On Improved Statistical Accuracy of Low-Order Polynomial Chaos Approximations

Polynomial chaos expansion is a popular way to develop surrogate models for stochastic systems with arbitrary random variables. Standard techniques such as Galerkin projection, stochastic collocation, and least squares approximation, are applied to determine polynomial chaos coefficients, which define the surrogate model. Since the surrogate models are developed from a function approximation perspective, there is no reason to expect accuracy of statistics from these models. The statistical moments estimated from the surrogate model may significantly differ from the true moments, especially for lower order approximations. Often arbitrary high orders are required to recover, for example, the second moment. In this paper, we present modifications of standard techniques and determine polynomial chaos coefficients by solving a constrained optimization problem. We present this new approach for algebraic functions and differential equations with random parameters, and demonstrate that the surrogate models from the new approach are able to recover the first two moments exactly.