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Análise de alta precisão em modelos tridimensionais de elementos de contorno utilizando técnicas avançadas de integração numérica.; Advanced numerical integration in three-dimensional boundary elements analysis.

Souza, Calebe Paiva Gomes de
Fonte: Biblioteca Digitais de Teses e Dissertações da USP Publicador: Biblioteca Digitais de Teses e Dissertações da USP
Tipo: Dissertação de Mestrado Formato: application/pdf
Publicado em 06/06/2007 PT
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Um dos principais problemas que o Método de Elementos de Contorno (MEC) apresenta encontra-se na avaliação de integrais singulares e quase-singulares que envolvem as soluções fundamentais de Kelvin em deslocamento e força. O processo de integração numérica em MEC tem sido o objetivo de inúmeras pesquisas, pois dele depende a qualidade das respostas quando se deseja obter uma excelente precisão numérica em uma análise. Este trabalho apresenta uma nova proposta de integração numérica para análise tridimensional com MEC. Esta técnica possui três características importantes. A primeira é a determinação da parcela efetiva de singularidade que ocorre na função raio, distância entre o ponto fonte e o elemento de contorno bidimensional. A correta obtenção desta parcela permite representar sem aproximações o comportamento da singularidade da função raio, que é a verdadeira fonte de singularidade e quase-singularidade nas soluções fundamentais. A segunda característica da técnica proposta é que ela baseia-se em um Método Semi-Analítico de avaliação de integrais, onde, para cada parcela efetiva de singularidade, utiliza-se uma quadratura numérica cujos pesos específicos são calculados analiticamente. A terceira característica da técnica proposta é apresentar um tratamento unificado para todos os tipos de integrais singulares...

Geometria do desacoplamento e integração numérica de equações diferenciais não lineares implícitas.; Decoupling geometry and numerical integration of differential equations implicit nonlinear systems.

Souza, Iderval Silva de
Fonte: Biblioteca Digitais de Teses e Dissertações da USP Publicador: Biblioteca Digitais de Teses e Dissertações da USP
Tipo: Dissertação de Mestrado Formato: application/pdf
Publicado em 24/11/2006 PT
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Existem métodos de integração de equações algébrico diferenciais não lineares (DAEs) considerados clássicos pela literatura. Porém, neste trabalho, através uma abordagem geométrica, apresenta-se um método de integração de DAEs. Tal método é inspirado na teoria de desacoplamento de sistemas não lineares explícitos, quando se considera que as saídas são restrições algébricas. Neste caso, a DAE pode ser identificada como dinâmica zero. O resultado principal desta abordagem é que, dada uma DAE, sob certas condições, é possível a construção de um sistema explícito, de tal maneira, que as soluções desse sistema explícito convergem para as soluções da DAE.; Classical methods for numerical integration of diferential algebraic equations (DAEs) can be formal in the literature. In this work, using a diferential geometric approach, a numerical method of integration of DAEs is established. This method is inspired in the decoupling theory of nonlinear explicit systems, when one considers that the outputs are algebraic constraints. The main result is the construction of an explicit system, whose solutions converge to the solutions of the DAE.

Comparison of methodologies for degree-day estimation using numerical methods

Souza, Adilson Pacheco de; Ramos, Clóvis Manoel Carvalho; Lima, Adriano Dawison de; Florentino, Helenice de Oliveira; Escobedo, João Francisco
Fonte: Editora da Universidade Estadual de Maringá (EDUEM) Publicador: Editora da Universidade Estadual de Maringá (EDUEM)
Tipo: Artigo de Revista Científica Formato: 391-400
ENG
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O desenvolvimento de projetos relacionados ao desempenho de diversas culturas tem recebido aperfeiçoamento cada vez maior, incorporado a modelos matemáticos sendo indispensável à utilização de equações cada vez mais consistentes que possibilitem previsão e maior aproximação do comportamento real, diminuindo o erro na obtenção das estimativas. Entre as operações unitárias que demandam maior estudo estão aquelas relacionadas com o crescimento da cultura, caracterizadas pela temperatura ideal para o acréscimo de matéria seca. Pelo amplo uso dos métodos matemáticos na representação, análise e obtenção de estimativas de graus-dia, juntamente com a grande importância que a cultura da cana-de-açúcar tem para a economia brasileira, foi realizada uma avaliação dos modelos matemáticos comumente usados e dos métodos numéricos de integração na estimativa da disponibilidade de graus-dia para essa cultura, na região de Botucatu, Estado de São Paulo. Os modelos de integração, com discretização de 6 em 6 h, apresentaram resultados satisfatórios na estimativa de graus-dia. As metodologias tradicionais apresentaram desempenhos satisfatórios quanto à estimativa de grausdia com base na curva de temperatura horária para cada dia e para os agrupamentos de três...

