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

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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.

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Authors and Affiliations

  1. Department of Mathematics, UCR, Riverside, CA, 92521, USA

    Yat Tin Chow

  2. Department of Mathematics, UCLA, Los Angeles, CA, 90095-1555, USA

    Wuchen Li, Stanley Osher & Wotao Yin

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  1. Yat Tin Chow
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  2. Wuchen Li
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  3. Stanley Osher
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  4. Wotao Yin
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Correspondence to Yat Tin Chow.

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Research supported by AFOSR MURI Proposal Numbers 18RT0073, ONR N000141712162 and NSF DMS-1720237, AFOSR MURI Proposal Number 18RT0073, ONR Grant: N00014-1-0444, N00014-16-1-2119, N00014-16-215-1, NSF Grant ECCS-1462398 and DOE Grant DE-SC00183838.

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Chow, Y.T., Li, W., Osher, S. et al. Algorithm for Hamilton–Jacobi Equations in Density Space Via a Generalized Hopf Formula. J Sci Comput 80, 1195–1239 (2019). https://doi.org/10.1007/s10915-019-00972-9

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  • DOI: https://doi.org/10.1007/s10915-019-00972-9

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