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We propose a new gradient projection algorithm that compares favorably withthe fastest algorithms available to date for $\ell_1$-constrained sparserecovery from noisy data, both in the compressed sensing and inverse problemframeworks. The method exploits a line-search along the feasible direction andan adaptive steplength selection based on recent strategies for the alternationof the well-known Barzilai-Borwein rules. The convergence of the proposedapproach is discussed and a computational study on both well-conditioned andill-conditioned problems is carried out for performance evaluations incomparison with five other algorithms proposed in the literature.

Accelerating gradient projection methods for $\ell_1$-constrained signal recovery by steplength selection rules