Karniadakis, Performance evaluation of generalized polynomial chaos

Abstract. In this paper we review some applications of generalized polynomial chaos expansion for uncertainty quantification. The math-ematical framework is presented and the convergence of the method is demonstrated for model problems. In particular, we solve the first-order and second-order ordinary differential equations with random parame-ters, and examine the efficiency of generalized polynomial chaos com-pared to Monte Carlo simulations. It is shown that the generalized poly-nomial chaos can be orders of magnitude more efficient than Monte Carlo simulations when the dimensionality of random input is low, e.g. for cor-related noise.