Periodic Reporting for period 1 - EYAWKAJKOS (Everything You Always Wanted to Know About the JKO Scheme)
Período documentado: 2023-09-01 hasta 2026-02-28
The project deals with the so-called Jordan-Kinderlehrer-Otto scheme, a time-discretization procedure consisting in a sequence of iterated optimization problems involving the Wasserstein distance W_2 between probability measures. This scheme allows to approximate the solutions of a wide class of PDEs (including many diffusion equations with possible aggregation effects) which have a variational structure w.r.t. the distance W2 but not w.r.t. Hilbertian distances. It has been used both for theoretical purposes (proving existence of solutions for new equations and studying their properties) and for numerical applications. Indeed, it naturally provides a time-discretization and, if coupled with efficient computational techniques for optimal transport problems, can be used for numerics. This project covers both equations which are well-studied (Fokker-Planck, for instance) and less classical ones (higher-order equations, crowd motion, cross-diffusion, sliced Wasserstein flow...). For the most classical ones, we mainly consider estimates and properties which are known for solutions of the continuous-in-time PDEs and try to prove sharp and equivalent analogues in the discrete setting: some of these results ($L^p$, Sobolev, BV...) have already been proven in the simplest cases ; the results in the classical case will provide techniques to be applied to the other equations, allowing to prove existence of solutions and to study their qualitative properties. Moreover, some estimates proven on each step of the JKO scheme can provide useful information for the numerical schemes, reducing the computational complexity or improving the quality of the convergence. During the project, the study of the JKO scheme is of course coupled with a deep study of the corresponding continuous-in-time PDEs, as in the case of the porous medium equation (new Lipschitz result, convergence rate to the incompressible limit...). A particular attention is devoted to the approximation of the variational problems in each step of the JKO scheme and their effect on the convergence.
The EYAWKAJKOS project has developed a broad and rigorous analysis of nonlinear diffusive partial differential equations (PDEs) through the lens of optimal transport and gradient flow theory,.
The work during the first two years of the project gave rise to important progress on the three research topics of he proposal.
In the (τ,x) part, i.e. the study of the time-discretization of gradient flows, the main achievements include new existing results for nonlinear gradient flows in W_p, new estimates for the solutions of the JKO scheme (first-order bounds on the Fisher information in the W_p case, but also second-order estimates in the form of Li-Yau or Aronson-Benilan bounds which will soon be announced), strong convergence results of the JKO scheme to the continuous equation including both linear Fokker-Planck and the 2D Keller-Segel system, and the comparison of the standard JKO with its entropic approximation.
In the (τ,h) part, i.e. the discretization of the variational problem of each step of the JKO, the main achievement consists in the proof of convergence under the sharp condition h/τ->0.
In the (t,x) part, i.e. the study of the corresponding continuous PDEs, many results have been obtained on the porous-medium equation with potentials, such as new Lipschitz bounds unifying a series of previous results, a sharp global convergence rate to the incompressible limit in the presence of convex potentials, and sharper Sobolev estimates on the pressure.
Some more exotic PDEs have also been considered, such as the sticky-reflecting diffusion which was proven to be a Wasserstein gradient flow of an entropy including a boundary part, or the sliced-Wassertein flow, for which the convergence in long-time when the target is the standard gaussian was proven, or the flow of the total variation, for which an existence result has been proven in a master thesis of the project and some new estimates are ongoing.
The work during the first two years of the project gave rise to important progress on the three research topics of he proposal.
In the (τ,x) part, i.e. the study of the time-discretization of gradient flows, the main achievements include new existing results for nonlinear gradient flows in W_p, new estimates for the solutions of the JKO scheme (first-order bounds on the Fisher information in the W_p case, but also second-order estimates in the form of Li-Yau or Aronson-Benilan bounds which will soon be announced), strong convergence results of the JKO scheme to the continuous equation including both linear Fokker-Planck and the 2D Keller-Segel system, and the comparison of the standard JKO with its entropic approximation.
In the (τ,h) part, i.e. the discretization of the variational problem of each step of the JKO, the main achievement consists in the proof of convergence under the sharp condition h/τ->0.
