| Abstract: |
| Here we present two approaches for computing the solution of Optimal Transport PDE on the sphere, for a wide variety of cost functions, such as the squared geodesic, logarithmic, and other exotic cost functions arising in various optics inverse problems. The first method is based on a monotone discretization that performs computations on wide-stencil neighborhoods projected on local tangent planes. This approach comes with convergence guarantees even for non-smooth $C^1(\mathbb{S}^2)$ solutions and is the most efficient provably convergent scheme over this class of problems. We show the success of this method in tackling moving mesh and reflector antenna problems and present work towards establishing convergence rates. The second method is derived by extending the Optimal Transport problem onto a thin shell containing the sphere, and then computing the solution for this new extended problem, which is consistent with the solution for the problem on the sphere. The benefit of this method is that discretization does not have to be done on point clouds, but rather on Cartesian grids, leading to a much simpler discretization. |
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