When modeling interest rates, using strong formulations of underlying differential equations is prone to bad numerical approximations and high computational costs, due to close to non-smoothness in the probability density function of the interest rate. To circumvent these problems, a weak formulation of the Fokker–Planck equation using Gaussian radial basis functions is suggested. This approach is used in a parameter estimation process for two interest rate models: the Vasicek model and the Cox–Ingersoll–Ross model. In this thesis, such an approach is shown to yield good numerical approximations at low computational costs.

This thesis explores the application of Radial Basis Functions (RBFs) to fluid dynamical problems. In particular, stationary Stokes and Navier-Stokes equations are solved using RBF collocation method. An existing approach from the literature, is enchanced by an additional polynomial basis and a new preconditioner. A faster method based on the partition of unity is introduced for stationary Stokes equations. Finally, a global method based on Picard linearization is introduced for stationary Navier-Stokes equations. / Denna avhandling utforskar tillämpningen av Radial Basis Functions (RBF) på dynamiska problem med vätskor. I synnerhet löses stationära Stokes och Navier-Stokes ekvationer lösas med hjälp av RBF-samlokaliseringsmetoden. En befintlig metod från litteraturen, förbättras genom en ytterligare polynombas och en ny förkonditionering. En snabbare metod baserad på enhetens partition introduceras för stationära Stokes-ekvationer. Slutligen introduceras en global metod baserad på Picard linjärisering för stationära Navier-Stokes ekvationer.

Species Distribution Modelling (SDM) plays a key role in a number of biological applications: assessment of temporal trends in distribution, environmental impact assessment and spatial conservation planning. From a statistical perspective, this thesis develops two methods for increasing the accuracy and reliability of maps of density surfaces and provides a solution to the problem of how to collate multiple density maps of the same region, obtained from differing sources. From a biological perspective, these statistical methods are used to analyse two marine mammal datasets to produce accurate maps for use in spatial conservation planning and temporal trend assessment. The first new method, Complex Region Spatial Smoother [CReSS; Scott-Hayward et al., 2013], improves smoothing in areas where the real distance an animal must travel (`as the animal swims') between two points may be greater than the straight line distance between them, a problem that occurs in complex domains with coastline or islands. CReSS uses estimates of the geodesic distance between points, model averaging and local radial smoothing. Simulation is used to compare its performance with other traditional and recently-developed smoothing techniques: Thin Plate Splines (TPS, Harder and Desmarais [1972]), Geodesic Low rank TPS (GLTPS; Wang and Ranalli [2007]) and the Soap lm smoother (SOAP; Wood et al. [2008]). GLTPS cannot be used in areas with islands and SOAP can be very hard to parametrise. CReSS outperforms all of the other methods on a range of simulations, based on their fit to the underlying function as measured by mean squared error, particularly for sparse data sets. Smoothing functions need to be flexible when they are used to model density surfaces that are highly heterogeneous, in order to avoid biases due to under- or over-fitting. This issue was addressed using an adaptation of a Spatially Adaptive Local Smoothing Algorithm (SALSA, Walker et al. [2010]) in combination with the CReSS method (CReSS-SALSA2D). Unlike traditional methods, such as Generalised Additive Modelling, the adaptive knot selection approach used in SALSA2D naturally accommodates local changes in the smoothness of the density surface that is being modelled. At the time of writing, there are no other methods available to deal with this issue in topographically complex regions. Simulation results show that CReSS-SALSA2D performs better than CReSS (based on MSE scores), except at very high noise levels where there is an issue with over-fitting. There is an increasing need for a facility to combine multiple density surface maps of individual species in order to make best use of meta-databases, to maintain existing maps, and to extend their geographical coverage. This thesis develops a framework and methods for combining species distribution maps as new information becomes available. The methods use Bayes Theorem to combine density surfaces, taking account of the levels of precision associated with the different sets of estimates, and kernel smoothing to alleviate artefacts that may be created where pairs of surfaces join. The methods were used as part of an algorithm (the Dynamic Cetacean Abundance Predictor) designed for BAE Systems to aid in risk mitigation for naval exercises. Two case studies show the capabilities of CReSS and CReSS-SALSA2D when applied to real ecological data. In the first case study, CReSS was used in a Generalised Estimating Equation framework to identify a candidate Marine Protected Area for the Southern Resident Killer Whale population to the south of San Juan Island, off the Pacific coast of the United States. In the second case study, changes in the spatial and temporal distribution of harbour porpoise and minke whale in north-western European waters over a period of 17 years (1994-2010) were modelled. CReSS and CReSS-SALSA2D performed well in a large, topographically complex study area. Based on simulation results, maps produced using these methods are more accurate than if a traditional GAM-based method is used. The resulting maps identified particularly high densities of both harbour porpoise and minke whale in an area off the west coast of Scotland in 2010, that might be a candidate for inclusion into the Scottish network of Nature Conservation Marine Protected Areas.

