Evans-Gangbo输运密度系统的一个凸数值格式（一维、二维和三维）
A convex numerical scheme for the Evans-Gangbo transport-density system, in one, two, and three dimensions
- University of Cheikh Anta Diop（谢赫·安塔·迪奥普大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
提出一种基于凸优化的数值格式，求解一至三维Evans-Gangbo输运密度系统，通过Kantorovich势和密度构建最优输运映射，并经强对偶性等验证，扩展至三维。
AI中文摘要：
我们针对一维、二维和三维的Monge最优输运问题给出了一种构造性数值处理方法，遵循Evans-Gangbo方法：映射由Kantorovich势$u$和密度$a$构建，它们满足$-\text{div}(a\nabla u) = f^+-f^-$，通过$\dot T = -a\nabla u(T)/[(1-t)f^+(T)+tf^-(T)]$，$T(0,x)=x$。在一维情形下，这给出了闭式解。在二维情形下——先前报告中未完成的部分——我们通过一个凸的Beckmann流问题（二阶锥规划）获得$(u,a)$，并与对偶线性规划交叉验证，然后通过直接积分或基于射线的1D“时钟”常微分方程（每条射线具有显式密度）构建映射。验证使用强对偶性、质量守恒和正则化消失极限，在一个平移不变情形和两个二维例子（径向锥、棋盘格）上通过对称性检查；两者都显示出边界集中的质量和支撑接触处的非唯一通量。我们将其扩展到三维（理论不变），与二维答案匹配到六位数字，并在径向球上重现相同现象，揭示了二维可视化风格的局限性。展示输运映射的动画可作为本提交arXiv页面上的辅助文件获取。
英文摘要：
We give a constructive numerical treatment of Monge's optimal transport problem in 1D, 2D, 3D, following Evans-Gangbo: the map is built from a Kantorovich potential $u$ and density $a$ solving $-\mathrm{div}(a\nabla u) = f^+-f^-$, via $\dot T = -a\nabla u(T)/[(1-t)f^+(T)+tf^-(T)]$, $T(0,x)=x$. In 1D this gives a closed form. In 2D--left unfinished in an earlier report--we obtain $(u,a)$ via a convex Beckmann flow problem (second-order-cone programming), cross-checked against a dual LP, then build the map by direct integration or a ray-based 1D "clock" ODE with explicit density per ray. Validation uses strong duality, mass conservation, and vanishing-regularization limits, on a translation-invariant case and two 2D examples (a radial cone, a checkerboard) checked by symmetry; both show boundary-concentrated mass and non-unique flux where supports touch. We extend to 3D (theory unchanged), matching the 2D answer to six digits and reproducing the same phenomena on a radial ball, exposing a limit of the 2D visualization style. Movies illustrating the transport maps are available as ancillary files on the arXiv page of this submission.
