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/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Cranley and Patterson put forward the following randomization as the basisfor the estimation of the error of a lattice rule for an integral of aone-periodic function over the unit cube in s dimensions. The lattice rule israndomized using independent random shifts in each coordinate direction thatare uniformly distributed in the interval [0,1]. This randomized lattice ruleresults in an unbiased estimator of the multiple integral. However, inpractice, random variables that are independent and uniformly distributed on[0,1] are not available, since this would require an infinite number of randomindependent bits. A more realistic practical implementation of the Cranley andPatterson randomization uses rs independent random bits, in the following way.The lattice rule is randomized using independent random shifts in eachcoordinate direction that are uniformly distributed on {0, 1/2^r, ...,(2^r-1)/2^r}, where r may be large. For a rank-1 lattice rule with 2^mquadrature points and r >= m, we show that this randomized lattice rule leadsto an estimator of the multiple integral that typically has a large bias. Wetherefore propose that these rs independent random bits be used to perform anew randomization that employs an extension, in the number of quadraturepoints, to a lattice rule with 2^(m+sr) quadrature points (leading to embeddedlattice rules).This new randomization is shown to lead to an estimator of themultiple integral that has much smaller bias.

A new method of randomization of lattice rules for multiple integration