557 research outputs found
Inner product computations using periodized daubechies wavelets
International audienceInner products of wavelets and their derivatives are presently known as connection coefficients. The numerical calculation of inner products of periodized Daubechies wavelets and their derivatives is reviewed, with the aim at providing potential users of the publicly-available numerical scheme, details of its operation. The numerical scheme for the calculation of connection coefficients is evaluated in the context of approximating differential operators, information which is useful in the solution of partial differential equations using wavelet-Galerkin techniques. Specific details of the periodization of inner products in the solution differential equations are included in this presentation
Superconvergence of Iterated Solutions for Linear and Nonlinear Integral Equations: Wavelet Applications
In this dissertation, we develop the Petrov-Galerkin method and the iterated Petrov-Galerkin method for a class of nonlinear Hammerstein equation. We also investigate the superconvergence phenomenon of the iterated Petrov-Galerkin and degenerate kernel numerical solutions of linear and nonlinear integral equations with a class of wavelet basis. The Fredholm integral equations and the Hammerstein equations are considered in linear and nonlinear cases respectively. Alpert demonstrated that an application of a class of wavelet basis elements in the Galerkin approximation of the Fredholm equation of the second kind leads to a system of linear equations which is sparse. The main concern of this dissertation is to address the issue of how this sparsity manifests itself in the setting of nonlinear equations, particularly for Hammerstein equations. We demonstrate that sparsity appears in the Jacobian matrix when one attempts to solve the system of nonlinear equations by the Newton\u27s iterative method. Overall, the dissertation generalizes the results of Alpert to nonlinear equations setting as well as the results of Chen and Xu, who discussed the Petrov-Galerkin method for Fredholm equation, to nonlinear equations setting
Wavelet Radiosity
Radiosity methods have been shown to be an effective means to solve the global illumination problem in Lambertian diffuse environments. These methods approximate the radiosity integral equation by projecting the unknown radiosity function into a set of basis functions with limited support resulting in a set of n linear equations where n is the number of discrete elements in the scene. Classical radiosity methods required the evaluation of n2 interaction coefficients. Efforts to reduce the number of required coefficients without compromising error bounds have focused on raising the order of the basis functions, meshing, accounting for discontinuities, and on developing hierarchical approaches, which have been shown to reduce the required interactions to O(n). In this paper we show that the hierarchical radiosity formulation is an instance of a more general set of methods based on wavelet theory. This general framework offers a unified view of both higher order element approaches to radiosity and the hierarchical radiosity methods. After a discussion of the relevant theory, we discuss a new set of linear time hierarchical algorithms based on wavelets such as the multiwavelet family and a flatlet basis which we introduce. Initial results of experimentation with these basis sets are demonstrated and discussed.Engineering and Applied Science
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