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DTSTART:20171105T020000
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UID:calendar.96041.field_date.0@www.wpi.edu
DTSTAMP:20191213T034012Z
CREATED:20180202T155228Z
DESCRIPTION:Description of Event: \n\n\n\nGuannan Zhang\nOak Ridge National
Laboratory\n\n\n\nTitle: A domain-decomposition model reduction method fo
r linear convection-diffusion equations with random coefficients\n\n\n\nAb
stract: \nWe will focus on linear steady-state convection-diffusion equati
ons 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\, and convection-dominated transport
equations with random velocity fields. For each of them\, we investigate t
wo types of random fields\, i.e.\, the colored noise and the discrete whit
e noise. We developed a new domain-decomposition-based model reduction (DD
MR) method\, which can exploit the low-dimensional structure of local solu
tions from various perspectives. We divide the physical domain into a set
of non-overlapping sub-domains\, generate local random fields and establis
h the correlation structure among local fields. We generate a set of reduc
ed bases for the PDE solution within sub-domains and on interfaces\, then
define reduced local stiffness matrices by multiplying each reduced basis
by the corresponding blocks of the local stiffness matrix. After that\, we
establish sparse approximations of the entries of the reduced local stiff
ness matrices in low-dimensional subspaces\, which finishes the offline pr
ocedure. In the online phase\, when a new realization of the global random
field is generated\, we map the global random variables to local random v
ariables\, evaluate the sparse approximations of the reduced local stiffne
ss matrices\, assemble the reduced global Schur complement matrix and solv
e the coefficients of the reduced bases on interfaces\, and then assemble
the reduced local Schur complement matrices and solve the coefficients of
the reduced bases in the interior of the sub-domains. The advantages and c
ontributions of our method lie in the following three aspects. First\, the
DDMR method has the online-offline decomposition feature\, i.e.\, the onl
ine computational cost is independent of the finite element mesh size. Sec
ond\, the DDMR method can handle the PDEs of interest with non-affine high
-dimensional random coefficients. The challenge caused by non-affine coeff
icients is resolved by approximating the entries of the reduced stiffness
matrices. The high-dimensionality is handled by the DD strategy. Third\, t
he DDMR method can avoid building polynomial sparse approximations to loca
l PDE solutions. This feature is useful in solving the convection-dominate
d PDE\, whose solution has a sharp transition caused by the boundary condi
tion. We will demonstrate the performance of our method based on the diffu
sion equation and convection-dominated equation with colored noises and di
screte white noises.
DTSTART;TZID=America/New_York:20180315T110000
DTEND;TZID=America/New_York:20180315T120000
LAST-MODIFIED:20180313T193845Z
LOCATION:Stratton Hall
SUMMARY:Mathematical Sciences - Seminar on Numerical Methods - 'A domain-de
composition model reduction method for linear convection-diffusion equatio
ns with random coefficients' by Guannan Zhang (Oak Ridge National Laborato
ry)
URL;TYPE=URI:https://www.wpi.edu/news/calendar/events/mathematical-sciences
-seminar-numerical-methods-domain-decomposition-model
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