  This paper describes a new efficient conjugate subgradient algorithm which
minimizes a convex function containing a least squares fidelity term and an
absolute value regularization term. This method is successfully applied to the
inversion of ill-conditioned linear problems, in particular for computed
tomography with the dictionary learning method. A comparison with other
state-of-art methods shows a significant reduction of the number of iterations,
which makes this algorithm appealing for practical use.
