The question how to rigorously derive continuum mechanics models of solids from atomistic models remains poorly understood.
This is because from an atomistic point of view, key phenomena such as elasticity, plasticity, fracture are ``small`` or ``localized`` deviations from crystalline order; but we do not really understand why crystalline order emerges in the first place. I will review recent advances in the following two areas:
(i) Crystallization. One would like to show that, under appropriate conditions, minimizers $(x_1,..,x_N)$ of atomistic energy functionals such as $E(x_1,..,x_N)=\sum_{1\le i (ii) Shape problems. One would like to show that in many situations, atoms assemble into specific shapes, governed by preferred solid-vapour interface orientations. In particular, I will explain the recent result that atomistic ground states of the 2D Heitmann-Radin model form regular hexagons as $N\to\infty$.
This follows by showing that in a suitable scaling limit, the atomistic energy converges (in the sense of De Giorgi`s Gamma convergence) to a continuum Wulff-Herring surface energy.
(Joint work with Yuen Au Yeung and Bernd Schmidt, Calc.Var.PDE (Online First) 2011, arXiv 0909.0927v1.)

For the Darcy-Brinkman equations, which model porous media flow, we present a new H^1-conforming exactly divergence free finite element for approximating the velocity based on composite quadrilateral or hexahedral elements which consist of triangular or tetrahedral subelements.
The pressure is approximated by corresponding discontinuous finite elements in such a way that the divergence of a discrete velocity function is contained in the discrete pressure space. As a consequence, the discretely divergence free velocity functions are exactly divergence free. Moreover, the inf-sup-condition is satisfied uniformly with respect to the mesh-size.
Due to the exactly divergence free property of the velocity approximation, the discretization error of the velocity does not depend on the approximation error of the pressure and we get an error estimate which is uniform with respect to the viscosity coefficient in the model. This property is also satisfied if the proposed element pair is applied to the numerical solution of the Navier-Stokes equations.
We present optimal a priori error estimates for the velocity- and pressure- approximation of the Darcy-Brinkman model. Furthermore, we discuss the efficient solution of the arising mixed linear system for the nodal vectors of velocity and pressure.

Multilevel methods have been one class of powerful iterative methods for solving a linear system of equations. Some examples are multigrid and domain decomposition methods. They are often referred to as subspace correction methods, because of error reduction processes that are carried out in a subset of the solution subspace. Multigrid, for instance, employs coarse-grid correction performed at a coarse grid to reduce the components of error that cannot be effectively reduced by a fixed-point iterative method on the fine (actual) computational grid. These components typically correspond to the slow varying components of the error function in the Fourier space. The fast varying, highly oscillatory components can however be easily handled by a fixed-point iteration. An efficient multigrid method is obtained via an effective interplay between the reduction of error on the fine grid - called smoothing - and the coarse-grid correction.
In this seminar, another method to perform a multilevel iteration is presented. Different from multigrid, the method utilizes a Krylov method as the basis in the error reduction process. Krylov methods however do not have any smoothing property like the usual multigrid smoothers (Jacobi or Gauss-Seidel), and hence the fast and low varying components of error are not distinguishable. As a consequence, a simple replacement of a smoother by a Krylov method will not lead to a useful multilevel method.
We shall show how to build such an iteration properly, which leads to a method that is as effective as an efficient multigrid, and requires almost the same setting as multigrid. In fact, the method can achieve a fast convergence typical for multigrid for problems that standard multigrid methods do not even work. Numerical results for some class of problems including the Helmholtz equation, the biharmonic equation, and involving singular matrices will be presented.
Some parts of this talk are a join work with Reinhard Nabben (TU Berlin) and/or Kees Vuik (TU Delft)

One of the foremost difficulties in solving Mixed Integer Nonlinear Programs (MINLP), either with exact or heuristic methods, is to find a feasible point. We address this issue with a new feasibility pump algorithm tailored for nonconvex MINLPs. Feasibility pumps are successive projection algorithms that iterate between solving a continuous relaxation and a mixed-integer relaxation of the original problems; such approaches currently exist in the literature for Mixed-Integer Linear Programs and convex MINLPs. Both cases exhibit the distinctive property that the continuous relaxation can be solved in polynomial time. In nonconvex MINLP such a property does not hold and the main innovations in this paper are tailored algorithmic methods to overcome such a difficulty. In this talk, we review the Feasibility Pump evolution from the original one proposed by Fischetti, Glover, and Lodi (2005) to the one addressed to nonconvex Mixed Integer Nonlinear Programming problems, one of the first effective heuristics proposed for general MINLPs.

