We present an application of sparse regularization of ill-posed linear inverse problems to the reconstruction of the 3D distribution of dark matter in the Universe. By its very nature dark matter cannot be directly observed. Nevertheless, it can be studied through its gravitational effects can it be studied. In particular, the presence of dark matter induces small deformations to the shapes of background galaxies which is known as weak gravitational lensing. However, reconstructing the 3D distribution of dark matter from tomographic lensing measurements amounts to solving an ill-posed linear inverse problem. Considering that the 3D dark matter density is sparse in an appropriate wavelet based 3D dictionary, we propose an iterative thresholding algorithm to solve a penalized least-squares problem. We present our results on simulated dark matter halos and compare them to state of the art linear reconstruction techniques. We show that thanks to our 3D sparsity constraint the quality of the reconstructed maps can be greatly improved.