Infinite-Dimensional Integration in Weighted Hilbert Spaces: Anchored Decompositions, Optimal Deterministic Algorithms, and Higher Order Convergence

摘要

We study numerical integration of functions depending on an infinite numberof variables. We provide lower error bounds for general deterministic linearalgorithms and provide matching upper error bounds with the help of suitablemultilevel algorithms and changing dimension algorithms. More precisely, the spaces of integrands we consider are weighted reproducingkernel Hilbert spaces with norms induced by an underlying anchored functionspace decomposition. Here the weights model the relative importance ofdifferent groups of variables. The error criterion used is the deterministicworst case error. We study two cost models for function evaluation which dependon the number of active variables of the chosen sample points, and two classesof weights, namely product and order-dependent (POD) weights and the newlyintroduced weights with finite active dimension. We show for these classes ofweights that multilevel algorithms achieve the optimal rate of convergence inthe first cost model while changing dimension algorithms achieve the optimalconvergence rate in the second model. As an illustrative example, we discuss the anchored Sobolev space withsmoothness parameter $\alpha$ and provide new optimal quasi-Monte Carlomultilevel algorithms and quasi-Monte Carlo changing dimension algorithms basedon higher-order polynomial lattice rules.