Construire une surface lisse d'un objet 3D est un problème important dans de nombreuses applications graphiques. En particulier, les méthodes de subdivision permettent de passer facilement d'un maillage discret à une surface continue. Un problème général résultant de la subdivision de deux maillages initialement connectés le long d'un bord est l'apparition de fissures ou de trous entre eux. Ces fissures produisent non seulement des formes indésirables, mais induisent aussi des difficultés pour les traitements ultérieurs. Il faut donc réparer ces défauts de sorte que la surface obtenue soit lisse et puisse être subdivisée ou modifiée. Nous proposons de nouvelles méthodes pour relier deux maillages avec des résolutions différentes en utilisant une transformée en ondelettes B-splines et une approximation locale ou une interpolation locale à l'aide de fonctions de base radiales (RBF). Ces procédés génèrent un maillage de connexion où la continuité est contrôlée. La résolution du maillage est ajustable pour respecter le changement de résolution entre les zones grossières et fines. En outre, nous présentons des méthodes pour combler les trous à n-côtés, et le raffinement des maillages grâce à un schéma de subdivision adaptative. Nous avons conçu, implémenté et testé les algorithmes en MatLab pour illustrer nos méthodes et montrer des résultats expérimentaux. Ces algorithmes sont mis en oeuvre sur de nombreux modèles d'objets 3D avec des formes complexes. En outre, nous avons fourni des approches différentes pour chaque problème. Ainsi, les résultats des différentes approches sont comparés et évalués afin d'exploiter les avantages et les inconvénients de ces approches. / Constructing a smooth surface of a 3D object is an important problem in many graphical applications. In particular, subdivision methods permit to pass easily from a discrete mesh to a continuous surface. A generic problem arising from subdividing two meshes initially connected along a common boundary is the occurrence of cracks or holes between them. These cracks not only produce undesired shapes, but also bring serious trouble for further mesh processing. They must be removed or filled so that the produced surface is smooth and can be further subdivided or edited. In order to remove cracks, we propose new methods for joining two meshes with different resolutions using a Lifted B-spline wavelet transform and a local approximation or radial basis function (RBF) local interpolation. These methods generate a connecting mesh where continuity is controlled from one boundary to the other and the connecting mesh can change gradually in resolution between coarse and fine areas. Additionally, we introduce methods for filling n-sided holes, and refining meshes with an adaptive subdivision scheme. We have designed, implemented, and tested the algorithms in MatLab to illustrate our proposed methods and show experimental results. These algorithms are implemented on many 3D object models with complex shapes. Additionally, we have provided some different approaches for each problem. Thus, results from the different approaches are compared and evaluated to exploit the advantages and disadvantages of these approaches.

