{"@context":{"@vocab":"https://cir.nii.ac.jp/schema/1.0/","rdfs":"http://www.w3.org/2000/01/rdf-schema#","dc":"http://purl.org/dc/elements/1.1/","dcterms":"http://purl.org/dc/terms/","foaf":"http://xmlns.com/foaf/0.1/","prism":"http://prismstandard.org/namespaces/basic/2.0/","cinii":"http://ci.nii.ac.jp/ns/1.0/","datacite":"https://schema.datacite.org/meta/kernel-4/","ndl":"http://ndl.go.jp/dcndl/terms/","jpcoar":"https://github.com/JPCOAR/schema/blob/master/2.0/"},"@id":"https://cir.nii.ac.jp/crid/1361137045631290624.json","@type":"Article","productIdentifier":[{"identifier":{"@type":"DOI","@value":"10.1017/s0962492913000044"}},{"identifier":{"@type":"URI","@value":"https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0962492913000044"}}],"dc:title":[{"@value":"High-dimensional integration: The quasi-Monte Carlo way"}],"description":[{"type":"abstract","notation":[{"@value":"<jats:p>This paper is a contemporary review of QMC (‘quasi-Monte Carlo’) methods, that is, equal-weight rules for the approximate evaluation of high-dimensional integrals over the unit cube [0,1]<jats:italic><jats:sup>s</jats:sup></jats:italic>, where<jats:italic>s</jats:italic>may be large, or even infinite. After a general introduction, the paper surveys recent developments in lattice methods, digital nets, and related themes. Among those recent developments are methods of construction of both lattices and digital nets, to yield QMC rules that have a prescribed rate of convergence for sufficiently smooth functions, and ideally also guaranteed slow growth (or no growth) of the worst-case error as s increases. A crucial role is played by parameters called ‘weights’, since a careful use of the weight parameters is needed to ensure that the worst-case errors in an appropriately weighted function space are bounded, or grow only slowly, as the dimension s increases. Important tools for the analysis are weighted function spaces, reproducing kernel Hilbert spaces, and discrepancy, all of which are discussed with an appropriate level of detail.</jats:p>"}]}],"creator":[{"@id":"https://cir.nii.ac.jp/crid/1380017279850682371","@type":"Researcher","foaf:name":[{"@value":"Josef Dick"}]},{"@id":"https://cir.nii.ac.jp/crid/1381137045631290625","@type":"Researcher","foaf:name":[{"@value":"Frances Y. Kuo"}]},{"@id":"https://cir.nii.ac.jp/crid/1381137045631290624","@type":"Researcher","foaf:name":[{"@value":"Ian H. Sloan"}]}],"publication":{"publicationIdentifier":[{"@type":"PISSN","@value":"09624929"},{"@type":"EISSN","@value":"14740508"},{"@type":"PISSN","@value":"https://id.crossref.org/issn/09624929"}],"prism:publicationName":[{"@value":"Acta Numerica"}],"dc:publisher":[{"@value":"Cambridge University Press (CUP)"}],"prism:publicationDate":"2013-04-02","prism:volume":"22","prism:startingPage":"133","prism:endingPage":"288"},"reviewed":"false","dc:rights":["https://www.cambridge.org/core/terms"],"url":[{"@id":"https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0962492913000044"}],"createdAt":"2013-04-02","modifiedAt":"2024-05-08","relatedProduct":[{"@id":"https://cir.nii.ac.jp/crid/1360017279850682496","@type":"Article","resourceType":"学術雑誌論文(journal article)","relationType":["isReferencedBy"],"jpcoar:relatedTitle":[{"@value":"Polynomial tractability for integration in an unweighted function space with absolutely convergent Fourier series"}]},{"@id":"https://cir.nii.ac.jp/crid/1360021389826722048","@type":"Article","resourceType":"学術雑誌論文(journal