A monomial chaos approach for efficient uncertainty quantification on nonlinear problems

A monomial chaos approach is presented for efficient uncertainty quantification in nonlinear computational problems. Propagating uncertainty through nonlinear equations can be computationally intensive for existing uncertainty quantification methods. It usually results in a set of nonlinear equations which can be coupled. The proposed monomial chaos approach employs a polynomial chaos expansion with monomials as basis functions. The expansion coefficients are solved for using differentiation of the governing equations, instead of a Galerkin projection. This results in a decoupled set of linear equations even for problems involving polynomial nonlinearities. This reduces the computational work per additional polynomial chaos order to the equivalence of a single Newton iteration. Error estimates are derived, and monomial chaos is applied to uncertainty quantification of the Burgers equation and a two-dimensional boundary layer flow problem. The results are compared with results of the Monte Carlo method, the perturbation method, the Galerkin polynomial chaos method, and a nonintrusive polynomial chaos method.Aerospace Engineerin