimento de um novo esquema \"upwind\" de alta resolução e sua associação com modelagem da turbulência. O desempenho do esquema é investigado nas soluções da equação de advecção 1D com dados iniciais descontínuos e de problemas de Riemann 1D para as equações de Burgers, Euler e águas rasas. Além disso, são apresentados resultados numéricos de escoamentos incompressíveis 2D e 3D no regime laminar a altos números de Reynolds. O novo esquema é então associado à modelagem \'capa\' - \'epsilon\' da turbulência para a simulação numérica de escoamentos incompressíveis turbulentos 2D e 3D com superfícies livres móveis. Aplicação, verificação e validação dos métodos numéricos são também fornecidas / Althought the PDEs that model conservation laws and fluid dynamics problems are well established, their numerical solutions have presented a continuing challenge. In particular, there are two challenges associated with the computation and the understanding of these problems, namely, formation of shocks and turbulence. Both challenges can be attributed to the nonlinear advection terms of these transport equations. In this scenario, this thesis presents the study of the development of a new high-resolution upwind scheme and its association with turbulence modelling. The performance of the scheme is investigated by solving the 1D advection equation with discontinuous initial data 1D Riemann problems for Burgers, Euler and shallow water equations. Besides, numerical results for 2D and 3D incompressible laminar flows at high Reynolds number are presented. The new scheme is then associated with the \'capa - \' epsilon\' turbulence model for the simulation of 2D and 3D incompressible turbulent flows with moving free surfaces. Application, verification and validation of the numerical methods are also provided

Apesar de as EDPS que modelam leis de conservação e problemas em dinâmica dos fluídos serem bem estabelecidas, suas soluções numéricas continuam ainda desafiadoras. Em particular, há dois desafios associados à computação e ao entendimento desses problemas: um deles é a formação de descontinuidades (choques) e o outro é o fenômeno turbulência. Ambos os desafios podem ser atribuídos ao tratamento dos termos advectivos não lineares nessas equações de transporte. Dentro deste canário, esta tese apresenta o estudo do desenvolvimento de um novo esquema \"upwind\" de alta resolução e sua associação com modelagem da turbulência. O desempenho do esquema é investigado nas soluções da equação de advecção 1D com dados iniciais descontínuos e de problemas de Riemann 1D para as equações de Burgers, Euler e águas rasas. Além disso, são apresentados resultados numéricos de escoamentos incompressíveis 2D e 3D no regime laminar a altos números de Reynolds. O novo esquema é então associado à modelagem \'capa\' - \'epsilon\' da turbulência para a simulação numérica de escoamentos incompressíveis turbulentos 2D e 3D com superfícies livres móveis. Aplicação, verificação e validação dos métodos numéricos são também fornecidas / Althought the PDEs that model conservation laws and fluid dynamics problems are well established, their numerical solutions have presented a continuing challenge. In particular, there are two challenges associated with the computation and the understanding of these problems, namely, formation of shocks and turbulence. Both challenges can be attributed to the nonlinear advection terms of these transport equations. In this scenario, this thesis presents the study of the development of a new high-resolution upwind scheme and its association with turbulence modelling. The performance of the scheme is investigated by solving the 1D advection equation with discontinuous initial data 1D Riemann problems for Burgers, Euler and shallow water equations. Besides, numerical results for 2D and 3D incompressible laminar flows at high Reynolds number are presented. The new scheme is then associated with the \'capa - \' epsilon\' turbulence model for the simulation of 2D and 3D incompressible turbulent flows with moving free surfaces. Application, verification and validation of the numerical methods are also provided

The Black-Scholes model and its assumptions has endured its fair share of criticism. One problematic issue is the model’s assumption that market volatility is constant. The past decade has seen numerous publications addressing this issue by adapting the Black-Scholes model to incorporate stochastic volatility. In this dissertation, American put options are priced under the Heston stochastic volatility model using the Crank- Nicolson finite difference method in combination with the Projected Over-Relaxation method (PSOR). Due to the early exercise facility, the pricing of American put options is a challenging task, even under constant volatility. Therefore the pricing problem under constant volatility is also included in this dissertation. It involves transforming the Black-Scholes partial differential equation into the heat equation and re-writing the pricing problem as a linear complementary problem. This linear complimentary problem is solved using the Crank-Nicolson finite difference method in combination with the Projected Over-Relaxation method (PSOR). The basic principles to develop the methods necessary to price American put options are covered and the necessary numerical methods are derived. Detailed algorithms for both the constant and the stochastic volatility models, of which no real evidence could be found in literature, are also included in this dissertation. / MSc (Applied Mathematics), North-West University, Potchefstroom Campus, 2013

The Black-Scholes model and its assumptions has endured its fair share of criticism. One problematic issue is the model’s assumption that market volatility is constant. The past decade has seen numerous publications addressing this issue by adapting the Black-Scholes model to incorporate stochastic volatility. In this dissertation, American put options are priced under the Heston stochastic volatility model using the Crank- Nicolson finite difference method in combination with the Projected Over-Relaxation method (PSOR). Due to the early exercise facility, the pricing of American put options is a challenging task, even under constant volatility. Therefore the pricing problem under constant volatility is also included in this dissertation. It involves transforming the Black-Scholes partial differential equation into the heat equation and re-writing the pricing problem as a linear complementary problem. This linear complimentary problem is solved using the Crank-Nicolson finite difference method in combination with the Projected Over-Relaxation method (PSOR). The basic principles to develop the methods necessary to price American put options are covered and the necessary numerical methods are derived. Detailed algorithms for both the constant and the stochastic volatility models, of which no real evidence could be found in literature, are also included in this dissertation. / MSc (Applied Mathematics), North-West University, Potchefstroom Campus, 2013