On the optimal convergence rate of universal and nonuniversal algorithms for multivariate integration and approximation

Abstract

We study the maximal rate of convergence (mrc) of algorithms for (multivariate) integration and approximation of d-variate functions from reproducing kernel Hilbert spaces H(K d ). Here K d is an arbitrary kernel all of whose partial derivatives up to order r satisfy a Holder-type condition with exponent 2β. Algorithms use n function values and we analyze their rate of convergence as n tends to infinity. We focus on universal algorithms which depend on d, r, and β but not on the specific kernel K d , and nonuniversal algorithms which may depend additionally on K d . For universal algorithms the mrc is (r + β)/d for both integration and approximation, and for nonuniversal algorithms it is 1/2+(r+β)/d for integration and a + (r + β)/d with a ∈ [1/(4 + 4(r + β)/d), 1/2] for approximation. Hence, the mrc for universal algorithms suffers from the curse of dimensionality if d is large relative to r + β, whereas the mrc for nonuniversal algorithms does not since it is always at least 1/2 for integration, and 1/4 for approximation. This is the price we have to pay for using universal algorithms. On the other hand, ifr r+β is large relative to d, then the mrc for universal and nonuniversal algorithms is approximately the same. We also consider the case when we have the additional knowledge that the kernel K d has product structure, K d,r,β = ⊗ d j=1 K rj,βj . Here K rj,βj are some univariate kernels whose all derivatives up to order rj satisfy a Holder-type condition with exponent 2 βj . Then the mrc for universal algorithms is q:= min j=1,2...,d (r j +β j ) for both integration and approximation, and for nonuniversal algorithms it is 1/2 + q for integration and a + q with a ∈ [1/(4 + 4q), 1/2] for approximation. If r j ≥ 1 or β j ≥ β > 0 for all j, then the mrc is at least min(1, β), and the curse of dimensionality is not present. Hence, the product form of reproducing kernels breaks the curse of dimensionality even for universal algorithms.