Structures made of sheet metal are very common in engineering applications. To increase their stiffness, often branches are added to the sheet metals. In the CRC 666 the two new techniques “linear flow splitting” and “linear bend splitting” for integrally producing such branched structures are developed and researched. To assist in the development of the considered products, optimization methods are employed. This is carried out in two subprojects: The aim of Subproject A2 is the optimization of the topological structure and the geometry of the considered products, using discrete as well as nonlinear and PDE-constrained optimization techniques. Subproject A6 is concerned with optimization methods for the deep drawing of branched structures. Here, nonlinear and PDE-constrained optimization methods are used.

Contact: Benjamin Horn, Hendrik Lüthen, Anna Walter, Stefan Ulbrich, Marc Pfetsch

Former: Daniela Bratzke, Thea Göllner, Ute Günther, Katrin Herr, Wolfgang Hess

Subproject A3: Robust Optimization of Load Carrying Systems in Mechanical Engineering

The presence of uncertainty is a prevalent subject in mechanical engineering which can lead to severe economical and safety consequences. The aim of this project is to find -for load-bearing mechanical systems- the optimal robust design regarding uncertainty of parameters (material properties, loading szenarios, etc.) as well as uncertainty of the manufacturing quality. To this end we use simulation-based optimization of geometry, topology and the placement of piezoelectric actuators, whereat we want to apply and extend modern robust optimization techniques.

Contact: Philip Kolvenbach, Stefan Ulbrich

Former: Adrian Sichau

Subproject A4: Optimal Combination of Active and Passive Parts in Load Carrying Systems

Mechanical trusses are found in many applications (undercarriages of airplanes, bicycles, electrical towers, etc.). Those trusses are often overdimensioned to withstand given forces under several uncertainties in loadings, material, production processes, etc. Active parts (e.g. piezo-elements) can react on these uncertain effects and reduce the dimension of trusses. CRC 805 introduces new technologies to handle uncertainty in load carrying systems. Mathematically, this leads to a mixed-integer semi-definite programming problem.

Contact: Tristan Gally, Anja Kuttich, Marc Pfetsch, Stefan Ulbrich

Former: Kai Habermehl, Sonja Mars

Subproject B1: Optimization of Process Chains under Uncertainty

The aim of this subproject is to control uncertainty in process chains using mathematical models and optimization procedures in order to maximize the value of process chains. Uncertainty in the production of components emerges from random variations in the raw material, from unpredictable process behavior or because the customer's use can only be vaguely predicted. The optimization procedures are based on quantified (mixed-integer) linear programs.

Contact: Thorsten Ederer, Marc Pfetsch

Subproject B04: Simulation Based Optimization and Optimal Design of Experiments for Wetting Processes

The optimization algorithms to be developed are adjoint-based and build the foundation for numerous optimization problems within the proposed SFB. Example optimization parameters could be the geometry of surfaces, the contact line velocity, or material properties of fluids or surfaces. Thereby, among others, experimentally immeasurable parameters can be estimated with minimal variance using the simulation model. The results of this project should be particularly valuable for the future selection of generic experimental parameters and will make available quantities which are difficult to measure experimentally.

Contact: Elisabeth Diehl, Stefan Ulbrich

Adaptive Multigrid Methods for Fluid-Structure Interaction Optimization

Strong fluid structure coupling is a part of many technical systems. In recent years, encouraging progress has been made concerning the numerical simulation of Fluid-Structure Interaction (FSI) problems. The aim of this project is to combine methods for PDE constrained optimization, adaptivity and FSI simulation to develop an efficient adaptive multigrid method for Fluid-Structure Interaction optimization. Thus we go for an adjoint based Trust-Region SQP method that allows adaptive refinement of both, spatial and temporal grids.

