Journal of Scientific Computing

, Volume 80, Issue 2, pp 1195–1239

# Algorithm for Hamilton–Jacobi Equations in Density Space Via a Generalized Hopf Formula

• Yat Tin Chow
• Wuchen Li
• Stanley Osher
• Wotao Yin
Article

## Abstract

We design fast numerical methods for Hamilton–Jacobi equations in density space (HJD), which arises in optimal transport and mean field games. We proposes an algorithm using a generalized Hopf formula in density space. The formula helps transforming a problem from an optimal control problem in density space, which are constrained minimizations supported on both spatial and time variables, to an optimization problem over only one spatial variable. This transformation allows us to compute HJD efficiently via multi-level approaches and coordinate descent methods. Rigorous derivation of the Hopf formula is provided under restricted assumptions and for a relatively narrow case; meanwhile our practical investigation allows us to conjecture that the actual range of applicability should be wider, and therefore we conjecture the formula can be applied to a wider class of practical examples.

## Keywords

Hamilton–Jacobi equation in density space Generalized Hopf formula Mean field games Optimal transport

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## Authors and Affiliations

• Yat Tin Chow
• 1
Email author
• Wuchen Li
• 2
• Stanley Osher
• 2
• Wotao Yin
• 2
1. 1.Department of MathematicsUCRRiversideUSA
2. 2.Department of MathematicsUCLALos AngelesUSA