Cubature, Approximation, and Isotropy in the Hypercube

Algorithms that combat the curse of dimensionality take advantage of nonuniformity properties of the underlying functions, which may be rotational (e.g., grid alignment) or translational (e.g., near-singularities localized at certain points of the domain). The significance of such effects is explored for four different classes of algorithms: low-rank compression, quasi-Monte Carlo integration, sparse grids, and cubature. The exponentially pronounced computational consequences of the anisotropy of the hypercube are described, notably its mismatch with the isotropy of the set of multivariate polynomials of a fixed degree on which some cubature formulas are based, and it is observed that the tensor product Gauss quadrature rule in $[-1,1]^d$ requires up to $(\pi/2)^d$ times fewer points if it is transformed to a nonpolynomial basis.