The department is currently involved in the following collaborative projects:
International Research Training Group 1529 “Mathematical Fluid Dynamics”
The International Research Training Group (IGK 1529) (Internationales Graduiertenkolleg) Mathematical Fluid Dynamics funded by DFG and JSPS and associated with TU Darmstadt, Waseda University in Tokyo and University of Tokyo has started on June 1, 2009.
Our research focuses on analytical, stochastic, geometric and optimization as well as on aerodynamical aspects of Fluid Dynamics. The program mainly seeks to combine methods from several mathematical disciplines, as analysis, stochastics, geometry and optimization to pursue fundamental research in Fluid Dynamics.
The International Research Training Group distinguishes itself through joint teaching and supervision. The core program consists of interdisciplinary lectures and seminars and includes research and study periods in Tokyo.
CRC TRR 154: Mathematical Modelling, Simulation and Optimization using the Example of Gas Networks
The “turnaround in energy policy” is currently in the main focus of public opinion. It concerns social, political and scientific aspects as the dependence on a reliable, efficient and affordable energy supply becomes increasingly dominant. On the other side, the desire for a clean, environmentally consistent and climate-friendly energy production is stronger than ever. To balance these tendencies while making a transition to nuclear-free energy supply, gas becomes more and more important in the decades to come. Natural gas is and will be sufficiently available, is storable and can be traded. On the other side focusing on an efficient handling of gas transportation induces a number of technical and regulatory problems, also in the context of coupling to other energy carriers. As an example, energy transporters are required by law to provide evidence that within the given capacities all contracts defining the market are physically and technically feasible. Given the amount of data and the potential of stochastic effects, this is a formidable task all by itself, regardless from the actual process of distributing the proper amount of gas with the required quality to the customer. It is the goal of the Transregio-CRC to provide certified novel answers to these grand challenges, based on mathematical modeling, simulation and optimization. In order to achieve this goal new paradigms in the integration of these disciplines and in particular in the interplay between integer and nonlinear programming in the context of stochastic data have to be established and brought to bear. Clearly, without a specified underlying structure of the problems to face, such a breakthrough is rather unlikely. Thus, the particular network structure, the given hierarchical hybrid modeling in terms of switching algebraic, ordinary and partial differential-algebraic equations of hyperbolic type that is present in gas network transportation systems gives rise to the confidence that the challenges can be met by the team of the proposed Transregio-CRC. Moreover, the fundamental research conducted here will also be applicable in the context of other energy networks such as fresh- and waste water networks. In this respect the proposed research goes beyond the exemplary problem chosen and will provide, besides a cutting edge in enabling technologies, new mathematics in the emerging area of discrete, respectively, integer and continuous problems.
Vice spokesperson: Jens Lang
CRC TRR 146 “Multiscale Simulation Methods for Soft Matter Systems”
Multiscale modeling is a central topic in theoretical condensed matter physics and materials science. One prominent class of materials, whose properties can rarely be understood on one length scale and one time scale alone, is soft matter. The properties of soft materials are determined by an intricate interplay of energy and entropy, and minute changes of molecular interactions may lead to massive changes of the system’s macroscopic properties.
In our collaborative research center (CRC TRR 146), we plan to tackle some of the most pressing problems in multiscale modeling in a joint effort of physicists, chemists, applied mathematicians, and computer scientists. The TRR 146 receives funding from the german science foundation (DFG) since October 2014. We address three major challenges:
- Dynamics: In the past, multiscale coarse-graining approaches have to a large extent focused on static equilibrium properties. However, a thorough understanding of a coarse-grained model’s dynamical properties is necessary if one wants to apply multiscale concepts to the study of transport and nonequilibrium processes.
- Coarse-graining and mixed resolution: In many applications, selected small (e.g., functional) regions of a material must be treated in great detail, whereas the large bulk can be modeled at a coarse-grained level. Simulation schemes are desirable, where fine-grained and coarse-grained regions can dynamically be assigned to the current state of the system. In this context, we will also have to re-analyze fundamental aspects of coarse-graining from a mathematical point of view.
- Bridging the particle-continuum gap: So far, only few successful attempts have been made to combine particle models of soft matter with continuum models in a nontrivial fashion. Multiscale schemes for particle models have mostly been developed in the soft matter community, whereas schemes for treating continuum models with variable resolution are developed in the applied mathematics community. In the CRC-TR, we will bring these two communities together to advance the field as a whole.
