  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.
