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FAU Erlangen-Nuremberg

Projects (summaries)

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Projects of the second funding period

(may include some continued projects from the first funding period)

 Apel, Neubiberg 
 Rösch, Duisburg
 Vexler, Garching

Numerical Analysis and Discretization Strategies for Optimal Control Problems with Singularities

Optimization of technological processes plays an increasing role in science and engineering. This project deals with different types of optimal control problems governed by elliptic or parabolic partial differential equations and characterized by additional pointwise inequality constraints for control and state. Of particular interest are problems with all kinds of singularities including those due to reentrant corners and edges, nonsmooth coefficients, small parameters, and inequality constraints. The project targets two goals: First, starting from a priori error estimates, families of meshes are generated that ensure optimal approximation rates. Second, reliable posteriori error estimators are developed and used for adaptive mesh refinement. A challenge is the incorporation of pointwise inequality constraints for control and state. Both techniques can ensure efficient and reliable numerical results. With a successful strategy it is possible to calculate numerical solutions of the optimal control problems with given accuracy at low cost.

 Bänsch, Erlangen 
 Benner, Chemnitz

Optimal Control-Based Feedback Stabilization in Multi-Field Flow Problems

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The aim of this project is to develop numerical methods for the stabilization of solutions to flow problems. This is to be achieved by action of boundary control using feedback mechanisms. In very recent work by Raymond and earlier attempts by Barbu, Triggiani, Lasiecka, and others, it is shown analytically that it is possible to construct a linear-quadratic optimal control problem associated to the Oseen approximation of the Navier-Stokes equation so that the resulting feedback law, applied to the instationary Navier-Stokes equation, is able to exponentially stabilize unstable solution trajectories assuming a certain smallness of initial values. Until recently, the numerical solution of these linear-quadratic optimal control problem and the associated algebraic Riccati equations (AREs) was an unsolved numerical challenge due to the computational complexity and storage requirements of existing algorithms. Employing recent advances in reducing these complexities essentially to a cost proportional to the simulation of the forward problem, we plan to apply this methodology to multi-field problems where the flow is coupled with other field equations. In particular, we plan to apply recent ideas to compute the approximate feedback operator directly without ever forming the explicit solution of the ARE.

 Behr, Aachen 
 Bischof, Aachen

Robust Shape Optimization for Artifical Blood Pumps: Hematological Design, Large-scale Transient Simulations, and Influence of Constitutive Models, Sensitivity Analysis

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Analysis and design optimization of blood-handling mechanical devices, and in particular, miniature heart-assist blood pumps, presents a number of unique challenges. The micro-structural properties of blood affect both the choice of the design objectives, such as minimizing blood damage and clotting, and the choice of flow equations, which should account for the non-Newtonian nature of blood as a continuum. We propose to address, in the context of shape optimization of blood pumps, the issue of objective functions which can be correlated with the accumulation of blood damage along flow pathlines, and the influence of constitutive model (Newtonian, generalized Newtonian, and viscoelastic) on the optimal shapes. The entire optimization tool chain, based on analytically- derived sensitivities and adjoints, will be then subjected to sensitivity analysis with the help of automatic differentiation. It is expected that criteria for detecting inadequacies in constitutive modeling will be exemplified, e.g., by extreme sensitivity of the optimal shapes to model parameters. A sample shape optimization problem in an actual complex geometry of an axial blood pump is to be solved during this project.

 Blank, Regensburg
 Garcke, Regensburg 

Optimization problems governed by Allen-Cahn and Cahn-Hilliard variational inequalities






The Allen-Cahn or Cahn-Hilliard variational inequalities can be derived as gradient flow, based on the Ginzburg-Landau energy with an obstacle potential. This project will be concerned with the multicomponent models resulting in vector valued problems. Starting from the gradient flow structure we will address the following main issues.

