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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:High-Dimensional Collocation for Lognormal Diffusi
on Problems - Oliver Ernst (Technische Universität
Chemnitz)
DTSTART;TZID=Europe/London:20180209T113000
DTEND;TZID=Europe/London:20180209T123000
UID:TALK100390AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/100390
DESCRIPTION:Co-authors: Bjö\;rn Sprungk (Universitä\;t
Mannheim)\, Lorenzo Tamellini (IMATI-CNR Pavia)
Many UQ models consist of random differential equa
tions in which one or more data components are unc
ertain and modeled as random variables. When the l
atter take values in a separable function space\,
their representation typically requires a large or
countably infinite number of random coordinates.
Numerical approximation methods for such functions
of an infinite number of parameters based on best
N-term approximation have recently been proposed
and shown to converge at an algebraic rate. Colloc
ation methods have a number of computational advan
tages over best N-term approximation\, and we show
how ideas successful there can be used to show a
similar convergence rate for sparse collocation of
Hilbert-space-valued functions depending on count
ably many Gaussian random variables. Such functio
ns appear as solutions of elliptic PDEs with a log
normal diffusion coefficient. We outline a general
L2-convergence theory based on previous work by B
achmayr et al. and Chen and establish an algebraic
convergence rate for sufficiently smooth function
s assuming a mild growth bound for the univariate
hierarchical surpluses of the interpolation scheme
applied to Hermite polynomials. We verify specifi
cally for Gauss-Hermite nodes that this assumption
holds and also show algebraic convergence with re
spect to the resulting number of sparse grid point
s for this case. Numerical experiments illustrate
the dimension-independent convergence rate.
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:info@newton.ac.uk
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