AI 中文总结
本文针对动态最优传输数值求解的不稳定与内存开销大问题，提出结合线性化交替方向乘子法的重构方案与分布式变量划分方法，通过实验验证了方法的稳定性、鲁棒性与可扩展性。
AI 中文摘要
本文针对动态最优传输（OT）问题数值求解中的两个核心挑战展开研究。第一个挑战是当初始或终端密度趋近于零且无正下界时，传统方法可能变得不稳定或计算效率低下，因为目标函数的Lipschitz常数会随密度的立方的倒数缩放，导致近零区域收敛缓慢甚至发散。第二个挑战是动态形式的内存开销巨大，在精细时空网格上离散时需存储整个时空域的变量，随着网格加密或空间维度提升，存储负担会迅速变得难以承受。为克服第一个困难，我们对经典离散动态OT问题进行重构，使所得目标函数具有精确的邻近映射，结合线性化交替方向乘子法（LADMM）后，得到一种高效且鲁棒的算法，即使在密度可能消失的复杂场景中仍保持稳定。为降低内存负担，我们进一步引入分布式形式，将优化变量划分到多个智能体，大幅降低每个智能体的存储需求，还可实现计算加速。我们在一维和二维空间设置下，针对不同难度水平开展数值实验验证所提框架，结果表明，与传统方法相比，该方法具有稳定性、鲁棒性和可扩展性。
英文摘要
In this paper, we address two fundamental challenges in the numerical solution of dynamic opti- mal transport (OT) problems. The first challenge arises when the initial and/or terminal densities approach zero and no positive lower bound is available. In this regime, conventional methods may become unstable or computationally inefficient, since the Lipschitz constant of the objective can scale like the reciprocal of the cube of the density. As a result, near-zero regions may lead to slow convergence or even divergence. The second challenge concerns the substantial memory cost of the dynamic formulation, whose discretization over fine spatial and temporal grids requires storing vari- ables across the entire space-time domain. This storage burden quickly becomes prohibitive as the grid is refined or the spatial dimension increases. To overcome the first difficulty, we reformulate the classical discretized dynamic OT problem so that the resulting objective admits an exact proximal mapping. When combined with a linearized alternating direction method of multipliers (LADMM), this reformulation yields an efficient and robust algorithm that remains stable even in challenging settings where the density may vanish. To reduce the memory burden, we further introduce a dis- tributed formulation in which the optimization variables are partitioned across multiple agents. This design substantially lowers the storage requirement for each agent and can also lead to computational acceleration. We validate the proposed framework through numerical experiments in one- and two- dimensional spatial settings under varying levels of difficulty. The results demonstrate the stability, robustness, and scalability of the proposed method in comparison with conventional approaches.
