Crash Course on Optimal Transport (continued)
summary
Defines gradient flow on a Hilbert space by matching differential and gradient through a chosen inner product and shows the Dirichlet energy’s L2 subdifferential is −Δu, so its gradient flow is the heat equation. Derives the Benamou–Brenier formula: W2^2 is the infimum of the action ∫∫ |v|^2 ρ dx dt subject to the continuity equation. Optimal v is ∇ψ solving a Neumann elliptic problem, which yields the W2 scalar product and the rule gradW2 F = −div(ρ ∇(δF/δρ)). Dirichlet energy and entropy give the heat flow.
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chapters (10)
- 0:00Start
- 0:07Guardian Flows
- 1:22Gradient Flow
- 7:06Define a Gradient Flow
- 16:54The Dirichlet Energy
- 20:48The Differential Structure on the Space of Probability Measures
- 23:34Associate a Flow to the Vector Field
- 24:17How You Define the Vector Field inside the outside the Boundary
- 26:58Definition of Action
- 28:22Venomous Formula
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