dc.contributor.advisor Joe, Stephen dc.contributor.author Kuo, Frances Y. dc.date.accessioned 2021-01-11T22:51:35Z dc.date.available 2021-01-11T22:51:35Z dc.date.issued 2002 dc.identifier.citation Kuo, F. Y. (2002). Constructive approaches to quasi-Monte Carlo methods for multiple integration (Thesis, Doctor of Philosophy (PhD)). The University of Waikato, Hamilton, New Zealand. Retrieved from https://hdl.handle.net/10289/14052 en dc.identifier.uri https://hdl.handle.net/10289/14052 dc.description.abstract Recently, quasi-Monte Carlo methods have been successfully used for approximating multiple integrals in hundreds of dimensions in mathematical finance, and were significantly more efficient than Monte Carlo methods. To understand the apparent success of quasi-Monte Carlo methods for multiple integration, one popular approach is to study worst-case error bounds in weighted function spaces in which the importance of the variables is moderated by some sequences of weights. Ideally, a family of quasi-Monte Carlo methods in some weighted function space should be strongly tractable. Strong tractability means that the minimal number of quadrature points n needed to reduce the initial error by a factor of ε is bounded by a polynomial in ε⁻¹ independently of the dimension d. Several recent publications show the existence of lattice rules that satisfy the strong tractability error bounds in weighted Korobov spaces of periodic integrands and weighted Sobolev spaces of non-periodic integrands. However, those results were non-constructive and thus give no clues as to how to actually construct these lattice rules. In this thesis, we focus on the construction of quasi-Monte Carlo methods that are strongly tractable. We develop and justify algorithms for the construction of lattice rules that achieve strong tractability error bounds in weighted Korobov and Sobolev spaces. The parameters characterizing these lattice rules are found ‘component-by-component’: the (d + 1)-th components are obtained by successive 1-dimensional searches, with the previous d components kept unchanged. The cost of these algorithms vary from O(nd²) to O(n³d²) operations. With currently available technology, they allow construction of rules easily with values of n up to several million and dimensions d up to several hundred. dc.format.mimetype application/pdf dc.language.iso en dc.publisher The University of Waikato dc.rights All items in Research Commons are provided for private study and research purposes and are protected by copyright with all rights reserved unless otherwise indicated. dc.title Constructive approaches to quasi-Monte Carlo methods for multiple integration dc.type Thesis thesis.degree.grantor The University of Waikato thesis.degree.level Doctoral thesis.degree.name Doctor of Philosophy (PhD) dc.date.updated 2021-01-11T22:50:35Z pubs.place-of-publication Hamilton, New Zealand en_NZ
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