  • Topology optimisation with non-local Allen-Cahn variational inequalities. In structural topology optimisation problem one tries to find the material distribution in a given design domain, such that the design objective is optimised. This project is mainly concerned with two aspects thereof, more precisely with:
    • Mean compliance optimisation using Allen-Cahn variational inequalities, where the Ginzburg-Landau energy has to be extended by an elastic contribution.
    • Relating phase field approaches to sharp interface approaches by means of asymptotic expansion techniques for phase field equations.
  • Optimal control of Allen-Cahn and Cahn-Hilliard variational inequality systems using a force term in the equations. This results into MPEC problems in function spaces. Applications are for example the control of quantum dot formation or the morphing problem, where the goal is the transformation of a given interface into another one. We want to derive
    • First order optimality conditions for the continuous setting.
    • Efficient computational tools to solve the MPECs on the discrete level.
    • A relation to an optimisation problem with the sharp interface model as constraint by means of asymptotic expansion techniques.
 Bock, Heidelberg 
 Engell, Dortmund
 Sager, Heidelberg

Optimal Control of Periodic Adsorption Processes

Periodic adsorption processes are widely established in process engineering as a separation procedure, e.g., for fine chemicals or pharmaceuticals. Recent variants also involve a combination of adsorption and reaction processes. Characteristic of such processes are moving concentrations profiles of different species in a solid fixed bed, and periodic controls that involve switching between different types of operation.

The dynamics of each phase can be modeled by non-stationary partial differential equations (PDE) in one or two spatial dimensions, so that the overall system is described by periodically switched PDE. Following start-up, a periodic attractor, or stable Cyclic Steady State (CSS), is finally reached and used for production, which should be optimal with respect to operational costs and product specifications. Recent years have seen the development of a variety of process operation variants to improve efficiency. Two of these are investigated in this project: the simulated moving bed with variable inlet concentrations ('ModiCon-SMB') and the novel fixed bed catalytic reactor with desorptive cooling ('DC process').

Due to the complexity of the process models and their periodic operation, only few model based online optimization approaches exist; and their use for large-scale applications is still limited by prohibitive computation times. The aim of this project is to develop new efficient numerical methods for the optimization of periodic adsorption processes described by periodically switched and non-stationary PDE, which are capable of online application to the process in the presence of perturbations. These methods combine Newton type optimization methods with a Newton-Picard approach to cope efficiently with the periodicity constraints.

The following features characterize the new methods, the development of which is driven by the requirements of the two processes investigated in Dortmund:

  • An adequate modeling of the ModiCon-SMB and DC processes as 1D and 2D non-stationary periodic PDE, and of the corresponding constrained process optimization problems
  • A simultaneous and direct optimization framework, in which the discretized model equations enter the optimization problem as non-linear constraints
  • Novel SQP and Gauss-Newton methods for the resulting inequality constrained non-linear programming problem, which can handle coarse approximations of the constraint Jacobians of Newton-Picard type that capture only derivativeswith respect to slow or unstable modes.
  • A time domain decomposition by multiple shooting, and an adaptation of the discretization mesh based on dual weighted residual (DWR) a-posteriori error estimates
  • A generalization of non-linear feedback control algorithms of Non-linear Model Predictive Control (NMPC) type to optimal periodic control problems to cope with process perturbations and model inaccuracies
  • Ultimately, the optimization and experimental validation of the optimal operating regime for the simulated moving bed and the desorptive cooling process.
 Braack, Kiel 
 Prohl, Tübingen

Consistent Finite Elements for Optimal Control Problems in Computational Fluid Dynamics

The focal point of this project proposal is the analysis of discretization methods and solving aspects in the field of fluid dynamics and their corresponding optimization problems. In particular, we examine, the crossover from the continuous to the discrete level, so that beneficial properties became maintained. The variety of fluid dynamical equations starts with the incompressible case and reaches compressible multiphase fluids. On the long-term, we have optimization problems including multi-physics in mind, as e.g. the optimization of aluminum production.



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 Büskens, Bremen 
 Peitgen, Bremen
 Preusser, Bremen

Optimization and Optimal Control of Therapy Parameters for Radio-Frequency Ablation

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The focus of our project is the RF-ablation of tumors/lesions in the human liver with bipolar systems: a probe, internally cooled and containing two electrodes, is placed in the vicinity of the malignant tissue. The electrodes are connected to a generator, and an electric current warms the tissue close to the probe up to temperatures of more than 60° C. Consequently the proteins of the heated tissue denaturate and its cells die. The treatment is successful, if all tumor cells are destroyed by the denaturation of their proteins. In areas distant from the probe the critical temperatures can only be achieved by propagation of heat away from the source. Thereby perfusion (blood flow) of the surrounding tissue by large vessels and capillary blood flow has a significant effect. To enlarge the volume of coagulated tissue (i.e. denaturated proteins) and to decrease the influence of perfusion the tumor can be penetrated with multiple probes (at most 2-3) simultaneously.

