Kernel interpolation in Sobolev spaces of hybrid regularity

TL;DR

This paper demonstrates that optimized sparse grids enable kernel interpolation in Sobolev spaces of hybrid regularity to avoid the usual curse of dimensionality, leading to more efficient high-dimensional approximation.

Contribution

It introduces a method to eliminate the logarithmic complexity factor in kernel interpolation using optimized sparse grids for Sobolev spaces of hybrid regularity.

Findings

Complexity avoids the curse of dimension in specific Sobolev spaces.

Optimized sparse grids improve interpolation efficiency.

Theoretical complexity estimates are refined for high-dimensional cases.

Abstract

Kernel interpolation in tensor product reproducing kernel Hilbert spaces allows for the use of sparse grids to mitigate the curse of the dimension. Typically, besides the generic constant, only a dimension dependent power of a logarithm term enters here into complexity estimates. We show that optimized sparse grids can avoid this logarithmic factor when the interpolation error is measured with respect to Sobolev spaces of hybrid regularity. Consequently, in such a situation, the complexity of kernel interpolation does not suffer from the curse of dimension.