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A stochastic collocation method for partial differential equations
with random input data
Raul Tempone
Abstract
We present a Stochastic-Collocation method to solve Partial Differential
Equations with random
coefficients and forcing terms (input data of the model). The input data
are assumed to depend on a finite number of random variables. The method
consists in a Galerkin approximation in space and a collocation in the
zeros of suitable tensor product orthogonal polynomials (Gauss points) in
the probability space and naturally leads to the solution of uncoupled
deterministic problems as in the Monte Carlo approach. It can be seen as a
generalization of the Stochastic Galerkin method proposed in
[Babuska-Tempone-Zouraris, SIAM J. Num. Anal. 42(2004)] and allows one to
treat easily a wider range of situations, such as: input data that depend
non-linearly on the random variables, diffusivity coefficients with
unbounded second moments, random variables that are correlated or have
unbounded support.
We give a complete convergence analysis in case of a linear elliptic
operator and demonstrate exponential convergence of the "probability
error" with respect of the number of Gauss points in each direction in
the probability space, under some regularity assumptions on the random
input data.
Numerical examples showing the effectiveness of the method will be
presented as well.