ICTS

We introduce Gamma-convergence, covering its definition, basic properties, and compactness results.

We introduce Gamma-convergence, covering its definition, basic properties, and compactness results

We introduce the theory of large deviations, beginning with Cramer's theorem and its historical significance. Particular emphasis will be placed on the analytic structures underlying the theory, especially convex duality and the Fenchel–Legendre transform, and on how these ideas arise naturally in the study of rare events. We will discuss fundamental results such as the Sanov's theorem, and elements of Donsker–Varadhan theory, highlighting the unifying role of analysis in understanding probabilistic phenomena.

We introduce Gamma-convergence, covering its definition, basic properties, and compactness results.

This lecture series will explore the fundamentals of jump processes and their associated large-deviation principles as a motivation for generalized gradient structures. It will also cover the definitions and preliminary results related to generalized gradient structures, along with existence and uniqueness results for generalized gradient flows. Finally, the series will discuss applications of these concepts to reversible interacting population dynamics.

We introduce Gamma-convergence, covering its definition, basic properties, and compactness results.

This series of lectures intends to cover

(1) Introduction to the optimal transport theory. Monge and Kantorovich problems, Wasserstein distances, connections with the Monge Ampère equation.

(2) The Jordan-Kinderlehrer-Otto scheme: brief reminders about gradient flows in linear and metric spaces, Implicit Euler scheme, minimizing movements and the JKO scheme; Variants corresponding to x'=Dh*(-DF(x)) for convex h (power-like or not); the energy dissipation principle in the Wasserstein space; first properties of the JKO scheme.

(3) Convergence of the JKO scheme via the limit of the PDE (piecewise constant and piecewise geodesic interpolations, identification of the limits) and via the energy dissipation principle (the geodesically convex case via the flow interchange and the general case via the variational interpolation).

(4) Higher-order estimates and applications to the PDE (decrease of the Fisher information), to functional inequalities (proof of  the log-Sobolev inequality using the heat flow and using the JKO scheme, and generalizations), and to the strong convergence of the JKO scheme (strong L2H2 convergence via second-order estimates for the JKO scheme in the case of the Fokker-Planck equation).

We will discuss that the porous medium equation can be read as a gradient flow on the space of probability densities equipped with the Wasserstein–Otto metric. We also discuss some associated inverse problems as well.

This lecture series will study interacting particle systems through the lens of gradient flows, free energies, and their large-particle limits. The central theme is that many systems of weakly interacting particles possess a finite-dimensional gradient-flow structure, and that, as the number of particles tends to infinity, this structure converges to a macroscopic gradient flow for a limiting free energy. We will begin with mean-field particle systems and their associated Gibbs measures. The empirical measure will provide the bridge between microscopic and macroscopic descriptions. At the particle level, the evolution dissipates a finite-(N) free energy; in the large-(N) limit, one obtains nonlinear Fokker–Planck or McKean–Vlasov equations, which are themselves gradient flows of the limiting free energy. A main goal of the course will be to explain how this variational structure can be used not only to identify the limiting equation, but also to prove convergence towards it.  In regimes where the free-energy landscape is stable, one can obtain quantitative convergence estimates, uniform-in-time propagation of chaos, and functional inequalities such as logarithmic Sobolev inequalities. These inequalities express, in finite dimensions, the same stability and dissipation properties that are visible in the limiting gradient flow. A second theme will be fluctuations. The law of large numbers corresponds to convergence towards minimisers or solutions of the limiting gradient flow, while fluctuations are governed by the second-order structure of the free energy. Around a stable equilibrium, the Hessian of the limiting free energy determines the Gaussian fluctuation field; dynamically, fluctuations are described by the linearised gradient flow together with the noise inherited from the particle system. Finally, we will discuss phase transitions from this gradient-flow viewpoint. A phase transition corresponds to a qualitative change in the free-energy landscape: loss of convexity, degeneration of the Hessian, loss of uniqueness of minimisers, symmetry breaking, metastability, or failure of uniform logarithmic Sobolev inequalities. Thus phase transitions mark precisely the regimes where the simple stability theory behind the mean-field limit and its fluctuations begins to break down.

This series of lectures intends to cover

(1) Introduction to the optimal transport theory. Monge and Kantorovich problems, Wasserstein distances, connections with the Monge Ampère equation.

(2) The Jordan-Kinderlehrer-Otto scheme: brief reminders about gradient flows in linear and metric spaces, Implicit Euler scheme, minimizing movements and the JKO scheme; Variants corresponding to x'=Dh*(-DF(x)) for convex h (power-like or not); the energy dissipation principle in the Wasserstein space; first properties of the JKO scheme.

(3) Convergence of the JKO scheme via the limit of the PDE (piecewise constant and piecewise geodesic interpolations, identification of the limits) and via the energy dissipation principle (the geodesically convex case via the flow interchange and the general case via the variational interpolation).