The project aims at several targets: The design of appropriate objective functionals which measure the quality of an ablation process. Furthermore the identification of the patient-individual material properties. These include the electric and thermal conductivities as well as densities and heat capacities. Moreover the project aims at an optimization and optimal control of the RF-probe's positioning and electric potential. Since the mathematical model describing the ablation process is a system of PDEs which are coupled in a complex way, increasing levels of modeling, discretization and optimization will be considered. Finally the postoptimal calculation of so-called parametric sensitivity differentials of the optimal solutions with respect to model data and perturbations will be considered at the different levels. This information plays an important role in the analysis of partial results as aforementioned and might be a helpful tool in the assessment of optimal solutions for users.

 Burger, Münster 
 Pinnau, Kaiserslautern

Optimal Control of Self-Consistent Classical and Quantum Particle Systems

 Conti, Bonn 
 Rumpf, Bonn
 Schultz, Duisburg

Multi-Scale Shape Optimization under Uncertainty

Our project deals with stochastic shape optimization for elastic materials where stochasticity enters the problem both via stochastic loading and stochastic geometry. We intend to bring together the analytical treatment of multi-scale problems via homogenization, the reliable numerical solution of PDE problems on complex domains described via level sets or phase field models, and two-stage stochastic programming approaches using different perceptions of risk aversion.

 Deckelnick, Magdeburg 
 Hinze, Hamburg

Structure Exploiting Galerkin Schemes for Optimization Problems with PDE Constraints

This project is concerned with the development and the analysis of discrete concepts and algorithms for pde constrained optimization problems including control and state constraints. We propose a tailored discrete concept for optimization problems with nonlinear pdes including control constraints and develop a new discrete concept in pde constrained optimization involving state constraints. The key idea consists in conserving as much as possible the structure of the infinite-dimensional KKT (Karush-Kuhn-Tucker) system on the discrete level, and to appropriately mimic the functional analytic relations of the KKT system through suitably chosen Ansätze for the variables involved.

For both cases we provide numerical analysis, including convergence proofs and adapted numerical algorithms. As a class of model problems we consider optimization with (nonlinear) elliptic and parabolic pde's. This allows to validate and compare the new concepts to be developed in this project against existing approaches for the class of elliptic control problems.

 Eppler, Dresden 
 Harbrecht, Basel

Shape Calculus for the Efficient Solution of Shape Optimization Problems with Elliptic and Parabolic State Equation

The scope of the present project is the analysis and efficient solution of shape optimization problems with elliptic and parabolic state equation. The main ingredient is the shape sensitivity analysis, i.e., the analytic computation of the shape gradient and Hessian. In particular, by analyzing the shape Hessian at stationary domains we can distinguish well-posed and ill-posed shape optimization problems. Well-posedness implies convergence of approximate shapes towards the optimal shape if the number of design parameters is increased. Regarding the efficient numerical solution of the state equation, we apply the whole range of available methods from boundary element to finite element methods.



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 Frank, Aachen 
 Herty, Aachen

Optimal Treatment Planning in Radiotherapy based on Boltzmann Transport Equations

We plan to investigate optimal control problems used in clinical external beam radiotherapy. The starting point is the formulation of the treatment planning problem as an optimal control problem for the Boltzmann Continuous Slowing Down model. Mathematically, the model consists of an integro-partial differential equation in energy, spatial and angular variables. Besides analytical investigations on existence and regularity of solutions and optimal controls we study moment approximations to the high-dimensional phase space in order to devise efficient numerical schemes. The relation between optimization and moment approximation will also be investigated. We use the analytical and numerical results on real CT-patient data and compare the optimal treatment plans obtain by solving the control problem with standard Monte-Carlo methods.

