Seminar: Kernel methods for high-dimensional data approximation

Thời gian: 10:00 đến 12:00 Ngày 24/02/2026

Địa điểm: Phòng C101, VIASM

Báo cáo viên: Helmut Harbrecht (University of Basel)

Tóm tắt: Kernel interpolation is based on the theoretical framework provided by reproducing kernel Hilbert spaces, RKHS for short. An RKHS is a specific type of Hilbert space of functions where every function’s value at any given point can be reproduced via its inner product with the reproducing kernel. Mathematically, the reproducing kernel is the Riesz representer of the point evaluation. This feature amounts to a simple and efficient way to obtain kernel-based approximations for given data. In this talk, we consider scattered data approximation on product regions of equal and different dimensionality. On each of these regions, we assume quasi-uniform but unstructured data sites and construct optimal sparse grids for scattered data interpolation on the product region. An efficient algorithm to solve the underlying linear system of equations is proposed. The algorithm is based on the sparse grid combination technique, where a sparse direct solver is used for the elementary anisotropic tensor product kernel interpolation problems. The application of the sparse direct solver is facilitated by applying a samplet matrix compression to each univariate kernel matrix, resulting in an essentially sparse representation of the latter. In this way, we obtain a method that is able to deal with large problems up to billions of interpolation points, especially in case of reproducing kernels of nonlocal nature. Numerical results are presented to qualify and quantify the approach.