Lower bounds for the complexity of linear functionals in the randomized setting

Abstract. Hinrichs [3] recently studied multivariate integration defined over reproducing kernel Hilbert spaces in the randomized setting and for the normalized error criterion. In particular, he showed that such problems are strongly polynomially tractable if the reproducing kernels are pointwise nonnegative and integrable. More specifically, let nran (ε, INTd) be the minimal number of randomized function samples that is needed to compute an ε-approximation for the d-variate case of multivariate integration. Hinrichs proved that nran (ε, INTd) ≤ π 2 1 ε 2 for all ε ∈ (0, 1) and d ∈ N. In this paper we prove that the exponent 2 of ε−1 is sharp for tensor product Hilbert spaces whose univariate reproducing kernel is decomposable and univariate integration is not trivial for the two parts of the decomposition. More specifically we have nran (ε, INTd) ≥ 1 8 1 ε 2 for all ε ∈ (0, 1) and d ≥ 2 ln ε−1 − ln 2 ln α−1 , where α ∈ [1/2, 1) depends on the par...

• Type: Journal
• Year: 2011
• Where: JC
• Authors: Erich Novak, Henryk Wozniakowski
• Keywords: Decomposable, JC 2011, Multivariate Integration, Reproducing Kernel