Comparison of methodologies for degree-day estimation using numerical methods

Souza,Adilson Pacheco de; Ramos,Clóvis Manoel Carvalho; Lima,Adriano Dawison de; Florentino,Helenice de Oliveira; Escobedo,João Francisco
Fonte: Editora da Universidade Estadual de Maringá - EDUEM Publicador: Editora da Universidade Estadual de Maringá - EDUEM
Tipo: Artigo de Revista Científica Formato: text/html
Publicado em 01/09/2011 EN
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The development of projects related to the yield of various crops has been greatly enhanced with the incorporation of mathematical models as well as essential and more consistent equations which enable a prediction and greater approximation to their actual behavior, thus reducing error in estimate. Among the operations requiring further investigation are those related to crop growth, characterized by the ideal temperature for addition of dry matter. Due to the wide use of mathematical methods for representing, analyzing and attaining degree-day estimation as well as the great importance of sugarcane in the Brazilian economy, we carried out an evaluation of the mathematical models and numerical integration methods commonly used for estimating the availability of degrees-day for this crop in the region of Botucatu, in São Paulo State, Brazil. Integration models with discretization every 6 hours have shown satisfactory results in degree-day estimation. Conventional methodologies have shown satisfactory results when the estimation of degrees-day was based on the time-temperature curve for each day and for groups of 3, 7, 15 and 30 days. Through numerical integration method, the region of Botucatu showed a annual thermal availability average from 1...

13.024 Numerical Marine Hydrodynamics, Spring 2003; Numerical Marine Hydrodynamics

Milgram, Jerome H.
Fonte: MIT - Massachusetts Institute of Technology Publicador: MIT - Massachusetts Institute of Technology
EN-US
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Introduction to numerical methods: interpolation, differentiation, integration, systems of linear equations. Solution of differential equations by numerical integration, partial differential equations of inviscid hydrodynamics: finite difference methods, panel methods. Fast Fourier Transforms. Numerical representation of sea waves. Computation of the motions of ships in waves. Integral boundary layer equations and numerical solutions.

On an infinite integral arising in the numerical integration of stochastic differential equations

Stump, D.; Hill, J.
Fonte: Royal Soc London Publicador: Royal Soc London
Tipo: Artigo de Revista Científica
Publicado em //2005 EN
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We study a stochastic integral that arises during the implementation of the Milstein method for the numerical integration of systems of stochastic differential equations. The distribution of the integral can be written as the inverse Fourier transform of a characteristic function with essential singularities. This leads to a generalized integral that can be expressed as an infinite series involving the derivatives of Meixner polynomials. The generating function of the polynomials in combination with the Mittag–Leffler expansion theorem is used to obtain a novel series representation for the integral and the motivating problem in particular. This new form is rapidly convergent and, therefore, well suited to numerical work.; David M. Stump and James M. Hill

Splitting and composition methods in the numerical integration of differential equations

Blanes, Sergio; Casas, Fernando; Murua, Ander
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 01/12/2008
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We provide a comprehensive survey of splitting and composition methods for the numerical integration of ordinary differential equations (ODEs). Splitting methods constitute an appropriate choice when the vector field associated with the ODE can be decomposed into several pieces and each of them is integrable. This class of integrators are explicit, simple to implement and preserve structural properties of the system. In consequence, they are specially useful in geometric numerical integration. In addition, the numerical solution obtained by splitting schemes can be seen as the exact solution to a perturbed system of ODEs possessing the same geometric properties as the original system. This backward error interpretation has direct implications for the qualitative behavior of the numerical solution as well as for the error propagation along time. Closely connected with splitting integrators are composition methods. We analyze the order conditions required by a method to achieve a given order and summarize the different families of schemes one can find in the literature. Finally, we illustrate the main features of splitting and composition methods on several numerical examples arising from applications.; Comment: Review paper; 56 pages...

