{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T07:13:46Z","timestamp":1776842026482,"version":"3.51.2"},"reference-count":22,"publisher":"American Mathematical Society (AMS)","issue":"240","license":[{"start":{"date-parts":[[2002,8,2]],"date-time":"2002-08-02T00:00:00Z","timestamp":1028246400000},"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 Dimensionally unbounded problems are frequently encountered in practice, such as in simulations of stochastic processes, in particle and light transport problems and in the problems of mathematical finance. This paper considers quasi-Monte Carlo integration algorithms for weighted classes of functions of infinitely many variables, in which the dependence of functions on successive variables is increasingly limited. The dependence is modeled by a sequence of weights. The integrands belong to rather general reproducing kernel Hilbert spaces that can be decomposed as the direct sum of a series of their subspaces, each subspace containing functions of only a finite number of variables. The theory of reproducing kernels is used to derive a quadrature error bound, which is the product of two terms: the generalized discrepancy and the generalized variation. Tractability means that the minimal number of function evaluations needed to reduce the initial integration error by a factor\n \n \n \n \n \u03b5\n \n <\/mml:mi>\n \\varepsilon<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is bounded by\n \n \n \n \n C<\/mml:mi>\n \n \n \u03b5\n \n <\/mml:mi>\n \n \n \u2212\n \n <\/mml:mo>\n p<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n C \\varepsilon ^{-p}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n for some exponent\n \n \n \n p<\/mml:mi>\n p<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and some positive constant\n \n \n \n C<\/mml:mi>\n C<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . The\n \n \n \n \n \u03b5\n \n <\/mml:mi>\n \\varepsilon<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -exponent of tractability is defined as the smallest power of\n \n \n \n \n \n \u03b5\n \n <\/mml:mi>\n \n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n \\varepsilon ^{-1}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n in these bounds. It is shown by using Monte Carlo quadrature that the\n \n \n \n \n \u03b5\n \n <\/mml:mi>\n \\varepsilon<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -exponent is no greater than 2 for these weighted classes of integrands. Under a somewhat stronger assumption on the weights and for a popular choice of the reproducing kernel it is shown constructively using the Halton sequence that the\n \n \n \n \n \u03b5\n \n <\/mml:mi>\n \\varepsilon<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -exponent of tractability is 1, which implies that infinite dimensional integration is no harder than one-dimensional integration.\n <\/p>","DOI":"10.1090\/s0025-5718-01-01377-1","type":"journal-article","created":{"date-parts":[[2002,9,20]],"date-time":"2002-09-20T14:12:48Z","timestamp":1032531168000},"page":"1641-1661","source":"Crossref","is-referenced-by-count":51,"title":["The error bounds and tractability of quasi-Monte Carlo algorithms in infinite dimension"],"prefix":"10.1090","volume":"71","author":[{"given":"Fred","family":"Hickernell","sequence":"first","affiliation":[]},{"given":"Xiaoqun","family":"Wang","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2001,8,2]]},"reference":[{"key":"1","first-page":"285","article-title":"Sur les inverses des \u00e9l\u00e9ments d\u00e9rivables dans un anneau abstrait","volume":"209","author":"Hebroni, P.","year":"1939","journal-title":"C. R. Acad. Sci. Paris","ISSN":"https:\/\/id.crossref.org\/issn\/0001-4036","issn-type":"print"},{"key":"2","doi-asserted-by":"crossref","unstructured":"[CMO97] R. E. Caflisch, W. Morokoff, and A. Owen, Valuation of mortgage backed securities using Brownian bridges to reduce effective dimension, J. Comput. Finance 1 (1997), 27\u201346.","DOI":"10.21314\/JCF.1997.005"},{"key":"3","series-title":"Computer Science and Applied Mathematics","isbn-type":"print","volume-title":"Methods of numerical integration","author":"Davis, Philip J.","year":"1984","ISBN":"https:\/\/id.crossref.org\/isbn\/0122063600","edition":"2"},{"key":"4","unstructured":"[Duf96] D. Duffie, Dynamic asset pricing theory, Princeton University Press, Princeton, New Jersey, 1996."},{"issue":"4","key":"5","doi-asserted-by":"publisher","first-page":"337","DOI":"10.4064\/aa-41-4-337-351","article-title":"Discr\u00e9pance de suites associ\u00e9es \u00e0 un syst\u00e8me de num\u00e9ration (en dimension \ud835\udc60)","volume":"41","author":"Faure, Henri","year":"1982","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"key":"6","series-title":"Monographs on Statistics and Applied Probability","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4899-3095-8","volume-title":"Number-theoretic methods in statistics","volume":"51","author":"Fang, K.