Adaptive Mesh Methods on Compact Manifolds via Optimal Transport and Optimal Information Transport
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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 a diffeomorphic mapping between density measures. In applications, an off-the-shelf grid needs to be restructured to have higher grid density in some regions than others. This should be done in a way that avoids tangling, hence, the attractiveness of diffeomorphic mapping techniques. For exact diffeomorphic mapping on the sphere a major tool used is Optimal Transport, which allows for diffeomorphic mapping between even non-continuous source and target densities. However, recently Optimal Information Transport was rigorously developed allowing for exact and inexact diffeomorphic mapping and the solving of a simpler partial differential equation. In this manuscript, we perform the first side-by-side comparison of using Optimal Transport and Optimal Information Transport on the sphere for adaptive mesh problems. We introduce how to generalize these computations to more general manifolds. In this manuscript, we choose to perform this comparison with provably convergent solvers, which is generally challenging for either problem due to the lack of boundary conditions and lack of comparison principle in the partial differential equation formulation. It appears that Optimal Information Transport produces better results and is more easily generalizable for the moving mesh problem. In order to use Optimal Transport for moving mesh methods, further work on the accuracy of Optimal Transport solvers on the sphere and a framework for generalization to other manifolds must be done before this becomes a recommended method in challenging cases.
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