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

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... on. 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.