In this paper, we characterize the extremal points of the unit ball of the Benamou–Brenier energy and of a coercive generalization of it, both subjected to the homogeneous continuity equation constraint. We prove that extremal points consist of pairs of measures concentrated on absolutely continuous curves which are characteristics of the continuity equation. Then, we apply this result to provide a representation formula for sparse solutions of dynamic inverse problems with finite-dimensional data and optimal-transport based regularization.

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关于 Benamou-Brenier 能量球的极值点

在本文中，我们描述了 Benamou-Brenier 能量单位球的极值点及其强制推广，两者均受齐次连续性方程约束。我们证明极值点由集中在绝对连续曲线上的成对测度组成，这些曲线是连续性方程的特征。然后，我们应用该结果为具有有限维数据和基于最优传输的正则化的动态逆问题的稀疏解提供表示公式。