Contact: Sarah Essert, Stefan Ulbrich, Michael Schäfer

Optimization with Complementarity Constraints

Complementarity constraints require that at most one of two variables is nonzero. In discrete optimization, complementarity constraints have an important significance for the modeling of logical relations. With their help one can express that from a set of possible events not more than one is allowed to occur. The applications of such relations are abundant, e.g., in machine learning, communication systems, capital budgeting or scheduling. The aim of this project, is to investigate a branch-and-cut algorithm for complementarity constrained optimization problems, including presolving techniques, branching rules, primal heuristics and cutting planes. The implemented software has to deal with problem instances involving large data in a robust manner. Furthermore, it should recognize and exploit special structures of a given problem instance automatically. As a tool, we use the software SCIP (see scip.zib.de) which provides a framework for solving discrete and combinatorial optimization problems. The purpose is to include further components to SCIP and to make them freely available for academic use.

Contact: Tobias Fischer, Marc Pfetsch

Subproject A01: Global Methods for Stationary Gastransport

This project will develop adaptive methods for the global solution of nonlinear integer optimization problemes with ODEs, on the example of stationary gastransport. Main issues are the construction of relaxations based on a priori error bounds, the combination of adaptive discretizations and branching methods, the handling of integral decisions via branch-and-bound as well as infeasibility cuts, the development of primal heuristics with a posteriori error bounds, and model reduction techniques.

Contact: Oliver Habeck, Marc Pfetsch, Stefan Ulbrich

Subproject A02: Adaptive Multilevel Method for the Optimal Control of Hyperbolic Equations in Gas Networks

In this project we analyze the optimal control of hyperbolic PDE systems with state constraints on the example of gas networks. Through the time-dependent control of compressors and valves, the pressure and velocity distribution of the transported gas in the network has to be optimized under constraints, e.g. such that the pressure lies within a specified tolerance range. The constraints of the resulting optimal control problem (P) consist of coupled systems of one-dimensional isothermal Euler equations describing the gas flow, node conditions and state constraints. We plan to use Moreau-Yosida regularizations to approximate (P) in order to derive optimality conditions. The main goal of this project is to provide an optimization theory, which will form the basis of adaptive multilevel methods.

Contact: Johann Michael Schmitt, Stefan Ulbrich

The field of computer algebra allows one to compute in and with a multitude of mathematical structures. It is interdisciplinary in nature, with links to quite a number of areas in mathematics, with applications in mathematics and other branches of science, and with constantly new and often surprising developments. Particular fruitful interactions unfold between computer algebra and algebraic geometry, number theory, and group theory. Algebraic algorithms open up new ways of accessing subareas of these key disciplines of mathematics, and they are fundamental to practical applications of the disciplines. Conversely, challenges arising in algebraic geometry, number theory, and group theory quite often lead to algorithmic breakthroughs which, in turn, open the door for new theoretical and practical applications of computer algebra. The goal of the DFG Priority Program SPP 1489 is to considerably further the algorithmic and experimental methods in the afore mentioned disciplines, to combine the different methods where needed, and to apply them to central questions in theory and praxis. Moreover, the programme is meant to support the further development of free computer algebra systems which are (co-)based in Germany, and which in the framework of different projects, may require crosslinking on different levels. Of particular interest are interactions with application areas inside and outside of mathematics such as system- and control theory, coding theory, cryptography, CAD, algebraic combinatorics, and algebraic statistics as well as hybrid methods which combine numerical and symbolic approaches.

Contact: Michael Joswig, Andreas Paffenholz

Exploiting Structure in Compressed Sensing Using Side Constraints (EXPRESS)

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.

Contact: Andreas Tillmann, Marc Pfetsch, Tobias Fischer

This project investigates optimization methods for mathematical programs with equilibrium constraints (MPECs) in function space that adaptively control the accuracy of the underlying discretization and of inexact subproblem solves in such a way that convergence is ensured. This enables the use of adaptive discretizations, reduced order models, and low rank tensor methods, thus making the solution of MPECs with high dimensional equilibrium constraints tractable and efficient. Two prototype classes of MPECs in function space are considered in the project: One with a family of parametric variational inequalities as constraints and the other constrained by a parabolic variational inequality. Based on a rigorous analytical foundation in function space, the project will develop and analyze inexact bundle methods combined with an implicit programming approach. In addition, inexact all-at-once methods will be considered. In both cases, the evaluation of cost function, constraints, and derivatives is carried out on discretizations which are adaptively refined during optimization and can further be approximated by reduced order models or low rank tensor methods. We will develop implementable control mechanisms for the inexactness, which are tailored to the needs of the optimization methods and can be based on a posteriori error estimators. The algorithms will be implemented and tested for the considered prototype classes of MPECs.

