Subsurface processes are usually characterized by rare field experiments, sparse measurements,multi-resolution interpretations, stochastic description, related uncertainties and computational complexity. Over the last few decades, different computational techniques and strategies have become indispensable tools for flow and solute transport prediction in heterogeneous porousmedia. This thesis develops a multi-resolution approach based on Fup basis functions with compactsupport, enabling the use of an efficient and adaptive procedure, closely related to currentunderstood physical interpretation. All flow and transport variables, as well as intrinsic heterogeneity,are described in a multi-resolution representation, in the form of a linear combination ofFup basis functions. Each variable is represented on a particular adaptive grid with a prescribedaccuracy. The methodology is applied to solving problems with sharp fronts, and to solving flowand advective transport in highly heterogeneous porous media, under mean uniform flow conditions.The adaptive Fup collocation method, through the well known method of lines, efficientlytracks solutions with sharp fronts, resolving locations and frequencies at all spatial and/or temporalscales. The methodology yields continuous velocity fields and fluxes, enabling accurate andreliable transport analysis. Analysis of the advective transport proves the robustness of the firstordertheory for low and mild heterogeneity. Moreover, due to the accuracy of the improved Monte-Carlo methodology, this thesis presents the effects of high heterogeneity on ensembleflow and travel time statistics. The difference between Eulerian and Lagrangian velocity statisticsand the importance of higher travel time moments are indicative of high heterogeneity. The thirdtravel time moment mostly describes a peak and late arrivals, while higher moments are requiredfor early arrivals which are linked with the largest uncertainty. A particular finding is the linearityof all travel time moments, which implies that in the limit an advective transport in multi-Gaussian field becomes Fickian. By comparison, the transverse displacement pdf converges to aGaussian distribution around 20 integral scales after injection, even for high heterogeneity. Thecapabilities of the presented multi-resolution approach, and the quality of the obtained results,open new areas for further research. / Markprocesser karakteriseras ofta av fåtaliga fältexperiment, glesa mätningar, heterogenitet påolika skalor, slumpmässighet och relaterade osäkerheter, samt beräkningsmässiga svårigheter.Under de senaste årtiondena har olika beräkningstekniker och strategier blivit ovärderliga verktygför att förutspå vattenflöde och ämnestransport i heterogena porösa medier. Denna doktorsavhandling utvecklar ett angreppssätt med flerskaliga upplösningar baserat på Fup basis funktionermed kompakt stöd, som möjliggör en effektiv och anpassningsbar procedur, nära relaterad tillrådande fysiska tolkningar. Alla flödes- och transportvariabler, så väl som heterogeniteten, beskrivsav en flerskaligt upplöst representation, i form av linjära kombinationer av Fup basis funktioner.Varje variabel representeras på ett speciellt anpassningsbar gridnät med given noggrannhet.Metoden appliceras för att lösa problem med skarpa fronter, samt vattenflöde och advektivämnestransport i starkt heterogena porösa medier. Adaptive Fup collocation metoden tillsammansmed den välkända Method of lines, spårar effektivt lösningar med skarpa fronter och löserupp positioner och frekvenser på alla rums- och/eller tidsskalor. Metoden ger kontinuerliga hastighetsfältoch flöden, och möjliggör noggrann och tillförlitlig transportanalys. Analys av advektivtransport understöder stabiliteten i första-ordningens transport teori för låg och mild heterogenitet.Utöver detta, som resultat av noggrannheten i den förbättrade Monte-Carlo metodiken, visardenna avhandling effekten av hög heterogenitet på ensemble statistiken för flöden och transporttider.Skillnaden mellan Eulerisk och Lagrangian hastighetsstatistik och betydelsen av högrestatistiska moment för transporttider, indikerar hög heterogenitet. Det tredje transporttidsmomentetbeskriver huvudsakligen sannolikhetspiken och de långa transporttiderna, medan högremoment behövs för de korta transporttiderna, som har den största osäkerheten. En speciell upptäcktär linjäariteten i transporttidsmoment, som indikerar att advektiv transport i multi-Gaussiska fält blir Gaussisk i gränsen. Som jämförelse konvergerar sannolikhetsfunktioner förden transversella transportförflyttningen mot en Gaussisk fördelning vid runt 20 korrelationslängder efter injektion, även för hög heterogenitet. Förmågan i det presenterade angreppssättet med flerskalig upplösning, och resultatens noggrannhet, öppnar nya områden för fortsatt forskning. / QC 20100714

