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Gerasimenko, V. I. (Institute of Math., Kyiv, Ukraine)  WPI, Seminarroom C714  Tue, 8. May 07, 15:00 
Mathematical problems of the derivation of kinetic equations.  
We argue possible approaches to the problem of the rigorous derivation of kinetic equations from underlying manyparticle 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 nparticle marginal distributions which is a solution of the initialvalue problem to the BBGKY hierarchy (see Cercignani C., Gerasimenko V., Petrina D. ManyParticle Dynamics and Kinetic Equations. Dordrecht:Kluwer Acad. Publ., 1997) then the evolution can be effectively described by the oneparticle 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 oneparticle distribution function then all possible states of infinite particle systems at arbitrary moment of time can be described within the framework of the oneparticle distribution function without any approximations. For that we construct a new representation of a solution of the initialvalue 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 initialvalue 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.  

Wolfgang Baatz (Academy of Fine Arts, Vienna)  WPI Seminarroom C714  Mon, 11. Jun 07, 10:15 
Inpainting and presentation  views of the the conservatorrestorer.  
During the 20th century various concepts for conservationrestoration 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.  

Mike Kostner (Academy of Fine Arts, Vienna)  WPI Seminarroom C714  Mon, 11. Jun 07, 11:00 
Interrelation aspects between artist, tool and image  
In the last 15 years the artists working process comes more and more in the influence by computercontrolled surfaces like screens, mouses, keyboards and specialdesigned 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 computeraided 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.  

Arjan Kuijper (RICAM Linz)  WPI Seminarroom C714  Mon, 11. Jun 07, 11:45 
Inpainting with higher order energies  
Second order variational inpainting methods, like total variation inpainting (cf. RudinOsherFatemi), 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 BertozziEsedogluGillette. In this talk I will present some analytic and numerical results for CahnHilliard inpainting. Besides this also other variations of this higher order approach for grayvalue images will be discussed.  

Darya Apushkinskaya (Saarland University)  WPI Seminarroom C714  Mon, 11. Jun 07, 14:45 
Regularity of free boundaries in parabolic problems  
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.  

Harald Grossauer (University of Innsbruck)  WPI Seminarroom C714  Mon, 11. Jun 07, 15:30 
Inpainting Algorithms for Every Purpose  
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, nondigitally 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 GinzburgLandau (GL) energy we derive an algorithm which can be used to inpaint levelsets. We show two possible applications: levelsetwise 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.  

Maria P. Gualdani (University of Texas, Austin)  WPI Seminarroom C714  Tue, 12. Jun 07, 10:00 
Diffusion type equations for price formation  
A meanfield approach to modeling in economics and finance is presented. The model consists on a system of nonlinear diffusion equation related to a freeboundary 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.  

Kangyu Ni (University of California, Los Angeles)  WPI Seminarroom C714  Tue, 12. Jun 07, 11:45 
A Texture Synthesis Approach to Elastica Inpainting  
We present a new, fully automatic technique for wire and scratch removal that works well in both textured and nontextured areas of an image. Chan, Shen and Kang introduced a technique for inpainting using an Euler's elastica energybased 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.  

Massimo Fornasier (RICAM Linz)  WPI Seminarroom C714  Tue, 12. Jun 07, 12:15 
Sparse recovery, freediscontinuity problems and image inpainting  

Matthes, Daniel (University of Mainz)  WPI Seminarroom C714  Tue, 19. Jun 07, 15:00 
Entropies and Energies for Nonlinear Equations of 4th Order.  
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. entropyentropy 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 multiperiodic boundary conditions.  

Rosado Linares, Jesus (UAB Barcelona)  WPI Seminarroom C714  Tue, 19. Jun 07, 15:45 
Global Existence of solution for FokkerPlanck equations for Fermions  
We show the existence and uniqueness of solution in any dimension for a nonlinear FokkerPlanck 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 $N$.  

Burger, Martin (University of Muenster)  WPI seminar room C714  Wed, 8. Aug 07, 11:30 
MicroMacro Transition in the Wasserstein Metric  
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 / Meanfield 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 stepflow 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 fourthorder 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.  

