Variable-preconditioned transformed primal-dual method for generalized Wasserstein Gradient Flows

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.