The time-dependent Schrödinger equation (TDSE) models the quantum nature of molecular processes.  Numerical simulations based on the TDSE help in understanding and predicting the outcome of chemical reactions. This thesis is dedicated to the derivation and analysis of efficient and reliable simulation tools for the TDSE, with a particular focus on models for the interaction of molecules with time-dependent electromagnetic fields. Various time propagators are compared for this setting and an efficient fourth-order commutator-free Magnus-Lanczos propagator is derived. For the Lanczos method, several communication-reducing variants are studied for an implementation on clusters of multi-core processors. Global error estimation for the Magnus propagator is devised using a posteriori error estimation theory. In doing so, the self-adjointness of the linear Schrödinger equation is exploited to avoid solving an adjoint equation. Efficiency and effectiveness of the estimate are demonstrated for both bounded and unbounded states. The temporal approximation is combined with adaptive spectral elements in space. Lagrange elements based on Gauss-Lobatto nodes are employed to avoid nondiagonal mass matrices and ill-conditioning at high order. A matrix-free implementation for the evaluation of the spectral element operators is presented. The framework uses hybrid parallelism and enables significant computational speed-up as well as the solution of larger problems compared to traditional implementations relying on sparse matrices. As an alternative to grid-based methods, radial basis functions in a Galerkin setting are proposed and analyzed. It is found that considerably higher accuracy can be obtained with the same number of basis functions compared to the Fourier method. Another direction of research presented in this thesis is a new algorithm for quantum optimal control: The field is optimized in the frequency domain where the dimensionality of the optimization problem can drastically be reduced. In this way, it becomes feasible to use a quasi-Newton method to solve the problem. / eSSENCE

The objective of this PhD research programme is to investigate the effectiveness of a finite volume method incorporating radial basis functions for simulating nonlinear transport processes. The finite volume method is the favoured numerical technique for solving the advection-diffusion equations that arise in transport simulation. The method transforms the original problem into a system of nonlinear, algebraic equations through the process of discretisation. The accuracy of this discretisation determines to a large extent the accuracy of the final solution. A new method of discretisation is presented that employs radial basis functions (rbfs) as a means of local interpolation. When combined with Gaussian quadrature integration methods, the resulting finite volume discretisation leads to accurate numerical solutions without the need for very fine meshes, and the additional overheads they entail. The resulting nonlinear, algebraic system is solved efficiently using a Jacobian-free Newton-Krylov method. By employing the new method as an extension of existing shape function-based approaches, the number of nonlinear iterations required to obtain convergence can be reduced. Furthermore, information obtained from these iterations can be used to increase the efficiency of subsequent rbf-based iterations, as well as to construct an effective parallel reconditioner to further reduce the number of nonlinear iterations required. Results are presented that demonstrate the improved accuracy offered by the new method when applied to several test problems. By successively refining the meshes, it is also possible to demonstrate the increased order of the new method, when compared to a traditional shape function basedmethod. Comparing the resources required for both methods reveals that the new approach can be many times more efficient at producing a solution of a given accuracy.

Thesis (M.S.) -- Mississippi State University. Department of Electrical and Computer Engineering. / Title from title screen. Includes bibliographical references.

