  We present an enriched formulation of the Least Squares (LSQ) regression
method for Uncertainty Quantification (UQ) using generalised polynomial chaos
(gPC). More specifically, we enrich the linear system with additional equations
for the gradient (or sensitivity) of the Quantity of Interest with respect to
the stochastic variables. This sensitivity is computed very efficiently for all
variables by solving an adjoint system of equations at each sampling point of
the stochastic space. The associated computational cost is similar to one
solution of the direct problem. For the selection of the sampling points, we
apply a greedy algorithm which is based on the pivoted QR decomposition of the
measurement matrix. We call the new approach sensitivity-enhanced generalised
polynomial chaos, or se-gPC. We apply the method to several test cases to test
accuracy and convergence with increasing chaos order, including an aerodynamic
case with $40$ stochastic parameters. The method is found to produce accurate
estimations of the statistical moments using the minimum number of sampling
points. The computational cost scales as $\sim m^{p-1}$, instead of $\sim m^p$
of the standard LSQ formulation, where $m$ is the number of stochastic
variables and $p$ the chaos order. The solution of the adjoint system of
equations is implemented in many computational mechanics packages, thus the
infrastructure exists for the application of the method to a wide variety of
engineering problems.
