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.

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In this study, the effect of a passive trapped-vortex cell on lift to drag (L/D) ratio of an FFA-W3-301 airfoil is studied. The upper surface of the airfoil was modified to incorporate a cavity defined by seven parameters. The L/D ratio of the airfoil is modeled using a radial basis function metamodel. This model is used to find the optimal design parameter values that give the highest L/D. The numerical results indicate that the L/D ratio is most sensitive to the position on an airfoil’s upper surface at which the cavity starts, the position of the end point of the cavity, and the vertical distance of the cavity end point relative to the airfoil surface. The L/D ratio can be improved by locating the cavity start point at the point of separation for a particular angle of attack. The optimal cavity shape (o19_aXX) is also tested for a NACA0024 airfoil.

A finite element human body model (FE HBM) is a detailed virtual model of the human body that, for example, is used for simulating traffic accidents. A problem with HBMs is that there is no simple way to position the HBMs in non-standard positions. As different postures during an impact will affect the body in different ways it is vital to have the ability to position the HBMs. In this project it was investigated if it is possible to position a HBM from THUMS, by first positioning only the skin and skeleton, as control points, in the 3D-graphics program Blender. Thereafter a radial basis function interpolation is utilised to morph the rest of the HBM into the new position. The results indicate that in theory, it is possible to position a HBM using a 3D-graphics software. However, the method developed in this project resulted in a disfigurement of the morphed model. The disfigurement is possibly due to the change in distance between the skin and skeleton when positioning those body parts in Blender. / En finit element människokroppsmodell (FE HBM) är en detaljerad virtuell modell av människokroppen som exempelvis används för att simulera trafikolyckor. Ett problem med HBM:er är att det inte finns något enkelt sätt att positionera dem i annat än standardpositioner. Eftersom olika kroppsställningar påverkar kroppen på olika sätt under en kollision är det viktigt att ha möjlighet att kunna positionera en HBM. I detta projekt undersöktes om det är möjligt att positionera en HBM från THUMS, genom att först positionera endast huden och skelettet, som kontrollpunkter, i 3D-grafikprogrammet Blender. Därefter användes en radiell basfunktionsinterpolation för att flytta resten av HBM till den nya positionen. Resultaten indikerar att det är möjligt att positionera en HBM med hjälp av ett 3D-grafikprogram. Metoden som utvecklades i detta projekt resulterade dock i en deformering av den positionerade modellen. Deformeringen beror möjligen på att avståndet mellan hud och skelett ändrades vid positioneringen av dessa kroppsdelar i Blender.

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An Automated Method for Hot-to-Cold Geometry Mapping.

Numerical methods for solving partial differential equations are commonplace in the engineering community and their popularity can be attributed to the rapid performance improvement of modern workstations and desktop computers. The ubiquity of computer technology has allowed all areas of engineering to have access to detailed thermal, stress, and fluid flow analysis packages capable of performing complex studies of current and future designs. The rapid pace of computer development, however, has begun to outstrip efforts to reduce analysis overhead. As such, most commercially available software packages are now limited by the human effort required to prepare, develop, and initialize the necessary computational models. Primarily due to the mesh-based analysis methods utilized in these software packages, the dependence on model preparation greatly limits the accessibility of these analysis tools. In response, the so-called meshless or mesh-free methods have seen considerable interest as they promise to greatly reduce the necessary human interaction during model setup. However, despite the success of these methods in areas demanding high degrees of model adaptability (such as crack growth, multi-phase flow, and solid friction), meshless methods have yet to gain notoriety as a viable alternative to more traditional solution approaches in general solution domains. Although this may be due (at least in part) to the relative youth of the techniques, another potential cause is the lack of focus on developing robust methodologies. The failure to approach development from a practical perspective has prevented researchers from obtaining commercially relevant meshless methodologies which reach the full potential of the approach. The primary goal of this research is to present a novel meshless approach called MIMS (Model Integrated Meshless Solver) which establishes the method as a generalized solution technique capable of competing with more traditional PDE methodologies (such as the finite element and finite volume methods). This was accomplished by developing a robust meshless technique as well as a comprehensive model generation procedure. By closely integrating the model generation process into the overall solution methodology, the presented techniques are able to fully exploit the strengths of the meshless approach to achieve levels of automation, stability, and accuracy currently unseen in the area of engineering analysis. Specifically, MIMS implements a blended meshless solution approach which utilizes a variety of shape functions to obtain a stable and accurate iteration process. This solution approach is then integrated with a newly developed, highly adaptive model generation process which employs a quaternary triangular surface discretization for the boundary, a binary-subdivision discretization for the interior, and a unique shadow layer discretization for near-boundary regions. Together, these discretization techniques are able to achieve directionally independent, automatic refinement of the underlying model, allowing the method to generate accurate solutions without need for intermediate human involvement. In addition, by coupling the model generation with the solution process, the presented method is able to address the issue of ill-constructed geometric input (small features, poorly formed faces, etc.) to provide an intuitive, yet powerful approach to solving modern engineering analysis problems.

This project presents the development and implementation of a GPU-accelerated meshless two-phase incompressible fluid flow solver. The solver uses a variant of the Generalized Finite Difference Meshless Method presented by Gerace et al. [1]. The Level Set Method [2] is used for capturing the fluid interface. The Compute Unified Device Architecture (CUDA) language for general-purpose computing on the graphics-processing-unit is used to implement the GPU-accelerated portions of the solver. CUDA allows the programmer to take advantage of the massive parallelism offered by the GPU at a cost that is significantly lower than other parallel computing options. Through the combined use of GPU-acceleration and a radial-basis function (RBF) collocation meshless method, this project seeks to address the issue of speed in computational fluid dynamics. Traditional mesh-based methods require a large amount of user input in the generation and verification of a computational mesh, which is quite time consuming. The RBF meshless method seeks to rectify this issue through the use of a grid of data centers that need not meet stringent geometric requirements like those required by finite-volume and finite-element methods. Further, the use of the GPU to accelerate the method has been shown to provide a 16-fold increase in speed for the solver subroutines that have been accelerated.

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