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dc.contributor.author
Graham, I.G.
dc.contributor.author
Kuo, Frances Y.
dc.contributor.author
Nichols, J.A.
dc.contributor.author
Scheichl, R.
dc.contributor.author
Schwab, Christoph
dc.contributor.author
Sloan, Ian H.
dc.date.accessioned
2017-06-11T03:54:47Z
dc.date.available
2017-06-11T03:54:47Z
dc.date.issued
2013
dc.identifier.uri
http://hdl.handle.net/20.500.11850/79176
dc.description.abstract
In this paper we analyze the numerical approximation of diffusion problems over polyhedral domains in R d ( d = 1 ; 2 ; 3), with diffusion coefficient a ( x ;ω ) given as a lognormal random field, i.e., a ( x ;ω ) = exp( Z ( x ;ω )) where x is the spatial variable and Z ( x ; · ) is a Gaussian random field. The analysis presents particular challenges since the corresponding bilinear form is not uniformly bounded away from 0 or ∞ over all possible realizations of a . Fo- cusing on the problem of computing the expected value of linear functionals of the solution of the diffusion problem, we give a rigorous error analysis for methods constructed from (i) standard continuous and piecewise linear finite element approximation in physical space; (ii) truncated Karhunen-Lo ́eve expansion for computing realizations of a (leading to a possi- bly high-dimensional parametrized deterministic diffusion problem); and (iii) lattice-based Quasi-Monte Carlo (QMC) quadrature rules for computing integrals over parameter space which define the expected values. The paper contains novel error analysis which accounts for the effect of all three types of approximation. The QMC analysis is based on a recent result on randomly shifted lattice rules for high-dimensional integrals over the unbounded domain of Euclidean space, which shows that (under suitable conditions) the quadrature error decays with O ( n − 1+ δ ) with respect to the number of quadrature points n , where δ > 0 is arbitrarily small and where the implied constant in the asymptotic error bound is independent of the dimension of the domain of integration.
dc.language.iso
en
dc.publisher
ETH Zürich, Seminar für Angewandte Mathematik
dc.title
Quasi-Monte Carlo finite element methods for elliptic PDEs with log-normal random coefficient
dc.type
Report
ethz.journal.title
Research Report
ethz.journal.volume
2013-14
ethz.size
33 p.
ethz.publication.place
Zürich
ethz.publication.status
unpublished
ethz.leitzahl
ETH Zürich::00002 - ETH Zürich::00012 - Lehre und Forschung::00007 - Departemente::02000 - Dep. Mathematik / Dep. of Mathematics::02501 - Seminar für Angewandte Mathematik / Seminar for Applied Mathematics::03435 - Schwab, Christoph / Schwab, Christoph
ethz.leitzahl.certified
ETH Zürich::00002 - ETH Zürich::00012 - Lehre und Forschung::00007 - Departemente::02000 - Dep. Mathematik / Dep. of Mathematics::02501 - Seminar für Angewandte Mathematik / Seminar for Applied Mathematics::03435 - Schwab, Christoph / Schwab, Christoph
ethz.identifier.url
http://www.sam.math.ethz.ch/sam_reports/reports_final/reports2013/2013-14.pdf
ethz.date.deposited
2017-06-11T03:57:52Z
ethz.source
ECIT
ethz.identifier.importid
imp593651891b0f368187
ethz.ecitpid
pub:124464
ethz.eth
yes
ethz.availability
Metadata only
ethz.rosetta.installDate
2017-07-18T09:01:39Z
ethz.rosetta.lastUpdated
2018-11-02T13:05:12Z
ethz.rosetta.versionExported
true
ethz.COinS
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