Optimal Transport and PDE: Gradient Flows in the Wasserstein Metric (continued)
summary
Time discretization uses the implicit Euler/proximal map as a minimization problem and yields existence and uniqueness under lower semi continuity and λ‑convexity, with energy dissipation and contraction rates recovering e^{−λt} in the limit and τ‑order convergence of the scheme. The metric generalization defines a curve of maximal slope via the metric derivative and metric slope and constructs it by minimizing movements (JKO). In the Wasserstein metric AC2 curves supply a velocity solving the continuity equation and the velocity equals −∇(first variation), so the curve of maximal slope is the PDE solution.
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chapters (12)
- 0:00Start
- 0:26Time Discretization of the Gradient Flow
- 1:34Implicit Boiler Scheme
- 5:37Energy Dissipation
- 12:58Metric Derivative
- 14:26How Do You Define the Gradient
- 14:41The Metric Slope
- 17:21Curve of Maximal Slope
- 25:06What Is a Geodesic
- 28:04Minimizing Movement Scheme
- 44:21Chain Rule
- 52:50Definition of a Curve of Maximal Slope
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