Wasserstein gradient flows

$$\partial_t\rho-\mathrm{div}(\rho\nabla\phi)=0,$$

$$\phi=\delta U(\rho)$$

on a domain $\Omega\subset\mathbb{R}^2$, with initial condition $\rho(t=0)=\rho_0$.

JKO scheme

Dual formulation

$$\sup_{\phi,\psi}\,\langle\psi,\mu\rangle - U^*(\phi)$$

over $(\phi,\psi)$ s.t.

$$\psi(x)-\phi(y)\le\frac{|x-y|^2}{2\tau}.$$

$$\inf_\rho U(\rho)+\frac{1}{2\tau}W_2^2(\mu,\rho).$$

Dual formulations

Lemma: $U^*$ is increasing, i.e.

$$\phi_1\le\phi_2 \implies U^*(\phi_1)\le U^*(\phi_2).$$

Dual formulations

with $\phi^c(x)=\inf_y\phi(y)+\frac{|x-y|^2}{2\tau}$,

        $\psi^c(y)=\sup_x\psi(x)-\frac{|x-y|^2}{2\tau}$.

☇

$$\sup_\phi\langle\phi^c,\mu\rangle-U^*(\phi)=:J(\phi)$$

$$\sup_\psi\langle\psi,\mu\rangle-U^*(\psi^c)=:I(\psi)$$

Dual formulations

→ unconstrained concave maximization problems

→ Recover $\rho^*$ from $\phi^*$ by

$$\rho^*=\delta U^*(\phi^*)$$

$$\sup_\phi\langle\phi^c,\mu\rangle-U^*(\phi)=:J(\phi)$$

$$\sup_\psi\langle\psi,\mu\rangle-U^*(\psi^c)=:I(\psi)$$

The power of duality

$$U(\rho)=\int_\Omega u_\infty(\rho(x))+V(x)\rho(x)\,dx.$$

Back-and-forth algorithm

$H$ is the Sobolev space

$$\|h\|_H^2=\int_\Omega\Theta_2|\nabla h(x)|^2+\Theta_1|h(x)|^2\,dx$$

What is $\nabla_{\!H}J(\phi)$ ?

Why $H$ ?

Gradient ascent of $J$ → get Hessian bound

$$0\le -\delta^2\!J(\phi)(h,h)\le \|h\|_H^2$$

Why $H$ ?

$$U^*(\phi)=\int_\Omega u^*_\infty(\phi(x)-V(x))\,dx$$

Movies

Slow diffusion (porous medium eq)

$V(x)=-\sin(5\pi x_1)\sin(3\pi x_2)$

$512\times 512$ points

$m=2$

$m=4$

Slow diffusion

$m=4$

$V(x)=\|x-a\|^2$

$512\times 512$ points

Incompressible

$V(x)=\|x-a\|^2$

$1024\times 1024$ points

Aggregation-diffusion

$$U(\rho)=\int \rho(x)^3dx+\iint |x-y|^2\rho(x)\rho(y)\,dxdy$$

Thanks!

Paper: https://arxiv.org/pdf/2011.08151.pdf

Code: https://github.com/wonjunee/wgfBFMcodes

Doc: https://wasserstein-gradient-flows.netlify.app/