The conventional delay-and-sum beamforming technique results in blurred source maps due to its poor spatial resolution and high side-lobe levels. To overcome these limitations, the deconvolution approach for the mapping of acoustic sources (DAMAS) has been proposed as a postprocessing stage for image enhancement. DAMAS solves an inverse problem in the form of a system of linear equations. However, this is computationally intensive. This paper presents an approach that imposes two additional constraints to the inverse problem, namely sparsity and nonnegativity of the solution. The resulting constrained problem is solved within the Krylov projection framework. Moreover, the mapping of the sparsity penalty into the Krylov subspace is approximated by a sequence of l2-norm problems via the iteratively reweighted norm (IRN) approach. Experimental results are presented which demonstrate the merits of the proposed method compared to several state-of-the-art approaches in terms of reconstruction accuracy and computation time.