Kernel-Based Approximation of Functions: Three Problems and Their Efficient Solutions

Scattered data approximation in multiple dimensions based on the positive definite kernels, especially Radial Basis Functions (RBFs), is frequently used in the modern approximation theory. RBF methods are known for their strong performance in function approximation and interpolation problems. This research addresses key challenges in multivariate approximation by utilizing the properties of RBFs, offering new insights and methodologies that improve the accuracy and conditioning of approximation processes. Specifically, it is known that the accuracy of the approximation depends on the selected set of bases with which the underlying function is reconstructed. Therefore, in this thesis, we follow the idea of finding a new set of bases that can improve the accuracy of the approximation, either through improving the conditioning of the interpolation matrix or by incorporating the features of the underlying function into the selected bases, such as discontinuities. To be more detailed, to overcome the ill-conditioning of the interpolation matrix resulting from the conditionally positive definite kernels, we present various sets of bases constructed by different types of decomposition applied in the interpolation matrix. Another class of bases is constructed using the Mercer expansion of the reproducing kernel corresponding to any conditionally positive definite kernels. Secondly, we design a moving least-squares approach for scattered data approximation that incorporates the discontinuities of the underlying functions into the weight functions. Thus, the newly constructed basis measures the influence of the data sites on the approximant, not only with regard to their distance from the evaluation point but also with respect to the discontinuities of the underlying function. Eventually, we use Direct RBF Partition of Unity to set up the differentiation matrix to solve the time-dependent PDEs intrinsic to the surface. Taking this approach, the bases are modified depending on the patches that include the evaluation points. Combined with the closest point representation of the surface, the preliminary results show an improvement in terms of accuracy. The findings of this research contribute to the broader understanding of kernel methods in approximation theory, offering valuable perspectives for future research and development. By advancing the methodologies and applications of RBFs, this thesis paves the way for more accurate and efficient approximation techniques that can be applied across various scientific and engineering disciplines.