The chemical master equation, formulated on the Markov assumption of underlying chemical kinetics, offers an accurate stochastic description of general chemical reaction systems on the mesoscopic scale. The chemical master equation is especially useful when formulating mathematical models of gene regulatory networks and protein-protein interaction networks, where the numbers of molecules of most species are around tens or hundreds. However, solving the master equation directly suffers from the so called "curse of dimensionality" issue. This thesis first tries to study the numerical properties of the master equation using existing numerical methods and parallel machines. Next, approximation algorithms, namely the adaptive aggregation method and the radial basis function collocation method, are proposed as new paths to resolve the "curse of dimensionality". Several numerical results are presented to illustrate the promises and potential problems of these new algorithms. Comparisons with other numerical methods like Monte Carlo methods are also included. Development and analysis of the linear Shepard algorithm and its variants, all of which could be used for high dimensional scattered data interpolation problems, are also included here, as a candidate to help solve the master equation by building surrogate models in high dimensions. / Ph. D.

Made available in DSpace on 2016-12-23T14:08:15Z (GMT). No. of bitstreams: 1 Atila Lupim Cruz.pdf: 1394501 bytes, checksum: 0954b2c5b1fdcb864ee81cef7d14e9e5 (MD5) Previous issue date: 2012-10-19 / A pesquisa envolvida na presente dissertação se baseou no uso de funções de base radial para gerar uma nova formulação integral, que interpola diretamente o termo não homogêneo da equação diferencial de governo, no contexto do Método dos Elementos de Contorno (MEC). Emprega-se o uso de funções primitivas das funções de interpolação originais no núcleo da integral de domínio, permitindo a transformação desta última numa integral de contorno, evitando assim a discretização do domínio por meio de células, semelhante ao realizado na Dupla Reciprocidade. Para melhor avaliação das potencialidades da formulação, os testes numéricos apresentados abordaram apenas a solução de problemas governados pela Equação de Poisson. Os problemas escolhidos dentro desta categoria possuem solução analítica, o que permitiu aferir com mais rigor a precisão dos resultados. Para melhor balizamento da eficiência da formulação proposta, todos os problemas abordados também foram resolvidos pela formulação com Dupla Reciprocidade. O custo computacional dispendido para cada uma dessas formulações também foi comparado. Para ambas as formulações também foram testados esquemas de ajuste da interpolação realizada, visando avaliar seus efeitos na precisão dos resultados e também propositando obter economia computacional em futuras aplicações em simulações na área de propagações de ondas / This research was based on the use of radial basis functions to generate a new integral formulation that interpolates directly the domain action, related to the inhomogeneous term of the governing differential equation, using the Boundary Element Method (BEM). The use of primitive functions of the original interpolation functions in the kernel of the inhomogeneous integral is proposed, allowing its transformation into a boundary integral, thus avoiding the domain discretization through cells, similar to that conducted in the Dual Reciprocity. To better evaluation of the capability of the proposed formulation, the numerical tests presented only solved problems governed by the Poisson Equation. Test problems chosen have known analytical solution, which allowed a better evaluation of the numerical accuracy. To better check the efficiency of the proposed formulation, all the problems were also solved by the Dual Reciprocity Boundary Element Formulation. The computational cost expended for each of these formulations was also compared. Fitting interpolation schemes for both formulations were also tested in order to evaluate their effects on the accuracy of the results and also looking for economy in future computational applications related to wave propagation problems

