OT methods for parabolic diffusion equations: the JKO scheme

This course has been given in two summer schools in summer 2024, in Rome La Sapienza (Vito Volterra Meeting, June 24-28) and in Chania (Festum Pi festival, July 9-13). In room it spanned across 2+2 hours, in Chania 1h45+1h30+1h45. The program I cover is more or less the same

Content and references

  • 1) Introduction to gradient flows and variants
    Implicit Euler, minimizing movements and the JKO. EDI and EVI definitions in metric spaces. Variants corresponding to x'=Dh*(-DF(x)) for convex h (power-like or not).
    See Sections 2 and 3 of the survey SGF.
  • 2) Introduction to OT, the Wasserstein space, and the PDE of Wasserstein gradient flows
    Monge and Kantorovich problems, optimality conditions, Wasserstein distances. The JKO scheme and the resulting PDEs.
    See Sections 4.1-4.2-4.3 of the survey SGF and, for the case in Wp, the paper CS1.
  • 3) Convergence of the JKO scheme via the limit of the PDE
    Piecewise constant and piecewise geodesic interpolations. Identification of the limits.
    See Section 4.4 of the survey SGF or Chapter 8 in the book OTAM.
  • 4) Convergence of the JKO scheme via an EDI formulation
    Back to the EDI formulation to characterize the solutions. The geodesically convex case (not covered in Rome): flow interchange. The general case via the variational interpolation.
    See the paper CS1. The flow interchange technique was introduced in MMS.
  • 5) The five-gradients inequality and BV estimates
    Proof with a rest in the quadratic case, only the inequality in the general case. Decrease of the BV norm along porous-medium type equations.
    See the paper DMSV for the quadratic case and the BV estimate. See the paper Cai for the non-quadratic case of the five-gradients inequality. See the last section in CS1 for BV estimates in the doubly nonlinear case.
  • 6) Fisher information estimates
    Decrease of fisher-like quantities along gradient flows of the entropy. Linear case with a potential, and nonlinear but 1-homogeneous case. Application to Lipschitz and continuity estimates.
    See the paper DS and CS2.
  • 7) Application to the log-Sobolev inequality
    (not covered in Rome)
    Rate of decrease of the entropy and of the Fisher information along the steps of the JKO scheme.
    Unofrtunately these results are not written yet...
  • 8) Strong L2H2 convergence of the JKO scheme
    Integral second-order estimates for the JKO scheme of the Fokker-Planck equation and applications to its strong convergence.
    See the paper ST.

    • Links to references

    SGF: a survey on gradient flows written in 2017, see here.
    CS1: a paper with Thibault Caillet on the Wp case, see here. This paper generalizes previous results by Agueh and Otto.
    OTAM: My book Optimal Transport for Applied Mathematicians (2015), see here or here for a non-official version.
    MMS: a paper by Matthes, McCann and Savaré on 4th equations, where they first introduce the flow-interchange technique, see here.
    DMSV: the paper with De Philippis, Mészáros and Velichkov on the five-gradients inequality and its application to BV estimates, see here.
    Cai: a paper by Thibault which generalizes the five-gradients inequality to other costs, see here.
    DS: a paper with Simone Di Marino where we prove Sobolev-like estimates (using Fisher-like quantities) on the JKO scheme for equations with linear diffusion, see here.
    CS2: a paper with Thibault on Fisher-like estimates for doubly nonlinear but 1-homogeneous equations. See here.
    ST: a paper with Gayrat Toshpulatov on the strong convergence of the JKO scheme for the linear Fokker-Planck equation. See here.