Spatially adaptive multiwavelet representations on unstructured grids with applications to multidimensional computational modeling

In this thesis, we develop wavelet surface 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 multiwavelets in R' 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 encountered 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 a model 3D potential problem using a surface wavelet basis with linear interpolating properties. We show both theoretically and experimentally that an O(h') convergence rate, h, being the mesh size, can be obtained by retaining only O((logN)2N) entries in the discrete operator matrix, where N is the number of unknowns. Moreover our theoretical proof of accuracy vs compression is applicable to a large class of Calder6n-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. Thesis Supervisor: Kevin S. Amaratunga Title: Assistant Professor Acknowledgments I have been fortunate for completing my education at MIT. I thank my advisor Kevin Amaratunga for his mentoring and guidance. I am also grateful to Bernie Lesieutre, Jacob White and Jerome Connor for their diligence in following my work and as teachers. I acknowledge the support given to me from the National Science Foundation for funding my research. I am also thankful for the faculty, student and friends during my long stay at school. Foremost, I am blessed with a loving and supportive family. My parents Samuel Castrill6n and Maria Teresa Candis. My brother Samuel Castrill6n Candas and sister Maria Evelyn Talei Castrill6n Candas. Me da dau doka ka vinakata na vanua List of Symbols See [22] for many of these definitions. 1. Multiindex notation for Dau(x): Let o = (a1,. .. , a2), where ai E Z+ and let