The problems addressed in the TRR 146 require a massive interdisciplinary effort at the level of fundamental science and algorithmic development. The TRR 146 brings together scientists with a complementary expertise in a wide range of modeling methods.
LOEWE research unit USAG Uniformized Structures in Arithmetic and Geometry
The new LOEWE research unit Uniformized Structures in Arithmetic and Geometry aims at joining the broad expertise of TU Darmstadt and GU Frankfurt in the fields of number theory and arithmetic/algebraic geometry.
Our research program focusses on the following three research areas:
A. Special Subvarieties
B. Automorphic Forms
C. Variation of Geometry
In research area A we explore Orthogonal Shimura Varieties and the Kudla Conjecture, in research area B we investigate Borcherds-Products as well as Vertex Algebras, and in research area C we study the Uniformization of Spherical Varieties, the Anabelian Section Conjecture, as well as Tropical Moduli Spaces.
The research areas A, B, and C are mutually interconnected and techniques of uniformization are crucial in our research approaches.
Coordinator: Jan Hendrik Bruinier
Graduate School of Excellence Computational Engineering
The Graduate School of Excellence Computational Engineering (GSC 233) at Technische Universität Darmstadt enables PhD students to develop their scientific skills in a focused way, and to cooperate under optimal conditions in a highly stimulating interdisciplinary environment.
Based on the interaction of Computer Science, Mathematics, and Engineering Sciences three key scientific areas are identified. Partnerships with well established research organizations as well as cooperation with industry increase the impact of the Graduate School.
Graduate School of Excellence Energy Science and Engineering
The mission of the Darmstadt Graduate School of Excellence Energy Science and Engineering is to educate tomorrow’s leading Energy Engineers in a multidisciplinary field of expertise needed to identify and master the most demanding scientific, engineering, economic and social challenges in an interdisciplinary approach. The main challenge is viewed to be a continuous transition from the carbon-based, non-renewable primary energy sources of today to renewable and environmentally friendly energy resources of tomorrow.
The optimal strategy to meet this challenge is on the one hand to improve conventional energy technologies and render them progressively more efficient, to meet the ever more stringent demands on pollutant emissions, and on the other hand to simultaneously develop innovative, advanced renewable energy technologies, which must be brought to a competitive technological readiness level and provide safe, reliable and cost-effective solutions.
CRC 805: Control of Uncertainty of Load Carrying Structures in Mechanical Engineering
At the Collaborative Research Center (CRC) 805, about 40 researchers from the fields of mechanical engineering, mathematics and law work closely together. Our common goal is to control uncertainty in load carrying systems. Together, we investigate new methods for product development under uncertainty, enhance procedures in manufacturing, and establish advanced mechatronic and adaptronic technologies to stabilize load-bearing systems during usage.
The social and economic impact of uncertainty is tremendous. This is reflected in the alarmingly high number of product recalls. At times, more than one million vehicles a year are recalled by manufacturers in the automotive industry. By controlling uncertainty, product failures can be limited, safety factors can be minimized, current oversizing can be avoided, and thus resources can be saved.
Focus of our research is a holistic approach in order to control uncertainty along all phases of the product life: from product development to production to usage. A total of 19 subprojects from the three project areas work together:
In the project area product development, special design principles and mathematical optimization methods are developed in order to control uncertainty as early as in the design phase of technical systems.
In the Projects focusing on the production phase, forming and cutting production processes are designed flexibly while maintaining the same production quality, and functional materials for active components are integrated synchronously with the shaping.
In the project area usage, methods as well as passive, semi-active and active technologies are developed and tested in order to control uncertainty during the usage phase of load-carrying structures.
CRC 1194: Interaction between Transport and Wetting Processes
The Collaborative Research Center (CRC) 1194 involves researchers from the TU Darmstadt and the Max Planck Institute for Polymer Research Mainz. Their common goal is the fundamental analysis of the interaction between transport and wetting processes – particularly when, parallel to momentum transport, also heat and mass transport, complex fluids or complex surfaces are involved.
What happens when surfaces are printed and coated with different liquids? What processes occur when a liquid meets a solid? How do wetting and dewetting depend on the local momentum, heat and mass transport processes? The underlying mechanisms of the interaction between these processes have to a large extent not been understood to date and are the focus of the Collaborative Research Centre (CRC) 1194. Although the physical phenomena take place only in a range of nanometres or micrometres, they often determine the efficiency of the overall process and the resulting product quality.
