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The starting point is a 2D model for the dynamics of \(n\) dislocations, which are modelled as point particles with a positive or negative ’charge’. In the celebrated engineering paper by Groma and Balogh in 1999, the limit passage n → ∞ of these dislocation dynamics is performed in a statistical mechanics framework, which relies on a phenomenological closure assumption. In my talk, I present how to pass rigorously to the limit n → ∞ by using the theory of Wasserstein gradient flows and using advanced functional analysis on the weak form of the evolution equation. Interestingly, our conclusion for the limiting dynamics of the dislocation density differs from the conclusion in the paper by Groma and Balogh.
Over the last six years, blockchains have developed into a 'mainstream' technology that entire industry sectors are talking about. The latest generation even supports smart contracts - programs that are executed by all participants and that may govern everything from simple transactions to the setup of organisations. Taking a closer look, however, we find that there is very little deployment beyond the two most prominent examples, Bitcoin and Ethereum. In this talk, we are going to look at some of the reasons: the problem of dependability and abortion of transactions, which is crucial for enterprises; the influence of the underlying network structure on transaction execution; and the problem of exploitable smart contracts. Correspondingly, we discuss some research directions that could prove fruitful in a number of systems, blockchains or beyond.
Bio: Ralph Holz is Lecturer in Networks and Security at the School of IT at the University of Sydney, where he leads the Node for Cybersecurity and Usable Security inside the Human-Centred Technologies cluster. He works closely with Data61|CSIRO, Australia's prime innovation body, and is a Visiting Fellow at the University of New South Wales. Ralph's research interest is empirical security, in particular measuring the deployment properties of critical infrastructure (including blockchains) and the effects and causes of network and routing incidents. He led the research efforts that culminated in the world's first large-scale, long- term analysis of the deployment of the Web Public Key Infrastructure. Most recently, he has turned his attention to analysing the security and dependability of blockchain networks. Ralph received his PhD from Technical University of Munich (TUM) in May 2014.
The (global) defect correction method combines two discretisation scheme. One may be easy solvable, the other may be better (e.g. or higher consistency order) but more complicated. Even if the second discretisation is unstable, the defect correction is a finite process leading to a solution inheriting the features of the better discretisation. In a similar way, the local defect correction improves the solution locally. This can be used instead of local grid refinement. Different from usual discretisations, the obtained solution is neither the solution of the first nor of the second discretisation. On the other hand it is much more flexible than usual (finite element) methods.
It is a well-known fact that the presence of re-entrant corners, i.e. corner with angle $\Theta > \pi$, in polygonal domains leads to the loss of regularity of solutions of elliptic problems [Kondratiev 1967]. This, in turn, means that only a suboptimal order of convergence of their standard piecewise linear finite element approximation can be obtained. Recently, an effective method of recovering the full second-order convergence for elliptic equations on domains with re-entrant corners, when measured in locally modified $L_2$ and $H^1$ norms, known as energy-correction, has been proposed [Egger, R\"ude, Wohlmuth 2014]. This method is based on a modification of a fixed number of entries in the system's stiffness matrix. In this talk, we present two applications of the energy-correction method.\\ Firstly, we show how the energy-correction method can be applied to finding an approximation of optimal Dirichlet boundary control problem on non-convex domains. We present the saddle-point structure of the problem and investigate the convergence properties of the method building on the work conducted in [Of, Phan, Steinbach 2015].\\ Secondly, we show how the energy-correction method can be applied to regain optimal convergence in weighted norms for parabolic problems and introduce a post-processing strategy yielding optimal convergence order in standard Sobolev norms. Standard discretization approach involving graded meshes results in a very restrictive form of a CFL condition, making the use of explicit time stepping practically impossible. On the other hand, the energy-correction can be used on uniform meshes, allowing for application of explicit time stepping scheme with relatively large time steps. This, combined with mass-lumping strategy, leads to a very efficient discretization of parabolic problems, where at each time step only one vector multiplication with a scaled stiffness matrix needs to be performed. Finally, we extend this idea to higher-order finite element methods.\\ All theoretical results are confirmed by the numerical tests.
The two most studied elliptic PDEs are probably the torsion problem, also known as St-Venant problem, and the Dirichlet eigenvalue problem. For these classical problems, many estimates and qualitative properties have been obtained, see for example works by Polya, Szego, Schiffer, Payne, Hersch, Bandle, and many others. In this seminar I present some recent results about upper and lower bounds of two shape functionals involving the maximum of the torsion function: I consider the ratio \(T(\Omega)\lambda_1(\Omega)/|\Omega|\) and the product \(M(\Omega)\lambda_1(\Omega)\), where \(\Omega\) is bounded open set with finite Lebesgue measure \(|\Omega|\), \(T(\Omega)\) denotes the torsion, and \(\lambda_1(\Omega)\) the first Dirichlet eigenvalue. Particular attention is devoted to the subclass of convex sets.
The main subject of the workshop is the mathematical investigation of equations describing the behaviour of quantum systems at macroscopic scales, and their rigorous emergence from microscopic dynamics. The workshop also serves as an occasion to celebrate the 70th birthday of Herbert Spohn and to congratulate Herbert on the reception of the Max Planck Medal 2017.
Invited Speakers: Jürg Fröhlich, Marcel Griesemer, Christian Hainzl, Mathieu Lewin, Jani Lukkarinen, Marcin Napiórkowski, Peter Pickl, Alessandro Pizzo, Wojciech De Roeck, Chiara Saffirio, Benjamin Schlein, Stefan Teufel, Juan J.L. Velázquez
More information and registration at https://www-m5.ma.tum.de/Allgemeines/MacroscopicLimitsWorkshop