We argue possible approaches to the problem of the rigorous derivation of kinetic equations from underlying many-particle dynamics. The current hypothesis on this problem consists in the following. Since the evolution of states of infinitely many particles is generally described by the sequence of n-particle marginal distributions which is a solution of the initial-value problem to the BBGKY hierarchy (see Cercignani C., Gerasimenko V., Petrina D. Many-Particle Dynamics and Kinetic Equations. Dordrecht:Kluwer Acad. Publ., 1997) then the evolution can be effectively described by the one-particle distribution function which satisfies the kinetic equation only in a suitable scaling limit. We demonstrate that in fact, if initial data are completely defined by the one-particle distribution function then all possible states of infinite particle systems at arbitrary moment of time can be described within the framework of the one-particle distribution function without any approximations. For that we construct a new representation of a solution of the initial-value problem to the BBGKY hierarchies as an expansion in terms of particle clusters whose evolution are described by the corresponding order cumulant of evolution operators of finitely many particle systems. For the initial data from the space of integrable functions satisfying the "chaos" property we prove that the Cauchy problem to the BBGKY hierarchy is equivalent to the corresponding initial-value problem for certain generalized kinetic equation and an infinite sequence of explicitly defined functionals depending from a solution of this generalized kinetic equation. We extend this result also on the quantum systems. The specific kinetic equations such as the Boltzmann equation and other ones, can be derived from the constructed generalized kinetic equation in appropriate scaling limits.

• Thematic program: Optimal transportation structures, gradient flows and entropy methods for Applied PDEs (2007)
• WK seminar

During the 20th century various concepts for conservation-restoration and in particular for presentation and retouching were postulated, according to the respective phases of development of the discipline. An overview over a series of relevant aspects, methods and problems is given with the aim of facilitating definitions forming the basis of automated virtual completion of works of art.

• Thematic program: Optimal transportation structures, gradient flows and entropy methods for Applied PDEs (2007)
• Event: Mini-workshop on "PDE's and Variational Tools in Image Inpainting" (2007)

In the last 15 years the artists working process comes more and more in the influence by computer-controlled surfaces like screens, mouses, keyboards and special-designed software. Most of the software and hardware was developed from classical analog image generation tools like the use of pencils or photographic methods. This main focus often clouds new capacities in the interaction between artist and computer-aided methods in image editing and image generation. So the proposal is to research new aspects in interrelation between artist and tool using transdiscipline science methods.

• Thematic program: Optimal transportation structures, gradient flows and entropy methods for Applied PDEs (2007)
• Event: Mini-workshop on "PDE's and Variational Tools in Image Inpainting" (2007)

Second order variational inpainting methods, like total variation inpainting (cf. Rudin-Osher-Fatemi), have drawbacks as in the connection of edges over large distances or the continuous propagation of level lines into the damaged domain. In an attempt to solve both the connectivity principle and the so called staircasing effect resulting from second order image diffusions, a number of third and fourth order diffusions have been suggested for image inpainting. A new approach in the class of fourth order inpainting algorithms is inpainting of binary images using the Cahn- Hilliard equation proposed in Bertozzi-Esedoglu-Gillette. In this talk I will present some analytic and numerical results for Cahn-Hilliard inpainting. Besides this also other variations of this higher order approach for grayvalue images will be discussed.

• Thematic program: Optimal transportation structures, gradient flows and entropy methods for Applied PDEs (2007)
• Event: Mini-workshop on "PDE's and Variational Tools in Image Inpainting" (2007)

In this talk we discuss resent results on the regularity of the free boundaries in a certain type of parabolic free boundary problems. Mathematically the problem is formulated as follows. Let a function $u$ and an open set $\Omega \subset \mathbb{R}^{n+1}_+=\{(x,t): x \in \mathbb{R}^n, t \in \mathbb{R}, x_1>0\}, n \geqslant 2$ solve the following problem: $$ H(u)=\chi_{\Omega } \quad \text{in} \quad Q_1^+, \qquad u=|Du|=0 \quad \text{in} \quad Q_1^+ \setminus \Omega, \qquad u=0 \quad \text{on} \quad \Pi, $$ where $H=\Delta -\partial_t$ is the heat operator, $\chi_{\Omega }$ denotes the characteristic function of $\Omega $, $Q_1$ is the unit cylinder in $\mathbb{R}^{n+1}$, $Q_1^+=Q_1 \cap \mathbb{R}^{n+1}_+$, $\Pi =\{(x,t): x_1=0\}$, and the first equation is understood in the weak (distributional) sense.