Made available in DSpace on 2014-12-17T14:55:44Z (GMT). No. of bitstreams: 1 AllanDGA_DISSERT.pdf: 1992696 bytes, checksum: 87d8b1dbc6fe4df6df2f85f90481f9be (MD5) Previous issue date: 2010-01-13 / Coordena??o de Aperfei?oamento de Pessoal de N?vel Superior / This study aims to seek a more viable alternative for the calculation of differences in images of stereo vision, using a factor that reduces heel the amount of points that are considered on the captured image, and a network neural-based radial basis functions to interpolate the results. The objective to be achieved is to produce an approximate picture of disparities using algorithms with low computational cost, unlike the classical algorithms / O presente trabalho visa buscar uma alternativa mais vi?vel para o c?lculo das disparidades em imagens de vis?o est?reo, utilizando um fator de salto que reduz a quantidade de pontos que s?o considerados da imagem capturada, e uma rede neural baseada em fun??es de base radial para interpolar os resultados obtidos. O objetivo a ser alcan?ado ? produzir uma imagem de disparidades aproximada da real com algoritmos de baixo custo computacional, diferentemente dos algoritmos tradicionais

Made available in DSpace on 2014-12-17T14:55:44Z (GMT). No. of bitstreams: 1 JoilsonBAR_DISSERT.pdf: 5903616 bytes, checksum: bee0d51eb1c54833e1d9a19364c80c76 (MD5) Previous issue date: 2010-07-30 / The present work describes the use of a mathematical tool to solve problems arising from control theory, including the identification, analysis of the phase portrait and stability, as well as the temporal evolution of the plant s current induction motor. The system identification is an area of mathematical modeling that has as its objective the study of techniques which can determine a dynamic model in representing a real system. The tool used in the identification and analysis of nonlinear dynamical system is the Radial Basis Function (RBF). The process or plant that is used has a mathematical model unknown, but belongs to a particular class that contains an internal dynamics that can be modeled.Will be presented as contributions to the analysis of asymptotic stability of the RBF. The identification using radial basis function is demonstrated through computer simulations from a real data set obtained from the plant / O presente trabalho descreve a utiliza??o de uma ferramenta matem?tica na solu??o de problemas decorrentes da teoria de controle, incluindo a identifica??o, a an?lise do retrato de fase e a estabilidade, bem como a evolu??o temporal da planta de corrente do motor de indu??o. A identifica??o de sistemas ? uma ?rea da modelagem matem?tica que tem como objetivo o estudo de t?cnicas que possam determinar um modelo din?mico na representa??o de um sistema real. A ferramenta utilizada na identifica??o e an?lise do sistema din?mico n?o linear ser? as Fun??es de Base Radial (RBF). O processo ou a planta que ser? utilizada possui um modelo matem?tico desconhecido, mas pertence a uma determinada classe que cont?m uma din?mica interna que pode ser modelada. Ser? apresentada como contribui??es a an?lise da estabilidade assint?tica da RBF. A identifica??o utilizando Fun??es de Base Radial ? demonstrada atrav?s de simula??es computacionais a partir de um conjunto de dados reais obtidos da planta de corrente do motor de indu??o