 Gauger, Braunschweig 
 Griewank, Berlin
 Slawig, Kiel

Optimal design with bounded retardation for problems with nonseparable adjoints

We study mathematical methods and algorithmic techniques for the transition from sim- ulation to optimization. We focus on applications in aerodynamics and climate studies. The methodology is applicable to all areas of scientific computing, where large scale PDEs are treated by fixed point solvers. To exploit the domain specific experience and expertise invested in these simulation tools we extend them in a semi-automated fashion. The op- timization steps are determined by a design space preconditioner. A principle of bounded retardation of the convergence rate can be achieved. On a classical example this factor can be determined explicitly. We plan to extend these theoretical results to problems where the adjoint equation is no longer the sum of a term on the states and the design. Such non- separability arises typically when the design variables enter in a truly nonlinear fashion, especially in shape optimization, parameter optimization, and on non-stationary problems to which our methods are already applied successfully.

 Herzog, Chemnitz 
 Meyer, Darmstadt

Analysis and Numerical Techniques for Optimal Control Problems Involving Variational Inequalities Arising in Elastoplasticity

Solid bodies depart from their rest shape under the influence of applied loads. In case the applied loads or stresses are sufficiently small, many solids exhibit a linearly elastic and reversible behavior. If, however, the stress induced by the applied loads exceeds a certain threshold (the yield stress), the material behavior switches from the elastic to the so-called plastic regime. In this state, the overall loading process is no longer reversible and permanent deformations remain even after the loads are withdrawn. Mathematically, this leads to a description involving variational inequalities.

Plastic deformation is desired for instance as an industrial shaping technique of metal workpieces, as e.g. by deep-drawing of body sheets in the automotive industry. The task of finding appropriate time-dependent loads which effect a desired final deformation leads to optimal control problems for elastoplasticity systems. These are also motivated by the desire to reduce the amount of springback, i.e., the partial reversal of the final material deformation due to a release of the stored elastic energy once the loads are removed.

The project targets optimal control problems for static and quasi-static models of infinitesimal elastoplasticity with hardening. Its main goals are

  • to investigate these optimization problems,
  • to quantify the error due to discretization,
  • and to develop fast algorithms for their solution.

Models of elastoplasticity involve non-smooth features due to their description by variational inequalities and pointwise projections. The mathematical treatment of associated optimal control problems is therefore highly challenging and it requires a substantial extension of the established techniques.

 Gugat, Erlangen 
 Herty, Aachen

Control of System Dynamics in Gas and Water Networks

We are interested in gas and water transportation networks where the dynamics is governed by hyperbolic balance laws. Typically, the transient dynamics inside a pipe or channel $j$ is described by an spatial one--dimensional system of balance laws for which as a prototype one considers the $p-$system.

Typically $\rho_j$ being either the density in case gas flow or the water height in case of flow in open channels and $q_j$ the mass flux. A simple model for the pressure term in gas networks is $p(\rho)=a^2 \rho$ where as in the case of water networks we have the typical $p(\rho)= g/2 \rho^2.$ The main difference is the source term: The major physical effect in gas networks is pipe--wall friction where as in water networks also the gravity force due to the slope of the pipe is essential. Both terms can be included in $g(x, \rho,\rho u).$ In order to pose a complete problem we need to pose additional algebraic coupling conditions at the vertices. These induce (nonlinear) boundary conditions for the $p-$system and couple the dynamics on the connected pipes. Depending on the type of the intersection (pipe--to--pipe, compressor, valve, gate, $\dots$), different conditions are imposed, in general, they can be written as 
\psi ( (\rho_{i})_{i\in\delta^\pm}, (\rho_{i} u_{i})_{i\in\delta^\pm} , U(t) ) = 0, 
where $U(t)$ is a possible control with the interpretation of either a compressor power or the height of an underflow gate or the position of a valve.

Three main problems can be identified as tasks for gas and water net providers: {\em Under minimal costs they should (1.) satisfy the costumer's demands (pressure, water height), (2.) avoid the occurence of shock waves (in order to prevent pipe breakes) and (3.) provide a gas (resp. water) mixture of a certain quality}. The latter problem is due to the fact the providers procure gas (water) from several sources with different quality bu need to provide customers with a desired composition.

