Many problems in financial world are being modeled by means of differential equation. These problems are time dependent, highly nonlinear, stochastic and heavily depend on the previous history of time. A variety of financial products exists in the market, such as forwards, futures, swaps and options. Our main focus in this thesis is to use the numerical analysis tools to solve some option pricing problems. Depending upon the inter-relationship of the financial derivatives, the dimension of the associated problem increases drastically and hence conventional methods (for example, the finite difference methods or finite element methods) for solving them do not provide satisfactory results. To resolve this issue, we use a special class of numerical methods, namely, the mesh free methods. These methods are often better suited to cope with changes in the geometry of the domain of interest than classical discretization techniques. In this thesis, we apply these methods to solve problems that price standard and non-standard options. We then extend the proposed approach to solve Heston' volatility model. The methods in each of these cases are analyzed for stability and thorough comparative numerical results are provided.

Philosophiae Doctor - PhD / Many problems in financial world are being modeled by means of differential equation. These problems are time dependent, highly nonlinear, stochastic and heavily depend on the previous history of time. A variety of financial products exists in the market, such as forwards, futures, swaps and options. Our main focus in this thesis is to use the numerical analysis tools to solve some option pricing problems. Depending upon the inter-relationship of the financial derivatives, the dimension of the associated problem increases drastically and hence conventional methods (for example, the finite difference methods or finite element methods) for solving them do not provide satisfactory results. To resolve this issue, we use a special class of numerical methods, namely, the mesh free methods. These methods are often better suited to cope with changes in the geometry of the domain of interest than classical discretization techniques. In this thesis, we apply these methods to solve problems that price standard and non-standard options. We then extend the proposed approach to solve Heston' volatility model. The methods in each of these cases are analyzed for stability and thorough comparative numerical results are provided. / South Africa

In the process of engineering design of structural shapes, the flat plate analysis results can be generalized to predict behaviors of complete structural shapes. In this case, the purpose of this project is to analyze a thin flat plate under conductive heat transfer and to simulate the temperature distribution, thermal stresses, total displacements, and buckling deformations. The current approach in these cases has been using the Finite Element Method (FEM), whose basis is the construction of a conforming mesh. In contrast, this project uses the mesh-free Scan Solve Method. This method eliminates the meshing limitation using a non-conforming mesh. I implemented this modeling process developing numerical algorithms and software tools to model thermally induced buckling. In addition, convergence analysis was achieved, and the results were compared with FEM. In conclusion, the results demonstrate that the method gives similar solutions to FEM in quality, but it is computationally less time consuming.

Philosophiae Doctor - PhD / Many problems in financial world are being modeled by means of differential equation. These problems are time dependent, highly nonlinear, stochastic and heavily depend on the previous history of time. A variety of financial products exists in the market, such as forwards, futures, swaps and options. Our main focus in this thesis is to use the numerical analysis tools to solve some option pricing problems. Depending upon the inter-relationship of the financial derivatives, the dimension of the associated problem increases drastically and hence conventional methods (for example, the finite difference methods or finite element methods) for solving them do not provide satisfactory results. To resolve this issue, we use a special class of numerical methods, namely, the mesh free methods. These methods are often better suited to cope with changes in the geometry of the domain of interest than classical discretization techniques. In this thesis, we apply these methods to solve problems that price standard and non-standard options. We then extend the proposed approach to solve Heston's volatility model. The methods in each of these cases are analyzed for stability and thorough comparative numerical results are provided.

