Abstract: We propose an efficient algorithm for computing large-scale bounded distortion maps of triangular and tetrahedral meshes. Specifically, given an initial map, we compute a similar map whose differentials are orientation preserving and have bounded condition number. Inspired by alternating optimization and Gauss-Newton approaches, we devise a first order method which combines the advantages of both. On the one hand, its iterations are as computationally efficient as those of alternating optimization. On the other hand, it enjoys preferable convergence properties, associated with Gauss-Newton like approaches. We demonstrate the utility of the proposed approach in efficiently solving geometry processing problems, focusing on challenging large-scale problems.