Structure Analysis of Nonstandard Kernels for Multivariate Reconstructions

Abstract

This dissertation concerns adaptive kernel-based approximation methods. We create a toolbox for adapting kernels to underlying problems, focusing on the interpolation of multivariate scattered data with an emphasis on anisotropies. By developing five nonstandard classes of flexible kernels – transformation, summation, and product kernels, as well as the anisotropic versions of the latter two orthogonal summation, and tensor product kernels – significant limitations of traditional radially symmetric kernels are addressed. These classes, some entirely new and others building on existing structures, provide the flexibility to select and combine kernels tailored to specific problems. Thus, they extend the variety of interpolation methods. The theoretical analysis conducted on each kernel class’s native space not only expands the understanding of native spaces in general but also enlightens underlying (name-giving) structures and their associated benefits. We investigate the interpolation method for each kernel, including impacts on accuracy and stability. Numerical tests confirm the theoretical findings and show which kernel class is suitable for specific problem adaptations: We propose transformation or tensor product kernels for adapting to the point set; transformation kernels for adapting to the domain; and summation, transformation, or orthogonal summation kernels for adapting to the target function.