Two-step hybrid methods adapted to the numerical integration of perturbed oscillators

Van de Vyver, Hans
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 21/12/2006
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Two-step hybrid methods specially adapted to the numerical integration of perturbed oscillators are obtained. The formulation of the methods is based on a refinement of classical Taylor expansions due to Scheifele [{\em Z. Angew. Math. Phys.}, {\bf 22}, 186--210 (1971)]. The key property is that those algorithms are able to integrate exactly harmonic oscillators with frequency $\omega$ and that, for perturbed oscillators, the local error contains the (small) perturbation parameter as a factor. The methods depend on a parameter $\nu=\omega h$, where $h$ is the stepsize. Based on the B2-series theory of Coleman [{\em IMA J. Numer. Anal.}, {\bf 23}, 197--220 (2003)] we derive the order conditions of this new type of methods. The linear stability and phase properties are examined. The theory is illustrated with some fourth- and fifth-order explicit schemes. Numerical results carried out on an assortment of test problems (such as the integration of the orbital motion of earth satellites) show the relevance of the theory.; Comment: 34 pages, 7 figures

Numerical integration for high order pyramidal finite elements

Nigam, Nilima; Phillips, Joel
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 01/03/2010
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We examine the effect of numerical integration on the convergence of high order pyramidal finite element methods. Rational functions are indispensable to the construction of pyramidal interpolants so the conventional treatment of numerical integration, which requires that the finite element approximation space is piecewise polynomial, cannot be applied. We develop an analysis that allows the finite element approximation space to include rational functions and show that despite this complication, conventional rules of thumb can still be used to select appropriate quadrature methods on pyramids. Along the way, we present a new family of high order pyramidal finite elements for each of the spaces of the de Rham complex.; Comment: 28 pages

A Family of Multistep Methods with Zero Phase-Lag and Derivatives for the Numerical Integration of Oscillatory ODEs

Anastassi, Z. A.; Vlachos, D. S.; Simos, T. E.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 18/07/2008
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In this paper we develop a family of three 8-step methods, optimized for the numerical integration of oscillatory ordinary differential equations. We have nullified the phase-lag of the methods and the first r derivatives, where r=1,2,3. We show that with this new technique, the method gains efficiency with each derivative of the phase-lag nullified. This is the case for the integration of both the Schrodinger equation and the N-body problem. A local truncation error analysis is performed, which, for the case of the Schrodinger equation, also shows the connection of the error and the energy, revealing the importance of the zero phase-lag derivatives. Also the stability analysis shows that the methods with more derivatives vanished, have a bigger interval of periodicity.; Comment: 34 pages, 4 figures

Numerical integration of Heath-Jarrow-Morton model of interest rates

Krivko, M.; Tretyakov, M. V.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 12/09/2011
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We propose and analyze numerical methods for the Heath-Jarrow-Morton (HJM) model. To construct the methods, we first discretize the infinite dimensional HJM equation in maturity time variable using quadrature rules for approximating the arbitrage-free drift. This results in a finite dimensional system of stochastic differential equations (SDEs) which we approximate in the weak and mean-square sense using the general theory of numerical integration of SDEs. The proposed numerical algorithms are computationally highly efficient due to the use of high-order quadrature rules which allow us to take relatively large discretization steps in the maturity time without affecting overall accuracy of the algorithms. Convergence theorems for the methods are proved. Results of some numerical experiments with European-type interest rate derivatives are presented.; Comment: 48 pages

Geometric Numerical Integration of Inequality Constrained, Nonsmooth Hamiltonian Systems

Kaufman, Danny M.; Pai, Dinesh K.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
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We consider the geometric numerical integration of Hamiltonian systems subject to both equality and "hard" inequality constraints. As in the standard geometric integration setting, we target long-term structure preservation. We additionally, however, also consider invariant preservation over persistent, simultaneous and/or frequent boundary interactions. Appropriately formulating geometric methods to include such conditions has long-remained challenging due to the inherent nonsmoothness they impose. To resolve these issues we thus focus both on symplectic-momentum preserving behavior and the preservation of additional structures, unique to the inequality constrained setting. Leveraging discrete variational techniques, we construct a family of geometric numerical integration methods that not only obtain the usual desirable properties of momentum preservation, approximate energy conservation and equality constraint preservation, but also enforce multiple simultaneous inequality constraints, obtain smooth unilateral motion along constraint boundaries and allow for both nonsmooth and smooth boundary approach and exit trajectories. Numerical experiments are presented to illustrate the behavior of these methods on difficult test examples where both smooth and nonsmooth active constraint modes persist with high frequency.; Comment: added new section...

