施蒂费尔流形上瓦瑟斯坦梯度流的动态逼近方案
A Dynamical Approximation Scheme on the Stiefel manifold for Wasserstein Gradient Flows
AI总结：
该研究提出无网格拉格朗日动力学方法逼近瓦瑟斯坦梯度流，通过在特定线性子空间中逼近传输映射，利用狄拉克 - 弗伦克尔原理在施蒂费尔流形上演化正交框架，证明其能诱导概率测度曲线，保留能量耗散结构，数值实验验证了方法的有效性。
AI中文摘要：
我们提出了一种用于逼近瓦瑟斯坦梯度流（WGFs）的无网格拉格朗日动力学方法。演化测度通过加权希尔伯特空间\(L^2_{\mu_0}\)中的传输映射表示为初始测度\(\mu_0\)的推送。我们在\(L^2_{\mu_0}\)的时间相关线性子空间中逼近此映射，其正交框架通过狄拉克 - 弗伦克尔动力学原理在受限于有限维背景空间的施蒂费尔流形上演化，通过WGF速度场的局部逼近自适应构建。我们证明所得传输映射在瓦瑟斯坦空间中诱导出概率测度的绝对连续曲线，其速度通过将精确的WGF速度投影到背景空间获得，并且表明该逼近在速度投影误差范围内保留能量耗散结构。此外，对于测地凸能量，我们通过背景空间的自适应构建推导出控制此类投影误差的后验估计，还给出了推送测度在瓦瑟斯坦度量下的逼近误差界。对线性和非线性福克 - 普朗克方程、多孔介质扩散和相互作用能的数值实验证明了该方法的准确性、能量耗散特性以及自适应构建的优势。
英文摘要：
We propose a meshless Lagrangian dynamical method for approximating Wasserstein gradient flows (WGFs). The evolving measure is represented as the pushforward of the initial measure $μ_0$ through a transport map in the weighted Hilbert space $L^2_{μ_0}$. We approximate this map in time-dependent linear subspaces of $L^2_{μ_0}$, whose orthonormal frames are evolved by a Dirac--Frenkel dynamical principle on a Stiefel manifold constrained to a finite-dimensional background space, adaptively constructed via local approximations of the WGF velocity field. We prove that the resulting transport map induces an absolutely continuous curve of probability measures in Wasserstein space, whose velocity is obtained by projecting the exact WGF velocity onto the background space, and we show that the approximation preserves the energy dissipation structure up to the projection error of the velocity. Moreover, for geodesically convex energies, we derive an a posteriori estimate controlling such projection error through the adaptive construction of the background space, yielding as well a bound on the approximation error of the pushforward measure in the Wasserstein metric. Numerical experiments on linear and nonlinear Fokker--Planck equations, porous-medium diffusion, and interaction energies demonstrate the accuracy of the method, its energy-dissipation properties, and the advantages of the adaptive construction.
