Max-Planck-Gesellschaft

2018

2018, Max Planck Institute for Mathematics in the Sciences

Michałek, M.; Sturmfels, B.

Tensors are higher-dimensional matrices. They provide an efficient way to store big data sets and to analyze them statistically. Using the decomposition into tensors of rank 1, hidden structures and correlations can be discovered. The geometry of tensors plays an important role in this research area.

2017

2017, Max Planck Institute for Mathematics in the Sciences

Laux, Tim; Otto, Felix

Grain growth is an intricate process during which the grain structure of a polycrystal coarsens. Efficient numerical schemes shed light on the statistical behavior of the overall structure. The underlying differential equation for the interfaces is mean curvature flow. The mathematical structure of the equation as a steepest descent in an energy landscape gives new insights and allows to develop and to analyze numerical algorithms.

2016

2016, Max Planck Institute for Mathematics in the Sciences

Pfante, Oliver; Bertschinger, Nils; Olbrich, Eckehard; Ay, Nihat; Jost, Jürgen

Analyzing complex systems, the question occurs what one needs to know at the detail level in order to understand the dynamics at the system level. Ideally, one could use a higher level of description at which the system dynamics unfolds autonomously. That means once the initial state is known, one no longer needs to check the details at the lower level in order to predict the system. Formal methods have been developed to analyze the issues involved. In particular, the question of the flow of information between levels has been linked to the question of memory effects at the respective levels.

2015

2015, Max Planck Institute for Mathematics in the Sciences

Tikhomirov, Sergey

Hysteresis often appears in nature as a mechanism for self-organisation. A mathematical description of this phenomenon is given and discussed in some example cases motivated by real life applications. These examples are decomposed into simple (transverse) and difficult (non transverse) cases. The solution for the transverse case is given, the non transverse case is partially solved and open problems and questions are discussed.

2014

2014, Max Planck Institute for Mathematics in the Sciences

Marahrens, Daniel; Otto, Felix

In applications, it is often desirable to obtain macroscopic properties of heterogeneous materials with microscopic structures. It is possible to obtain these by simulating a representative volume element. In order to do so efficiently, one requires accurate error estimates which are derived by combining ideas from analysis and probability theory.

2013

2013, Max Planck Institute for Mathematics in the Sciences

Kabalak, Alihan

That a subject knows something is conceptualized as it ruling out certain facts while considering other facts possible. Thus the distinction between ‘ruled out’ and ‘possible’ draws upon a given set of conceivable facts. If in a model the knowledge of several subjects is based on the same conceivable facts, everyone receives the same information about what anyone can possibly know (identical ‘awareness’). Overcoming this restriction allows more differentiated analyses of economic decision making.

2012

2012, Max Planck Institute for Mathematics in the Sciences

Banjai, Lehel; Gruhne, Volker

The understanding and prediction of the behaviour of, for example, acoustic, electromagnetic, and elastic waves in different media is of great importance in a number of applications. The propagation of waves is often investigated in unbounded spatial domains. To obtain a numerical approximation in such situations the method of boundary integral equations is particularly attractive. The analysis and development of numerical methods for boundary integral equations for wave equations, in contrast to the time-harmonic Helmholtz equation, has until recently been in its infancy.

2011

2011, Max Planck Institute for Mathematics in the Sciences

Döring, Lukas; Otto, Felix; Steiner, Jutta

The magnetization of a ferromagnetic material forms a variety of patterns, but the scale separation between large domains of almost constant magnetization and narrow transition layers, so-called domain walls, hinders the numerical simulation of these structures. However, rigorous asymptotic analysis makes it possible to derive reduced models that explain the observed phenomena and make them accessible to numerical simulation, as we will illustrate using the example of the deformed Landau state, asymmetric domain walls and the concertina pattern.

2010

2010, Max Planck Institute for Mathematics in the Sciences

Bertschinger, Nils

Many living systems can survive thanks to sophisticated and flexible information processing capabilities enabling them to act autonomously. Even though this ability is increasingly demanded from technical systems, it is still largely unknown which properties are required for a system to obtain autonomy. To fill this gap, an information theoretic measure of autonomy is proposed here. For illustration the autonomy of simple systems is measured and a close relationship between autonomy and memory is established.

2009

2009, Max Planck Institute for Mathematics in the Sciences

Grasedyck, Lars

In many applications one has to solve large-scale (linear) systems. The task to solve these as fast as possible is the goal of “fast solvers”. At the beginning only sparse systems were considered, but due to the recently at the MPI-MIS developed “hierarchical matrices” the scope has become much broader. However, even fast solvers are limited by the dimensionality of the problem under consideration – a challenge for modern mathematical methods which the group “Scientific Computing” at the MPI-MIS wants to meet.

2007

2007, Max Planck Institute for Mathematics in the Sciences

Börm, Steffen; Hackbusch, Wolfgang

The numerical approximation of physical or biological models leads to large systems of equations that have to be solved as rapidly as possible by a computer. If these systems are approximated by *H*^{2}-matrices, they can be handled far more efficiently than by standard methods. Under certain conditions it is even possible to reach the *optimal* order of complexity, i.e., to ensure that the number of operations is proportional to the size of the solution vector.

2007, Max Planck Institute for Mathematics in the Sciences

Dirr, Nicolas

Models for materials which are valid on the length scales of day-to-day experience describe quantities which are derived by averaging over many degrees of freedom of a finer length scale. The deviations from this average due to thermal effects or impurities are often negligible. In some situations, however, they can trigger effects which are observable on the rough length scale. This is exemplified by the mathematical investigation of a prototypical model.

2006

2006, Max Planck Institute for Mathematics in the Sciences

Fleischhack, Christian

General relativity and quantum theory have not been merged into a consistent theory of quantum gravity yet. Unfortunately, to date, there are no experiments available that may disclose parts of the unified theory. Nevertheless, mathematics is already in a position to provide us with rigorous statements on how quantum gravity may look like.

2006, Max Planck Institute for Mathematics in the Sciences

Ay, Nihat

At the MPI for Mathematics in the Sciences general principles that underlie the robustness of evolved systems are studied and formalized. The aim is to formulate a mathematical theory of robustness that can be used for developing artificial adaptive systems.

2005

2005, Max Planck Institute for Mathematics in the Sciences

Atay, Fatihcan; Jost, Jürgen

We investigate the formation of structure in coupled dynamical systems. Although the individual elements may possess complex dynamics of their own, the system can globally synchronize. This is even possible when the activities between elements are transmitted with temporal delays, that is, when the system does no longer have an intrinsic notion of simultaneity. For suitable parameter constellations, it can also be observed that the restriction of the individual degrees of freedom of the participating elements caused by the coupling leads to the emergence of global dynamical patterns on a longer time scale. This suggests a new mathematical approach to the formation of structure in coupled systems.

2004

2004, Max Planck Institute for Mathematics in the Sciences

Flad, Heinz-Jürgen; Hackbusch, Wolfgang

The vision to perform chemical experiments routinely on a computer has been realized nowadays, at least to some extent, by modern techniques in quantum chemistry. For future applications, however, it will be a major challenge to better incorporate the multiscale character of quantum chemical problems. Wavelets provide local and systematic decompositions of the quantities involved into their characteristic length- and energy-scales. Furthermore these decompositions lead to numerical algorithms with almost optimal complexity. We have studied within our research project various aspects of this approach for concrete model problems in many-particle theory. Thereby we are aiming to bridge the gap between quantum chemistry and the relevant disciplines in applied and numerical mathematics.