2012/04/18 by Emmanuil H. Georgoulis, Georgoulis, Emmanuil H., Jeremy Levesley +3 · 1 citation
Engineering · #Advanced Numerical Analysis Techniques #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in engineering
paper · pdf · doi:10.48550/arxiv.1204.4153
openalex publication_date 2012/04/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
A multilevel kernel-based interpolation method, suitable for moderately high-dimensional function interpolation problems, is proposed. The method, termed multilevel sparse kernel-based interpolation (MLSKI, for short), uses both level-wise and direction-wise multilevel decomposition of structured (or mildly unstructured) interpolation data sites in conjunction with the application of kernel-based interpolants with different scaling in each direction. The multilevel interpolation algorithm is based on a hierarchical decomposition of the data sites, whereby at each level the detail is added to the interpolant by interpolating the resulting residual of the previous level. On each level, anisotropic radial basis functions are used for solving a number of small interpolation problems, which are subsequently linearly combined to produce the interpolant. MLSKI can be viewed as an extension of d-boolean interpolation (which is closely related to ideas in sparse grid and hyperbolic crosses literature) to kernel-based functions, within the hierarchical multilevel framework to achieve accelerated convergence. Numerical experiments suggest that the new algorithm is numerically stable and efficient for the reconstruction of large data in ℝd× ℝ, for d = 2, 3, 4, with tens or even hundreds of thousands data points. Also, MLSKI appears to be generally superior over classical radial basis function methods in terms of complexity, run time and convergence at least for large data sets.