{"paper":{"title":"Optimal quasi-Monte Carlo rules on order 2 digital nets for the numerical integration of multivariate periodic functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Aicke Hinrichs, Jens Oettershagen, Lev Markhasin, Tino Ullrich","submitted_at":"2015-01-08T11:32:11Z","abstract_excerpt":"We investigate quasi-Monte Carlo rules for the numerical integration of multivariate periodic functions from Besov spaces $S^r_{p,q}B(\\mathbb{T}^d)$ with dominating mixed smoothness $1/p<r<2$. We show that order 2 digital nets achieve the optimal rate of convergence $N^{-r} (\\log N)^{(d-1)(1-1/q)}$. The logarithmic term does not depend on $r$ and hence improves the known bound provided by J. Dick for the special case of Sobolev spaces $H^r_{\\text{mix}}(\\mathbb{T}^d)$. Secondly, the rate of convergence is independent of the integrability $p$ of the Besov space, which allows for sacrificing inte"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.01800","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"}