Nonlinear Approximation in Geometric, Harmonic, and Anisotropic Settings with Applications

University Of South Carolina At Columbia, Columbia SC

Investigators

Abstract

1714369 Petrushev Many areas ranging from physics, geodesy and geomagnetism to cosmology and to data analysis require efficient representation and approximation of the underlying functions in the natural topology of the targeted application. The capturing of physical phenomena and data structure occurring at various scales requires approximation from locally supported multiscale systems relative to the application domains. Moreover, these approximation methods should be amenable to fast and accurate computation. Thus project aims at increasing our fundamental understanding of nonlinear approximation theory and its applications in three main directions. The first objective is to develop nonlinear approximation theory in various geometric and nonclassical settings from multiscale systems that are well adapted to the targeted applications. The second aim is to study the approximation of harmonic functions from shifts of the Newtonian potential, with targeted applications to geodesy, geomagnetism, and physics. The third goal is to approximate functions that are smooth on domains in space separated by smooth curves or surfaces. Here the idea is to use adaptively anisotropic multiscale dilations of the space, which enable the approximation tool to adjust to curved singularities. A core objective of this project is the development of nonlinear n-term approximation from frames and other systems in various geometric and nonclassical settings, such as on the sphere, ball, box, and simplex with weights, as well as in the context of Lie groups and Riemannian manifolds. All these settings are covered by the general framework of Dirichlet spaces with heat kernel having Gaussian bounds. The key point of the approach is to give us the freedom of dealing with (a) different geometries, (b) compact and noncompact spaces, and (c) spaces with nontrivial weights, and at the same time to allow for the development and frame decomposition of Besov and Triebel-Lizorkin spaces with complete range of indices. The development of the underlying heat kernel theory and nonlinear n-term approximation from localized systems are basic aspects of this theory. Another goal of this project is the development of nonlinear approximation of harmonic functions on the d-dimensional ball from linear combinations of shifts of the Newtonian potential. This includes the complete characterization of the rates of approximation and the development of an effective algorithm that achieves the rates of best approximation. Anisotropic phenomena appear in various contexts in analysis, partial differential equations, and in applications. For instance, functions are frequently very smooth on domains in space separated by smooth curves or manifolds. The project aims at resolving this kind of singularity of functions by using the framework of anisotropic multiscale dilations, 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 an algorithm for rapid construction of best or near best dilation matrices leading to optimal sparsity, (ii) the construction of highly localized anisotropic frames and their use in nonlinear approximation of functions.