English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Open

Presents the pdf analysis as premium feature

PDF Analysis

Insights

Presents the summarisation as premium feature

Summarisation

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

We study the numerical integration problem for functions with infinitely manyvariables. The function spaces of integrands we consider are weightedreproducing kernel Hilbert spaces with norms related to the ANOVA decompositionof the integrands. The weights model the relative importance of differentgroups of variables. We investigate randomized quadrature algorithms andmeasure their quality by estimating the randomized worst-case integrationerror.  In particular, we provide lower error bounds for a very general class ofrandomized algorithms that includes non-linear and adaptive algorithms.Furthermore, we propose new randomized changing dimension algorithms andpresent favorable upper error bounds. For product weights andfinite-intersection weights our lower and upper error bounds match and showthat our changing dimension algorithms are optimal in the sense that theyachieve convergence rates arbitrarily close to the best possible convergencerate. As more specific examples, we discuss unanchored Sobolev spaces ofdifferent degrees of smoothness and randomized changing dimension algorithmsthat use as building blocks scrambled polynomial lattice rules.  Our analysis extends the analysis given in [J. Baldeaux, M. Gnewuch. Optimalrandomized multilevel algorithms for infinite-dimensional integration onfunction spaces with ANOVA-type decomposition. arXiv:1209.0882v1 [math.NA],Preprint 2012]. In contrast to the previous article we now investigate adifferent cost model for algorithms. With respect to that cost model,randomized multilevel algorithms cannot, in general, achieve optimalconvergence rates, but, as we prove for important classes of weights, changingdimension algorithms actually can.

Optimal randomized changing dimension algorithms for infinite-dimensional integration on function spaces with ANOVA-type decomposition