(4) Higher-order estimates and applications to the PDE (decrease of the Fisher information), to functional inequalities (proof of  the log-Sobolev inequality using the heat flow and using the JKO scheme, and generalizations), and to the strong convergence of the JKO scheme (strong L2H2 convergence via second-order estimates for the JKO scheme in the case of the Fokker-Planck equation).

We will discuss a variety of approaches to structure preserving spatial discretization of Wasserstein gradient flows, mainly nonlinear Fokker-Planck and fourth order thin film equations. Here "structure preserving" means that the discretized evolution is still a gradient flow, but not on all probability measures but only on a finite-dimensional subspace, with respect to a genuine Riemannian metric that mimics the L2-Wasserstein distance. We will start with one physical space dimension, where the natural discretization of the inverse distribution function leads to a Lagrangian scheme, which provides surprising geometric insights in the dynamics. In the last part, we will discuss graph-based discretizations that can be seen as finite-volume schemes. Analytical questions that we will address are, naturally, convergence, but also the preservation of qualitative properties, like global equilibration, metric contractivity, or the waiting time phenomenon.

This lecture series will study interacting particle systems through the lens of gradient flows, free energies, and their large-particle limits. The central theme is that many systems of weakly interacting particles possess a finite-dimensional gradient-flow structure, and that, as the number of particles tends to infinity, this structure converges to a macroscopic gradient flow for a limiting free energy. We will begin with mean-field particle systems and their associated Gibbs measures. The empirical measure will provide the bridge between microscopic and macroscopic descriptions. At the particle level, the evolution dissipates a finite-(N) free energy; in the large-(N) limit, one obtains nonlinear Fokker–Planck or McKean–Vlasov equations, which are themselves gradient flows of the limiting free energy. A main goal of the course will be to explain how this variational structure can be used not only to identify the limiting equation, but also to prove convergence towards it.  In regimes where the free-energy landscape is stable, one can obtain quantitative convergence estimates, uniform-in-time propagation of chaos, and functional inequalities such as logarithmic Sobolev inequalities. These inequalities express, in finite dimensions, the same stability and dissipation properties that are visible in the limiting gradient flow. A second theme will be fluctuations. The law of large numbers corresponds to convergence towards minimisers or solutions of the limiting gradient flow, while fluctuations are governed by the second-order structure of the free energy. Around a stable equilibrium, the Hessian of the limiting free energy determines the Gaussian fluctuation field; dynamically, fluctuations are described by the linearised gradient flow together with the noise inherited from the particle system. Finally, we will discuss phase transitions from this gradient-flow viewpoint. A phase transition corresponds to a qualitative change in the free-energy landscape: loss of convexity, degeneration of the Hessian, loss of uniqueness of minimisers, symmetry breaking, metastability, or failure of uniform logarithmic Sobolev inequalities. Thus phase transitions mark precisely the regimes where the simple stability theory behind the mean-field limit and its fluctuations begins to break down.

We will discuss a variety of approaches to structure preserving spatial discretization of Wasserstein gradient flows, mainly nonlinear Fokker-Planck and fourth order thin film equations. Here "structure preserving" means that the discretized evolution is still a gradient flow, but not on all probability measures but only on a finite-dimensional subspace, with respect to a genuine Riemannian metric that mimics the L2-Wasserstein distance. We will start with one physical space dimension, where the natural discretization of the inverse distribution function leads to a Lagrangian scheme, which provides surprising geometric insights in the dynamics. In the last part, we will discuss graph-based discretizations that can be seen as finite-volume schemes. Analytical questions that we will address are, naturally, convergence, but also the preservation of qualitative properties, like global equilibration, metric contractivity, or the waiting time phenomenon.

This lecture series will study interacting particle systems through the lens of gradient flows, free energies, and their large-particle limits. The central theme is that many systems of weakly interacting particles possess a finite-dimensional gradient-flow structure, and that, as the number of particles tends to infinity, this structure converges to a macroscopic gradient flow for a limiting free energy. We will begin with mean-field particle systems and their associated Gibbs measures. The empirical measure will provide the bridge between microscopic and macroscopic descriptions. At the particle level, the evolution dissipates a finite-(N) free energy; in the large-(N) limit, one obtains nonlinear Fokker–Planck or McKean–Vlasov equations, which are themselves gradient flows of the limiting free energy. A main goal of the course will be to explain how this variational structure can be used not only to identify the limiting equation, but also to prove convergence towards it.  In regimes where the free-energy landscape is stable, one can obtain quantitative convergence estimates, uniform-in-time propagation of chaos, and functional inequalities such as logarithmic Sobolev inequalities. These inequalities express, in finite dimensions, the same stability and dissipation properties that are visible in the limiting gradient flow. A second theme will be fluctuations. The law of large numbers corresponds to convergence towards minimisers or solutions of the limiting gradient flow, while fluctuations are governed by the second-order structure of the free energy. Around a stable equilibrium, the Hessian of the limiting free energy determines the Gaussian fluctuation field; dynamically, fluctuations are described by the linearised gradient flow together with the noise inherited from the particle system. Finally, we will discuss phase transitions from this gradient-flow viewpoint. A phase transition corresponds to a qualitative change in the free-energy landscape: loss of convexity, degeneration of the Hessian, loss of uniqueness of minimisers, symmetry breaking, metastability, or failure of uniform logarithmic Sobolev inequalities. Thus phase transitions mark precisely the regimes where the simple stability theory behind the mean-field limit and its fluctuations begins to break down.

