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Moving mesh methods are designed to redistribute a mesh in a regular way.This applied problem can be considered to overlap with the problem of finding adiffeomorphic mapping between density measures. In applications, anoff-the-shelf grid needs to be restructured to have higher grid density in someregions than others. This should be done in a way that avoids tangling, hence,the attractiveness of diffeomorphic mapping techniques. For exact diffeomorphicmapping on the sphere a major tool used is Optimal Transport, which allows fordiffeomorphic mapping between even non-continuous source and target densities.However, recently Optimal Information Transport was rigorously developedallowing for exact and inexact diffeomorphic mapping and the solving of asimpler partial differential equation.  In this manuscript, we perform the first side-by-side comparison of usingOptimal Transport and Optimal Information Transport on the sphere for adaptivemesh problems. We introduce how to generalize these computations to moregeneral manifolds. In this manuscript, we choose to perform this comparisonwith provably convergent solvers, which is generally challenging for eitherproblem due to the lack of boundary conditions and lack of comparison principlein the partial differential equation formulation. It appears that OptimalInformation Transport produces better results and is more easily generalizablefor the moving mesh problem. In order to use Optimal Transport for moving meshmethods, further work on the accuracy of Optimal Transport solvers on thesphere and a framework for generalization to other manifolds must be donebefore this becomes a recommended method in challenging cases.

Adaptive Mesh Methods on Compact Manifolds via Optimal Transport and Optimal Information Transport