{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:15:40Z","timestamp":1787328940361,"version":"build-2736575974"},"reference-count":22,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","funder":[{"DOI":"10.13039\/501100001711","name":"Swiss National Science Foundation","doi-asserted-by":"crossref","award":["SNF 159940"],"award-info":[{"award-number":["SNF 159940"]}],"id":[{"id":"10.13039\/501100001711","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100001711","name":"Swiss National Science Foundation","doi-asserted-by":"crossref","award":["SNF 149819"],"award-info":[{"award-number":["SNF 149819"]}],"id":[{"id":"10.13039\/501100001711","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Numer. Anal."],"published-print":{"date-parts":[[2018,1]]},"abstract":"<jats:p>We analyze convergence rates of first-order quasi--Monte Carlo (QMC) integration with randomly shifted lattice rules and for higher-order, interlaced polynomial lattice rules for a class of countably parametric integrands that result from linear functionals of solutions of linear, elliptic diffusion equations with affine-parametric, uncertain coefficient function $a(x,{y}) = \\bar{a}(x) + \\sum_{j\\geq 1} y_j \\psi_j(x)$ in a bounded domain $D\\subset \\mathbb{R}^d$. Extending the result in [F. Y. Kuo, C. Schwab, and I. H. Sloan, SIAM J. Numer. Anal., 50 (2012), pp. 3351--3374], where $\\psi_j$ was assumed to have global support in the domain $D$, we assume in the present paper that ${supp}(\\psi_j)$ is localized in $D$ and that we have control on the overlaps of these supports. Under these conditions we prove dimension-independent convergence rates in [1\/2,1) of randomly shifted lattice rules with product weights and corresponding higher-order convergence rates by higher-order, interlaced polynomial lattice rules with product weights. The product structure of the QMC weights facilitates work bounds for the fast, component-by-component constructions of [D. Nuyens and R. Cools, Math. Comp., 75 (2006), pp. 903--920] which scale linearly with respect to the parameter dimension $s$. The dimension-independent convergence rates are only limited by the degree of digit interlacing used in the construction of the higher-order QMC quadrature rule and, for locally supported coefficient functions, by the summability of the locally supported coefficient sequence in the affine-parametric coefficient.<\/jats:p>","DOI":"10.1137\/16m1082597","type":"journal-article","created":{"date-parts":[[2018,1,5]],"date-time":"2018-01-05T16:23:22Z","timestamp":1515169402000},"page":"111-135","source":"Crossref","is-referenced-by-count":26,"title":["Quasi--Monte Carlo Integration for Affine-Parametric, Elliptic PDEs: Local Supports and Product Weights"],"prefix":"10.1137","volume":"56","author":[{"given":"Robert N.","family":"Gantner","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3402-6420","authenticated-orcid":true,"given":"Lukas","family":"Herrmann","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Christoph","family":"Schwab","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2018,1,5]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1051\/m2an\/2016045"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1051\/m2an\/2012027"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1007\/s10208-010-9072-2"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1142\/S0219530511001728"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1137\/130943984"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1137\/151005518"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1137\/16M1078690"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492913000044"},{"key":"atypb9","volume-title":"Technical report 2017-03, Seminar for Applied Mathematics","author":"Gantner R. 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