Invariant integrated convexity profiles for Hamilton-Jacobi-Bellman equations and applications | Vidéo | Carmin.tv

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Invariant integrated convexity profiles for Hamilton-Jacobi-Bellman equations and applications

By Giovanni Conforti

Appears in collection : PDE & Probability in interaction: functional inequalities, optimal transport and particle systems / Interactions EDP/Probabilité: inégalités fonctionnelles, transport optimal et systèmes de particules

It has been known for a long time that Hamilton-Jacobi-Bellman (HJB) equations preserve convexity, namely if the terminal condition is convex, the solution stays convex at all times. Equivalently, log-concavity is preserved along the heat equation, namely if one starts with a log-concave density, then the solution stays log-concave at all times. Both these facts are a direct consequence of Prékopa-Leindler inequality. In this talk, I will illustrate how a careful second-order analysis on coupling by reflection on the characteristics of the HJB equation reveals the existence of weaker notions of convexity that propagate backward along HJB. More precisely, by introducing the notion of integrated convexity profile, we are able to construct families of functions that fail to be convex, but are still invariant under the action of the HJB equation. In the second part of the talk I will illustrate some applications of these invariance results to the exponential convergence of learning algorithms for entropic optimal transport.

Information about the video

Date of recording 23/01/2024
Date of publication 14/02/2024
Institution CIRM
Licence CC BY NC ND
Language English
Audience Researchers, Graduate Students
Director(s) Luca Recanzone
Format MP4

Citation data

DOI 10.24350/CIRM.V.20129103
Cite this video Conforti, Giovanni (23/01/2024). Invariant integrated convexity profiles for Hamilton-Jacobi-Bellman equations and applications. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.20129103
URL https://dx.doi.org/10.24350/CIRM.V.20129103

Domain(s)

Probability

Bibliography

CONFORTI, Giovanni. Weak semiconvexity estimates for Schrödinger potentials and logarithmic Sobolev inequality for Schrödinger bridges. arXiv preprint arXiv:2301.00083, 2022. - https://doi.org/10.48550/arXiv.2301.00083

MSC codes

49Q22 Optimal transportation

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