To date research in this area has focused on the dependence of the wetting process on the local fluid velocity at the contact line, i.e. wetting coupled with momentum transport. However serious deficits exist in our understanding when, parallel to momentum transport, also heat and mass transport are involved. These deficits become even more blatant when complex fluids are involved, e.g. suspensions or mixtures, or when complex surfaces are examined, e.g. rough or porous.
Fundamental processes and phenomena are examined over a wide range of length scales (nano-micro-macro) and the transfer of basic research to applications is an integral part of the research program. Overall the subject demands the use of very diverse methods and expertise from a multitude of disciplines.
The CRC is grouped into three research areas:
(A) Generic Experiments
(B) Modelling and Simulation
(C) New and Improved Applications.
As an overarching and integrating factor two generic configurations have been specified (Immersed Body, Sessile Drop). Furthermore, OpenFOAM has been selected as a common software platform.
Deputy Speaker: Dieter Bothe
DFG-Forschergruppe 1920 Heidelberg/Darmstadt: “Symmetrie, Geometrie und Arithmetik”
The DFG Research Unit 1920 “Symmetry, Geometry and Arithmetic” examines current issues in modern arithmetic. An important and key theme is the investigation of absolute Galois groups and their generalisations. These elegantly code arithmetic information which can be extracted through the study of these groups and their representations. The researchers, who are based in Heidelberg and Darmstadt, are hoping that by dovetailing motivic homotopy theory, deformation theory, Iwasawa theory, the theory of automorphic forms and L-functions, they will be able to draw interesting conclusions from new insight into one of these areas which they can apply to the others, in a contemporary vision and modern understanding of basic mathematical research.
As a principal investigator Jan Bruinier is part of this research unit with a project centered around special cycles on the moduli space of abelian surfaces and their connections with L-functions. The spokesperson is Alexander Schmidt from the Universität Heidelberg.
Proof Mining in Convex Optimization
In this Project we aim at using proof-theoretic methods from logic for the extraction of new data (such as effective bounds, “proof mining”) from prima facie noneffective proofs in convex optimization and related areas. We need to tailor the proof-theoretic methods to the specific domain of applications and will then apply them for the extraction of rates of asymptotic regularity, metastability (in the sense of T. Tao) and convergence of central iterative procedures used in convex optimization. In particular, we will study convergence proofs which make use of facts from the abstract theory of set-valued operators (e.g. maximally monotone operators).
Konstruktionen und Modelltheorie für Hypergraphen kontrollierter Azyklizität
Azyklizitätskriterien spielen in vielen Bereichen der algorithmischen Modelltheorie und der Logik in der Informatik eine Rolle. Azyklizität erweist sich als nützlich für die Komplexität algorithmischer Probleme wie für die modelltheoretische Analyse. Oft können ideale Azyklizitätsbedingungen durch Abwicklungs- und Überlagerungskonstruktionen erreicht werden; typische Konstruktionen (wie Baumabwicklungen) stehen aber i.d.R. nicht zur Verfügung wenn man sich aufgrund der Problemstellung auf endliche Strukturen beschränken muss. Hier werden qualitativ und quantitativ eingeschränkte Approximationen wichtig und es geht darum(i) geeignete approximative Azyklizitätsbegriffe zu isolieren, die entsprechende endliche Überlagerungs- oder Abwicklungskonstruktionen erlauben, und(ii) Methoden zu gewinnen, die gute algorithmische oder logische Eigenschaften bei solchermaßen kontrollierter Azyklizität verfügbar machen.Neue Konstruktionsmethoden, neue Techniken zur Analyse und neue Anwendungsbereiche sollen in einem weiteren Kontext – ausgehend von den entscheidenden Durchbrüchen in [4, 35] – systematisch erforscht und entwickelt werden.
DFG-PP 1798: Compressed Sensing in Information Processing
The project EXPRESS is funded within the framework of the DFG program of emphasis “Compressed Sensing in Information Processing” (CoSIP).