Currently, there exist results in the following directions. At first, existing commercial tools as well as most parts of the literature use only simplified models such as quasi--static and heuristic approximations , linearizations or coarse grid approximations to treat problem (1). However, these results are too simple since several crucial effects of the system dynamics, such as the occurrence of shock waves or the finite speed of wave propagation, are lost. Furthermore, the construction of controls or stabilization cannot be discussed in this context.

Second, in the previous funding period of the DFG SPP1253 within the project 'Optimal Nodal Control of Networked Hyperbolic Equations' the problem (1) has been studied in the case of weak solutions including shock waves. Several results concerning (1) could be achieved and have been published in a series of papers. In particular, in well--posedness of the problem under the expected assumption of a small $TV$--bound of the initial data could be established. The result is a well--posedeness and regularity result for coupled nonlinear system of balance laws and valid for the $p-$system under a subsonic condition and rather weak assumptions on the coupling function $\Psi.$ The obtained solutions do only posses BV--regularity in space, but may well include discontinuous solutions (shock waves). However, a result on the dependence of the solution on the control $U(t)$ could be established. This in turn has been used to prove existence of weak entropic and possibly discontinuous optimal controls and states.

These results partly solve problem (1) in the sense of an optimal control problem, but do not yet extend to controllability or stabilitzation questions.

Last, there has been intense research concerning controllability, i.e., solving problem (1) and (2) using classical solutions. The existence of classical solutions for quasilinear hyperbolic systems on a given finite time interval (so-called semi-global solutions) has been investigated by Li Ta-Tsien and his group. For the case without source terms, also classical solutions on networks have been considered. However, for the system the influence of the source term is essential and, for example, the form of the equilibria is determined by the source term: Due to the source term, for each fixed non-zero flow rate at most one constant equilibrium exists. The existence of classical semi-global solutions for systems with source term has been analysed only recently, and the corresponding case of networked systems with source-terms has not yet been analyzed. We have a similar situation for problem (1) concerning stabilization of classical solutions at equilibria. There is some material for the case without source terms, also for networks where feedback laws are constructed using strict Lyapunov functions, but the case of systems with source terms is still open.


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 Hintermüller, Berlin 

Elliptic Mathematical Programs with Equilibrium Constraints (MPECs) in Function Space: Optimality Conditions and Numerical Realization

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Many phenomena in engineering, life sciences, mathematical finance or physics result in a mathematical model of variational or quasi-variational inequality type. Applications comprise contact problems (with friction) in elasticity, torsion problems in plasticity, option pricing in finance, the magnetization of superconductors or ionization problems in electrostatics. Often, one is interested in influencing the system under consideration by some control means in order to optimize a certain output quantity. The resulting optimization problem falls into the realm of Mathematical Programs with Equilibrium Constraints (MPECs), which are challenging due to constraint degeneracy.

The project work concentrates on the development of a suitable optimality theory as well as the design and implementation of efficient solution algorithms for classes of MPECs in function space which are governed by elliptic (quasi)variational inequalities. Among others, the results will be applied to the following processes:

   * Stationary magnetization of type-II superconductors;
   * Control of torsion phenomena with variable plasticity threshold;
   * Ionization problems in electrostatics.

Concerning stationarity conditions, the project aims at developing new mathematical techniques for deriving and categorizing adequate optimality conditions in function space for such problems. The employed methodology relies on constraint relaxation to satisfy constraint qualifications, so called path-following approaches for constraints with low multiplier regularity and a subsequent asymptotic study for deriving a first order system for the original problem.

Since discretized MPECs result in large scale problems, tailored numerical solution techniques relying on adaptive finite element methods, semismooth Newton and multilevel techniques are developed within this project. The semi- or non-smoothness aspect arises due to the equivalence of the first order optimality systems to non-smooth operator equations.