Mathematical definitions provide a precise, unambiguous way to formulate concepts. They also provide a common language between disciplines. Thus, they are the basis for a well-founded scientific discussion. In addition, mathematical definitions allow for deeper insights into the defined subject based on mathematical theorems that are incontrovertible under the given definition. Besides their value in mathematics, mathematical definitions are indispensable in other sciences like physics, chemistry, and computer science. In computer science, they help to derive the expected behavior of a computer program and provide guidance for the design and testing of software. Therefore, mathematical definitions can be used to design and implement advanced algorithms. One class of widely used algorithms in computer science is the class of particle-based algorithms, also known as particle methods. Particle methods can solve complex problems in various fields, such as fluid dynamics, plasma physics, or granular flows, using diverse simulation methods, including Discrete Element Methods (DEM), Molecular Dynamics (MD), Reproducing Kernel Particle Methods (RKPM), Particle Strength Exchange (PSE), and Smoothed Particle Hydrodynamics (SPH). Despite the increasing use of particle methods driven by improved computing performance, the relation between these algorithms remains formally unclear. In particular, particle methods lack a unifying mathematical definition and precisely defined terminology. This prevents the determination of whether an algorithm belongs to the class and what distinguishes the class. Here we present a rigorous mathematical definition for determining particle methods and demonstrate its importance by applying it to several canonical algorithms and those not previously recognized as particle methods. Furthermore, we base proofs of theorems about parallelizability and computational power on it and use it to develop scientific computing software. Our definition unified, for the first time, the so far loosely connected notion of particle methods. Thus, it marks the necessary starting point for a broad range of joint formal investigations and applications across fields.:1 Introduction 1.1 The Role of Mathematical Definitions 1.2 Particle Methods 1.3 Scope and Contributions of this Thesis 2 Terminology and Notation 3 A Formal Definition of Particle Methods 3.1 Introduction 3.2 Definition of Particle Methods 3.2.1 Particle Method Algorithm 3.2.2 Particle Method Instance 3.2.3 Particle State Transition Function 3.3 Explanation of the Definition of Particle Methods 3.3.1 Illustrative Example 3.3.2 Explanation of the Particle Method Algorithm 3.3.3 Explanation of the Particle Method Instance 3.3.4 Explanation of the State Transition Function 3.4 Conclusion 4 Algorithms as Particle Methods 4.1 Introduction 4.2 Perfectly Elastic Collision in Arbitrary Dimensions 4.3 Particle Strength Exchange 4.4 Smoothed Particle Hydrodynamics 4.5 Lennard-Jones Molecular Dynamics 4.6 Triangulation refinement 4.7 Conway's Game of Life 4.8 Gaussian Elimination 4.9 Conclusion 5 Parallelizability of Particle Methods 5.1 Introduction 5.2 Particle Methods on Shared Memory Systems 5.2.1 Parallelization Scheme 5.2.2 Lemmata 5.2.3 Parallelizability 5.2.4 Time Complexity 5.2.5 Application 5.3 Particle Methods on Distributed Memory Systems 5.3.1 Parallelization Scheme 5.3.2 Lemmata 5.3.3 Parallelizability 5.3.4 Bounds on Time Complexity and Parallel Scalability 5.4 Conclusion 6 Turing Powerfulness and Halting Decidability 6.1 Introduction 6.2 Turing Machine 6.3 Turing Powerfulness of Particle Methods Under a First Set of Constraints 6.4 Turing Powerfulness of Particle Methods Under a Second Set of Constraints 6.5 Halting Decidability of Particle Methods 6.6 Conclusion 7 Particle Methods as a Basis for Scientific Software Engineering 7.1 Introduction 7.2 Design of the Prototype 7.3 Applications, Comparisons, Convergence Study, and Run-time Evaluations 7.4 Conclusion 8 Results, Discussion, Outlook, and Conclusion 8.1 Problem 8.2 Results 8.3 Discussion 8.4 Outlook 8.5 Conclusion

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A couple of discretization schemes, based on an FE-like tessellation of the domain and polynomial reproducing, globally smooth shape functions, are considered and numerically explored to a limited extent. The first one among these is an existing scheme, the smooth DMS-FEM, that employs Delaunay triangulation or tetrahedralization (as approximate) towards discretizing the domain geometry employs triangular (tetrahedral) B-splines as kernel functions en route to the construction of polynomial reproducing functional approximations. In order to verify the numerical accuracy of the smooth DMS-FEM vis-à-vis the conventional FEM, a Mindlin-Reissner plate bending problem is numerically solved. Thanks to the higher order continuity in the functional approximant and the consequent removal of the jump terms in the weak form across inter-triangular boundaries, the numerical accuracy via the DMS-FEM approximation is observed to be higher than that corresponding to the conventional FEM. This advantage notwithstanding, evaluations of DMS-FEM based shape functions encounter singularity issues on the triangle vertices as well as over the element edges. This shortcoming is presently overcome through a new proposal that replaces the triangular B-splines by simplex splines, constructed over polygonal domains, as the kernel functions in the polynomial reproduction scheme. Following a detailed presentation of the issues related to its computational implementation, the new method is numerically explored with the results attesting to a higher attainable numerical accuracy in comparison with the DMS-FEM.

