Felipe Wisniewski | Universidade Federal de Santa Catarina - UFSC (Federal University of Santa Catarina) (original) (raw)

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Papers by Felipe Wisniewski

Research paper thumbnail of Funções wavelets de haar e aplicações

Resumo: Neste trabalho, apresenta-se um estudo de bases ortogonais wavelets, com ênfase no sistem... more Resumo: Neste trabalho, apresenta-se um estudo de bases ortogonais wavelets, com ênfase no sistema de Haar e suas aplicações. Inicialmente é feita uma breve revisão de conceitos, partindo em seguida para resultados da análise de multirresolução para wavelets em geral. Feito isso, passa-se a estudar o caso particular das wavelets de Haar unidimensionais: suas principais propriedades e um algoritmo que pode ser usado no cálculo de aproximações de funções com suporte contido no intervalo [0,1]. A teoria estudada para funções de uma dimensão é estendida para funções bidimensionais. Com isto, será vista a implementação de métodos de aproximação de funções utilizando a base de Haar 2D e a aproximação da solução da equação integral de Fredholm, tanto homogênea como não-homogênea. Utilizando o pacote computacional Matlab são feitos experimentos numéricos a fim de ilustrar tais aproximações

Research paper thumbnail of Numerical Approximation of Fredholm Integral Equation (Fie) of 2nd Kind using Galerkin and Collocation Methods

GANIT: Journal of Bangladesh Mathematical Society, 2019

In this research work, Galerkin and collocation methods have been introduced for approximating th... more In this research work, Galerkin and collocation methods have been introduced for approximating the solution of FIE of 2nd kind using LH (product of Laguerre and Hermite) polynomials which are considered as basis functions. Also, a comparison has been done between the solutions of Galerkin and collocation method with the exact solution. Both of these methods show the outcome in terms of the approximate polynomial which is a linear combination of basis functions. Results reveal that performance of collocation method is better than Galerkin method. Moreover, five different polynomials such as Legendre, Laguerre, Hermite, Chebyshev 1st kind and Bernstein are also considered as a basis functions. And it is found that all these approximate solutions converge to same polynomial solution and then a comparison has been made with the exact solution. In addition, five different set of collocation points are also being considered and then the approximate results are compared with the exact anal...

Research paper thumbnail of Weighted Gaussian Quadratures for Orthogonal Wavelet Functions in the Interval

International Journal of Apllied Mathematics, 2015

We study the numerical evaluation of integrals involving scaling functions from the Cohen-Daubech... more We study the numerical evaluation of integrals involving scaling functions from the Cohen-Daubechies-Vial (CDV) family of compactly supported orthogonal wavelets on the interval. The computation of the wavelet coefficients is performed by a weighted Gaussian quadrature, in conjunction with the Chebyshev and modified Chebyshev algorithms. We validate the proposed quadratures with the numerical approximation of a Fredholm integral equation of second kind by the Galerkin method with CDV scaling functions as basis functions.

Research paper thumbnail of A wavelet Galerkin approximation of Fredholm integral eigenvalue problems with bidimensional Haar functions

Proceeding Series of the Brazilian Society of Computational and Applied Mathematics, 2014

Resumo: We consider the numerical approximation of homogeneous Fredholm integral equations of sec... more Resumo: We consider the numerical approximation of homogeneous Fredholm integral equations of second kind. We employ the wavelet Galerkin method with 2D Haar wavelets as shape functions. We thoroughly describe the derivation of the shape functions and present a preliminary numerical experiment illustrating the computation of eigenvalues for a particular covariance kernel.

Research paper thumbnail of A Galerkin method with two-dimensional Haar basis functions for the computation of the Karhunen–Loève expansion

Computational and Applied Mathematics, 2017

We study the numerical approximation of a homogeneous Fredholm integral equation of second kind a... more We study the numerical approximation of a homogeneous Fredholm integral equation of second kind associated with the Karhunen-Loève expansion of Gaussian random fields. We employ the Galerkin method with two-dimensional Haar wavelets as basis functions. The shape functions are constructed from the orthogonal decomposition of tensor product spaces of one-dimensional Haar functions, and a recursive algorithm is employed to compute the matrix of the discrete eigenvalue system without the explicit calculation of integrals. Numerical experiments confirm the convergence rate of the method and assess the approximation error and the sparsity of the eigenvalue system when the wavelet expansion is truncated. We also illustrate the numerical solution of a diffusion problem with random input data with the present method.

