发表机构
Tsinghua University（清华大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对曲面平均场规划问题，提出一种结合时间有限差分、空间曲面有限元及多项式保持恢复的梯度增强近端算法，并集成于ISTA、FISTA和Douglas-Rachford分裂中，推导了残差恒等式和扰动界，数值实验验证了其有效性。
AI 中文摘要
曲面上的平均场规划规定了初始和终态密度，并在满足连续性方程的前提下最小化输运能量。针对该问题的近端算法需要反复求解一个时间-空间泊松方程，其时间导数和曲面梯度决定了密度和动量的修正量。我们研究了一种该约束投影的梯度增强近似方法，在时间上采用有限差分，在空间上采用曲面有限元，并结合时间多项式保持恢复和空间参数多项式保持恢复。相同的构造被纳入ISTA、FISTA和Douglas-Rachford分裂算法中。我们将恢复更新与精确离散投影区分开来，并推导了残差恒等式以及条件有限迭代扰动界，这些界保留了数据、边界和线性求解器误差。现有的导数恢复估计在适当的正则性和网格假设下识别出高阶贡献；但它们本身并不能确立外部优化迭代的收敛性。我们讨论了球面上以及一个更复杂的代数曲面上的可用数值示例，并指出了记录的细化数据的局限性。
英文摘要
Mean field planning on a surface prescribes initial and terminal densities and minimizes a transport energy subject to the continuity equation. Proximal algorithms for this problem repeatedly solve a time--space Poisson equation, whose temporal derivative and surface gradient determine the density and momentum corrections. We study a gradient-enhanced approximation of this constraint projection using finite differences in time, surface finite elements in space, temporal polynomial preserving recovery, and spatial parametric polynomial preserving recovery. The same construction is incorporated into ISTA, FISTA, and Douglas--Rachford splitting. We distinguish the recovered update from an exact discrete projection and derive residual identities and conditional finite-iteration perturbation bounds that retain data, boundary, and linear-solver errors. Existing derivative-recovery estimates identify a higher-order contribution under suitable regularity and mesh assumptions; they do not by themselves establish convergence of the outer optimization iteration. Available numerical illustrations on the sphere and a more complicated algebraic surface are discussed together with the limits of the recorded refinement data.
