Kernel interpolation in Sobolev spaces of hybrid regularity

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.