A new flow dynamic approach for Wasserstein gradient flows

Abstract

We develop in this paper a new regularized flow dynamic approach to construct efficient numerical schemes for Wasserstein gradient flows in Lagrangian coordinates. Instead of approximating the Wasserstein distance which needs to solve constrained minimization problems, we reformulate the problem using the Benamou-Brenier's flow dynamic approach, leading to algorithms which only need to solve unconstrained minimization problem in $L^2$ distance. Our schemes automatically inherit some essential properties of Wasserstein gradient systems such as positivity-preserving, mass conservative and energy dissipation. We present ample numerical simulations of Porous-Medium equations, Keller-Segel equations and Aggregation equations to validate the accuracy and stability of the proposed schemes. Compared to numerical schemes in Eulerian coordinates, our new schemes can capture sharp interfaces for various Wasserstein gradient flows using relatively smaller number of unknowns.

Figures (32)

• Figure 1: Relationship between the strong formula and the variational formula for Wasserstein gradient flows.
• Figure 2: A schematic illustration of a flow map $\bm x(\bm X,t)$ at a fixed time $t$: $\bm x(\bm X,t)$ maps $\Omega_0^{\bm X}$ to $\Omega_t^{\bm x}$. $\bm X$ is the Lagrangian coordinate while $\bm x$ is the Eulerian coordinate, and $F(\bm X,t)=\frac{\partial \bm x(\bm X,t)}{\partial \bm X}$ represents the deformation associated with the flow map.
• Figure 3: The evolutions of energy and mass with respective to time under $m=2$ with $M=800$, $\delta t=1/6400$.
• Figure 4: The evolutions of density, energy and mass for the Barenblatt solution with $M=800$, $\delta t=1/6400$.
• Figure 5: Density plots for the initial value \ref{['initial:wt']} with $m=2$, $\theta=0.25$, $M=800$, $\delta t=1/800$.

Theorems & Definitions (29)

• Remark 2.1
• Proposition 2.1
• proof
• Remark 2.2
• Remark 2.3
• Theorem 2.4
• proof
• Proposition 2.2
• Remark 2.5
• Theorem 2.6