Abstract: In this paper we study mathematically the eigenvalue
problem for stochastic elliptic partial differential equation of Wick
type. Using the Wick-product and the Wiener-Itô chaos expansion,
the stochastic eigenvalue problem is reformulated as a system of an
eigenvalue problem for a deterministic partial differential equation
and elliptic partial differential equations by using the Fredholm
alternative. To reduce the computational complexity of this system,
we shall use a decomposition method using the Wiener-Itô chaos
expansion. Once the approximation of the solution is performed using
the finite element method for example, the statistics of the numerical
solution can be easily evaluated.
Abstract: We present in this paper a useful strategy to solve stochastic partial differential equations (SPDEs) involving stochastic coefficients. Using the Wick-product of higher order and the Wiener-Itˆo chaos expansion, the SPDEs is reformulated as a large system of deterministic partial differential equations. To reduce the computational complexity of this system, we shall use a decomposition-coordination method. To obtain the chaos coefficients in the corresponding deterministic equations, we use a least square formulation. Once this approximation is performed, the statistics of the numerical solution can be easily evaluated.