Primal Dual Methods for Wasserstein Gradient Flows
Primal Dual Methods for Wasserstein Gradient Flows
复制标题
Wasserstein 梯度流的原始对偶方法
DOI：
10.1007/s10208-021-09503-1
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发表时间：
2021
影响因子：
3
通讯作者：
Wei, Chaozhen
中科院分区：
文献类型：
--
作者：
Carrillo, José A.;Craig, Katy;Wang, Li;Wei, Chaozhen
结合经典的最优输运理论和现代算子分裂技术，我们开发了一种新的非线性非局部偏微分方程的数值方法，出现在多孔介质、材料科学和生物群模型中。我们的方法如下：首先，我们通过经典的JKO格式或通过我们引入的新颖的crank - nicolson型方法进行时间离散。接下来，我们使用Wasserstein距离的Benamou-Brenier动态表征来减少离散时间方程解的计算，以解决具有严格凸目标函数和线性约束的完全离散最小化问题。第三，我们通过应用最近引入的，可证明收敛的三个算子的原始对偶分裂方案来计算最小值（Yan in J Sci compuput 1 - 20,2018）。通过利用偏微分方程潜在的变分结构，我们的方法克服了以往建立在显式时间离散化上的数值工作中存在的稳定性问题，这些问题由于方程的强非线性和退化而受到影响。我们的方法也是自然的正性和质量守恒，在JKO方案的情况下，能量是递减的。我们证明了随着空间离散化的细化，完全离散问题的极小值收敛于空间连续离散时间问题的极小值。最后，我们对一维和二维非线性偏微分方程和Wasserstein测地线进行了模拟，说明了我们的方法的关键特性，包括与经典JKO方法相比，我们的新crank - nicolson型方法的高阶收敛性。
Combining the classical theory of optimal transport with modern operator splitting techniques, we develop a new numerical method for nonlinear, nonlocal partial differential equations, arising in models of porous media, materials science, and biological swarming. Our method proceeds as follows: first, we discretize in time, either via the classical JKO scheme or via a novel Crank–Nicolson-type method we introduce. Next, we use the Benamou–Brenier dynamical characterization of the Wasserstein distance to reduce computing the solution of the discrete time equations to solving fully discrete minimization problems, with strictly convex objective functions and linear constraints. Third, we compute the minimizers by applying a recently introduced, provably convergent primal dual splitting scheme for three operators (Yan in J Sci Comput 1–20, 2018). By leveraging the PDEs’ underlying variational structure, our method overcomes stability issues present in previous numerical work built on explicit time discretizations, which suffer due to the equations’ strong nonlinearities and degeneracies. Our method is also naturally positivity and mass preserving and, in the case of the JKO scheme, energy decreasing. We prove that minimizers of the fully discrete problem converge to minimizers of the spatially continuous, discrete time problem as the spatial discretization is refined. We conclude with simulations of nonlinear PDEs and Wasserstein geodesics in one and two dimensions that illustrate the key properties of our approach, including higher-order convergence our novel Crank–Nicolson-type method, when compared to the classical JKO method.