Lagrangian Discretization of Crowd Motion and Linear Diffusion
Lagrangian Discretization of Crowd Motion and Linear Diffusion
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人群运动的拉格朗日离散化和线性扩散
DOI：
10.1137/19m1274201
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发表时间：
2019
期刊：
影响因子：
--
通讯作者：
Federico Stra
中科院分区：
文献类型：
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作者：
H. Leclerc;Quentin M'erigot;F. Santambrogio;Federico Stra
我们研究了一个梯度矢量场下的人群运动模型，可能有额外的相互作用项，如吸引力/排斥力，并通过拉格朗日离散给出了其解的数值格式。所得粒子的密度约束是通过在每个时间步上的部分最优输运问题来实现的。我们证明了描述一维人群运动的连续PDE解的离散测度的收敛性。在第二部分中，我们展示了如何使用类似的方法来构造线性平流扩散方程的拉格朗日离散化，解释为Wasserstein空间中的梯度流。我们还提供了一个二维的数值实现来证明计算的可行性。
We study a model of crowd motion following a gradient vector field, with possibly additional interaction terms such as attraction/repulsion, and we present a numerical scheme for its solution through a Lagrangian discretization. The density constraint of the resulting particles is enforced by means of a partial optimal transport problem at each time step. We prove the convergence of the discrete measures to a solution of the continuous PDE describing the crowd motion in dimension one. In a second part, we show how a similar approach can be used to construct a Lagrangian discretization of a linear advection-diffusion equation, interpreted as a gradient flow in Wasserstein space. We provide also a numerical implementation in 2D to demonstrate the feasibility of the computations.