Made available in DSpace on 2016-12-23T14:08:09Z (GMT). No. of bitstreams: 1 Parte Inicial.pdf: 580643 bytes, checksum: 1783483d80317ac5307ad55e7cbdb752 (MD5) Previous issue date: 2013-03-25 / O propósito da pesquisa aqui elaborada é mostrar a viabilidade da aplicação de Funções de Base Radial de Suporte Compacto (FBRSC) no processo de aproximação direta do núcleo da ação de domínio através de integração de contorno. Essa formulação utilizada no tratamento da integral de domínio é denominada como (Método dos Elementos de Contorno com Integração Direta de Contorno) MECIC. Com o intuito de se avaliar a efetividade das FBRSC como funções de interpolação, serão realizados diversos testes numéricos, onde se deseja calcular o volume de superfícies. Então, serão realizados testes bidimensionais de aproximação, variando-se o suporte das FBRSCs, a fim de analisar o comportamento dessas funções. Depois de verificar a efetividade e a precisão das FBRSCs no processo de interpolação, desenvolvem-se programas, no ambiente do Método dos Elementos de Contorno (MEC), para a solução de problemas governados pela Equação de Poisson com a Formulação MECIC associada ao conceito de interpolação com FBRSC com suporte devidamente otimizados. A aferição das soluções numéricas obtidas se dá a partir da comparação com as suas respectivas soluções analíticas, facilmente encontradas na literatura especializada. Assim, possibilita-se estimar o erro relativo e então a eficácia da Formulação MECIC com FBRSC. Uma vez comprovado a sua eficácia, a Formulação MECIC com FBRSC é testada também com o esquema de interpolação com ajuste de pontos. Durante todo o desenvolvimento, atenta-se para a importância do custo computacional da formulação, a partir da geração de tabelas com o tempo de processamento dos programas implementados no MEC. Dessa forma, avalia-se qualitativamente o desempenho das FBRSC na Formulação MECIC, visando futuras aplicações na área de propagação de ondas sísmicas / The purpose of this research is to show the viability of application of Compactly Supported Radial Basis Function (CSRBF) in the process of direct approximation of the core of the domain action through boundary integration. This formulation is termed as (Boundary Elements Method with Directs Boundary Integration) MECIC, and is used in the treatment of the domain integration. By evaluating the effectiveness of CSRBF as interpolation functions, it performed several numerical tests to calculate the volume of surfaces. Also; by varying the support of CSRBFs, it performed two-dimensional approximation tests to examine the behavior of these functions. After verifying the effectiveness and accuracy of CSRBFs in the interpolation process, it developed computational programs to solve physical problems using the MECIC formulation, which is governed by Poisson s Equation. That formulation is associated with the concept of CSRBF in which the support is properly optimized. The calibration of the numerical solutions is given by the confrontation with their respective analytical solutions, easily found in the specialized literature. In this way, it is possible to estimate the relative error and the effectiveness of the MECIC formulation in association with the CSRBF concept. It is tested also with the curve fitting interpolation scheme. Owing the importance of the computational cost of that formulation, it is generated several time tables showing the processing time of those Boundary Elements Method computational programs. Therefore; aiming future applications in the seismic propagation wave area, it was finally evaluated the qualitative performance of the CSRBF in MECIC s formulation

在許多的研究和應用之中都需要預測的技巧。本論文中, 我們建構了一個 新的神經網路模式--動態徑向基底函數 (dynamical radial basis function) 網路 (DRBF網路) , 並且用這種模式的神經網路作為「函數近 似子」(function approximator) 去處理預測上的問題。另外我們也設計 幾種不同的學習演算法以測試DRBF網路的功能。 / The forecasting technique is important for many researches and applications. In this paper, we shall construct a new model of neural networks -- the dynamical radial basis function (DRBF) networks and use the DRBF networks as "function approximators" to solve some forecasting problems. Different learning algorithms are used to test the capability of DRBF networks.

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.

