Adaptive mesh methods on compact manifolds via Optimal Transport and Optimal Information Transport

Abstract

Moving mesh methods were devised to redistribute a mesh in a smooth way, while keeping the number of vertices of the mesh and their connectivity unchanged. A fruitful theoretical point-of-view is to take such moving mesh methods and think of them as an application of the diffeomorphic density matching problem. Given two probability measures μ0 and μ1, the diffeomorphic density matching problem consists of finding a diffeomorphic pushforward map T such that T#μ0=μ1. Moving mesh methods are seen to be an instance of the diffeomorphic density matching problem by treating the probability density as the local density of nodes in the mesh. It is preferable that the restructuring of the mesh be done in a smooth way that avoids tangling the connections between nodes, which would lead to numerical instability when the mesh is used in computational applications. This then suggests that a diffeomorphic map T is desirable to avoid tangling. The first tool employed to solve the moving mesh problem between source and target probability densities on the sphere was Optimal Transport (OT). Recently Optimal Information Transport (OIT) was rigorously derived and developed allowing for the computation of a diffeomorphic mapping by simply solving a Poisson equation. Not only is the equation simpler to solve numerically in OIT, but with Optimal Transport there is no guarantee that the mapping between probability density functions defines a diffeomorphism for general 2D compact manifolds.In this manuscript, we perform a side-by-side comparison of using Optimal Transport and Optimal Information Transport on the sphere for adaptive mesh problems. We choose to perform this comparison with recently developed provably convergent solvers, but these are, of course, not the only numerical methods that may be used. We believe that Optimal Information Transport is preferable in computations due to the fact that the partial differential equation (PDE) solve step is simply a Poisson equation. For more general surfaces M, we show how the Optimal Transport and Optimal Information Transport problems can be reduced to solving on the sphere, provided that there exists a diffeomorphic mapping Φ:M→S2. This implies that the Optimal Transport problem on M with a special cost function can be solved with regularity guarantees, while computations for the problem are performed on the unit sphere.