On polynomial chaos expansion via gradient enhanced l1 minimization
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, 1 minimization can identify the PCE coefficients with a relatively small number of samples. In this work, we investigate a gradient enhanced 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. An approach for sparse polynomial chaos (PC) expansions including derivative information is presented.Method relies on 1 minimization to solve for the PC coefficients.Formal analysis is presented to guarantee the stability and convergence of gradient enhanced 1 minimization.Three examples presented to illustrate (cost/accuracy) improvements achieved by gradient enhanced 1 minimization.
J. Comput. Phys.