论文标题
最佳运输中的强大C- con虫和稳定性
Strong c-concavity and stability in optimal transport
论文作者
论文摘要
从数学角度来看，解决方案在措施变化下对最佳运输问题的稳定性是基本的：它与数值方法的收敛密切相关，以解决最佳运输问题并证明最佳运输的许多应用是合理的。在本文中，我们介绍了强大的C型腔的概念，我们表明，它在证明稳定性的过程中起着重要作用，从而导致一般成本功能的最佳运输c。然后，我们引入了一个差异标准，以证明函数是强烈的C-Concave，这是关于Ma-Trudinger-Wang最初用于建立最佳运输图的规律性的成本的假设。最后，我们提供了两个示例，可以在其中应用这种稳定性结果，以供成本函数，在球体上具有值 +$ \ infty $：反射器问题和高斯曲率测量处方问题。
The stability of solutions to optimal transport problems under variation of the measures is fundamental from a mathematical viewpoint: it is closely related to the convergence of numerical approaches to solve optimal transport problems and justifies many of the applications of optimal transport. In this article, we introduce the notion of strong c-concavity, and we show that it plays an important role for proving stability results in optimal transport for general cost functions c. We then introduce a differential criterion for proving that a function is strongly c-concave, under an hypothesis on the cost introduced originally by Ma-Trudinger-Wang for establishing regularity of optimal transport maps. Finally, we provide two examples where this stability result can be applied, for cost functions taking value +$\infty$ on the sphere: the reflector problem and the Gaussian curvature measure prescription problem.
