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Gradient-enhanced Uncertainty Quantification (UQ) has received recentattention, in which the derivatives of a Quantity of Interest (QoI) withrespect to the uncertain parameters are utilized to improve the surrogateapproximation. Polynomial chaos expansions (PCEs) are often employed in UQ, andwhen the QoI can be represented by a sparse PCE, $\ell_1$-minimization canidentify the PCE coefficients with a relatively small number of samples. Inthis work, we investigate a gradient-enhanced $\ell_1$-minimization, wherederivative information is computed to accelerate the identification of the PCEcoefficients. For this approach, stability and convergence analysis arelacking, and thus we address these here with a probabilistic result. Inparticular, with an appropriate normalization, we show the inclusion ofderivative information will almost-surely lead to improved conditions, e.g.related to the null-space and coherence of the measurement matrix, for asuccessful solution recovery. Further, we demonstrate our analysis empiricallyvia three numerical examples: a manufactured PCE, an elliptic partialdifferential equation with random inputs, and a plane Poiseuille flow withrandom boundaries. These examples all suggest that including derivativeinformation admits solution recovery at reduced computational cost.

On Polynomial Chaos Expansion via Gradient-enhanced $\ell_1$-minimization