Jordan–Kinderlehrer–Otto scheme

Remarks

Example: let $\beta > 0$, let $\Psi \colon \R^n \to [0, \infty)$ be smooth with $\abs{\nabla \Psi(x)} \le C(\Psi(x) + 1)$ for a constant $C$, and let $$F(\rho) = \int \Psi \rho \, dx + \beta^{-1} \int \rho \log \rho \, dx$$ for $\rho$ with a density, and $F(\rho) = +\infty$ otherwise. Jordan, Kinderlehrer and Otto proved that minimizers exist at each step, and that $\rho^\tau$ converges, as $\tau \to 0$, to the solution of the Fokker–Planck equation $\partial_t \rho = \operatorname{div}(\rho \nabla \Psi) + \beta^{-1} \Delta \rho$ with initial datum $\rho_0$. With $\Psi = 0$ and $\beta = 1$ this gives the heat equation $\partial_t \rho = \Delta \rho$.

Non-example: replacing $W_2$ by the $L^2$ distance between densities gives a different scheme, the implicit Euler scheme for the $L^2$ gradient flow. For the entropy $\int \rho \log \rho$ this flow is $\partial_t \rho = -(\log \rho + 1)$, which is not the heat equation.

The step is the Wasserstein analogue of the implicit Euler step $x_{k+1} \in \operatorname{argmin}_x \left( f(x) + \frac{1}{2\tau} \abs{x - x_k}^2 \right)$ for the gradient flow $\dot x = -\nabla f(x)$ in $\R^n$.