article)","relationType":["isReferencedBy"],"jpcoar:relatedTitle":[{"@value":"A Universal Median Quasi-Monte Carlo Integration"}]},{"@id":"https://cir.nii.ac.jp/crid/1360285711087005696","@type":"Article","resourceType":"学術雑誌論文(journal article)","relationType":["isReferencedBy"],"jpcoar:relatedTitle":[{"@value":"Richardson Extrapolation of Polynomial Lattice Rules"}]},{"@id":"https://cir.nii.ac.jp/crid/1360286995087696896","@type":"Article","resourceType":"学術雑誌論文(journal article)","relationType":["isReferencedBy"],"jpcoar:relatedTitle":[{"@value":"Exploring Uplink Achievable Rate for HPO MIMO Through Quasi-Monte Carlo and Variance Reduction Techniques"}]},{"@id":"https://cir.nii.ac.jp/crid/1360306904386089088","@type":"Article","resourceType":"学術雑誌論文(journal article)","relationType":["isReferencedBy"],"jpcoar:relatedTitle":[{"@value":"Dimension reduction for Quasi-Monte Carlo methods via quadratic regression"}]},{"@id":"https://cir.nii.ac.jp/crid/1360567181908491520","@type":"Article","resourceType":"学術雑誌論文(journal article)","relationType":["isReferencedBy"],"jpcoar:relatedTitle":[{"@value":"Quasi-Monte Carlo point sets with small t-values and WAFOM"}]},{"@id":"https://cir.nii.ac.jp/crid/1360580230605876864","@type":"Article","resourceType":"学術雑誌論文(journal article)","relationType":["isReferencedBy"],"jpcoar:relatedTitle":[{"@value":"Construction-Free Median Quasi-Monte Carlo Rules for Function Spaces with Unspecified Smoothness and General Weights"}]},{"@id":"https://cir.nii.ac.jp/crid/1360580232394797056","@type":"Article","resourceType":"学術雑誌論文(journal article)","relationType":["isReferencedBy"],"jpcoar:relatedTitle":[{"@value":"A note on concatenation of quasi-Monte Carlo and plain Monte Carlo rules in high dimensions"}]},{"@id":"https://cir.nii.ac.jp/crid/1360580232394805248","@type":"Article","resourceType":"学術雑誌論文(journal article)","relationType":["isReferencedBy"],"jpcoar:relatedTitle":[{"@value":"Improved bounds on the gain coefficients for digital nets in prime power base"}]},{"@id":"https://cir.nii.ac.jp/crid/1360848656102705664","@type":"Article","resourceType":"学術雑誌論文(journal article)","relationType":["isReferencedBy"],"jpcoar:relatedTitle":[{"@value":"Lattice rules in non-periodic subspaces of Sobolev spaces"}]},{"@id":"https://cir.nii.ac.jp/crid/1360848656261040768","@type":"Article","resourceType":"学術雑誌論文(journal article)","relationType":["isReferencedBy"],"jpcoar:relatedTitle":[{"@value":"Construction of scrambled polynomial lattice rules over $\\mathbb{F}_{2}$ with small mean square weighted $\\mathcal{L}_{2}$ discrepancy"}]},{"@id":"https://cir.nii.ac.jp/crid/1360848662788446080","@type":"Article","resourceType":"学術雑誌論文(journal article)","relationType":["isReferencedBy"],"jpcoar:relatedTitle":[{"@value":"Comparison of Sobol’ sequences in financial applications"}]},{"@id":"https://cir.nii.ac.jp/crid/1360861707139939072","@type":"Article","resourceType":"学術雑誌論文(journal article)","relationType":["isReferencedBy"],"jpcoar:relatedTitle":[{"@value":"Component-by-component construction of randomized rank-1 lattice rules achieving almost the optimal randomized error rate"}]},{"@id":"https://cir.nii.ac.jp/crid/1360869855578773248","@type":"Article","resourceType":"学術雑誌論文(journal article)","relationType":["isReferencedBy"],"jpcoar:relatedTitle":[{"@value":"How Sharp Are Error Bounds? 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