论文标题
最佳运输的$ l^1 $ - 收集原则
The $L^1$-contraction principle in optimal transport
论文作者
论文摘要
在这项工作中，我们使用JKO方案来近似达西定律产生的一般扩散问题。尽管该方案现在是经典的，但是如果能量密度在空间上是不均匀的或不规则的，则许多标准方法无法应用于在连续限制中建立收敛。为了克服这些困难，我们通过其双重问题分析了该方案，并为密度变量建立了一种新颖的$ l^1 $ contraction原理。值得注意的是，收缩原理仅依赖于最佳传输图的存在和能量的凸结构。结果，该原理在非常通用的环境中，并为使用基于最佳传播的变异方案打开了大门，以研究一类较大的非线性不均匀抛物线方程。
In this work, we use the JKO scheme to approximate a general class of diffusion problems generated by Darcy's law. Although the scheme is now classical, if the energy density is spatially inhomogeneous or irregular, many standard methods fail to apply to establish convergence in the continuum limit. To overcome these difficulties, we analyze the scheme through its dual problem and establish a novel $L^1$-contraction principle for the density variable. Notably, the contraction principle relies only on the existence of an optimal transport map and the convexity structure of the energy. As a result, the principle holds in a very general setting, and opens the door to using optimal-transport-based variational schemes to study a larger class of non-linear inhomogeneous parabolic equations.