This thesis makes two noteworthy contributions in the are of mesh-free methods: a Pseudo-Divergence-Free (PDF) Element Free Galerkin (EFG) method which alleviates the volumetric locking and a Stabilized Updated Lagrangian formulation which allows to solve fast-transient dynamic problems involving large distortions. The thesis is organized in the following way. First of all, this thesis dedicates one chapter to the state of the art of mesh-free methods. The main reason is because there are many mesh-free methods that can be found in the literature which can be based on different ideas and with different properties. There is a real need of classifying, ordering and comparing these methods: in fact, the same or almost the same method can be found with different names in the literature. Secondly, a novel improved formulation of the (EFG) method is proposed in order to alleviate volumetric locking. It is based on a pseudo-divergence-free interpolation. Using the concept of diffuse derivatives an a convergence theorem of these derivatives to the ones of the exact solution, the new approximation proposed is obtained imposing a zero diffuse divergence. In this way is guaranteed that the method verifies asymptotically the incompressibility condition and in addition the imposition can be done a priori. This means that the main difference between standard EFG and the improved method is how is chosen the interpolation basis. Modal analysis and numerical results for two classical benchmark tests in solids corroborate that, as expected, diffuse derivatives converge to the derivatives of the exact solution when the discretization is refined (for a fixed dilation parameter) and, of course, that diffuse divergence converges to the exact divergence with the expected theoretical rate. For standard EFG the typical convergence rate is degrade as the incompressible limit is approached but with the improved method good results are obtained even for a nearly incompressible case and a moderately fine discretization. The improved method has also been used to solve the Stokes equations. In this case the LBB condition is not explicitly satisfied because the pseudo-divergence-free approximation is employed. Reasonable results are obtained in spite of the equal order interpolation for velocity and pressure. Finally, several techniques have been developed in the past to solve the well known tensile instability in the SPH (Smooth Particle Hydrodynamics) mesh-free method. It has been proved that a Lagrangian formulation removes completely the instability (but zero energy modes exist). In fact, Lagrangian SPH works even better than the Finite Element Method in problems involving distortions. Nevertheless, in problems with very large distortions a Lagrangian formulation will need of frequent updates of the reference configuration. When such updates are incorporated then zero energy modes are more likely to be activated. When few updates are carried out the error is small but when updates are performed frequently the solution is completely spoilt because of the zero energy modes. In this thesis an updated Lagrangian formulation is developed. It allows to carry out updates of the reference configuration without suffering the appearance of spurious modes. To update the Lagrangian formulation an incremental approach is used: an intermediate configuration will be the new reference configuration for the next time steps. It has been observed that this updated formulation suffers from similar numerical fracture to the Eulerian case. A modal analysis has proven that there exist zero energy modes. In the paper the updated Lagrangian method is exposed in detail, a stability analysis is performed and finally a stabilization technique is incorporated to preclude spurious modes.

Among the presently known numerical solvers of integral equations, two main categories of approaches can be traced: mesh-free approaches, mesh-based approaches. We will propose some techniques to process geometric data so that they can be efficiently used in subsequent numerical treatments of integral equations. In order to prepare geometric information so that the above two approaches can be automatically applied, we need the following items: (1) Splitting a given surface into several four-sided patches, (2) Generating a diffeomorphism from the unit square to a foursided patch, (3) Generating a mesh M on a given surface, (4) Patching of a given triangulation. In order to have a splitting, we need to approximate the surfaces first by polygonal regions. We use afterwards quadrangulation techniques by removing quadrilaterals repeatedly. We will generate the diffeomorphisms by means of transfinite interpolations of Coons and Gordon types. The generation of a mesh M from a piecewise Riemannian surface will use some generalized Delaunay techniques in which the mesh size will be determined with the help of the Laplace-Beltrami operator. We will describe our experiences with the IGES format because of two reasons. First, most of our implementations have been done with it. Next, some of the proposed methodologies assume that the curve and surface representations are similar to those of IGES. Patching a mesh consists in approximating or interpolating it by a set of practical surfaces such as B-spline patches. That approach proves useful when we want to utilize a mesh-free integral equation solver but the input geometry is represented as a mesh.

