Accelerated projected gradient method for linear inverse problems with sparsity constraints

Regularization of ill-posed linear inverse problems via ℓ₁ penalization has been proposed for cases where the solution is known to be (almost) sparse. One way to obtain the minimizer of such an ℓ₁ penalized functional is via an iterative soft-thresholding algorithm. We propose an alternative implementation to ℓ₁-constraints, using a gradient method, with projection on ℓ₁-balls. The corresponding algorithm uses again iterative soft-thresholding, now with a variable thresholding parameter. We also propose accelerated versions of this iterative method, using ingredients of the (linear) steepest descent method. We prove convergence in norm for one of these projected gradient methods, without and with acceleration.