On convergence of moments in uncertainty quantification based on direct quadrature

Theoretical results for the convergence of statistical moments in numerical quadrature based polynomial chaos computational uncertainty quantification are presented in this work. This is accomplished by considering the computation of the moments through a direct numerical quadrature method, which is shown to be equivalent to stochastic collocation. For problems which involve output variables which have a polynomial dependence on the random input variables, lower bound expressions are derived for the number of quadrature points required for convergence of arbitrary order moments. In addition, an error expression is derived for when this lower bound is used for problems which have a higher degree of continuity than what was assumed when the bounds are computed. The theoretical results are demonstrated through a simple random algebraic problem and a nonlinear plate problem. The results presented in this work provide further insight into the widely used polynomial chaos expansion method of uncertainty quantification along with presenting simple expressions which can be used for uncertainty quantification code verification.