包含Monge-Ampère算子的泛函的离散化
Discretization of functionals involving the Monge-Ampère operator
摘要：
在Jordan，Kinderlehrer和Otto（JKO）的开创性工作之后，Wasserstein空间中的梯度流已经成为分析扩散方程的有力工具。这个公式的数值应用受到了Wasserstein距离在维数>= 2时难以计算的限制。JKO格式的一个步骤等价于凸函数空间上的一个变分问题，它涉及到Monge-Ampère算子。凸性约束在数值上很难处理，但在我们的设置中，内能对这些约束起着障碍的作用。这使我们能够引入一个一致的离散化，它继承了连续变分问题的凸性。我们证明了我们的方法对非线性扩散和人群运动模型的有效性。
英文摘要：
Gradient flows in the Wasserstein space have become a powerful tool in the analysis of diffusion equations, following the seminal work of Jordan, Kinderlehrer and Otto (JKO). The numerical applications of this formulation have been limited by the difficulty to compute the Wasserstein distance in dimension >= 2. One step of the JKO scheme is equivalent to a variational problem on the space of convex functions, which involves the Monge-Ampère operator. Convexity constraints are notably difficult to handle numerically, but in our setting the internal energy plays the role of a barrier for these constraints. This enables us to introduce a consistent discretization, which inherits convexity properties of the continuous variational problem. We show the effectiveness of our approach on nonlinear diffusion and crowd-motion models.
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