{"paper":{"title":"Lattice rules with random $n$ achieve nearly the optimal $\\mathcal{O}(n^{-\\alpha-1/2})$ error independently of the dimension","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Dirk Nuyens, Frances Y. Kuo, Mario Ullrich, Peter Kritzer","submitted_at":"2017-06-14T14:03:36Z","abstract_excerpt":"We analyze a new random algorithm for numerical integration of $d$-variate functions over $[0,1]^d$ from a weighted Sobolev space with dominating mixed smoothness $\\alpha\\ge 0$ and product weights $1\\ge\\gamma_1\\ge\\gamma_2\\ge\\cdots>0$, where the functions are continuous and periodic when $\\alpha>1/2$. The algorithm is based on rank-$1$ lattice rules with a random number of points~$n$. For the case $\\alpha>1/2$, we prove that the algorithm achieves almost the optimal order of convergence of $\\mathcal{O}(n^{-\\alpha-1/2})$, where the implied constant is independent of the dimension~$d$ if the weig"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.04502","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1706.04502/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}