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 solve adaptive mesh problems using Optimal Transport and Optimal Information Transport on the sphere and introduce how to generalize these computations to more general manifolds. 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.
翻译:移动网格方法旨在定期重新分配网格。 这个应用问题可以被视为与在密度测量之间找到异己型绘图的问题重叠。 在应用中, 需要调整现成网格, 以便某些区域网格密度高于其它区域。 这样做的方式应该避免相交, 从而避免二己型绘图技术的吸引力。 对于在球体上精确的二异己型绘图来说, 一个主要工具是优化交通, 允许甚至在非连续源和目标密度之间进行异己型绘图。 然而, 最近, 优化信息运输得到了严格开发, 允许精确和不完全的异己型绘图, 并解决了较简单的部分差异方程式。 在手稿中, 我们用优化交通和最佳信息传输方法解决适应型网格问题, 并介绍如何将这些计算概括到更一般的元件。 我们选择与精准的趋同式解决方案进行这种比较, 但由于缺乏边界条件和部分差异方程式的比较原则, 通常对两种问题都具有挑战性。