Wolfram, MarieTherese (University of Muenster)  WPI seminar room C714  Wed, 8. Aug 07, 15:00 
Numerical Methods for NonLinear FokkerPlanck Equations.  
In this talk we present mixed finiteelement methods for nonlinear FokkerPlanck 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 porousmedium 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 twodimensional BarenblattPattle profiles. Furthermore we briefly review the analysis of the PatlakKellerSegel model, especially the blowup 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 onedimensional relativistic heat equations. We present a finite element discretization and discuss its numerical results.  

Brenier, Yann (Université de NiceSophiaAntipolis)  Seminarroom C206 + C207  Mon, 10. Sep 07, 10:00 
Optimal transportation for conservative evolution equations (1/3)  
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, HamiltonJacobi, ideal MHD...) will be discussed.  

Figalli, Alessio (Scuola Normale di Pisa  École Normale Supérieure de Lyon)  Seminarroom C206 + C207  Mon, 10. Sep 07, 15:00 
Incompressible models for the incompressible Euler's equations (1/2)  
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.  

Brenier, Yann (Université de NiceSophiaAntipolis)  Seminarroom C206 + C207  Tue, 11. Sep 07, 10:00 
Optimal transportation for conservative evolution equations (2/3)  
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, HamiltonJacobi, ideal MHD...) will be discussed.  

Blanchet, Adrien (Université des Sciences et Technologies de Lille)  Seminarroom C206 + C207  Tue, 11. Sep 07, 15:00 
Entropy methods applied to the KellerSegel system (1/2)  

Brenier, Yann (Université de NiceSophiaAntipolis)  Seminarroom C206 + C207  Wed, 12. Sep 07, 10:00 
Optimal transportation for conservative evolution equations (3/3)  
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, HamiltonJacobi, ideal MHD...) will be discussed.  

Figalli, Alessio (Scuola Normale di Pisa  École Normale Supérieure de Lyon)  Seminarroom C206 + C207  Wed, 12. Sep 07, 15:00 
Incompressible models for the incompressible Euler's equations (2/2)  
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.  

Blanchet, Adrien (Université des Sciences et Technologies de Lille)  Seminarroom C206 + C207  Thu, 13. Sep 07, 10:00 
Entropy methods applied to the KellerSegel system (2/2)  

Di Francesco, Marco (Università di L'Aquila)  Seminarroom C206 + C207  Thu, 13. Sep 07, 14:30 
A contraction result in Wasserstein distance for 1d scalar conservation laws.  

Vázquez, Juan Luis (Universidad Autónoma de Madrid)  Seminarroom C206 + C207  Mon, 17. Sep 07, 10:00 
Recent progress in fast diffusion and geometrical diffusion (1/3)  

Simeoni, Chiara (University of Nice  Sophia Antipolis)  WPI seminar room, Nordbergstrasse 15  Mon, 17. Sep 07, 15:00 
Upwind Interface Source method for hyperbolic conservation laws with a source term on nonuniform mesh  

Toscani, Giuseppe (Università di Pavia)  Seminarroom C206 + C207  Tue, 18. Sep 07, 10:00 
Kinetic models of Maxwell type with applications (1/3)  
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.  

Toscani, Giuseppe (Università di Pavia)  Seminarroom C206 + C207  Tue, 18. Sep 07, 15:00 
Kinetic models of Maxwell type with applications (2/3)  
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.  

Toscani, Giuseppe (Università di Pavia)  Seminarroom C206 + C207  Wed, 19. Sep 07, 10:00 
Kinetic models of Maxwell type with applications (3/3)  
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.  

Vázquez, Juan Luis (Universidad Autónoma de Madrid)  Seminarroom C206 + C207  Wed, 19. Sep 07, 15:00 
Recent progress in fast diffusion and geometrical diffusion (2/3)  

Vázquez, Juan Luis (Universidad Autónoma de Madrid)  Seminarroom C206 + C207  Thu, 20. Sep 07, 10:00 
Recent progress in fast diffusion and geometrical diffusion (3/3)  
 

Carrillo, José Antonio (Universitat Autònoma de Barcelona)  Seminarroom C206 + C207  Fri, 21. Sep 07, 10:00 
Contractions in Wasserstein distances and applications  

Carrillo, Josè Antonio (UAB Barcelona)  Seminarroom C206 + C207  Mon, 24. Sep 07, 9:15 
Opening of the Workshop "Optimal transportation structures, gradient flows and entropy methods for applied PDE's"  
 

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