New families of symplectic splitting methods for numerical integration in dynamical astronomy

Blanes, Sergio; Casas, Fernando; Farres, Ariadna; Laskar, Jacques; Makazaga, Joseba; Murua, Ander
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
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We present new splitting methods designed for the numerical integration of near-integrable Hamiltonian systems, and in particular for planetary N-body problems, when one is interested in very accurate results over a large time span. We derive in a systematic way an independent set of necessary and sufficient conditions to be satisfied by the coefficients of splitting methods to achieve a prescribed order of accuracy. Splitting methods satisfying such (generalized) order conditions are appropriate in particular for the numerical simulation of the Solar System described in Jacobi coordinates. We show that, when using Poincar\'e Heliocentric coordinates, the same order of accuracy may be obtained by imposing an additional polynomial equation on the coefficients of the splitting method. We construct several splitting methods appropriate for each of the two sets of coordinates by solving the corresponding systems of polynomial equations and finding the optimal solutions. The experiments reported here indicate that the efficiency of our new schemes is clearly superior to previous integrators when high accuracy is required.; Comment: 24 pages, 2 figures. Revised version, accepted for publication in Applied Numerical Mathematics

Quasi-Monte Carlo numerical integration on $\mathbb{R}^s$: digital nets and worst-case error

Dick, Josef
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
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Quasi-Monte Carlo rules are equal weight quadrature rules defined over the domain $[0,1]^s$. Here we introduce quasi-Monte Carlo type rules for numerical integration of functions defined on $\mathbb{R}^s$. These rules are obtained by way of some transformation of digital nets such that locally one obtains qMC rules, but at the same time, globally one also has the required distribution. We prove that these rules are optimal for numerical integration in fractional Besov type spaces. The analysis is based on certain tilings of the Walsh phase plane.

New Iterative Methods for Interpolation, Numerical Differentiation and Numerical Integration

Muthumalai, Ramesh Kumar
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
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Through introducing a new iterative formula for divided differnce using Neville's and Aitken's algorithms,we study new iterative methods for interpolation,numerical differentiation and numerical integration formulas with arbitrary order of accuracy for evanly or unevanly spaced data. Basic computer algorithms for new methods are given; Comment: 11 pages, comments are invited

Change of variable in spaces of mixed smoothness and numerical integration of multivariate functions on the unit cube

Nguyen, Van Kien; Ullrich, Mario; Ullrich, Tino
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 06/11/2015
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In a recent article by two of the present authors it turned out that Frolov's cubature formulae are optimal and universal for various settings (Besov-Triebel-Lizorkin spaces) of functions with dominating mixed smoothness. Those cubature formulae go well together with functions supported inside the unit cube $[0,1]^d$. The question for the optimal numerical integration of multivariate functions with non-trivial boundary data, in particular non-periodic functions, arises. In this paper we give a general result that the asymptotic rate of the minimal worst-case integration error is not affected by boundary conditions in the above mentioned spaces. In fact, we propose two tailored modifications of Frolov's cubature formulae suitable for functions supported on the cube (not in the cube) which provide the same minimal worst-case error up to a constant. This constant involves the norms of a ``change of variable'' and a ``pointwise multiplication'' mapping, respectively, between the function spaces of interest. In fact, we complement, extend and improve classical results by Bykovskii, Dubinin and Temlyakov on the boundedness of change of variable mappings in Besov-Sobolev spaces of mixed smoothness. Our proof technique relies on a new characterization via integral means of mixed differences and maximal function techniques...