Les objectifs de la thèse sont d’élaborer des systèmes d’instrumentation électronique qui permettent une analyse et un diagnostic fins de l’état d’intégrité et du processus de vieillissement des composants de puissance à semi-conducteur. Ces travaux visent à évaluer la variation de la conductivité de la métallisation à l’aide de capteurs à Courant Foucault (CF) mais aussi à estimer l’effet du vieillissement des puces et de leur assemblage sur la distribution de courant dans les puces afin de mieux comprendre les mécanismes de défaillance. Des éprouvettes simplifiées mais également des modules de puissance représentatifs ont été vieillis par les cyclages thermique. Les capteurs développés ont été utilisés afin, d’une part de suivre le vieillissement, mais aussi d’autre part afin de comprendre l’effet de ce vieillissement sur le comportement des puces de puissance. Un banc d’instrumentation dédié a été élaboré et exploité pour la mesure locale de la conductivité électrique par le capteur à courants de Foucault, et l’estimation de la distribution de courants à partir de la mesure de cartographies de champ magnétique par capteurs de champ, ou à partir de la cartographie de la distribution de tension sur la métallisation de source. Ce banc a permis en premier lieu d’évaluer la pertinence et les performances de différents types de capteurs exploitables. Le travail s’est également appuyé sur des techniques de traitement de signal, à la fois pour estimer de manière quantitative les informations de conductivité des métallisations issues des capteurs à courant de Foucault, mais aussi pour l’analyse de la distribution de courant à partir des informations fournies par des capteurs de champ magnétiques. Les modèles utilisés exploitent des techniques de modélisation comportementale (le modèle approché de « transformateur analogique » modélisant capteurs à CF ou bien d’inversion de modèle semi-analytique dans le cas l’estimation de la distribution de courant). Les résultats obtenus à partir de ces modèles nous permettrons, d’une part de mieux comprendre certains mécanismes de défaillance, mais également de proposer une implantation et des structures de capteurs pour le suivi « in situ » de l’intégrité des composants. / This thesis is dedicated to develop electronic instrumentation systems that allow to analyse the ageing process and to make a diagnosis the failure mechanisms of power semiconductor device. The research objectives were to evaluate the electrical conductivity variation of metallization layer using the eddy current technique but also to estimate the ageing effect of the semiconductor dies and their module packaging on the current distribution in the die, to better understand the mechanism failures. The specimens simplified and the power semiconductor modules were aged by thermal cycles. The various sensors have been used (eddy current sensor, Hall sensor), to follow the ageing process, and to understand the ageing effect on the power semiconductor die. The experimental instrumentation system has been developed and used, to realize the non destructive evaluation by the eddy current technique on the metallization layer and to measure the map of magnetic field induced above the die by the magnetic sensor, the potential distribution. In the first time, this system allowed to evaluate the relevance and the performance of different type sensors used for the local measure on the electrical conductivity by eddy current sensors and on the currents distribution by Hall sensors or the potential distribution of the source metallization layer. This work was also supported by the signal processing techniques. To estimate quantitatively the electrical conductivity of metallization layer by the eddy current sensors, a model using the two-winding transformer analogy simulate the electromagnetic interaction between the sensor and the conducting plate. And, the current distribution from the measured data is given by inverting a mesh-free modeling of the induced magnetic field. The results obtained from these models can allow us to firstly understand certain failure mechanisms, but also to propose the integrated circuit with the sensors for monitoring "in situ" the state ageing of power semiconductor device.