Orientadores: Wagner Caradori do Amaral, Ricardo Jose Gabrielli Barreto Campello / Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Eletrica e de Computação / Made available in DSpace on 2018-08-14T00:00:28Z (GMT). No. of bitstreams: 1 Rosa_Alexda_D.pdf: 1534572 bytes, checksum: 9100bf7dc7bd642daebdac3e973c668c (MD5) Previous issue date: 2009 / Resumo: Este trabalho enfoca a modelagem de sistemas não-lineares usando modelos de Volterra com funções de base ortonormal (Orthonormal Basis Functions - OBF). Os modelos de Volterra representam uma generalização do modelo de resposta ao impulso para a descrição de sistemas não-lineares e, em geral, exigem um elevado número de termos para representar os kernels de Volterra. Esta desvantagem pode ser superada representando-se os kernels usando um conjunto de funções ortonormais. O modelo resultante, conhecido como modelo OBF-Volterra, pode ser truncado em um n'umero menor de termos se as funções da base forem projetadas adequadamente. O problema central é como selecionar os polos livres que completamente parametrizam estas funções, particularmente as funções de Kautz e as funções ortonormais generalizadas (Generalized Orthonormal Basis Functions - GOBF). Uma das abordagens adotadas para resolver este problema envolve a minimização de um limitante superior para o erro resultante do truncamento da expansao do kernel. Cada kernel multidimensional é decomposto em um conjunto de bases de Kautz independentes, em que cada base é parametrizada por um par individual de pólos complexos conjugados com a intenção de representar a dinamica dominante do kernel ao longo de uma dimensão particular. Obtem-se uma solução analítica para um dos parâmetros de Kautz, válida para modelos de Volterra de qualquer ordem. Outra abordagem envolve a otimização numerica das bases de funções ortonormais usadas para a aproximação de sistemas dinamicos. Esta estrategia e baseada no cálculo de expressões analíticas para os gradientes da sa?da dos filtros ortonormais com relação aos pólos da base. Estes gradientes fornecem direções de busca exatas para otimizar os pólos de uma dada base ortonormal. As direções de busca, por sua vez, podem ser usadas como parte de um procedimento de otimização para obter o mínimo de uma função de custo que leva em consideração o erro de estimação da saída do sistema. As expressões relativas à base de Kautz e à base GOBF são obtidas. A metodologia proposta conta somente com dados entrada-sa'?da medidos do sistema a ser modelado, isto é, não se exige nenhuma informação prévia sobre os kernels de Volterra. Exemplos de simulação ilustram a aplicação desta abordagem para a modelagem de sistemas lineares e não-lineares, incluindo um sistema real de levitação magnética com comportamento oscilatorio. Por ultimo, estuda-se a representação de sistemas dinâmicos incertos baseada em modelos com incerteza estruturada. A incerteza de um conjunto de kernels de Volterra e mapeada em intervalos de pertinência que definem os coeficientes da expansão ortonormal. Condições adicionais são propostas para garantir que todos os kernels do processo sejam representados pelo modelo, o que permite estimar os limites das incertezas / Abstract: This work is concerned with the modeling of nonlinear systems using Volterra models with orthonormal basis functions (OBF). Volterra models represent a generalization of the impulse response model for the description of nonlinear systems and, in general, require a large number of terms for representing the Volterra kernels. Such a drawback can be overcome by representing the kernels using a set of orthonormal functions. The resulting model, so-called OBF-Volterra model, can be truncated into fewer terms if the basis functions are properly designed. The underlying problem is how to select the free-design poles that fully parameterize these functions, particularly the two-parameter Kautz functions and the Generalized Orthonormal Basis Functions (GOBF). One of the approaches adopted to solve this problem involves minimizing an upper bound for the error resulting from the truncation of the kernel expansion. Each multidimensional kernel is decomposed into a set of independent Kautz bases, in which every basis is parameterized by an individual pair of complex conjugate poles intended to represent the dominant dynamic of the kernel along a particular dimension. An analytical solution for one of the Kautz parameters, valid for Volterra models of any order, is derived. Other approach involves the numerical optimization of orthonormal bases of functions used for approximation of dynamic systems. This strategy is based on the computation of analytical expressions for the gradients of the output of the orthonormal filters with respect to the basis poles. These gradients provide exact search directions for optimizing the poles of a given orthonormal basis. Such search directions can, in turn, be used as part of an optimization procedure to locate the minimum of a cost-function that takes into consideration the error of estimation of the system output. The expressions relative to the Kautz basis and to the GOBF are addressed. The proposed methodology relies solely on input-output data measured from the system to be modeled, i.e., no previous information about the Volterra kernels is required. Simulation examples illustrate the application of this approach to the modeling of linear and nonlinear systems, including a real magnetic levitation system with oscillatory behavior. At last, the representation of uncertain systems based on models having structured uncertainty is studied. The uncertainty of a set of Volterra kernels is mapped on to intervals defining the coefficients of the orthonormal expansion. Additional conditions are proposed to guarantee that all the process kernels to be represented by the model, which allows estimating the uncertainty bounds / Doutorado / Automação / Doutor em Engenharia Elétrica

