The Multiscale Nonlocal-in-Time Schrödinger Bridge Problem
Yiqun Li, Wuchen Li, Hong Wang · Multiscale Modeling and Simulation · 2025
Abstract. We formulate the multiscale nonlocal-in-time Schrödinger bridge problem (SBP) constrained by a time-fractional Wasserstein gradient flow with the advective flux. A strongly coupled nonlinear system of a time-fractional equation and a time-fractional Hamilton–Jacobi equation is derived based on the first-order optimality condition. After changing of variables, we reformulate the nonlocal SBP as a related minimization problem constrained by a time-fractional transport equation. The corresponding cost function introduces an additional intricate term involving the time-fractional derivative compared with its classical analogue, and this reformulation again yields a coupled nonlinear system equivalent to the original system. The widely used general-proximal primal-dual hybrid gradient (G-prox PDHG) algorithm is extended to solve the nonlocal SBP, and we apply a preconditioner induced by the discretization scheme of the time-fractional model to accelerate the convergence of the algorithm. Numerical experiments between Gaussian distributions are performed to investigate the performance of the nonlocal SBP, which demonstrate the reliability, efficiency, and effectiveness of our proposed algorithm and the multiscale feature of the nonlocal SBP.