creator Zhou, Xuan advisor Hickernell, Fred J. This dissertation studies the problem of approximating functions of d variables in a separable Banach space Fd. In particular we are interested in convergence and tractability results in the worst case setting and in the average case setting. The symmetric positive definite kernel in both settings is of a product form Kd(x, t) := d =1 1 − α2 + α2 Kγ (x , t ) for all x, t ∈ Rd. The kernel Kd generalizes the anisotropic Gaussian kernel, whose tractability properties have been established in the literature. For a fixed d, we study rates of convergence, which indicate how quickly approximation errors decay. Since rates of convergence can deteriorate quickly as d increases, it is desirable to have dimension-independent convergence rates, which corresponds to the concept of strong polynomial tractability. We present sufficient conditions on {α }∞ =1 and {γ }∞ =1 under which strong polynomial tractability holds for function approximation problems in Fd. Numerical examples are presented to support the theory and guaranteed automatic algorithms are provided to solve the function approximation problem in a straightforward and efficient way. viii Submitted by Erma Thomas (thomase@iit.edu) on 2016-03-28T21:42:42Z No. of bitstreams: 1 etdadmin_upload_386080.zip: 294775 bytes, checksum: 2d861b2170885d543aea19666826a6db (MD5) Made available in DSpace on 2016-03-28T21:42:42Z (GMT). No. of bitstreams: 1 etdadmin_upload_386080.zip: 294775 bytes, checksum: 2d861b2170885d543aea19666826a6db (MD5) Previous issue date: 2015-12 Ph.D. in Applied Mathematics, December 2015 2015 2015-12 http://hdl.handle.net/10560/3757 en big data complexity function approximation kernel machine learning tractability Dissertation born digital application/pdf In Copyright http://rightsstatements.org/page/InC/1.0/ Restricted Access MATH / Applied Mathematics Illinois Institute of Technology Affiliated department