Anisotropic Kernel-based Multilevel Interpolation of High-Dimensional Functions on Sparse Grids
TL;DR
Using a recent nodal representation of the low-dimensional multilevel operators, an explicit, computable representation of the high-dimensional interpolant is derived, and the full range of anisotropy weights that attain the optimal rate against the number of degrees of freedom are characterised.
Abstract
The tensor product multilevel method (TPML) approximates high-dimensional functions from scattered data on general bounded domains by combining Smolyak's sparse grid construction with the kernel-based multilevel interpolation, coupling the kernel scale to the fill distance at each level so that condition numbers stay uniformly bounded. Its original formulation depended only implicitly on the target function, which precluded numerical evaluation. Using a recent nodal representation of the low-dimensional multilevel operators, we derive an explicit, computable representation of the high-dimensional interpolant. On this basis we prove error estimates in mixed-regularity Sobolev and continuity norms, extending the existing $ L_2 $-theory, and we characterise the full range of anisotropy weights that attain the optimal rate against the number of degrees of freedom. Three explicit weight strategies follow, equilibrating the accuracy, the degrees of freedom, or the cost-benefit ratio. Numerical experiments in up to ten dimensions illustrate the theory.