• Thematic program: Optimal transportation structures, gradient flows and entropy methods for Applied PDEs (2007)
• Event: Mini-workshop on "PDE's and Variational Tools in Image Inpainting" (2007)

In many image acquisition processes only incomplete data is recorded for a variety of reasons, like for example defective sensors, superimposed texts or logos, or occlusions by other objects. Further, non-digitally stored image data like photographs or celluloid movies may suffer from fading, mechanical stress or simply from aging. Typical symptoms are blotches or torn out pieces, which both lead to image regions which contain no information. To restore the missing image data a multitude of inpainting algorithms has been devised in recent years. We present several inpainting algorithms, each suitable for completion of a different type of image data. Starting from the Ginzburg--Landau (GL) energy we derive an algorithm which can be used to inpaint levelsets. We show two possible applications: levelset-wise inpainting of images, and surface inpainting. Thereafter we show how to embed images into complex valued functions, such that GL inpainting can be directly applied to image functions, without the indirection using levelsets. The GL inpainting algorithm -- like most inpainting algorithms based on variational methods or PDEs -- does not handle textured images very well. Therefore we add a texture synthesis algorithm based on Markov Random Fields to supplement the missing texture information. Finally, we consider inpainting of movies. Since consecutive frames of a movie usually differ very little, one may find the image information missing in one frame in another frame nearby. To identify corresponding regions of different frames, we employ a kind of optical flow approach with a piecewise continuity constraint. If possible, the missing frame region is copied from undamaged frames, otherwise a still image inpainting is performed.

• Thematic program: Optimal transportation structures, gradient flows and entropy methods for Applied PDEs (2007)
• Event: Mini-workshop on "PDE's and Variational Tools in Image Inpainting" (2007)

A mean-field approach to modeling in economics and finance is presented. The model consists on a system of nonlinear diffusion equation related to a free-boundary value problem. It describes the idealized situation of two groups of people, buyers and vendors, trading one good; the resulting price of the good comes from a dynamical equilibrium.

• Thematic program: Optimal transportation structures, gradient flows and entropy methods for Applied PDEs (2007)
• Event: Mini-workshop on "PDE's and Variational Tools in Image Inpainting" (2007)

We present a new, fully automatic technique for wire and scratch removal that works well in both textured and non-textured areas of an image. Chan, Shen and Kang introduced a technique for inpainting using an Euler's elastica energy-based variational model that works well for repairing smooth areas of the image while maintaining edge detail. The technique is very slow, due to a stiff, 4th order PDE. Efros and Leung used texture synthesis techniques for inpainting and hole filling. This works well for areas of an image that contain repeating patterns. We have combined these two techniques to accelerate and constrain the solution of the 4th order PDE. Instead of a stiff minimization, we have a combinatorial optimization problem that is much quicker to solve.

• Thematic program: Optimal transportation structures, gradient flows and entropy methods for Applied PDEs (2007)
• Event: Mini-workshop on "PDE's and Variational Tools in Image Inpainting" (2007)

• Thematic program: Optimal transportation structures, gradient flows and entropy methods for Applied PDEs (2007)
• Event: Mini-workshop on "PDE's and Variational Tools in Image Inpainting" (2007)

A systematic approach to determine Lyapunov functionals of entropy and energy type for nonlinear evolutionary PDEs has been developed by Juengel and the speaker. The core of the method is to prove associated functional inequalities, i.e. entropy-entropy dissipation relations, in an algorithmic way. The use of computer algebra allows to determine constants in these inequalities, yielding explicit bounds on the relaxation rates for solutions to the PDE. We apply this method to the Quantum Diffusion and the Thin Film equation with multi-periodic boundary conditions.

• Thematic program: Optimal transportation structures, gradient flows and entropy methods for Applied PDEs (2007)

We show the existence and uniqueness of solution in any dimension for a nonlinear Fokker-Planck equation modeling the relaxation of a fermions gas. We use entropy methods to show that in one dimension the solution can be extended to all positive time, and see how under more restrictive initial conditions, this can be done in $\mathbb{R}^N$ for any $N$.