Numerical integration in $\log$-Korobov and $\log$-cosine spaces

Dick, Josef; Kritzer, Peter; Leobacher, Gunther; Pillichshammer, Friedrich
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 11/11/2014
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QMC rules are equal weight quadrature rules for approximating integrals over $[0,1]^s$. One line of research studies the integration error of functions in the unit ball of so-called Korobov spaces, which are Hilbert spaces of periodic functions on $[0,1]^s$ with square integrable partial mixed derivatives of order $\alpha$. Using Parseval's identity, this smoothness can be defined for all real numbers $\alpha > 1/2$. This condition is necessary as otherwise the Korobov space contains discontinuous functions for which function evaluation is not well defined. This paper is concerned with more precise endpoint estimates of the integration error using QMC rules for Korobov spaces with $\alpha$ arbitrarily close to $1/2$. To obtain such estimates we introduce a $\log$-scale for functions with smoothness close to $1/2$, which we call $\log$-Korobov spaces. We show that lattice rules can be used to obtain an integration error of order $\mathcal{O}(N^{-1/2} (\log N)^{-\mu(1-\lambda)/2})$ for any $1/\mu <\lambda \le 1$, where $\mu>1$ is a power in the $\log$-scale. We also consider tractability of numerical integration for weighted Korobov spaces with product weights $(\gamma_j)_{j \in \mathbb{N}}$. It is known that if $\sum_{j=1}^\infty \gamma_j^\tau < \infty$ for some $1/(2\alpha) < \tau \le 1$ one can obtain error bounds which are independent of the dimension. In this paper we give a more refined estimate for the case where $\tau$ is close to $1/(2 \alpha)$...

New Formulas and Methods for Interpolation, Numerical Differentiation and Numerical Integration

Muthumalai, Ramesh kumar
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 02/09/2008
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We present a new formula for divided difference and few new schemes of divided difference tables in this paper. Through this, we derive new interpolation, numerical differentiation and numerical integration formulas with arbitrary order of accuracy for evanly and unevanly spaced data. First, we study the new interpolation formula which generalizes both Newton's and Lagrange's interpolation formula with the new divided difference table for unevanly spaced points and using this; we derive other interpolation formulas, in terms of differences for evanly spaced data. Second, we study two new different methods of numerical differentiation for both evanly and unevanly spaced points without differentiating the interpolating polynomials or the use of operators. Third, we derive new numerical integration formulas using new differentiation formulas and Taylor formula for both evanly and unevanly spaced data. Basic computer algorithms for few new formulas are given. In Comparison to former polynomial interpolation, numerical differentiation and numerical integration formuals, these new formulas have some featured advantages for approximating functional values, numerical derivatives of higher order and approximate integral values for evanly and unevanly spaced data.; Comment: 34 pages

Hopf algebras of formal diffeomorphisms and numerical integration on manifolds

Lundervold, Alexander; Munthe-Kaas, Hans
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
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B-series originated from the work of John Butcher in the 1960s as a tool to analyze numerical integration of differential equations, in particular Runge-Kutta methods. Connections to renormalization theory in perturbative quantum field theory have been established in recent years. The algebraic structure of classical Runge-Kutta methods is described by the Connes-Kreimer Hopf algebra. Lie-Butcher theory is a generalization of B-series aimed at studying Lie-group integrators for differential equations evolving on manifolds. Lie-group integrators are based on general Lie group actions on a manifold, and classical Runge-Kutta integrators appear in this setting as the special case of R^n acting upon itself by translations. Lie--Butcher theory combines classical B-series on R^n with Lie-series on manifolds. The underlying Hopf algebra combines the Connes-Kreimer Hopf algebra with the shuffle Hopf algebra of free Lie algebras. We give an introduction to Hopf algebraic structures and their relationship to structures appearing in numerical analysis, aimed at a general mathematical audience. In particular we explore the close connection between Lie series, time-dependent Lie series and Lie--Butcher series for diffeomorphisms on manifolds. The role of the Euler and Dynkin idempotents in numerical analysis is discussed. A non-commutative version of a Faa di Bruno bialgebra is introduced...

Numerical integration of H\"older continuous, absolutely convergent Fourier-, Fourier cosine-, and Walsh series

Dick, Josef
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
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We introduce quasi-Monte Carlo rules for the numerical integration of functions $f$ defined on $[0,1]^s$, $s \ge 1$, which satisfy the following properties: the Fourier-, Fourier cosine- or Walsh coefficients of $f$ are absolutely summable and $f$ satisfies a H\"older condition of order $\alpha$, for some $0 < \alpha \le 1$. We show a convergent rate of the integration error of order $\max((s-1) N^{-1/2}, s^{\alpha/2} N^{-\alpha} )$. The construction of the quadrature points is explicit and is based on Weil sums.; Comment: Added literature review and tractability discussion; Minor corrections