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In this paper we are concerned with the solution of compressed sensing (CS) problems where the signals to be recovered are sparse in coherent and redundant dictionaries. We extend the primal-dual Newton Conjugate Gradient method (pdNCG) in [T. F. Chan, G. H. Golub, and P. Mulet, SIAM J. Sci. Comput., 20 (1999), pp. 1964--1977] to CS problems. We provide an inexpensive and provably effective preconditioning technique for linear systems using pdNCG. Numerical results are presented on CS problems which demonstrate the performance of pdNCG with the proposed preconditioner compared to state-of-the-art existing solvers.

A Preconditioner for A Primal-Dual Newton Conjugate Gradient Method for Compressed Sensing Problems