 Hinze, Hamburg 
 Turek, Dortmund

Hierarchical Solution Concepts for Flow Control Problems

The goal of the project is the development of a generally applicable hierarchical (multigrid) solution framework for flow control problems which allows their numerical solution requiring a computational effort of only a small multiple of that of the flow simulation itself. To achieve our goal we combine

  • High performance scientific computing techniques in flow simulation,
  • Discrete concepts in space and time which are tailored to the structure of the first-order necessary optimality conditions of the underlying optimization problem (Karush-Kuhn-Tucker system, short KKT system), and
  • Sophisticated optimization algorithms combined with multigrid concepts which allow to exploit the structure of the underlying optimization problem.

We investigate two different multigrid approaches. APPROACH I tackles the KKT system all-at-once, and APPROACH II exploits the structure of the KKT system and reduces it to a nonlinear integral equation (variational inequality) for the control u, which then is tackled by a multigrid approach tailored to integral equations.

Within this first application period we concentrate on methodic and algorithmic aspects which are validated at optimization problems for 2D time dependent flows with prototypical character.

 Hoffmann, Garching 

Optimal Control in Cryopreservation of Cells and Tissues

      

The project concerns the application of the theory of partial differential equations and optimal control techniques to the minimization of damaging factors in cryopreservation of living cells and tissues in order to increase the survival rate of frozen and subsequently thawed out cells.
   The objective of the proposal is the development of a hierarchical, coupled mathematical model that describes the most inuring effects of cryopreservation of living cells and especially tissues. The model should be basically governed by partial differential equations and should contain feasible controls that can essentially reduce the inuring effects of freeing and thawing, when chosen optimally. Such a model is expected to consist of several coupled submodels each of them covers a part of the problem.
   The objective index has to reflect the survival rate of tissue cells, which includes accounting for biological aspects of the problem. Analytical characterization, numerical implementation, and practical verification of optimal controls through experiments is the intention of the project. Project home page is located on: http://www-m6.ma.tum.de/~botkin/spp1253.html.

 Hoppe, Augsburg 
 Franke, Augsburg

PDE Constrained Optimization Based on Adaptive Model Reduction with Applications to Shape Optimization of Microfluidic Biochips and to Blood Flow in Microchannels

 Kostina, Marburg 

Towards Optimum Experimental Design for Partial Differential Equations



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 Lang, Darmstadt 
 Ulbrich, Darmstadt

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. To this end we want to 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.

 Leugering, Erlangen 
 Peukert, Erlangen

Optimization of Particle Synthesis

Disperse or particulate products, i.e. powders and suspensions, are of enormous economic relevance. E.g., they comprise around 70% of the production in chemical industry, covering a wide range of fields from gross to novel high performance products. As a special characteristic of disperse materials the attributes and consequently the value are governed to a large extent by their inherent disperse properties, i.e. particle size, primary particle size, morphology etc. Therefore, it is of great importance to control and optimize the synthesis process in order to get tailor-made products. This is especially true for nanosized particulate materials which become more and more important nowadays. Population balance equations represent the state-of-the-art for the modelling of polydisperse particulate processes. Describing the occurring phenomena like nucleation, growth and agglomeration leads to a nonlinear integro-differential equation, on which we will focus in this project.

 Wittum, Heidelberg 
 Queisser, Heidelberg

Estimation of Parameters in Three Dimensional Volume Models for Signal Processing in Neurons by Optimization Multigrid

We have developed mathematical models with which we can investigate functional properties of signal coding in brain cells, in particular neurons. Signal processing in neurons takes place on different levels. For one, electrical signals are propagated in the dendritic tree, towards or away from the soma. Secondly, electrical information that reaches the soma is partially re-coded as a calcium wave that propagates towards and into the cell nucleus, activating cascades which result in the expression of certain genes. To address both signal processing in the dendritic tree of the neuron and calcium signaling at the cell nucleus, we have developed three dimensional volume models of neurons and of neuron cell nuclei on which PDEs describe the underlying signaling processes. Embedded in these equations are a number of parameters, such as neuronal membrane capacitance, inner- and outer-neuronal capacitance or diffusion parameters for nuclear calcium. In some cases literature is rare concerning the actual values of these parameters. This is where our project steps in. The target of this project is to develop a method based on the multiple shooting technique for PDEs to master the full extent of parameter identification in signal processing in neurons.

