On Polynomial Chaos Expansion via Gradient-enhanced $\ell_1$-minimization
This work provides theoretical guarantees for a known method (gradient-enhanced ℓ1-minimization) in uncertainty quantification, addressing a gap in stability and convergence analysis.
The authors provide a probabilistic stability and convergence analysis for gradient-enhanced ℓ1-minimization in polynomial chaos expansions, showing that including derivative information almost-surely improves recovery conditions and reduces computational cost, as demonstrated on three numerical examples.
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.