Contact: Anne-Therese Rauls, Stefan Ulbrich

polymake

polymake is a (more and more versatile) tool for the algorithmic treatment of convex polyhedra and finite simplicial complexes.

Contact: Michael Joswig, Andreas Paffenholz, Silke Horn, Katrin Herr

SCIP - Solving Constraint Integer Programs

SCIP is a framework for Constraint Integer Programming and branch-cut-and-price. It is also currently one of the fastest non-commercial mixed integer programming (MIP) solvers. In Cooperation with the Zuse Institute Berlin and the University of Erlangen-Nürnberg we improve and extend SCIP.

Contact: Marc Pfetsch

SCIP-SDP

SCIP-SDP is a plugin for SCIP to solve mixed integer semidefinite programs. It combines the branch-and-bound framework of SCIP with interior-point SDP-solvers. It provides the data handling, some presolving and propagation as well as a reader for a modified sparse SDPA-format with additional lines for integrality constraints. Currently the SDP-Solvers DSDP and SDPA are interfaced.

Contact: Tristan Gally

Former: Sonja Mars

Multilevel Methods for PDE-constrained Optimization with State Constraints

We extend the adaptive multilevel SQP-method for control-constrained optimal control problems of Ziems and Ulbrich to state-constrained optimal control problems. To this end, we combine the Moreau Yosida regularization with the adaptive SQP method. The refinement conditions and the penalty parameter update are modified specifically. Based on the convergence theory for the Moreau Yosida regularization and the adaptive SQP method we deliver new convergence results for the output of the multilevel SQP method for state constraints. In order to reduce the computational effort we include a reduced-order model based on POD. We apply this theory to flow control problems.

Contact: Stefanie Bott, Stefan Ulbrich, Michael Schäfer

Optimal Flow Control based on Reduced Models

The aim of this research project is to explore the possibility of developing POD-based reduced-order models for active control of fluid flows governed by the Navier-Stokes equations. In particular, we consider the cancellation of Tollmien-Schlichting waves in the boundary layer of a flat plate by plasma actuators. By optimal control of the plasma actuator parameters it is possible to reduce or even cancel the Tollmien-Schlichting waves. The optimization is performed within the reduced system with a Model Predictive Control (MPC) approach.

Contact: Jane Ghighlieri, Stefan Ulbrich, Michael Schäfer

AdRIA (Adaptronik-Research, Innovation, Application)

Adaptronics is an interdisciplinary science dealing with mechanical structures which adjust autonomously to changing conditions. In order to realize such adaptive structures, actuator and sensor systems as well as real-time-capable controllers are developed. Adaptive structures have many possible fields of application among them active vibration control.

Contact: Carsten Schäfer, Stefan Ulbrich

Cocoon: Mixed-Integer Nonlinear Models in Wireless Networks

In this project we explore the utilization of mixed-integer optimization in wireless telecommunication networks. Typical for problems occurring in this context is the simultaneous consideration of continuous optimization variables, e.g., like beamforming vectors and combinatorial aspects, e.g., like the assignment of base stations to mobile users. Mathematical models are derived that account both for the requirements of the application and the solvability. Usually one has to deal with NP-hard problems in this context that cannot be solved by standard software. We investigate convex approximations as well as heuristics to derive reasonable good solutions. We use these approximations as well as techniques like cutting plane generation aiming to solve the mixed integer nonlinear model of the original problem. The global optimal solution can then also be used to evaluate heuristic and approximation approaches.

Contact: Anne Philipp, Stefan Ulbrich

Forschungskooperation Netzoptimierung

ForNe is an industry coorperation project concerned with long-term planning of gas networks.

