A Gradient-Enhanced L1 Approach for the Recovery of Sparse Trigonometric Polynomials

Zhiqiang Xu and
Tao Zhou

10.4208/cicp.OA-2018-0006
Commun. Comput. Phys., 24 (2018), pp. 286-308.

In this paper, we discuss a gradient-enhanced ℓ1 approach for the recovery
of sparse Fourier expansions. By gradient-enhanced approaches we mean that the
directional derivatives along given vectors are utilized to improve the sparse approximations.
We first consider the case where both the function values and the directional
derivatives at sampling points are known. We show that, under some mild conditions,
the inclusion of the derivatives information can indeed decrease the coherence
of measurement matrix, and thus leads to the improved the sparse recovery conditions
of the ℓ1 minimization. We also consider the case where either the function values or
the directional derivatives are known at the sampling points, in which we present a
sufficient condition under which the measurement matrix satisfies RIP, provided that
the samples are distributed according to the uniform measure. This result shows that
the derivatives information plays a similar role as that of the function values. Several
numerical examples are presented to support the theoretical statements. Potential applications
to function (Hermite-type) interpolations and uncertainty quantification are
also discussed.