• Thematic program: Optimal transportation structures, gradient flows and entropy methods for Applied PDEs (2007)

The derivation of macroscopic limits of stochastic interacting particle systems is an important problem in various applications and rigorous mathematical results are extremely challenging. In this talk we highlight some points where the use of the Wasserstein Metric can be benefitial or even rather natural in performing this limit. In some cases like Vlasov / Mean-field limits the usefulness of the Wasserstein metric has been demonstrated in the 70s in the work of Dobrushin, which however seems not have attracted as much attention as it would deserve. In this talk we shall discussion some preliminary results and open problems related to the extension of Wasserstein metric to applied problems of current interest. One of them are aggregation models with local repulsion, leading to nonlinear diffusion models in the limit. A second one are step-flow models, which lead to continuum surface evolution models in the limit. In this case a modification of the optimal transport theory from probability measure to more general measures should be used. The theory has a significant impact in particular in the case of surface diffusion phenomena, where fourth-order equations have often been derived by formal arguments. Since the latter lead to unphysical behaviour, modifications are needed. It turns out that the limit in the Wasserstein metric yields such modifications in a rather natural way. Based on joint work with Marco Di Francesco, Daniela Morale, Axel Voigt.

• Thematic program: Optimal transportation structures, gradient flows and entropy methods for Applied PDEs (2007)

In this talk we present mixed finite-element methods for non-linear Fokker-Planck equations of the form $\frac{\partial u}{\partial t} &= \nabla \cdot \left(u \nabla F\left(u,x\right)\right).$ We present a mixed finite element formulation for the porous-medium equations which originates from a special linearization. Based on this method we depict the behavior of the solution for different powers $m$ and compare them to the two-dimensional Barenblatt-Pattle profiles. Furthermore we briefly review the analysis of the Patlak-Keller-Segel model, especially the blow-up behavior of solutions with initial mass greater than $8 \pi$. We will motivate our numerical solving technique and illustrate the behavior of the solution with various examples in 2D. Finally we focus on the one-dimensional relativistic heat equations. We present a finite element discretization and discuss its numerical results.

• Thematic program: Optimal transportation structures, gradient flows and entropy methods for Applied PDEs (2007)

The relevence of optimal transportation techniques for dissipative evolution equations is now very well known (following the seminal paper of Jordan, Kinderlehrer and Otto on the heat equation viewed as a gradient flow on probability measures for the Boltzmann entropy). In this minicourse, applications to conservative evolution equations (such as the Euler equations, multidimensional scalar conservation laws, Hamilton-Jacobi, ideal MHD...) will be discussed.

• Thematic program: Optimal transportation structures, gradient flows and entropy methods for Applied PDEs (2007)
• Event: Summer School "Optimal transportation structures, gradient flows and entropy methods for applied PDE's" (2007)

Following Arnold's interpretation, Euler's equations can be seen as the geodesic equation in the space of measure preserving diffeomorphism. Thus, one can try to find solutions to Euler's equation minimizing the Energy functional with fixed endpoints (this is the usual way to find geodesics on a manifold). It turns out that the study of (a relaxed version of) this problem presents many links with optimal transportation. In this minicourse I will explain the problem in details, I will review the results of Brenier and Shnirelman, and I will present recent results obtained in collaboration with L.Ambrosio.

• Thematic program: Optimal transportation structures, gradient flows and entropy methods for Applied PDEs (2007)
• Event: Summer School "Optimal transportation structures, gradient flows and entropy methods for applied PDE's" (2007)

The relevence of optimal transportation techniques for dissipative evolution equations is now very well known (following the seminal paper of Jordan, Kinderlehrer and Otto on the heat equation viewed as a gradient flow on probability measures for the Boltzmann entropy). In this minicourse, applications to conservative evolution equations (such as the Euler equations, multidimensional scalar conservation laws, Hamilton-Jacobi, ideal MHD...) will be discussed.

• Thematic program: Optimal transportation structures, gradient flows and entropy methods for Applied PDEs (2007)
• Event: Summer School "Optimal transportation structures, gradient flows and entropy methods for applied PDE's" (2007)

• Thematic program: Optimal transportation structures, gradient flows and entropy methods for Applied PDEs (2007)
• Event: Summer School "Optimal transportation structures, gradient flows and entropy methods for applied PDE's" (2007)

The relevence of optimal transportation techniques for dissipative evolution equations is now very well known (following the seminal paper of Jordan, Kinderlehrer and Otto on the heat equation viewed as a gradient flow on probability measures for the Boltzmann entropy). In this minicourse, applications to conservative evolution equations (such as the Euler equations, multidimensional scalar conservation laws, Hamilton-Jacobi, ideal MHD...) will be discussed.

