  Gradient-enhanced Uncertainty Quantification (UQ) has received recent
attention, in which the derivatives of a Quantity of Interest (QoI) with
respect to the uncertain parameters are utilized to improve the surrogate
approximation. Polynomial chaos expansions (PCEs) are often employed in UQ, and
when the QoI can be represented by a sparse PCE, $\ell_1$-minimization can
identify the PCE coefficients with a relatively small number of samples. In
this work, we investigate a gradient-enhanced $\ell_1$-minimization, where
derivative information is computed to accelerate the identification of the PCE
coefficients. For this approach, stability and convergence analysis are
lacking, and thus we address these here with a probabilistic result. In
particular, with an appropriate normalization, we show the inclusion of
derivative information will almost-surely lead to improved conditions, e.g.
related to the null-space and coherence of the measurement matrix, for a
successful solution recovery. Further, we demonstrate our analysis empirically
via three numerical examples: a manufactured PCE, an elliptic partial
differential equation with random inputs, and a plane Poiseuille flow with
random boundaries. These examples all suggest that including derivative
information admits solution recovery at reduced computational cost.
