Algorithm for Hamilton-Jacobi equations in density space via a generalized Hopf formula
This provides a practical numerical method for optimal transport and mean field games, which are important in applied mathematics and economics.
The authors propose a generalized Hopf formula to solve Hamilton-Jacobi equations in density space, overcoming the curse of infinite-dimensionality and enabling efficient computation via multi-level and coordinate descent methods.
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.