{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T03:17:44Z","timestamp":1776827864529,"version":"3.51.2"},"reference-count":10,"publisher":"American Mathematical Society (AMS)","issue":"240","license":[{"start":{"date-parts":[[2003,3,20]],"date-time":"2003-03-20T00:00:00Z","timestamp":1048118400000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n We develop and justify an algorithm for the construction of quasi\u2013Monte Carlo (QMC) rules for integration in weighted Sobolev spaces; the rules so constructed are shifted rank-1 lattice rules. The parameters characterising the shifted lattice rule are found \u201ccomponent-by-component\u201d: the (\n \n \n \n \n d<\/mml:mi>\n +<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n d+1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n )-th component of the generator vector and the shift are obtained by successive\n \n \n \n 1<\/mml:mn>\n 1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -dimensional searches, with the previous\n \n \n \n d<\/mml:mi>\n d<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n components kept unchanged. The rules constructed in this way are shown to achieve a strong tractability error bound in weighted Sobolev spaces. A search for\n \n \n \n n<\/mml:mi>\n n<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -point rules with\n \n \n \n n<\/mml:mi>\n n<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n prime and all dimensions 1 to\n \n \n \n d<\/mml:mi>\n d<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n requires a total cost of\n \n \n \n \n O<\/mml:mi>\n (<\/mml:mo>\n \n n<\/mml:mi>\n 3<\/mml:mn>\n <\/mml:msup>\n \n d<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n O(n^3d^2)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n operations. This may be reduced to\n \n \n \n \n O<\/mml:mi>\n (<\/mml:mo>\n \n n<\/mml:mi>\n 3<\/mml:mn>\n <\/mml:msup>\n d<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n O(n^3d)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n operations at the expense of\n \n \n \n \n O<\/mml:mi>\n (<\/mml:mo>\n \n n<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n O(n^2)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n storage. Numerical values of parameters and worst-case errors are given for dimensions up to 40 and\n \n \n \n n<\/mml:mi>\n n<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n up to a few thousand. The worst-case errors for these rules are found to be much smaller than the theoretical bounds.\n <\/p>","DOI":"10.1090\/s0025-5718-02-01420-5","type":"journal-article","created":{"date-parts":[[2002,9,20]],"date-time":"2002-09-20T14:12:48Z","timestamp":1032531168000},"page":"1609-1640","source":"Crossref","is-referenced-by-count":65,"title":["On the step-by-step construction of quasi\u2013Monte Carlo integration rules that achieve strong tractability error bounds in weighted Sobolev spaces"],"prefix":"10.1090","volume":"71","author":[{"given":"I.","family":"Sloan","sequence":"first","affiliation":[]},{"given":"F.","family":"Kuo","sequence":"additional","affiliation":[]},{"given":"S.","family":"Joe","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2002,3,20]]},"reference":[{"issue":"221","key":"1","doi-asserted-by":"publisher","first-page":"299","DOI":"10.1090\/S0025-5718-98-00894-1","article-title":"A generalized discrepancy and quadrature error bound","volume":"67","author":"Hickernell, Fred J.","year":"1998","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"2","isbn-type":"print","doi-asserted-by":"publisher","first-page":"109","DOI":"10.1007\/978-1-4612-1702-2_3","article-title":"Lattice rules: how well do they measure up?","author":"Hickernell, Fred J.","year":"1998","ISBN":"https:\/\/id.crossref.org\/isbn\/0387985549"},{"key":"3","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1142\/4061","volume-title":"Recent advances in numerical methods and applications. II","year":"1999","ISBN":"https:\/\/id.crossref.org\/isbn\/9810238274"},{"key":"4","doi-asserted-by":"crossref","unstructured":"Novak, E. and Wo\u017aniakowski, H. (2001), Intractability results for integration and discrepancy, J. Complexity, 17, 388\u2013441.","DOI":"10.1006\/jcom.2000.0577"},{"key":"5","series-title":"CBMS-NSF Regional Conference Series in Applied Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611970081","volume-title":"Random number generation and quasi-Monte Carlo methods","volume":"63","author":"Niederreiter, Harald","year":"1992","ISBN":"https:\/\/id.crossref.org\/isbn\/0898712955"},{"key":"6","series-title":"Oxford Science Publications","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198534723.001.0001","volume-title":"Lattice methods for multiple integration","author":"Sloan, I. H.","year":"1994","ISBN":"https:\/\/id.crossref.org\/isbn\/0198534728"},{"key":"7","doi-asserted-by":"crossref","unstructured":"Sloan, I. H. and Reztsov, A. V. (2002). Component-by-component construction of good lattice rules, Math. Comp., 17, 263\u2013273.","DOI":"10.1090\/S0025-5718-01-01342-4"},{"issue":"1","key":"8","doi-asserted-by":"publisher","first-page":"65","DOI":"10.1016\/S0096-3003(97)10066-2","article-title":"Convergence on a deformed Newton method","volume":"94","author":"Han, Danfu","year":"1998","journal-title":"Appl. Math. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0096-3003","issn-type":"print"},{"key":"9","doi-asserted-by":"crossref","unstructured":"Sloan, I. H. and Wo\u017aniakowski, H. (2001). Tractability of multivariate integration for weighted Korobov classes, J. Complexity, 17, 697\u2013721.","DOI":"10.1006\/jcom.2001.0599"},{"key":"10","doi-asserted-by":"crossref","unstructured":"Wo\u017aniakowski, H. (1999), Efficiency of quasi-Monte Carlo algorithms for high-dimensional integrals, Monte Carlo and Quasi-Monte Carlo Methods 1998 (H. Niederreiter and J. 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