Wasserstein Gradient Flows

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This page briefly describes the paper, The back-and-forth method for Wasserstein gradient flows [1]. To see the details, check the PDF and Code links above.

In this project, we proposed a new algorithm to compute the following class of PDEs (also known as Darcy's law):

$$\begin{array}{rl}{\partial }_t\rho -\mathrm{\nabla }\cdot (\rho \mathrm{\nabla }\varphi )=0& \\ \varphi =\delta U(\rho )& \end{array}$$

where $\rho$ is a mass density, $\varphi$ is a pressure, and $\delta U(\rho )$ is the Frechet derivative of $U$ at $\rho$. This class of PDEs models various physical phenomena such as fluid flow, heat transfer, aggregation-diffusion, and crowd motion. For example, if

$$U(\rho )=\frac{1}{m-1}\int {\rho }^{m}dx,$$

then the PDE becomes the porous medium equation,

$${\partial }_t\rho -\mathrm{\Delta }({\rho }^m)=0.$$

We solve this PDE via the JKO scheme [2], a discrete-in-time variational formulation of the PDE. Given ${\rho }^{(0)}$, the scheme iterates

$${\rho }^{(n+1)}=\text{arg}\underset{\rho }{min}U(\rho )+\frac{1}{2\tau }{W}_2(\rho ,{\rho }^{(n)}{)}^2$$

where $\tau$ is a time step size, $W_2$ is a 2-Wasserstein distance, and ${\rho }^{(n)}$ is an approximate solution $\rho (n\tau ,\cdot )$ of the PDE. We solve developed an algorithm to solve the JKO scheme expanding upon the back-and-forth method (BFM) [3], a fast numerical method for optimal transport problems.

Below are some of our results solving difficult PDE problems including porous medium equations, incompressible flows, and aggregation-diffusion equations.

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Porous medium equations with an obstacle and potential

$$\begin{array}{rl}U(\rho )& =\int \frac{1}{3}{\rho }^4(x)+V(x)dx\\ V(x)& =\Vert x-a{\Vert }^2,a=(0.9,0.9).\end{array}$$

Aggregation-diffusion equations

$$U(\rho )=\int {\rho }^2(x)dx+\int \int \rho (x)\Vert x-y{\Vert }^2\rho (y)dxdy.$$

Incompressible flows (crowd motion) with an obstacle and potential

$$\begin{array}{rl}U(\rho )& =\int u_{\mathrm{\infty }}(\rho (x))+V(x)dx\\ V(x)& =\Vert x-a{\Vert }^2,a=(0.9,0.9)\end{array}$$

$$u_{\mathrm{\infty }}(t)=\{\begin{array}{ll}0& \text{if }0\le t\le 1\\ \mathrm{\infty }& \text{otherwise.}\end{array}$$

References

[1] Matt Jacobs, Wonjun Lee and Flavien Léger. The back-and-forth method for Wasserstein gradient flows. 2020.

[2] Richard Jordan and David Kinderlehrer and Felix Otto. The variational formulation of the Fokker-Planck equation. SIAM journal on mathematical analysis 29.1 (1998): 1–17.

[3] Matt Jacobs and Flavien Léger. A fast approach to optimal transport: The back-and-forth method. Numerische Mathematik (2020): 1-32.

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