## Publication: Quasi-Monte Carlo methods with applications to partial differential equations with random coefficients

 dc.contributor.advisor Sloan, Ian en_US dc.contributor.advisor Kuo, Frances en_US dc.contributor.author Nichols, James en_US dc.date.accessioned 2022-03-21T13:58:00Z dc.date.available 2022-03-21T13:58:00Z dc.date.issued 2014 en_US dc.description.abstract This thesis provides the theoretical foundation for the component-by-component (CBC) construction of randomly shifted lattice rules that are tailored to integrals over $\bbR^s$ arising from Darcy-flow PDE problems where the permeability coefficient is given by a lognormal random field. We focus on the problem of computing the expected value of linear functionals of the solution of the PDE, which gives rise to integrals of the form $\int_{\bbR^s} f(\bsy) \prod_{j=1}^s\phi(y_j) \,\rd\bsy$ with a univariate probability density $\phi$. Our general strategy is to first map the integral into the unit cube $[0,1]^s$ using the inverse of the cumulative distribution function of $\phi$, and then apply quasi-Monte Carlo (QMC) methods. However, the transformed integrand in the unit cube does not fall within the standard QMC settings from the literature. Therefore, a non-standard function space setting for integrands over $\bbR^s$ is required for the analysis. Such spaces were previously considered in Kuo et. al. 2010, however due to the needs of the PDE problem, we must extend the theory of the aforementioned paper in several nontrivial directions, including a new error analysis for the CBC construction of lattice rules with general non-product weights, the introduction of an unanchored weighted space for the setting, the use of coordinate-dependent weight functions in the norm, and the strategy for fast CBC construction with POD ("product and order dependent") weights. Our method of numerical approximation of this problem includes piecewise linear finite element approximation in physical space, the truncation of the parameterised expansion of the random field, and QMC quadrature rules for computing integrals over parameterised probability space which define the expected values. We give a rigorous error analysis for the effect of all three of these types of approximation. We show, using the non-standard function space setting developed in the thesis, that the quadrature error decays with $\mathcal{O}(n^{-1+\delta})$ with respect to the number of quadrature points $n$, where $\delta>0$ is arbitrarily small and where the implied constant in the asymptotic error bound is independent of the dimension of the domain of integration. en_US dc.identifier.uri http://hdl.handle.net/1959.4/53480 dc.language English dc.language.iso EN en_US dc.publisher UNSW, Sydney en_US dc.rights CC BY-NC-ND 3.0 en_US dc.rights.uri https://creativecommons.org/licenses/by-nc-nd/3.0/au/ en_US dc.subject.other Random coeffiecients en_US dc.subject.other QMC en_US dc.subject.other Partial differential equations en_US dc.subject.other Lattice rules en_US dc.subject.other High dimensional analysis en_US dc.subject.other Numerical analysis en_US dc.subject.other Computational mathematics en_US dc.title Quasi-Monte Carlo methods with applications to partial differential equations with random coefficients en_US dc.type Thesis en_US dcterms.accessRights open access dcterms.rightsHolder Nichols, James dspace.entity.type Publication en_US unsw.accessRights.uri https://purl.org/coar/access_right/c_abf2 unsw.identifier.doi https://doi.org/10.26190/unsworks/16796 unsw.relation.faculty Science unsw.relation.originalPublicationAffiliation Nichols, James, Mathematics & Statistics, Faculty of Science, UNSW en_US unsw.relation.originalPublicationAffiliation Sloan, Ian, Mathematics & Statistics, Faculty of Science, UNSW en_US unsw.relation.originalPublicationAffiliation Kuo, Frances, Mathematics & Statistics, Faculty of Science, UNSW en_US unsw.relation.school School of Mathematics & Statistics * unsw.thesis.degreetype PhD Doctorate en_US
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