O Método da Coordenada Geradora Hartree-Fock Polinomial (pMCG-HF), desenvolvido por R.C. Barbosa e A.B.F. da Silva [1], é uma ferramenta matemática valiosa que permite gerar funções de base (também conhecidas como conjuntos de base). As funções de base geradas por este método têm um bom comportamento e são capazes de calcular valores precisos de propriedades eletrônicas moleculares. Porém, depois de gerar funções de base do hidrogênio até o flúor [2], fez-se necessário a adição de expoentes à função de base, correspondentes a cada átomo, para melhor adaptação à realização dos cálculos moleculares. Estas funções adicionais são o que chamamos de funções de polarização. A adição de funções de polarização, através de otimização computacional, é muito custosa, deste modo o desenvolvimento de uma regra de polarização para se esquivar desta otimização é de grande importância e por isso se transforma na beleza e no objetivo deste trabalho. Portanto, nesta dissertação, estudar-se-á um procedimento para escolher funções de polarização que reduza drasticamente o tempo computacional, no sentido de permitir uma seleção, mais simples, de expoentes da própria função de base primitiva para serem usadas nas funções de polarização p, d, f, g, etc. para a obtenção de propriedades moleculares calculadas através de métodos químico-quânticos / The polynomial generate coordinate method pGCM developed by R.C. Barbosa and A.B.F. da Silva [1] is an remarkble mathematic tool for the generation of basis functions (also known as basis sets). The basis sets generated from this method have a good behavior and are able to produce accurate values for electronic molecular properties. In fact, after generating a basis set [2] we need to add a set of exponent functions in order to better adequate a basis set to perform molecular calculations. These sets of additional functions are called polarizations functions. This work provides a methodology where the polarization functions are obtained from the initial basis set (the primitive set) without optimizing them separately by using optimization algorithms that are, computationally speaking, very costly. This procedure reduces drastically the computational time used to find polarization functions to be used in molecular quantum chemical calculations. Our methodology permits to choose the polarization functions directly from the primitive orbital exponents of each atomic symmetry s, p, d, f etc. in a very simple manner. The finding of polarization functions using our methodology was performed with several quantum chemical methods.

As a result of the increase in accessibility of computational resources and the increase in the power of the computers during the last two decades, designers are able to create computer models to simulate the behavior of a complex products. To address global competitiveness, companies are forced to optimize their designs and products. Optimizing the design needs several runs of computationally expensive simulation models. Therefore, using metamodels as an efficient and sufficiently accurate approximate of the simulation model is necessary. Radial basis functions (RBF) is one of the several metamodeling methods that can be found in the literature. The established approach is to add a bias to RBF in order to obtain a robust performance. The a posteriori bias is considered to be unknown at the beginning and it is defined by imposing extra orthogonality constraints. In this thesis, a new approach in constructing RBF with the bias to be set a priori by using the normal equation is proposed. The performance of the suggested approach is compared to the classic RBF with a posteriori bias. Another comprehensive comparison study by including several modeling criteria, such as problem dimension, sampling technique and size of samples is conducted. The studies demonstrate that the suggested approach with a priori bias is in general as good as the performance of RBF with a posteriori bias. Using the a priori RBF, it is clear that the global response is modeled with the bias and that the details are captured with radial basis functions. Multi-objective optimization and the approaches used in solving such problems are briefly described in this thesis. One of the methods that proved to be efficient in solving multi-objective optimization problems (MOOP) is the strength Pareto evolutionary algorithm (SPEA2). Multi-objective optimization of a disc brake system of a heavy truck by using SPEA2 and RBF with a priori bias is performed. As a result, the possibility to reduce the weight of the system without extensive compromise in other objectives is found. Multi-objective optimization of material model parameters of an adhesive layer with the aim of improving the results of a previous study is implemented. The result of the original study is improved and a clear insight into the nature of the problem is revealed.

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