Artificial neural networks (ANNs) have been used successfully in various practical problems. Though extensive improvements on different types of ANNs have been made to improve their performance, each ANN design still experiences its own limitations. The existing digital human models are mature enough to provide accurate and useful results for different tasks and scenarios under various conditions. There is, however, a critical need for these models to run in real time, especially those with large-scale problems like motion prediction which can be computationally demanding. For even small changes to the task conditions, the motion simulation needs to run for a relatively long time (minutes to tens of minutes). Thus, there can be a limited number of training cases due to the computational time and cost associated with collecting training data. In addition, the motion problem is relatively large with respect to the number of outputs, where there are hundreds of outputs (between 500-700 outputs) to predict for a single problem. Therefore, the aforementioned necessities in motion problems lead to the use of tools like the ANN in this work. This work introduces new algorithms for the design of the radial-basis network (RBN) for problems with minimal available training data. The new RBN design incorporates new training stages with approaches to facilitate proper setting of necessary network parameters. The use of training algorithms with minimal heuristics allows the new RBN design to produce results with quality that none of the competing methods have achieved. The new RBN design, called Opt_RBN, is tested on experimental and practical problems, and the results outperform those produced from standard regression and ANN models. In general, the Opt_RBN shows stable and robust performance for a given set of training cases. When the Opt_RBN is applied on the large-scale motion prediction application, the network experiences a CPU memory issue when performing the optimization step in the training process. Therefore, new algorithms are introduced to modify some steps of the new Opt_RBN training process to address the memory issue. The modified steps should only be used for large-scale applications similar to the motion problem. The new RBN design proposes an ANN that is capable of improved learning without needing more training data. Although the new design is driven by its use with motion prediction problems, the consequent ANN design can be used with a broad range of large-scale problems in various engineering and industrial fields that experience delay issues when running computational tools that require a massive number of procedures and a great deal of CPU memory. The results of evaluating the modified Opt_RBN design on two motion problems are promising, with relatively small errors obtained when predicting approximately 500-700 outputs. In addition, new methods for constraint implementation within the new RBN design are introduced. Moreover, the new RBN design and its associated parameters are used as a tool for simulated task analysis. This work initiates the idea that output weights (W) can be used to determine the most critical basis functions that cause the greatest reduction in the network test error. Then, the critical basis functions can specify the most significant training cases that are responsible for the proper performance achieved by the network. The inputs with the most change in value can be extracted from the basis function centers (U) in order to determine the dominant inputs. The outputs with the most change in value and their corresponding key body degrees-of-freedom for a motion task can also be specified using the training cases that are used to create the network's basis functions.

In this work, a numerical approach based on meshless methods is proposed to obtain eigenmodes of Laplace-Beltrami operator on manifolds, and its performance is compared against existing alternative methods. Radial Basis Function (RBF)-based methods allow one to obtain interpolation and differentiation matrices easily by using scattered data points. We derive expressions for such matrices for the Laplace-Beltrami operator via so-called Reilly’s formulas and use them to solve the respective eigenvalue problem. Numerical studies of proposed methods are performed in order to demonstrate convergence on simple examples of one-dimensional curves and two-dimensional surfaces.

A solar and diesel-wind hybrid power control systems is proposed in the thesis. The system consists of solar power, wind power, diesel-engine, a static synchronous compensator and an intelligent power controller. MATLAB/Simulink was used to build the dynamic model and simulate the solar and diesel-wind hybrid power system. A static synchronous compensator was used to supply reactive power and regulate the voltage of the hybrid system. To achieve a fast and stable response for the real power control, an intelligent controller was proposed, which consists of the Radial Basis Function Network (RBFN) and the Elman Neural Network (ENN) for maximum power point tracking (MPPT). The pitch angle control of wind power uses ENN controller, and the output is fed to the wind turbine to achieve the MPPT. The solar system uses RBFN, and the output signal is used to control the DC / DC boost converters to achieve the MPPT.

In this paper, we use the newly developed method of particular solution (MPS) and one-stage method of fundamental solution (MFS-MPS) for solving partial differential equation (PDE). In the 1-D Poisson equation, we prove the solution of MFS-MPS is converge to Spectral Collocation Method using Polynomial, and show that the numerical solution similar to those of using the method of particular solution (MPS), Kansa's method, and Spectral Collocation Method using Polynomial (SCMP). In 2-D, we also test these results for the Poisson equation and find the error behaviors.

No description available.

Thesis (MSc (Applied Mathematics))--Stellenbosch University, 2008. / A popular method for interpolating multidimensional scattered data is using radial basis functions. In this thesis we present the basic theory of radial basis function interpolation and also regard the solvability and stability of the method. Solving the interpolant directly has a high computational cost for large datasets, hence using numerical methods to approximate the interpolant is necessary. We consider some recent numerical algorithms. Software to implement radial basis function interpolation and to display the 3D interpolants obtained, is developed. We present results obtained from using our implementation for radial basis functions on GIS and 3D face data as well as an image warping application.

