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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:A domain-decomposition-based model reduction metho
d for convection-diffusion equations with random c
oefficients - Guannan Zhang (Oak Ridge National
Laboratory)
DTSTART;TZID=Europe/London:20180206T143000
DTEND;TZID=Europe/London:20180206T153000
UID:TALK99943AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/99943
DESCRIPTION:We focuses on linear steady-state convection-diffu
sion equations with random-field coefficients. Our
particular interest to this effort are two types
of partial differential equations (PDEs)\, i.e.\,
diffusion equations with random diffusivities\, an
d convection-dominated transport equations with ra
ndom velocity fields. For each of them\, we invest
igate two types of random fields\, i.e.\, the colo
red noise and the discrete white noise. We develop
ed a new domain-decomposition-based model reductio
n (DDMR) method\, which can exploit the low-dimens
ional structure of local solutions from various pe
rspectives. We divide the physical domain into a s
et of non-overlapping sub-domains\, generate local
random fields and establish the correlation struc
ture among local fields. We generate a set of redu
ced bases for the PDE solution within sub-domains
and on interfaces\, then define reduced local stif
fness matrices by multiplying each reduced basis b
y the corresponding blocks of the local stiffness
matrix. After that\, we establish sparse approxima
tions of the entries of the reduced local stiffnes
s matrices in low-dimensional subspaces\, which fi
nishes the offline procedure. In the online phase\
, when a new realization of the global random fiel
d is generated\, we map the global random variable
s to local random variables\, evaluate the sparse
approximations of the reduced local stiffness matr
ices\, assemble the reduced global Schur complemen
t matrix and solve the coefficients of the reduced
bases on interfaces\, and then assemble the reduc
ed local Schur complement matrices and solve the c
oefficients of the reduced bases in the interior o
f the sub-domains. The advantages and contribution
s of our method lie in the following three aspects
. First\, the DDMR method has the online-offline d
ecomposition feature\, i.e.\, the online computati
onal cost is independent of the finite element mes
h size. Second\, the DDMR method can handle the PD
Es of interest with non-affine high-dimensional ra
ndom coefficients. The challenge caused by the non
-affine coefficients is resolved by approximating
the entries of the reduced stiffness matrices. The
high-dimensionality is handled by the DD strategy
. Third\, the DDMR method can avoid building polyn
omial sparse approximations to local PDE solutions
. This feature is useful in solving the convection
-dominated PDE\, whose solution has a sharp transi
tion caused by the boundary condition. We demonstr
ate the performance of our method based on the dif
fusion equation and convection-dominated equation
with colored noises and discrete white noises.
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:info@newton.ac.uk
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