{"paper":{"title":"On the Randomization of Frolov's Algorithm for Multivariate Integration","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"David Krieg","submitted_at":"2016-03-15T11:19:52Z","abstract_excerpt":"We are concerned with the numerical integration of functions from the Sobolev space $H^{r,\\text{mix}}([0,1]^d)$ of dominating mixed smoothness $r\\in\\mathbb{N}$ over the $d$-dimensional unit cube.\n  In 1976, K. K. Frolov introduced a deterministic quadrature rule whose worst case error has the order $n^{-r} \\, (\\log n)^{(d-1)/2}$ with respect to the number $n$ of function evaluations. This is known to be optimal. 39 years later, Erich Novak and me introduced a randomized version of this algorithm using $d$ random dilations. We showed that its error is bounded above by a constant multiple of $n^"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.04637","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}