Abstract
We construct Two-Point Flux Approximation (TPFA) finite volume schemes to solve the quadratic optimal transport problem in its dynamic form, namely the problem originally introduced by Benamou and Brenier. We show numerically that these type of discretizations are prone to form instabilities in their more natural implementation, and we propose a variation based on nested meshes in order to overcome these issues. Despite the lack of strict convexity of the problem, we also derive quantitative estimates on the convergence of the method, at least for the discrete potential and the discrete cost. Finally, we introduce a strategy based on the barrier method to solve the discrete optimization problem.
Highlights
The theory of optimal transport provides a robust way to define an interpolation between probability measures which takes into account the geometry of the space where they are defined
Several numerical methods are available to solve optimal transport problems and in particular to compute the associated interpolations between measures
Only few of these can be generalized to more complex settings which are relevant for numerical modelling, and their numerical analysis is often neglected
Summary
The theory of optimal transport provides a robust way to define an interpolation between probability measures which takes into account the geometry of the space where they are defined This theory is built around the problem of finding the optimal way of reallocating one given density into another, minimizing a total cost of displacement in space. We consider finite volume discretizations of the so-called dynamical formulation of such a problem, following the approach originally proposed by Benamou and Brenier [3]. This formulation has inspired some of the first numerical methods for optimal transport, but it is still one of the most general, since it can be adapted to very complex settings. We tackle the issue of the efficient computation of numerical solutions by applying and analyzing a classical interior point strategy adapted to our setting
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