Algorithmisches (oder auch Automatisches) Differenzieren - AD - ist eine Technologie der angewandten Mathematik und Informatik. Auf Basis eines Programmcodes, der eine Funktion auswertet, generiert AD-Technologie ein neues Programm, das die Funktion und ihre Ableitung auswertet. AD kann überall dort eingesetzt werden, wo Sensitivitäts- und Ableitungsinformationen wichtig sind. Dies ist z.B. bei der Lösung nichtlinearer Gleichungssysteme, bei der Optimierung und bei Unsicherheitsanalysen der Fall. AD-Softwrae ist für viele Programmiersprachen verfügbar, darunter Fortran, C++ und Matlab. Wesentliche Gesichtspunkte beim Einsatz von AD sind die problemangepasste Auswahl der richtigen Methode, um eine hohe Effizienz gerade bei hochdimensionalen Problemen zu erhalten. Als Beispiel betrachten wir Parameteridentifikations- oder -Optimierungsprobleme in marinen Ökosystemen, die zur Modellierung des globalen Kxohlenstoffkreislaufs und damit der CO2-Bilanz bedeutend sind. Diese Modelle basieren auf partiellen Differentialgleichungen, und klimatologisch sinnvolle Lösungen werden als periodische Lösungen in einer langen Fixpunktiteration berechnet. AD-Softwrae wird hier in speziellen Optimierungsverfahren eingesetzt, um den Zuwachs an Aufwand für eine Optimierung gegenüber der Simulation möglichst im Rahmen zu halten.

A challenging task in global optimization is to construct tight convex relaxations that provide reasonably globally valid bounds on a mixed-integer nonlinear program (MINLP). For a general MINLP, convex relaxations are usually obtained by replacing each non-linearity by convex under- and concave overestimators. The mathematical object studied to derive such estimators is given by the convex hull of the graph of the function over the relevant domain. To derive improved relaxations, we consider a finite set of given functions as a vector-valued function and study the convex hull of its graph. We establish a link between such a convex hull object and the convex hulls of the graphs of a certain family of real-valued functions. This link can be used to define improved relaxations. We especially focus on small sets of well-structured univariate functions. Moreover, we consider some classes of (real-valued) functions for which a simultaneous convexification with the set of multi-linear monomials leads to an explicit description for the tightest convex underestimating function. The extended formulation is based on the idea of the RLT-appraoch introduced by Sherali and Adams. Numerical examples are presented demonstrating the impact of the concepts.

With “fully anisotropic” I describe diffusion models of the form u_t=\nabla \nabla (D(x) u), where the diffusion tensor appears inside both derivatives. This model arises naturally in the modeling of brain tumor spread and wolf movement and other applications. Since this model does not satisfy a maximum principle, it can lead to interesting spatial pattern formation, even in the linear case. I will present a detailed derivation of this model and discuss its application to brain tumors and wolf movement. Furthermore, I will present an example where, in the linear case, the solution blows-up in infinite time; a quite surprising result for a linear parabolic equation
(joint work with K.J. Painter and M. Winkler).

Die nichtlineare Schrödinger-Gleichung (NLS) beschreibt klassische Wellen, wie z.B. Wasserwellen oder optische Wellen. Aber wie so und was tut das i da? Ist die Welle u komplex? Und wo ist die zweite Zeit-Ableitung in dieser ‘Wellen-Gleichung’? Wir werden diese Fragen antworten und zeigen, wie die NLS als ein asymptotiches Wellenmodell entsteht. Einige interessante Eigenschaften, wie die Existenz von Soliton-Lösungen, Hamiltonische Struktur, volle Integrabilität und Blow-up in endlicher Zeit werden besprochen. Auch ein einfaches numerisches Verfahren für die NLS wird erwähnt.

A new clarification is given for a seemingly paradoxical prediction of the behavior of an object with non-constant contact angle when dipped into a fluid. The method is then applied to the problem of determining the mutual attraction (or repulsion) of two parallel vertical plates dipped into an infinite liquid bath, subject to a downward gravity field. Criteria for attraction or repulsion of the plates are delineated, and exact expressions are obtained for the forces in all configurations. It is shown that the horizontal forces exerted on the plates by the fluid, and which act on the sides of the plates facing infinite domains, are independent of the contact angles on those sides; the horizontal force resultants on the plates can thus be characterized in terms only of the angles on the plate sides that face each other, and the plate separation distance. The behavior for fixed contact angles and varying plate separations can be exotic in some striking respects, and is suggestive for supporting experiments.

In diesem Vortrag diskutieren wir die mathematische Modellierung und die numerische Si- mulation von Erdbeben, d.h. die Ausbreitung seismischer Wellen in der Erde. Anwendungen finden sich in so verschiedenen Gebieten wie dem Katastrophenschutz und der Öl- und Gasexploration. Ausgehend von der allgemeinen Wellengleichung (einer partiellen Differen- tialgleichung) für elastische anisotrope Materialien beschreiben wir die notwendigen Erwei- terungen zur Berücksichtigung des inhomogenen Aufbaus der Erde mit ihren verschiedenen festen und flüssigen Schichten. Die Diskretisierung des Modells erfolgt im Ort mit Spektralen Elementen, einer speziellen Form von Finiten Elementen. In der Zeit beschränken wir uns auf ein explizites Verfahren. Zur Auflösung physikalisch relevanter Frequenzen ist eine sehr feine Diskretisierung mit Milliarden von Gitterpunkten erforderlich. Simulationsrechnungen erfordern also beliebig große Supercomputer.