Research paper thumbnail of Funções wavelets de haar e aplicações

Resumo: Neste trabalho, apresenta-se um estudo de bases ortogonais wavelets, com ênfase no sistem... more Resumo: Neste trabalho, apresenta-se um estudo de bases ortogonais wavelets, com ênfase no sistema de Haar e suas aplicações. Inicialmente é feita uma breve revisão de conceitos, partindo em seguida para resultados da análise de multirresolução para wavelets em geral. Feito isso, passa-se a estudar o caso particular das wavelets de Haar unidimensionais: suas principais propriedades e um algoritmo que pode ser usado no cálculo de aproximações de funções com suporte contido no intervalo [0,1]. A teoria estudada para funções de uma dimensão é estendida para funções bidimensionais. Com isto, será vista a implementação de métodos de aproximação de funções utilizando a base de Haar 2D e a aproximação da solução da equação integral de Fredholm, tanto homogênea como não-homogênea. Utilizando o pacote computacional Matlab são feitos experimentos numéricos a fim de ilustrar tais aproximações

Research paper thumbnail of Numerical Approximation of Fredholm Integral Equation (Fie) of 2nd Kind using Galerkin and Collocation Methods

GANIT: Journal of Bangladesh Mathematical Society, 2019

In this research work, Galerkin and collocation methods have been introduced for approximating th... more In this research work, Galerkin and collocation methods have been introduced for approximating the solution of FIE of 2nd kind using LH (product of Laguerre and Hermite) polynomials which are considered as basis functions. Also, a comparison has been done between the solutions of Galerkin and collocation method with the exact solution. Both of these methods show the outcome in terms of the approximate polynomial which is a linear combination of basis functions. Results reveal that performance of collocation method is better than Galerkin method. Moreover, five different polynomials such as Legendre, Laguerre, Hermite, Chebyshev 1st kind and Bernstein are also considered as a basis functions. And it is found that all these approximate solutions converge to same polynomial solution and then a comparison has been made with the exact solution. In addition, five different set of collocation points are also being considered and then the approximate results are compared with the exact anal...

Research paper thumbnail of Weighted Gaussian Quadratures for Orthogonal Wavelet Functions in the Interval

International Journal of Apllied Mathematics, 2015

We study the numerical evaluation of integrals involving scaling functions from the Cohen-Daubech... more We study the numerical evaluation of integrals involving scaling functions from the Cohen-Daubechies-Vial (CDV) family of compactly supported orthogonal wavelets on the interval. The computation of the wavelet coefficients is performed by a weighted Gaussian quadrature, in conjunction with the Chebyshev and modified Chebyshev algorithms. We validate the proposed quadratures with the numerical approximation of a Fredholm integral equation of second kind by the Galerkin method with CDV scaling functions as basis functions.

Research paper thumbnail of A wavelet Galerkin approximation of Fredholm integral eigenvalue problems with bidimensional Haar functions

Proceeding Series of the Brazilian Society of Computational and Applied Mathematics, 2014

Resumo: We consider the numerical approximation of homogeneous Fredholm integral equations of sec... more Resumo: We consider the numerical approximation of homogeneous Fredholm integral equations of second kind. We employ the wavelet Galerkin method with 2D Haar wavelets as shape functions. We thoroughly describe the derivation of the shape functions and present a preliminary numerical experiment illustrating the computation of eigenvalues for a particular covariance kernel.

Research paper thumbnail of A Galerkin method with two-dimensional Haar basis functions for the computation of the Karhunen–Loève expansion

Computational and Applied Mathematics, 2017

We study the numerical approximation of a homogeneous Fredholm integral equation of second kind a... more We study the numerical approximation of a homogeneous Fredholm integral equation of second kind associated with the Karhunen-Loève expansion of Gaussian random fields. We employ the Galerkin method with two-dimensional Haar wavelets as basis functions. The shape functions are constructed from the orthogonal decomposition of tensor product spaces of one-dimensional Haar functions, and a recursive algorithm is employed to compute the matrix of the discrete eigenvalue system without the explicit calculation of integrals. Numerical experiments confirm the convergence rate of the method and assess the approximation error and the sparsity of the eigenvalue system when the wavelet expansion is truncated. We also illustrate the numerical solution of a diffusion problem with random input data with the present method.