A gradient enhanced $\ell_1$-minimization for sparse approximation of polynomial chaos expansions
This work addresses the problem of efficient sparse approximation in polynomial chaos expansions for uncertainty quantification, offering an incremental improvement by incorporating gradient information.
The paper proposes a gradient-enhanced ℓ1-minimization method for sparse polynomial chaos expansions that incorporates gradient measurements to accelerate coefficient recovery, demonstrating stable and accurate recovery with reduced computational cost compared to standard ℓ1-minimization.
We investigate a gradient-enhanced $\ell_1$-minimization for constructing sparse polynomial chaos expansions. In addition to function evaluations, measurements of the function gradient is also included to accelerate the identification of expansion coefficients. By designing appropriate preconditioners to the measurement matrix, we show gradient-enhanced $\ell_1$ minimization leads to stable and accurate coefficient recovery. The framework for designing preconditioners is quite general and it applies to recover of functions whose domain is bounded or unbounded. Comparisons between the gradient enhanced approach and the standard $\ell_1$-minimization are also presented and numerical examples suggest that the inclusion of derivative information can guarantee sparse recovery at a reduced computational cost.