Numerical Methods and Smolyak Quadrature for Nonlinear Stochastic Partial Differential Equations

We describe methods for the numerical solution of nonlinear problems with stochastic uncertainties in the operator, boundary conditions, and right hand side. First, we compute statistics of the solution directly as high-dimensional integrals and compare their evaluation by sparse (Smolyak) quadrature and Monte Carlo integration. Subsequently, we employ a Galerkin method to obtain an expansion of the solution in a stochastic ansatz-space. This requires the numerical evaluation of the residual, which is again a high-dimensional integral, and we show that sparse quadrature may be an efficient technique for this. The large nonlinear system resulting from the Galerkin conditions is solved by quasi-Newton methods. Finally, we alternatively compute the expansion of the solution by direct orthogonal projection onto stochastic ansatz-functions. We apply the methods to a prototype nonlinear groundwater-flow situation (pressure equation).

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