• Thematic program: Optimal transportation structures, gradient flows and entropy methods for Applied PDEs (2007)
• Event: Summer School "Optimal transportation structures, gradient flows and entropy methods for applied PDE's" (2007)

Following Arnold's interpretation, Euler's equations can be seen as the geodesic equation in the space of measure preserving diffeomorphism. Thus, one can try to find solutions to Euler's equation minimizing the Energy functional with fixed endpoints (this is the usual way to find geodesics on a manifold). It turns out that the study of (a relaxed version of) this problem presents many links with optimal transportation. In this minicourse I will explain the problem in details, I will review the results of Brenier and Shnirelman, and I will present recent results obtained in collaboration with L.Ambrosio.

• Thematic program: Optimal transportation structures, gradient flows and entropy methods for Applied PDEs (2007)
• Event: Summer School "Optimal transportation structures, gradient flows and entropy methods for applied PDE's" (2007)

• Thematic program: Optimal transportation structures, gradient flows and entropy methods for Applied PDEs (2007)
• Event: Summer School "Optimal transportation structures, gradient flows and entropy methods for applied PDE's" (2007)

• Thematic program: Optimal transportation structures, gradient flows and entropy methods for Applied PDEs (2007)
• Event: Summer School "Optimal transportation structures, gradient flows and entropy methods for applied PDE's" (2007)

• Thematic program: Optimal transportation structures, gradient flows and entropy methods for Applied PDEs (2007)
• Event: Summer School "Optimal transportation structures, gradient flows and entropy methods for applied PDE's" (2007)

• Thematic program: Optimal transportation structures, gradient flows and entropy methods for Applied PDEs (2007)

In these lectures we will introduce and discuss various kinetic models of Boltzmann type, which have a collision kernel independent of the relative velocity (Maxwell type models). These kinetic equations possess an interesting mathematical structure, which in most cases allow to recover a precise rate for the relaxation to equilibrium in terms of different metrics. Applications both to economy and social sciences of these models are presented into details.

• Thematic program: Optimal transportation structures, gradient flows and entropy methods for Applied PDEs (2007)
• Event: Summer School "Optimal transportation structures, gradient flows and entropy methods for applied PDE's" (2007)

In these lectures we will introduce and discuss various kinetic models of Boltzmann type, which have a collision kernel independent of the relative velocity (Maxwell type models). These kinetic equations possess an interesting mathematical structure, which in most cases allow to recover a precise rate for the relaxation to equilibrium in terms of different metrics. Applications both to economy and social sciences of these models are presented into details.

• Thematic program: Optimal transportation structures, gradient flows and entropy methods for Applied PDEs (2007)
• Event: Summer School "Optimal transportation structures, gradient flows and entropy methods for applied PDE's" (2007)

In these lectures we will introduce and discuss various kinetic models of Boltzmann type, which have a collision kernel independent of the relative velocity (Maxwell type models). These kinetic equations possess an interesting mathematical structure, which in most cases allow to recover a precise rate for the relaxation to equilibrium in terms of different metrics. Applications both to economy and social sciences of these models are presented into details.

• Thematic program: Optimal transportation structures, gradient flows and entropy methods for Applied PDEs (2007)
• Event: Summer School "Optimal transportation structures, gradient flows and entropy methods for applied PDE's" (2007)

• Thematic program: Optimal transportation structures, gradient flows and entropy methods for Applied PDEs (2007)
• Event: Summer School "Optimal transportation structures, gradient flows and entropy methods for applied PDE's" (2007)

• Thematic program: Optimal transportation structures, gradient flows and entropy methods for Applied PDEs (2007)
• Event: Summer School "Optimal transportation structures, gradient flows and entropy methods for applied PDE's" (2007)

• Thematic program: Optimal transportation structures, gradient flows and entropy methods for Applied PDEs (2007)
• Event: Summer School "Optimal transportation structures, gradient flows and entropy methods for applied PDE's" (2007)

To see the schedule of the workshop, go to the event website.

• Thematic program: Optimal transportation structures, gradient flows and entropy methods for Applied PDEs (2007)
• Event: Workshop "Optimal transportation structures, gradient flows and entropy methods for applied PDE's" (2007)