The goal of this project is to develop new tools and algorithms for effective kernel approximation on Euclidean domains and certain compact manifolds. This work includes two important aspects. The first is to develop schemes to treat highly nonuniform arrangements of data (with approximation rates controlled by a parameter reflecting the local density of the data). The second is to devise nonlinear schemes that approximate using linear combinations of very few kernels. Schemes developed abide by two features of mainstream approximation theory (features that have generally been elusive for kernel-based approximation schemes): they provide approximation that is precise, by providing convergence rates dictated by the smoothness of the approximand, and they are universal, by treating approximands at all levels of smoothness.
The use of kernels to treat scattered, high-dimensional data is, by now, an established methodology in approximation theory. Kernels are especially prized for their ability to approximate in the absence of underlying geometrical structures, like meshes or triangulations. At this point there exist several algorithms employing kernels to treat large datasets that have been sampled almost uniformly. However, the approximation power of such algorithms -- judged in terms of the fidelity of the approximant to the approximand -- is rarely completely understood. Furthermore, the question of how to treat highly nonuniform data (data with large gaps, or with points that coalesce) using kernels is only beginning to be addressed. An important goal of this project is to develop kernel-based approximation methods that approximate from highly unstructured datasets and that approximate high-dimensional datasets with little computational overhead. Another goal is to acquire a precise understanding of the approximation power of such methods. Of particular interest are problems where there is some underlying geometric or algebraic structure to be exploited, as, for example, is the case in problems in geodesy, crystallography, and molecular biology.
Positive definite kernels have been shown to satisfactorily treat problems involving irregularly sampled data, data from complicated non-Euclidean structures, and, more generally, computational problems for which underlying uniform structures (like meshes, grids and triangulations) are not available. Such problems arise often in diverse scientific fields, including: geoscience, learning theory, medical imaging and many more. The research funded by this project fits broadly into three themes, each dealing with kernel computation/approximation: approximation on exotic spaces; construction of local bases suitable for treating large-scale and highly non-uniform problems; acquiring a precise mathematical understanding of the approximation power of certain kernel methods. As a result of his research in these topics, the PI and his coathors have developed a theory for kernel based interpolation on a very general type of mathematical object: compact Riemannian manifolds. These include spheres as well as many other structures which are both relevant to applications and mathematical research. He and coauthors have developed algorithms to construct highly localized bases from kernels (both on manifolds in conventional Euclidean settings) and have applied these localized bases to very large data fitting and numerical integration problems on spheres. These are now being used to solve partial differential equations numerically. The PI has studied the boundary value problems (Dirichlet problems) associated with the class of polyharmonic kernels -- this has yielded insights into the boundary effects phenomenon, where degradation of approximation power occurs when boundaries are present. Over the course of the project, the PI helped to organize two workshops (one in Tennessee and one in Hawaii) and co-organized one international conference in Hawaii involving participants from the US and abroad. The PI successfully mentored a graduate student who has gone on to teach mathematics in the state of Hawaii. In addition to this, he has given numerous research talks at international conferences and has completed about ten research papers, many co-authored with US and international collaborators. All have been published in, or are currently submitted for publication in, high quality, peer reviewed, international journals.
