Scientific Computing Seminar
Abstract: In this talk, we discuss the development of wavelet representations for complex surfaces, with the goal of demonstrating their potential for 3D scientific and engineering computing applications. Surface wavelets were originally developed for representing geometric objects in a multiresolution format in computer graphics. However, we further extend the construction of surface wavelets and prove the existence of a large class of Surface multiwavelets in R n with vanishing moments around corners that are well suited for complex geometries. These wavelets share all of the major advantages of conventional wavelets, in that they provide an analysis tool for studying data, functions and operators at different scales. However, unlike conventional wavelets, which are restricted to uniform grids, surface wavelets have the power to perform signal processing operations on complex meshes, such as those en-countered in finite element modeling. This motivates the study of surface wavelets as an efficient representation for the modeling and simulation of physical processes. We show how surface wavelets can be applied to partial differential equations, cast in the integral form. We analyze and implement the wavelet approach for several 3D potential problem using a surface wavelet basis with linear interpolating properties. We show both theoretically and experimentally that an O(h 2n) convergence rate, hn being the mesh size, can be obtained by retaining only O((logN) N) entries in the discrete operator matrix, where N is the number of unknowns and is a small positive number. Moreover our theoretical proof of accuracy vs compression is applicable to a large class of Calder on-Zygmund integral operators. In principle, this convergence analysis may be extended to higher order wavelets with greater vanishing moment. This results in higher convergence and greater compression. In addition, these novel wavelet constructions have been shown to be a generalization of hierarchal finite elements, thus they are also well suited for PDEs in the differential weak form.
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