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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Adaptive Stochastic Galerkin Finite Element Approx
imation for Elliptic PDEs with Random Coefficients
- Catherine Powell (University of Manchester)
DTSTART;TZID=Europe/London:20180205T113000
DTEND;TZID=Europe/London:20180205T123000
UID:TALK99862AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/99862
DESCRIPTION:Co-author: Adam Crowder (University of Man
chester)
We consider a stand
ard elliptic PDE model with uncertain coefficients
. Such models are simple\, but are well understood
theoretically and so serve as a canonical class o
f problems on which to compare different numerical
schemes (computer models).
Approximation
s which take the form of polynomial chaos (PC) exp
ansions have been widely used in applied mathemati
cs and can be used as surrogate models in UQ studi
es. When the coefficients of the approximation are
computed using a Galerkin method\, we use the ter
m &lsquo\;Stochastic Galerkin approximation&rsquo\
;. In statistics\, the term &lsquo\;intrusive PC a
pproximation&rsquo\; is also often used. In the Ga
lerkin approach\, the resulting PC approximation i
s optimal in that the energy norm of the error bet
ween the true model solution and the PC approximat
ion is minimised. This talk will focus on how to b
uild the approximation space (in a computer code)
in a computationally efficient way while also guar
anteeing accuracy.
In the stochastic
Galerkin finite element (SGFEM) approach\, an app
roximation is sought in a space which is defined t
hrough a chosen set of spatial finite element basi
s functions and a set of orthogonal polynomials in
the parameters that define the uncertain PDE coef
ficients. When the number of parameters is too hig
h\, the dimension of this space becomes unmanageab
le. One remedy is to use &lsquo\;adaptivity&rsquo\
;. First\, we generate an approximation in a low-d
imensional approximation space (which is cheap) an
d then use a computable a posteriori error estimat
or to decide whether the current approximation is
accurate enough or not. If not\, we enrich the app
roximation space\, estimate the error again\, and
so on\, until the final approximation is accurate
enough. This allows us to design problem-specific
polynomial approximations. We describe an error es
timation procedure\, outline the computational cos
ts\, and illustrate its use through numerical resu
lts. An improved multilevel implem entation will b
e outlined in a poster given by Adam Crowder.
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:INI IT
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