We propose a Variable-Preconditioned Transformed Primal-Dual (VPTPD) method for solving generalized Wasserstein gradient flows based on the structure-preserving JKO scheme. This is a nontrivial extension of the TPD method [Chen et al. (2025) SIAM J. Sci. Comput.] incorporating proximal splitting techniques to address the challenges arising from the nonsmoothness of the objective function. Our key contributions include: (i) a semi-implicit-explicit iterative scheme that combines proximal gradient steps with explicit gradient steps to treat the nonsmooth and smooth terms respectively; (ii) variable-dependent preconditioners constructed from the Hessian of a regularized objective to balance iteration count and per-iteration cost; (iii) a proof of existence and uniqueness of bounded solutions for the generalized proximal operator with the chosen preconditioner, along with a convergent and bound-preserving Newton solver; and (iv) an adaptive step-size strategy to improve robustness and accelerate convergence under poor Lipschitz conditions of the energy derivative. Comprehensive numerical experiments spanning from 1D to 3D settings demonstrate that our method achieves superior computational efficiency-achieving up to a 20$\times$ speedup over existing methods-thereby highlighting its broad applicability through several challenging simulations.

翻译：本文提出了一种基于结构保持JKO格式的变量预条件变换原始对偶（VPTPD）方法，用于求解广义Wasserstein梯度流。该方法是对TPD方法[Chen et al. (2025) SIAM J. Sci. Comput.]的重要扩展，通过引入邻近分裂技术处理目标函数非光滑性带来的挑战。我们的主要贡献包括：（i）提出半隐式-显式迭代格式，结合邻近梯度步与显式梯度步分别处理非光滑项与光滑项；（ii）基于正则化目标函数的Hessian矩阵构造变量依赖型预条件子，以平衡迭代次数与单步计算成本；（iii）证明所选预条件下广义邻近算子的有界解存在唯一性，并给出具有收敛性与保界性的牛顿求解器；（iv）设计自适应步长策略以增强鲁棒性，并在能量导数Lipschitz条件较差时加速收敛。从一维到三维的全面数值实验表明，本方法实现了卓越的计算效率——相比现有方法最高可达20倍加速比，并通过多个挑战性仿真案例凸显了其广泛的适用性。