Multilevel Sparse Kernel-Based Interpolation

Abstract

A multilevel kernel-based interpolation method, suitable for moderately high-dimensional function interpolation problems, is proposed. The method, termed multilevel sparse kernelbased 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 R d × 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.

Multilevel quasi-interpolation on a sparse grid with the Gaussian

“…This means that the approximation process is much faster. We compare our results with the analagous interpolation method, MuSIK, described in [9] and see that they give more accurate results in the same amount of time.…”

“…It can be seen that these methods are similar to the hyperbolic cross notion of Babenko [1]. In 2012, Georgoulis, Levesley and Subhan [9] introduced a new sparse grid kernel-based interpolation technique which circumvents both computational complication and conditioning problems using. The same basic algorithm was implemented by Schreiber [18], but was not effective because Gaussians with a fixed width were used.…”

“…Schemes which retain the same shape of basis function while the points get more dense are termed non-stationary, and non-stationary Gaussian interpolation is known to be ill-conditioned. The innovation in [9] was to use stationary interpolation with Gaussians (see the scheme in Section 2), mimicking the sparse-grid scheme with b-splines (this scheme is called SIK). However, SIK with Gaussians does not converge, in contract to the b-spline case.…”

“…In order to gain convergence they used an iterative correction scheme, interpolating the residual from the previous level at the next level. This scheme is called MuSIK, and is observed in [9] to work very well. In this paper we use the same scheme with quasi-interpolation Q-SIK and Q-MuSIK respectively instead of SIK and MuSIK, and observe similar good results in high dimensions.…”

An adaptive interpolation scheme for molecular potential energy surfaces

“…Sparse grid methods 32 combine results computed on a particular sequence of structured grids in order to get the final approximation. This type of technique has also been used together with RBFs 33 .…”