A Gradient-Enhanced $ℓ_1$ Approach for the Recovery of Sparse Trigonometric Polynomials (Communications in Computational Physics, 2018)
In this paper, we discuss a gradient-enhanced $ℓ_1$ approach for the recoveryof sparse Fourier expansions. By $gradient$-$enhanced$ approaches we mean that thedirectional derivatives along given vectors are utilized to improve the sparse approximations.We first consider the case where both the function values and the directionalderivatives at sampling points are known. We show that, under some mild conditions,the inclusion of the derivatives information can indeed decrease the coherenceof measurement matrix, and thus leads to the improved the sparse recovery conditionsof the $ℓ_1$ minimization. We also consider the case where either the function values orthe directional derivatives are known at the sampling points, in which we present asufficient condition under which the measurement matrix satisfies RIP, provided thatthe samples are distributed according to the uniform measure. This result shows thatthe derivatives information plays a similar role as that of the function values. Severalnumerical examples are presented to support the theoretical statements. Potential applicationsto function (Hermite-type) interpolations and uncertainty quantification arealso discussed.
Keywords: Gradient-enhanced $ℓ_1$ minimization; compressed sensing; sparse Fourier expansions; restricted isometry property; mutual incoherence.