We study a new variant of mathematical prediction-correction model for crowd motion. The prediction phase is handled by a transport equation where the vector field is computed via an eikonal equation $\Vert \nabla\varphi\Vert=f$, with a positive continuous function $f$ connected to the speed of the spontaneous travel. The correction phase is handled by a new version of the minimum flow problem. This model is flexible and can take into account different types of interactions between the agents, from gradient flow in Wassersetin space to granular type dynamics like in sandpile. Furthermore, different boundary conditions can be used, such as non-homogeneous Dirichlet (e.g., outings with different exit-cost penalty) and Neumann boundary conditions (e.g., entrances with different rates). Combining finite volume method for the transport equation and Chambolle-Pock's primal dual algorithm for the eikonal equation and minimum flow problem, we present numerical simulations to demonstrate the behavior in different scenarios.
翻译:我们为人群运动研究一个新的数学预测-校正模型。 预测阶段由迁移方程式处理, 该方程式通过电子方程式 $\ Vert \ nabla\ varphi\ Vert=f$计算矢量字段, 其正连续函数为f美元, 与自发旅行的速度相连接。 校正阶段由新版本的最低流量问题处理。 这个模型是灵活的, 可以考虑到各种物剂之间的不同类型互动, 从瓦塞塞廷空间的梯度流到颗粒型的动态, 如沙丘。 此外, 还可以使用不同的边界条件, 如非恒化的 Dirichlet (例如, 外出不同的离价罚款) 和纽曼边界条件( 例如, 入口与不同的费率) 。 将运输方程式的有限量法和Chambole- Pock 用于电子方程式和最小流量问题的初步双重算法结合起来, 我们提出数字模拟来演示不同情景中的行为。