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\title{}{\bf Stochastic collocation and polynomial chaos expansion for solving PDEs with random coefficients }
\author{}{Shalimova I.A. and Sabelfeld K.K. }
{\it Institute of Computational Mathematics and Mathematical Geophysics, \\ SBRAS, NSU, Novosibirsk}
{\it ias@osmf.sscc.ru, sabelfeld.karl@yahoo.de}
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Solution of PDEs with random inputs like random coefficients, fluctuating functions prescribed on the boundary,
stochastic sources, etc., is of high interest both from theoretical and practical viewpoints.
Direct attacking of this problem via calculating the ensemble of solutions is not efficient, especially
for large fluctuations which is the most interesting case, since slow fluctuations are often well treated by
the small perturbation technique (e.g., see \cite{1,2}). The statistical moment methods are used when there is some
prior information at hand, and one can derive a closed system of equations of such moments. In the Monte Carlo approach,
the double randomization technique is used in some cases when the Monte Carlo estimators can be found which is possible
for some simple examples only (e.g., see \cite{1}). Stochastic finite element based method which uses a
polynomial chaos expansion of the random processes is developed for solving a wide class of random equations (e.g., see \cite{3}).
A closely related approach is the stochastic collocation method which is based on the Karhunen-Lo\`eve expansion of the random
inputs (e.g., see \cite{4}). We develop this technique for solving the Darcy equation governing the flows in stochastically porous media,
and extend it to some other PDEs with random coefficients.
The work has been supported by RFBR under Grants 12-01-00635-a and 12-01-00727-a.
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\bibitem{1}
{\it Sabelfeld K.K.} Random Fields and Stochastic Lagrangian Models.
De Gruyter, Berlin/Boston, 2012, 399 pp.
\bibitem{2}
{\it
Sabelfeld K. and Kolyukhin D.}
Stochastic Eulerian model for the flow
simulation in porous media.
Monte Carlo Methods and Applications.
vol.9, N3, 271-290, 2003.
\bibitem{3}
{\it
Roger G. Ghanem, Pol D. Spanos.}
{Stochastic finite elements. A spectral approach}.
Courier Dover Publications, 2003.
\bibitem{4}
{\it Heng Li and Dongxiao Zhang}. Probabilistic collocation method for flow
in porous media: comparison with other stochastic methods.
Water Resources research, v. 43, W09409, 2007.
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