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An Anisotropic Sparse Grid Stochastic Collocation Method for Partial Differential Equations with High-Dimensional Random Input Data

This talk proposes and analyzes an anisotropic sparse grid stochastic collocation method for the approximation of statistical quantities related to the solution of partial differential equations with random coefficients and forcing terms (input data of the model). This method is an extension of the Sparse Grid Stochastic Collocation method analyzed in [7], which consists of a Galerkin approximation in the space variables and a collocation, in probability space, on sparse tensor product grids utilizing either Clenshaw-Curtis or Gaussian knots. Even in the presence of nonlinearities, the collocation approach leads to the solution of uncoupled deterministic problems, just as in the Monte Carlo method. Our previous sparse collocation procedure is very effective for problems whose input data depend on a moderate number of random variables, which “weigh equally” in the solution. For such an isotropic situation the displayed convergence is faster than standard collocation techniques built upon full tensor product spaces.