In this thesis, we use Green functions (kernels) to set up reproducing kernels such that their related reproducing kernel Hilbert spaces ... Show moreIn this thesis, we use Green functions (kernels) to set up reproducing kernels such that their related reproducing kernel Hilbert spaces (native spaces) are isometrically embedded into or even are isometrically equivalent to generalized Sobolev spaces. These generalized Sobolev spaces are set up with the help of a vector distributional operator P consisting of finitely or countably many elements, and possibly a vector boundary operator B. The above Green functions can be computed by the distributional operator L := P TP with possible boundary conditions given by B. In order to support this claim we ensure that the distributional adjoint operator P of P is well-defined in the distributional sense. The types of distributional operators we consider include not only di erential operators but also more general distributional operators such as pseudo-di erential operators. The generalized Sobolev spaces can cover even classical Sobolev spaces and Beppo-Levi spaces. The well-known examples covered by our theories include thin-plate splines, Mat´ern functions, Gaussian kernels, min kernels and others. As an application for high-dimensional approximations, we can use the Green functions to construct a multivariate minimum-norm interpolant s f;X to interpolate the data values sampled from an unknown generalized Sobolev function f at data sites X Rd. Moreover, we also use Green functions to set up reproducing kernel Banach spaces, which can be equivalent to classical Sobolev spaces. This is a new tool for support vector machines. Finally, we show that stochastic Gaussian fields can be well-defined on the generalized Sobolev spaces. According to these Gaussian-field constructions, we find that kernel-based collocation methods can be used to approximate the numerical solutions of high-dimensional stochastic partial differential equations. Ph.D. in Applied Mathematics Show less