Duration: Jul 01 2009 - Mar 31 2014

Contact: Imke Joormann, Marc Pfetsch

Analysis of Sparse Solutions of Underdetermined Linear Equation Systems

This research project deals with the problem to recover a sparse solution of an underdetermined linear (equality) system. This topic has many applications and is a very active research area. It is located at the border between analysis and combinatorial optimization. The main goal of our project is to obtain a better understanding of the conditions under which (efficiently) finding such a sparse solution, i.e., recovery, is possible. Our project is characterized by both theoretical and computational aspects as well as the interplay of continuous and discrete methods.

Duration: Apr 01 2011 - Jun 30 2013

Contact: Andreas Tillmann, Marc Pfetsch

This project is part of the BMWi project "Investigation of the technical capacities of gas networks", in which six research partners and one gas transportation company are involved. The technical capacities determine the upper bounds on the amount of gas that can be charged into or discharged from a gas network, which limit the revenue of a gas transportation company. Therefore, a central aspect is to compute the technical capacities.

Duration: Jul 01 2009 - Jun 30 2012

Contact: Imke Joormann, Marc Pfetsch

Advanced Numerical Methods for PDE Constrained Optimization with Application to Optimal Design and Control of a Racing Yacht in the America's Cup

The goal of this project is the development, analysis, and implementation of robust and efficient optimization algorithms for the optimal design and control of a racing yacht competing in the America's Cup. The project focuses on the optimization of the hull-keel-winglet configuration toward drag minimization. This involves optimization problems including very complex and highly coupled systems of PDE constraints.

The main research topics of the project are:

- Multilevel optimization methods based on inexact trust-region SQP techniques using a hierarchy of adaptive discretizations or models.
- Semismooth Newton and interior point methods to handle inequality constraints for design and state variables.
- Adaptivity in time and space based on the goal oriented approach and including the issue of inequality constraints.
- Parallel processing for the optimization schemes via space and time domain decomposition.

Contact: Christian Brandenburg, Stefan Ulbrich

Optimal Control of Switched Networks for Nonlinear Hyperbolic Conservation Laws

The aim of this project is the analytical study of optimal control problems for networks of nonlinear hyperbolic conservation laws under modal switching. Networks of this type arise for example in traffic flow models where the switching is considered in the coupling conditions at the nodes. Since entropy solutions of conservation laws may develop shocks, the analysis quite is difficult. Switching, for example introduced by traffic lights, may lead to additional discontinuities in the solution.

Contact: Sebastian Pfaff, Stefan Ulbrich

Adaptive Multilevel SQP-Methods for PDAE-Constrained Optimization with Restrictions on Control and State. Theory and Applications

The aim of this project is to develop, analyze and apply highly efficient optimization methods for optimal control problems with control- and state-constraints governed by time-dependent PDAEs. We combine in a modular way modern space-time adaptive multilevel finite elements methods with linearly implicit time integrators of higher order for time-dependent PDAEs and modern multilevel optimization techniques. The aim is to reduce the computational costs for the optimization process to the costs of only a few state solves. This can only be achieved by controlling the accuracy of the PDAE state solver and adjoint solver adaptively in such a way that most of the optimization iterations are performed on comparably cheap discretizations of the PDAE. We will focus on two exemplary applications.

Contact: Jan Carsten Ziems, Stefan Ulbrich

Subproject D5: Efficient Numerical Multilevel-Methods for Optimization of Gas Turbine Combustion Chambers

In modern gas turbine combustion chambers, there are still many different aspects that are not fully understood.

The CRC 568 was founded to make fundamental progress in this field of research. The main focus lies on the simulation of the flow and combustion that are important for the understanding of the different phenomena.

Even though the simulation itself is a challenging task, in this project we aim for the optimization of the combustion chamber with respect to various variables including the geometry. To match this purpose, we try to apply and develop mathematical techniques that make optimization applicable.

Contact: Rolf Roth, Stefan Ulbrich

Research Group Optimization

Ursula Röder

roeder (at) mathematik.tu-darmstadt.de

Phone: +49 (0) 6151 16-23444

Fax: +49 (0) 6151 16-23445

Monika Kammer

kammer (at) mathematik.tu-darmstadt.de

Phone: +49 (0) 6151 16-23448

Fax: +49 (0) 6151 16-23445