Multilevel Quasi-Interpolation With a Gaussian Kernel using Chebyshev Points - Numerical and theoretical studies using the radial basis function approach

Radial basis functions (RBFs) have been recently begun to see widespread use in fitting scattered data in Rd and, for this purpose, have been found to be highly effective. However, there are limitations to RBFs in the interpolation of two dimensions, three dimensions, or higher, with large datasets in Rd. In this thesis, we propose a novel multilevel quasi-interpolation method with Gaussian kernels and Chebyshev points associated with a sparse grid (Chapter 2 and Chapter 3).

The primary aim is to offer a method that uses a sparser grid with fast convergence, as shown in Section 2.3 and Section 3.3. In Chapter 1, the thesis lays down the mathematical groundwork, introducing key concepts like RBFs, quasiinterpolation, and anisotropic tensor product basis functions. It also discusses the challenges of working with multidimensional data in Section 1.8. Chapter 2 specifically delves into quasi-interpolation on Chebyshev points, introducing boundary corrections to improve the method’s performance (Section 2.2). The use of Gaussian kernels in this context, analysed to provide both smoothness and better locality, contribute to excellent approximative properties. This is supported by numerical experiments (Section 2.3).

The core contribution lies in Chapter 3, where a multilevel sparse-grid quasiinterpolation technique is developed. This approach, inspired by the work in [1, 2], is effective for large datasets in Rd with d = 2, 3 and higher. The algorithm described in Section 3.2 employs direction-wise multilevel decomposition and distinct scaling in each direction, resulting in quicker convergence. Numerical validation confirms that quasi-sparse interpolation with Chebyshev points has advantages in terms of complexity, runtime, and convergence when compared to classical methods (Sections 3.3). The thesis concludes with Chapter 4 by summarising the contributions made and proposing possible future work.