Environmental simulation models have been playing a key role in civil and environmental engineering decision making processes for decades. The utility of an environmental model depends on how well the model is structured and calibrated. Model calibration is typically in an automated form where the simulation model is linked to a search mechanism (e.g., an optimization algorithm) such that the search mechanism iteratively generates many parameter sets (e.g., thousands of parameter sets) and evaluates them through running the model in an attempt to minimize differences between observed data and corresponding model outputs. The challenge rises when the environmental model is computationally intensive to run (with run-times of minutes to hours, for example) as then any automatic calibration attempt would impose a large computational burden. Such a challenge may make the model users accept sub-optimal solutions and not achieve the best model performance. The objective of this thesis is to develop innovative strategies to circumvent the computational burden associated with automatic calibration of computationally intensive environmental models. The first main contribution of this thesis is developing a strategy called “deterministic model preemption” which opportunistically evades unnecessary model evaluations in the course of a calibration experiment and can save a significant portion of the computational budget (even as much as 90% in some cases). Model preemption monitors the intermediate simulation results while the model is running and terminates (i.e., pre-empts) the simulation early if it recognizes that further running the model would not guide the search mechanism. This strategy is applicable to a range of automatic calibration algorithms (i.e., search mechanisms) and is deterministic in that it leads to exactly the same calibration results as when preemption is not applied. One other main contribution of this thesis is developing and utilizing the concept of “surrogate data” which is basically a reasonably small but representative proportion of a full set of calibration data. This concept is inspired by the existing surrogate modelling strategies where a surrogate model (also called a metamodel) is developed and utilized as a fast-to-run substitute of an original computationally intensive model. A framework is developed to efficiently calibrate hydrologic models to the full set of calibration data while running the original model only on surrogate data for the majority of candidate parameter sets, a strategy which leads to considerable computational saving. To this end, mapping relationships are developed to approximate the model performance on the full data based on the model performance on surrogate data. This framework can be applicable to the calibration of any environmental model where appropriate surrogate data and mapping relationships can be identified. As another main contribution, this thesis critically reviews and evaluates the large body of literature on surrogate modelling strategies from various disciplines as they are the most commonly used methods to relieve the computational burden associated with computationally intensive simulation models. To reliably evaluate these strategies, a comparative assessment and benchmarking framework is developed which presents a clear computational budget dependent definition for the success/failure of surrogate modelling strategies. Two large families of surrogate modelling strategies are critically scrutinized and evaluated: “response surface surrogate” modelling which involves statistical or data–driven function approximation techniques (e.g., kriging, radial basis functions, and neural networks) and “lower-fidelity physically-based surrogate” modelling strategies which develop and utilize simplified models of the original system (e.g., a groundwater model with a coarse mesh). This thesis raises fundamental concerns about response surface surrogate modelling and demonstrates that, although they might be less efficient, lower-fidelity physically-based surrogates are generally more reliable as they to-some-extent preserve the physics involved in the original model. Five different surface water and groundwater models are used across this thesis to test the performance of the developed strategies and elaborate the discussions. However, the strategies developed are typically simulation-model-independent and can be applied to the calibration of any computationally intensive simulation model that has the required characteristics. This thesis leaves the reader with a suite of strategies for efficient calibration of computationally intensive environmental models while providing some guidance on how to select, implement, and evaluate the appropriate strategy for a given environmental model calibration problem.

Over the past decade there has been a growing demand for interactive multimedia applications deployed over public IP networks. To achieve acceptable Quality of Ser- vice (QoS) without significantly modifying the existing infrastructure, the end-to-end applications need to optimize their behavior and adapt according to network char- acteristics. Most existing application optimization techniques are based on reactive strategies, i.e. reacting to occurrences of congestion. We propose the use of predic- tive control to address the problem in an anticipatory manner. This research deals with developing models to predict end-to-end single flow characteristics of Wide Area Networks (WANs). A novel signal, in the form of single flow packet accumulation, is proposed for feedback purposes. This thesis presents a variety of effective predictors for the above signal using Auto-Regressive (AR) models, Radial Basis Functions (RBF) and Sparse Basis Functions (SBF). The study consists of three sections. We first develop time- series models to predict the accumulation signal. Since encoder bit-rate is the most logical and generic control input, a statistical analysis is conducted to analyze the effect of input bit-rate on end-to-end delay and the accumulation signal. Finally, models are developed using this bit-rate as an input to predict the resulting accu- mulation signal. The predictors are evaluated based on Noise-to-Signal Ratio (NSR) along with their accuracy with increasing accumulation levels. In time-series models, RBF gave the best NSR closely followed by AR models. Analysis based on accu- racy with increasing accumulation levels showed AR to be better in some cases. The study on effect of bit-rate revealed that bit-rate may not be a good control input on all paths. Models such as Auto-Regressive with Exogenous input (ARX) and RBF were used to develop models to predict the accumulation signal using bit-rate as a modeling input. ARX and RBF models were found to give comparable accuracy, with RBF being slightly better.

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.