-T.","year":"1994","ISBN":"https:\/\/id.crossref.org\/isbn\/0412465205"},{"key":"7","doi-asserted-by":"publisher","first-page":"84","DOI":"10.1007\/BF01386213","article-title":"On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals","volume":"2","author":"Halton, J. H.","year":"1960","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"221","key":"8","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"},{"issue":"1","key":"9","doi-asserted-by":"publisher","first-page":"25","DOI":"10.1023\/A:1018948631251","article-title":"Integration and approximation in arbitrary dimensions","volume":"12","author":"Hickernell, F. J.","year":"2000","journal-title":"Adv. Comput. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1019-7168","issn-type":"print"},{"issue":"6","key":"10","doi-asserted-by":"publisher","first-page":"1251","DOI":"10.1137\/0915077","article-title":"Quasi-random sequences and their discrepancies","volume":"15","author":"Morokoff, William J.","year":"1994","journal-title":"SIAM J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/1064-8275","issn-type":"print"},{"key":"11","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":"12","doi-asserted-by":"crossref","unstructured":"[NW00] E. Novak and H. Wo\u017aniakowski, When are integration and discrepancy tractable?, Foundations of Computational Mathematics (R. De Vore, A. Iserles, E. S\u00fcli, eds.), Chap. 8, Cambridge University Press, Cambridge, 2001.","DOI":"10.1017\/CBO9781107360198.009"},{"key":"13","isbn-type":"print","doi-asserted-by":"publisher","first-page":"269","DOI":"10.1017\/CBO9780511525988.022","article-title":"Quasirandom points and global function fields","author":"Niederreiter, Harald","year":"1996","ISBN":"https:\/\/id.crossref.org\/isbn\/052156736X"},{"key":"14","series-title":"Pitman Research Notes in Mathematics Series","isbn-type":"print","volume-title":"Theory of reproducing kernels and its applications","volume":"189","author":"Saitoh, Saburou","year":"1988","ISBN":"https:\/\/id.crossref.org\/isbn\/0582035643"},{"key":"15","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":"16","volume-title":"{\\cyr Mnogomernye kvadraturnye formuly i funktsii Khaara}","author":"Tsobol\u2032, I. M.","year":"1969"},{"issue":"2-5","key":"17","doi-asserted-by":"publisher","first-page":"103","DOI":"10.1016\/S0378-4754(98)00096-2","article-title":"On quasi-Monte Carlo integrations","volume":"47","author":"Sobol, I. M.","year":"1998","journal-title":"Math. Comput. Simulation","ISSN":"https:\/\/id.crossref.org\/issn\/0378-4754","issn-type":"print"},{"issue":"1","key":"18","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1006\/jcom.1997.0463","article-title":"When are quasi-Monte Carlo algorithms efficient for high-dimensional integrals?","volume":"14","author":"Sloan, Ian H.","year":"1998","journal-title":"J. Complexity","ISSN":"https:\/\/id.crossref.org\/issn\/0885-064X","issn-type":"print"},{"key":"19","series-title":"Lezioni Lincee. [Lincei Lectures]","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1080\/16073606.1997.9631861","volume-title":"Complexity and information","author":"Traub, J. F.","year":"1998","ISBN":"https:\/\/id.crossref.org\/isbn\/0521485061"},{"key":"20","series-title":"CBMS-NSF Regional Conference Series in Applied Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611970128","volume-title":"Spline models for observational data","volume":"59","author":"Wahba, Grace","year":"1990","ISBN":"https:\/\/id.crossref.org\/isbn\/0898712440"},{"issue":"1","key":"21","doi-asserted-by":"publisher","first-page":"185","DOI":"10.1090\/S0273-0979-1991-15985-9","article-title":"Average case complexity of multivariate integration","volume":"24","author":"Wo\u017aniakowski, H.","year":"1991","journal-title":"Bull. Amer. Math. Soc. (N.S.)","ISSN":"https:\/\/id.crossref.org\/issn\/0273-0979","issn-type":"print"},{"issue":"4","key":"22","doi-asserted-by":"publisher","first-page":"2071","DOI":"10.1063\/1.531493","article-title":"On tractability of path integration","volume":"37","author":"Wasilkowski, Grzegorz W.","year":"1996","journal-title":"J. Math. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0022-2488","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2002-71-240\/S0025-5718-01-01377-1\/S0025-5718-01-01377-1.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2002-71-240\/S0025-5718-01-01377-1\/S0025-5718-01-01377-1.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T23:04:15Z","timestamp":1776726255000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2002-71-240\/S0025-5718-01-01377-1\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2001,8,2]]},"references-count":22,"journal-issue":{"issue":"240","published-print":{"date-parts":[[2002,10]]}},"alternative-id":["S0025-5718-01-01377-1"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-01-01377-1","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2001,8,2]]}}}