In the EXPRESS project we study the compressed sensing (CS) problem in the presence of side information and additional constraints. Side information as well as constraints are due to a specific structure encountered in the system model and may originate from the structure of the measurement system or the sensing matrix (shift-invariance, subarray structure, etc.), the structure of the signal waveforms (integrality, box constraints, constellation constraints such as non-circularity, constant modulus, finite constellation size, etc.), the sparsity structure of the signal (block or group sparsity, rank sparsity, etc.) or the channel, as well as the structure of the measurements (quantization effects, K-bit measures, magnitude-only measurements, etc.). We will investigate in which sense structural information can be incorporated into the CS problem and how it affects existing algorithms and theoretical results. Based on this analysis, we will develop new algorithms and theoretical results particularly suited for these models. It is expected, on the one hand, that exploiting structure in the measurement system, i.e., the sensing matrix, can lead to fast CS algorithms with novel model identifiability conditions and perfect reconstruction/recovery results. In this sense, exploiting structure in the observed signal waveforms and the sparsity structure of the signal representation can lead to reduced complexity CS algorithms with simplified recovery conditions and provably enhanced convergence properties. On the other hand, we expect that quantized measurements, which are of great importance when considering cost efficient hardware and distributed measurement systems, will generally result in a loss of information for which new algorithms and perfect recovery conditions need to be derived.
As an application in this project, we consider collaborative (distributed) multi- dimensional spatial spectrum sensing, i.e., sensing along the frequency-, time-, and space-axes using a network of multi-antenna sensing devices. Depending on the signal model under consideration, the frequency-, time-, and space dependence of the measurements can emerge in several ways. For example, the sensing parameters of interest can include directions-of- arrival, carrier frequencies, and Doppler-shifts. The EXPRESS project aims at exploiting the underlying sparsity properties in the signal model for this application while incorporating the aforementioned various types of side information.
DFG-PP 1962: Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization
The short and mid-term goals of the DFG-PP consist in establishing a theoretical and numerical foundation as well as in developing new algorithmic paradigms for the treatment of non-smooth phenomena and associated parameter influences. Long-term goals involve the realization and further advance of these concepts in the context of robust and hierarchical optimization, partial differential games, and nonlinear partial differential complementarity problems as well as the validation in the context of complex applications. This DFG-PP is motivated by important applications and very recent advances in theory and numerics for non-smooth distributed parameter systems (such as semi-smoothness in function space) and their optimization as well as the beginning of the blending of such non-smooth systems with problems requiring robust solutions. Rooted in applied mathematics, the DFG-PP has an interdisciplinary outreach as its results will benefit computational scientists and engineers, who face challenging applications involving non-smooth components. Structurally, on the one hand, the DFG-PP requires basic research in single projects, while for the synthesis and fostering of ideas and techniques the networking of research groups is needed. Moreover, clustering the research projects around prototypical applications is important to achieve landmark results and to succeed in addressing challenging applications.
DFG-PP 1748: Reliable Simulation Techniques in Solid Mechanics. Development of Non-standard Discretization Methods, Mechanical and Mathematical Analysis
The main objective of this Priority Programme is the development of modern non-conventional discretisation methods, based on e.g. mixed (Galerkin or least-squares) finite element or discontinuous Galerkin formulations, including the mathematical analysis for geometrically as well as physically non-linear problems in the fields of e.g. incompressibility, anisotropies and discontinuities (cracks, contact). It is the aim to pool the expertise of mechanics and mathematics in Germany and to create new and strengthen existing networks. In the framework of this cooperation the experiences should be exchanged in between the different working groups to create synergies, save time and costs and raise the efficiency. Furthermore, it is intended to lead this research union to international excellence in the field of non-conventional discretisation techniques.In detail the Priority Programme will drive research towards the following directions concerning non-conventional finite element formulations:· deep mathematical understanding of the structural requirements of reliable non-conforming finite element method (FEM) approaches for finite deformations,· mathematically sound variational formulations,· robust and stiffening-free discretisations at finite deformations for (quasi-)incompressible, isotropic and anisotropic material behaviour as well as for domains with oscillating coefficients,· accurate approximation of all process variables in the latter mentioned extremal cases,· insensitive behaviour concerning significant mesh deformation, · convergence of adaptive mesh refinement,and discontinuities:· creation of a variational basis as well as suitable discretisation techniques for discontinuities: convergence, stability and approximation properties,· resolution of discontinuities based on isogeometric formulations,· novel crack growth and crack branching models,· contact formulations based on non-conventional discretisation techniques exceeding Mortar-methods.