Representation and approximation of functions in nonclassical and anisotropic settings with applications

University South Carolina Research Foundation, Columbia SC

Investigators

Abstract

This project centers on the development of multiscale representation systems in various nonclassical and/or anisotropic topological settings such as on Lie groups and Riemannian manifolds and in the framework of anisotropic dilations of the space. Furthermore, it will develop sophisticated new methods for approximation and numerical computation using these systems. The project lies at the interface between computational harmonic analysis, spectral decompositions, orthogonal polynomials, nonlinear approximation and numerical analysis. It is organized into two main directions of investigation with several specific goals. The first research objective of the project is to develop frames with elements of nearly exponential space localization in the general setting of strictly local regular Dirichlet spaces with doubling measure and local scale-invariant Poincare inequality which lead to a Markovian heat kernel with small time Gaussian bounds and Holder continuity. The key point of the proposed approach is to be able to deal with (a) different geometries, (b) compact and noncompact spaces, and (c) spaces with nontrivial weights, and at the same time to allow for the frame decomposition of Besov and other spaces with complete range of indices. This will facilitate the development of well localized frames in the context of Lie groups or homogeneous spaces with polynomial volume growth, Riemannian manifolds with Ricci curvature bounded from below and other new settings. The development of frames on the simplex and on graphs is another aim of this project. The second core objective of this project is the development of adaptive representations in the framework of anisotropic dilations of the space. Anisotropic phenomena appear in various contexts in analysis, PDEs and in applications. For instance, functions are frequently very smooth on subdomains of Rd separated by smooth curves or manifolds. This project aims at resolving this kind of singularities of functions (and more general singular behaviors) by utilizing the framework of anisotropic multiscale dilations of d-dimensional space or its subdomain, which may change rapidly from point to point at any level and in depth. The main strands of this approach are (i) the development of algorithms for rapid construction of best or near best dilation matrices leading to optimal sparsity, (ii) the construction of highly localized anisotropic frames and their utilization to representation and approximation of functions. Many scientific areas require efficient representation of the underlying functions in the natural topology of the targeted application. The capturing of physical phenomena occurring at various scales requires locally supported multiscale systems relative to the application domains. Moreover, these systems should be amenable to fast and accurate computation. Such systems (called needlets) have been recently developed by the investigator and his collaborators for the sphere and the ball. The needlets are the outcome of a complete rethinking of data representations in the context of classical orthogonal and spectral representations and break new conceptual and practical ground, going far beyond traditional multiscale ideas like wavelets. Spherical needlets have already had a significant impact in cosmology/astrophysics for the statistical study of the cosmic microwave background radiation data. This award will support the development of image and data processing techniques, which will lead to much novel representation systems in new mathematical settings, allowing the treatment of new data structures. It will also enhance our fundamental understanding of complicated processes through the development of innovative adaptive methods for efficient (sparse) representation and approximation of geometrical objects that have jumps or other sharp transitions along curves or surfaces. It has the potential to impact many areas ranging from image processing and edge detection to geopotential, oceanographic and atmospheric modeling, and to physics and cosmology.