Algorithm for Hamilton-Jacobi equations in density space via a generalized Hopf formula

TL;DR

This paper introduces a fast numerical method for Hamilton-Jacobi equations in density space, leveraging a generalized Hopf formula to reduce complex constrained problems to simpler spatial optimizations, enabling efficient computation.

Contribution

The paper proposes a novel generalized Hopf formula for density space Hamilton-Jacobi equations, overcoming high-dimensional challenges and enabling efficient numerical solutions.

Findings

Efficient computation of HJD using multi-level approaches.

Reduction of constrained minimization to spatial-only optimization.

Applicable to optimal transport and mean field games.

Abstract

We design fast numerical methods for Hamilton-Jacobi equations in density space (HJD), which arises in optimal transport and mean field games. We overcome the curse-of-infinite-dimensionality nature of HJD by proposing a generalized Hopf formula in density space. The formula transfers optimal control problems in density space, which are constrained minimizations supported on both spatial and time variables, to optimization problems over only spatial variables. This transformation allows us to compute HJD efficiently via multi-level approaches and coordinate descent methods.