
Abstract
An equation that arises in mathematical studies of the transport of
pollutants in groundwater and of oil recovery processes is of the
form: $\nabla_{x}\cdot(\kappa(x,\cdot)\nabla_{x}u(x,\omega))=f(x)$,
for $x\in D$, where $\kappa(x,\cdot)$, the permeability tensor, is
random and models the properties of the rocks, which are not know
with certainty. Further, geostatistical models assume
$\kappa(x,\cdot)$ to be a lognormal random field. The use of Monte
Carlo methods to approximate the expected value of $u(x,\cdot)$,
higher moments, or other functionals of $u(x,\cdot)$, require
solving similar system of equations many times as trajectories are
considered, thus it becomes expensive and impractical. In this
paper, we present and explain several advantages of using the
White Noise probability space as a natural framework for this
problem. Applying properly and timely the WienerItô Chaos
decomposition and an eigenspace decomposition, we obtain a symmetric
positive definite linear system of equations whose solutions are the
coefficients of a Galerkintype approximation to the solution of the
original equation. Moreover, this approach reduces the simulation of
the approximation to $u(x,\omega)$ for a fixed $\omega$, to the
simulation of a finite number of independent normally distributed
random variables.
Mathematics Subject Classification: Primary: 60H30, 60H35, 65C20; Secondary: 65C30, 65N30.
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