Approximation of random/stochastic partial differential equations

dc.contributor.advisor Sloan, Ian H. en_US
dc.contributor.advisor Le Gia, Quoc Thong en_US
dc.contributor.advisor Kuo, Frances Y. en_US Kazashi, Yoshihito en_US 2022-03-15T12:04:55Z 2022-03-15T12:04:55Z 2018 en_US
dc.description.abstract The overarching interest of this thesis lies in approximations of partial differential equations (PDEs) with randomness or stochasticity. We focus on three rather different problems: a study of random fields on spherical shells, and its applications to PDE problems; quasi-Monte Carlo (QMC) methods for a class of PDEs with random coefficients; and a discretisation for the solution of stochastic PDEs. First, we consider Gaussian random fields on spherical shells that are radially anisotropic and rotationally isotropic. The smoothness of the covariance function is connected to the sample continuity, partial differentiability, and the Sobolev smoothness. Based on the regularity results, convergence rates of filtered approximations are established: Gaussian and log-normal random fields approximated with filtering, and a class of elliptic PDEs with approximated random coefficients, are considered. Second, we consider QMC integration of output functionals of solutions of a class of PDEs with a log-normal random coefficient. The coefficient is assumed to be given by an exponential of a Gaussian random field that is represented by a series expansion in terms of some system of functions with local supports. A quadrature error decay rate almost 1 is established, and the theory developed here is applied to a wavelet stochastic model. It is shown that a wide class of path smoothness can be treated with this framework. Finally, we turn our attention to an approximation of stochastic parabolic PDEs. We consider three discretisations: temporal, spatial, and the truncation of the infinite-dimensional space-valued Wiener process. Temporally, we consider the implicit Euler–Maruyama method with a non-uniform time step. For the spatial discretisation, we consider the spectral method. Further, we truncate the Wiener process, which is assumed to admit a series representation. We establish a time discrete error estimate for this algorithm. Further, a discrete analogue of maximal L2-regularity of the scheme is established, which has the same form as their continuous counterpart. en_US
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 en_US
dc.subject.other Partial differential equations en_US
dc.subject.other Gaussian random fields en_US
dc.subject.other Spherical shells en_US
dc.subject.other Quasi-Monte Carlo en_US
dc.title Approximation of random/stochastic partial differential equations en_US
dc.type Thesis en_US
dcterms.accessRights open access
dcterms.rightsHolder Kazashi, Yoshihito
dspace.entity.type Publication en_US
unsw.accessRights.uri 2020-09-01 en_US
unsw.description.embargoNote Embargoed until 2020-09-01
unsw.relation.faculty Science
unsw.relation.originalPublicationAffiliation Kazashi, Yoshihito, Mathematics & Statistics, Faculty of Science, UNSW en_US
unsw.relation.originalPublicationAffiliation Sloan, Ian H., Mathematics & Statistics, Faculty of Science, UNSW en_US
unsw.relation.originalPublicationAffiliation Le Gia, Quoc Thong, Mathematics & Statistics, Faculty of Science, UNSW en_US
unsw.relation.originalPublicationAffiliation Kuo, Frances Y., Mathematics & Statistics, Faculty of Science, UNSW en_US School of Mathematics & Statistics *
unsw.thesis.degreetype PhD Doctorate en_US
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