We will discuss a variety of approaches to structure preserving spatial discretization of Wasserstein gradient flows, mainly nonlinear Fokker-Planck and fourth order thin film equations. Here "structure preserving" means that the discretized evolution is still a gradient flow, but not on all probability measures but only on a finite-dimensional subspace, with respect to a genuine Riemannian metric that mimics the L2-Wasserstein distance. We will start with one physical space dimension, where the natural discretization of the inverse distribution function leads to a Lagrangian scheme, which provides surprising geometric insights in the dynamics. In the last part, we will discuss graph-based discretizations that can be seen as finite-volume schemes. Analytical questions that we will address are, naturally, convergence, but also the preservation of qualitative properties, like global equilibration, metric contractivity, or the waiting time phenomenon.

This lecture series will study interacting particle systems through the lens of gradient flows, free energies, and their large-particle limits. The central theme is that many systems of weakly interacting particles possess a finite-dimensional gradient-flow structure, and that, as the number of particles tends to infinity, this structure converges to a macroscopic gradient flow for a limiting free energy. We will begin with mean-field particle systems and their associated Gibbs measures. The empirical measure will provide the bridge between microscopic and macroscopic descriptions. At the particle level, the evolution dissipates a finite-(N) free energy; in the large-(N) limit, one obtains nonlinear Fokker–Planck or McKean–Vlasov equations, which are themselves gradient flows of the limiting free energy. A main goal of the course will be to explain how this variational structure can be used not only to identify the limiting equation, but also to prove convergence towards it.  In regimes where the free-energy landscape is stable, one can obtain quantitative convergence estimates, uniform-in-time propagation of chaos, and functional inequalities such as logarithmic Sobolev inequalities. These inequalities express, in finite dimensions, the same stability and dissipation properties that are visible in the limiting gradient flow. A second theme will be fluctuations. The law of large numbers corresponds to convergence towards minimisers or solutions of the limiting gradient flow, while fluctuations are governed by the second-order structure of the free energy. Around a stable equilibrium, the Hessian of the limiting free energy determines the Gaussian fluctuation field; dynamically, fluctuations are described by the linearised gradient flow together with the noise inherited from the particle system. Finally, we will discuss phase transitions from this gradient-flow viewpoint. A phase transition corresponds to a qualitative change in the free-energy landscape: loss of convexity, degeneration of the Hessian, loss of uniqueness of minimisers, symmetry breaking, metastability, or failure of uniform logarithmic Sobolev inequalities. Thus phase transitions mark precisely the regimes where the simple stability theory behind the mean-field limit and its fluctuations begins to break down.

We will discuss a variety of approaches to structure preserving spatial discretization of Wasserstein gradient flows, mainly nonlinear Fokker-Planck and fourth order thin film equations. Here "structure preserving" means that the discretized evolution is still a gradient flow, but not on all probability measures but only on a finite-dimensional subspace, with respect to a genuine Riemannian metric that mimics the L2-Wasserstein distance. We will start with one physical space dimension, where the natural discretization of the inverse distribution function leads to a Lagrangian scheme, which provides surprising geometric insights in the dynamics. In the last part, we will discuss graph-based discretizations that can be seen as finite-volume schemes. Analytical questions that we will address are, naturally, convergence, but also the preservation of qualitative properties, like global equilibration, metric contractivity, or the waiting time phenomenon.

Many optimization and sampling algorithms make use of a gradient flow structure to define algorithm dynamics. In application settings, where gradients are not accessible, or too expensive to obtain, one is interested in derivative-free methods instead. A very popular family of derivative-free algorithms are ensemble Kalman methods, including Ensemble Kalman Inversion, the Ensemble Kalman Filter, and the Ensemble Kalman Sampler. These methods are especially well-adapted for Bayesian inverse problems, where complex science and engineering applications often force us to think of the objective function as a black-box. Despite their popularity, very little theoretical justification for these methods currently exist. We discuss recent results providing insights on the way ensemble Kalman methods are linked to an underlying covariance-modulated gradient flow structure, which is able to explain why the rate of convergence for these methods is independent of the target distribution. Time permitting, we will discuss other derivative-free optimization and sampling methods and how they are linked to gradient flows and ideas from optimal transport.