The aim of this paper is twofold. Based on the geometric Wasserstein tangent space, we first introduce Wasserstein steepest descent flows. These are locally absolutely continuous curves in the Wasserstein space whose tangent vectors point into a steepest descent direction of a given functional. This allows the use of Euler forward schemes instead of minimizing movement schemes introduced by Jordan, Kinderlehrer and Otto. For locally Lipschitz continuous functionals which are $\lambda$-convex along generalized geodesics, we show that there exists a unique Wasserstein steepest descent flow which coincides with the Wasserstein gradient flow. The second aim is to study Wasserstein flows of the maximum mean discrepancy with respect to certain Riesz kernels. The crucial part is hereby the treatment of the interaction energy. Although it is not $\lambda$-convex along generalized geodesics, we give analytic expressions for Wasserstein steepest descent flows of the interaction energy starting at Dirac measures. In contrast to smooth kernels, the particle may explode, i.e., a Dirac measure becomes a non-Dirac one. The computation of steepest descent flows amounts to finding equilibrium measures with external fields, which nicely links Wasserstein flows of interaction energies with potential theory. Finally, we provide numerical simulations of Wasserstein steepest descent flows of discrepancies.
翻译:本文的目的是双重的。 根据瓦森斯坦平坦的几何空间, 我们首先引入了瓦塞斯坦最陡峭的下降流。 这些是瓦塞斯坦空间中当地绝对连续的曲线, 瓦塞斯坦空间的变相矢量指向某个功能的最陡峭的下降方向。 这允许使用尤勒前期计划, 而不是将约旦、 基德勒元首 和奥托 引入的移动计划最小化。 当地利普施茨 连续的功能是 $\ lambda$- convex 和 通用的地德学。 对于本地的利普施茨 连续的功能, 我们从 Dirac 测量开始, 瓦瑟斯坦最陡峭的下降流流与瓦塞斯坦梯度流的流相吻合。 第二个目标是研究瓦塞斯坦最深的流, 与某些Riesz内核内核内核内核内核流的最大值差异值差异。 关键部分是处理交互能量的能量。 虽然它不是$lambda$- convexx, 我们用解表达了自Dirac rodeal rodealalal modeal modeal exdealdeal la.