In the (t,x) part, i.e. the study of the corresponding continuous PDEs, many results have been obtained on the porous-medium equation with potentials, such as new Lipschitz bounds unifying a series of previous results, a sharp global convergence rate to the incompressible limit in the presence of convex potentials, and sharper Sobolev estimates on the pressure.
Some more exotic PDEs have also been considered, such as the sticky-reflecting diffusion which was proven to be a Wasserstein gradient flow of an entropy including a boundary part, or the sliced-Wassertein flow, for which the convergence in long-time when the target is the standard gaussian was proven, or the flow of the total variation, for which an existence result has been proven in a master thesis of the project and some new estimates are ongoing.
The results obtained by the EYAWKAJKOS project have exceeded the state of the arts in many different aspects.
As a first example, we mention the fact that the JKO scheme was originally introduced to treat the linear Fokker-Planck equation, that it naturally provides convergence in the weak sense of measures to the solution of the PDE, and that the original paper by Jordan,Kinderlehrer and Otto proved strong L^1 convergence. Among the first published results of EYAWKAJKOS there is a paper where this strong convergence is transformed into strong L^2 convergence of the second-order spatial derivatives of the solution. Generalizations to more general equations are ongoing.
To cite other remarkable results from the (τ,x) part, we underline that the existence results for nonlinear gradient flows obtained in the project are the first ones which do not require the assumption of the geodesic convexity of the functional and that the comparison of the standard JKO with its entropic approximation allows now to obtain convergence to the limit equation as soon as the ration between the entropic parameter ε and the time step τ tends to zero, differently from a previous result requiring εlog ε < τ^2. This fact has strong numerical implications, as it allows to choose a value of ε which is easier to deal numerically.
In the (t,x) part, the new Lipschitz bounds for the porous medium equation have been able to provide a C^1 polynomial rate of convergence to the Barenblatt profile which was unknown before. In what concerns the total variation flow (a fourth-)order PDE used in image denoising), the existence result we provided is the first one valid in any space dimension. Similarly, the convergence in long-time to the Gaussian target of the sliced Wasserstein flow is the first proven convergence result of this type, despite being limited to a very particular situation, and despite the numerical experiments showing evidence of a much more general convergence.
On the other hand, the results from the (τ,h) part will only be considered as a complete improvement of the state of the art when they will be transformed into an efficient numerical method to solve these PDEs, which is planned for the next period of the project.
As a first example, we mention the fact that the JKO scheme was originally introduced to treat the linear Fokker-Planck equation, that it naturally provides convergence in the weak sense of measures to the solution of the PDE, and that the original paper by Jordan,Kinderlehrer and Otto proved strong L^1 convergence. Among the first published results of EYAWKAJKOS there is a paper where this strong convergence is transformed into strong L^2 convergence of the second-order spatial derivatives of the solution. Generalizations to more general equations are ongoing.
To cite other remarkable results from the (τ,x) part, we underline that the existence results for nonlinear gradient flows obtained in the project are the first ones which do not require the assumption of the geodesic convexity of the functional and that the comparison of the standard JKO with its entropic approximation allows now to obtain convergence to the limit equation as soon as the ration between the entropic parameter ε and the time step τ tends to zero, differently from a previous result requiring εlog ε < τ^2. This fact has strong numerical implications, as it allows to choose a value of ε which is easier to deal numerically.
In the (t,x) part, the new Lipschitz bounds for the porous medium equation have been able to provide a C^1 polynomial rate of convergence to the Barenblatt profile which was unknown before. In what concerns the total variation flow (a fourth-)order PDE used in image denoising), the existence result we provided is the first one valid in any space dimension. Similarly, the convergence in long-time to the Gaussian target of the sliced Wasserstein flow is the first proven convergence result of this type, despite being limited to a very particular situation, and despite the numerical experiments showing evidence of a much more general convergence.
On the other hand, the results from the (τ,h) part will only be considered as a complete improvement of the state of the art when they will be transformed into an efficient numerical method to solve these PDEs, which is planned for the next period of the project.