Motivated by the question of optimal functional approximation via compressed sensing, we propose generalizations of the Iterative Hard Thresholding and the Compressive Sampling Matching Pursuit algorithms able to promote sparse in levels signals. We show, by means of numerical experiments, that the proposed algorithms are successfully able to outperform their unstructured variants when the signal exhibits the sparsity structure of interest. Moreover, in the context of piecewise smooth function approximation, we numerically demonstrate that the structure promoting decoders outperform their unstructured variants and the basis pursuit program when the encoder is structure agnostic.