Optimal randomized multilevel algorithms for infinite-dimensional integration on function spaces with ANOVA-type decomposition
For researchers in numerical integration and high-dimensional approximation, this work provides theoretically optimal algorithms and tight lower bounds, advancing the understanding of randomized methods for infinite-dimensional problems.
The paper presents optimal randomized multilevel algorithms for infinite-dimensional integration on weighted reproducing kernel Hilbert spaces with ANOVA-type decomposition, proving upper and lower error bounds that demonstrate optimality. The algorithms substantially improve upon previous error bounds, with an illustrative example in the unanchored Sobolev space using scrambled polynomial lattice rules.
In this paper, we consider the infinite-dimensional integration problem on weighted reproducing kernel Hilbert spaces with norms induced by an underlying function space decomposition of ANOVA-type. The weights model the relative importance of different groups of variables. We present new randomized multilevel algorithms to tackle this integration problem and prove upper bounds for their randomized error. Furthermore, we provide in this setting the first non-trivial lower error bounds for general randomized algorithms, which, in particular, may be adaptive or non-linear. These lower bounds show that our multilevel algorithms are optimal. Our analysis refines and extends the analysis provided in [F. J. Hickernell, T. Müller-Gronbach, B. Niu, K. Ritter, J. Complexity 26 (2010), 229-254], and our error bounds improve substantially on the error bounds presented there. As an illustrative example, we discuss the unanchored Sobolev space and employ randomized quasi-Monte Carlo multilevel algorithms based on scrambled polynomial lattice rules.