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SUMMARY:A domain-decomposition-based model reduction method for convection
 -diffusion equations with random coefficients - Guannan  Zhang  (Oak Ridge
  National Laboratory)
DTSTART:20180206T143000Z
DTEND:20180206T153000Z
UID:TALK99943@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:We focuses on linear steady-state convection-diffusion equatio
 ns 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 e
 quations with random velocity fields. For each of them\, we investigate tw
 o types of random fields\, i.e.\, the colored noise and the discrete white
  noise. We developed a new domain-decomposition-based model reduction (DDM
 R) method\, which can exploit the low-dimensional structure of local solut
 ions from various perspectives. We divide the physical domain into a set o
 f non-overlapping sub-domains\, generate local random fields and establish
  the correlation structure among local fields. We generate a set of reduce
 d bases for the PDE solution within sub-domains and on interfaces\, then d
 efine reduced local stiffness matrices by multiplying each reduced basis b
 y the corresponding blocks of the local stiffness matrix. After that\, we 
 establish sparse approximations of the entries of the reduced local stiffn
 ess matrices in low-dimensional subspaces\, which finishes the offline pro
 cedure. In the online phase\, when a new realization of the global random 
 field is generated\, we map the global random variables to local random va
 riables\, evaluate the sparse approximations of the reduced local stiffnes
 s matrices\, assemble the reduced global Schur complement matrix and solve
  the coefficients of the reduced bases on interfaces\, and then assemble t
 he reduced local Schur complement matrices and solve the coefficients of t
 he reduced bases in the interior of the sub-domains. The advantages and co
 ntributions of our method lie in the following three aspects. First\, the 
 DDMR method has the online-offline decomposition feature\, i.e.\, the onli
 ne computational cost is independent of the finite element mesh size. Seco
 nd\, the DDMR method can handle the PDEs of interest with non-affine high-
 dimensional random coefficients. The challenge caused by the non-affine co
 efficients is resolved by approximating the entries of the reduced stiffne
 ss matrices. The high-dimensionality is handled by the DD strategy. Third\
 , the DDMR method can avoid building polynomial sparse approximations to l
 ocal PDE solutions. This feature is useful in solving the convection-domin
 ated PDE\, whose solution has a sharp transition caused by the boundary co
 ndition. We demonstrate the performance of our method based on the diffusi
 on equation and convection-dominated equation with colored noises and disc
 rete white noises.
LOCATION:Seminar Room 1\, Newton Institute
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