Made available in DSpace on 2014-06-11T19:30:50Z (GMT). No. of bitstreams: 0 Previous issue date: 2010-11-27Bitstream added on 2014-06-13T19:47:13Z : No. of bitstreams: 1 altran_ab_dr_ilha.pdf: 733564 bytes, checksum: 382a61569b0f5da4ceb6a2f45c0815a4 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / A previsão de carga, em sistemas de energia elétrica, constitui-se numa atividade de grande importância, tendo em vista que a maioria dos estudos realizados (fluxo de potência, despacho econômico, planejamento da expansão, compra e venda de energia, etc.) somente poderá ser efetivada se houver a disponibilidade de uma boa estimativa da carga a ser atendida. Deste modo, visando contribuir para que o planejamento e operação dos sistemas de energia elétrica ocorram de forma segura, confiável e econômica, foi desenvolvida uma metodologia para previsão de carga, a previsão multinodal, que pode ser entendida como um sistema inteligente que considera vários pontos da rede elétrica durante a realização da previsão. O sistema desenvolvido conta com o uso de uma rede neural artificial composta por vários módulos, sendo esta do tipo perceptron multicamadas, cujo treinamento é baseado no algoritmo retropropagação. Porém, foi realizada uma modificação na função de ativação da rede, em substituição à função usual, a função sigmoide, foram utilizadas as funções de base radial. Tal metodologia foi aplicada ao problema de previsão de cargas elétricas a curto-prazo (24 horas à frente) / Load forecasting in electric power systems is a very important activity due to several studies, e.g. power flow, economic dispatch, expansion planning, purchase and sale of energy that are extremely dependent on a good estimate of the load. Thus, contributing to a safe, reliable, economic and secure operation and planning this work is developed, which is an intelligent system for multinodal electric load forecasting considering several points of the network. The multinodal system is based on an artificial neural network composed of several modules. The neural network is a multilayer perceptron trained by backpropagation where the traditional sigmoide is substituted by radial basis functions. The methodology is applied to forecast loads 24 hours in advance

Made available in DSpace on 2014-12-17T14:56:03Z (GMT). No. of bitstreams: 1 OsmarADJ.pdf: 1159660 bytes, checksum: 65307a903dfe1a1f71297194d1c7e2a5 (MD5) Previous issue date: 2004-12-22 / Conselho Nacional de Desenvolvimento Cient?fico e Tecnol?gico / This dissertation presents a new proposal for the Direction of Arrival (DOA) detection problem for more than one signal inciding simultaneously on an antennas array with linear or planar geometry by using intelligent algorithms. The DOA estimator is developed by using techniques of Conventional Beam-forming (CBF), Blind Source Separation (BSS), and the neural estimator MRBF (Modular Structure of Radial Basis Functions). The developed MRBF estimator has its capacity extended due to the interaction with the BSS technique. The BSS makes an estimation of the steering vectors of the multiple plane waves that reach the array in the same frequency, that means, obtains to separate mixed signals without information a priori. The technique developed in this work makes possible to identify the multiple sources directions and to identify and to exclude interference sources / Esta disserta??o apresenta uma nova proposta para os problemas de detec??o de dire??o de chegada para mais de um sinal incidindo simultaneamente sobre um arranjo de antenas de geometria planar ou linear empregando algoritmos inteligentes. O estimador de DOA ? desenvolvido utilizando as t?cnicas de Conforma??o de Feixes Digital Convencional (CBF - Conventional Beamforming), de Separa??o Cega de Fontes (BSS {Blind Source Separation) e o estimador neural MRBF (Modular Structure of Radial Basis Functions). O estimador MRBF desenvolvido tem sua capacidade ampliada gra?as ?a intera??o com a t?cnica BSS, a qual faz uma estima??o dos vetores de guiamento das m?ltiplas ondas planas que alcan?am o arranjo na mesma freq??ncia, isto ?, consegue separar sinais misturados sem informa??es a priori. A t?cnica desenvolvida neste trabalho possibilita identificar a dire??o de m?ltiplas fontes e identificar e excluir as fontes de interfer?ncia