TY - EJOUR
DO - 10.1002/nme.2656
AB - We consider the problem of numerically approximating statistical moments of the solution of a time-dependent linear parabolic partial differential equation (PDE), whose coefficients and/or forcing terms are spatially correlated random fields. The stochastic coefficients of the PDE are approximated by truncated Karhunen–Loève expansions driven by a finite number of uncorrelated random variables. After approximating the stochastic coefficients, the original stochastic PDE turns into a new deterministic parametric PDE of the same type, the dimension of the parameter set being equal to the number of random variables introduced. After proving that the solution of the parametric PDE problem is analytic with respect to the parameters, we consider global polynomial approximations based on tensor product, total degree or sparse polynomial spaces and constructed by either a Stochastic Galerkin or a Stochastic Collocation approach. We derive convergence rates for the different cases and present numerical results that show how these approaches are a valid alternative to the more traditional Monte Carlo Method for this class of problems. Copyright © 2009 John Wiley & Sons, Ltd.
T1 - Analysis and implementation issues for the numerical approximation of parabolic equations with random coefficients
IS - 6-7
DA - 2009
AU - Nobile, Fabio
AU - Tempone, Raul
JF - INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING
SP - 979-1006
VL - 80
EP - 979-1006
ID - 180349
KW - PDEs with random data
KW - parabolic equations
KW - multivariate polynomial approximation
KW - Stochastic Galerkin methods
KW - Stochastic Collocation methods
KW - sparse grids
KW - Smolyak approximation
KW - Point Collocation
KW - Monte Carlo sampling
SN - 0029-5981
UR - http://infoscience.epfl.ch/record/180349/files/Analysis_and_implementation_issues.pdf
ER -