 Rannacher, Heidelberg 

Model Reduction by Adaptive Discretization in Optimal Control

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This project will employ the concept of 'goal-oriented' adaptivity for model 'reduction' in solving optimal control problems governed by partial differential equations (PDE). The underlying framework is the 'Dual Weighted Residual (DWR) method' wich was originally developed by R. Becker and R. Rannacher for adaptive discretization of PDE by the finite element Galerkin method. In this approach residual-based 'weighted' a posteriori error estimates are derived for quantities interest, where the weights are obtained by numerically solving an associated 'dual problem'. Due to the use of problem inherent sensitivity information, these a posteriori error estimates are tailored to the special needs of the computation. This allows for successively improved control of spatial and time discretization which eventually results in highly economical discretization. In this project the main emphasis is on nonstationary optimal control problems, which pose particularly high requirements on computational resources, and on problems invlolving additional constraints for controls and states. In these cases model reduction by adaptive discretization may prove most usefull. Research on these topics in the context of PDE-constrained optimal control has started only recently and there are still many theoretical as well as practical open questions.

 Sachs, Trier 

Adaptive Trust Region POD Algorithms

Dynamical systems described by general evolution equations, like a partial integro-differential equation, can be reduced in size through a proper orthogonal decomposition technique. If these systems are part of an optimization problem, one has to solve this problem numerically. While the optimization algorithm is working, the need for an update of the model might arise, which could prove to be a costly part in the overall computing budget. In order to alleviate this effect, a model management technique can be employed handling the coarse and fine grid models in the optimization phase. We base this on a multi-level trust region technique which is also able to deal with nonlinear models. Applications of this technique can be found in various fields of engineering (fluid flow problems) and finance (calibration of derivative products).



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 Rösch, Duisburg
 Siebert, Duisburg 

Design and Analysis of Adaptive Finite Element Discretizations for Optimal Control Problems

The aim of this project is the design and the mathematical analysis of adaptive finite element methods for the efficient solution of optimal control problems governed by partial differential equations. It is our goal to perform a thorough analysis of adaptive methods for constrained optimal control problems. This includes a revision of existing estimators, a revision of existing and design of new adaptive methods as well as a rigorous convergence analysis for the adaptive discretization.

 Leugering, Erlangen 
 Ulbrich, Darmstadt 

Optimal Control of Switched Networks for Nonlinear Hyperbolic Conservation Laws


Projects of the first funding period

 Agar, Dortmund
 Bock, Heidelberg
 Engell, Dortmund

Optimal Control of Periodic Adsorption Processes via a Reduced Newton-Picard Scheme

Goal of the project is the development of practically applicable numerical methods for periodic optimal control of large, switched PDAE systems, and their application to two processes (Simulated Moving Bed, Desorptive Cooling). The methods we have in mind shall be based on novel ideas for reduced Newton type methods that work with inexact jacobian matrices. In contrast to most existing approaches, only one accurate adjoint solve is required in each iteration, independent of the number of state inequalities. This framework shall allow to compute cheap inexact derivatives of the cyclic steady state constraint in some dominant directions only. A mixture of Newton and Picard iterates shall be used to exploit the fact that the jacobian matrix of the desired cyclic steady state constraint has most eigenvalues close to zero.

 Bänsch, Erlangen 
 Kaltenbacher, Erlangen 
 Leugering, Erlangen

Optimization of Electro-Mechanical Smart Structures

The project deals with shape and topology optimization of piezoelectric-mechanical-acoustic coupled PDE systems.

To achieve our goal, we have combined and will combine sophisticated engineering methods with numerical and analytical mathematical methods. Especially, due to the application of optimization methodologies to real world problems, we are forced to develop new theories and algorithms. Within the second period we will further enhance our mathematical methods and will focus on the application to real world problems in technical science. Furthermore, we will fabricate prototypes and perform measurements to validate our theoretical results.

 Gauger, Braunschweig 
 Schulz, Trier

Multilevel Parameterizations and Fast Multigrid Methods for Aerodynamic Shape Optimization

In this project, we are going to investigate multilevel shape parameterizations and fast optimization algorithms by exploiting the arising multilevel structures in the shape and in the flow problem. The algorithmic paradigm favored is that of an overall multigrid optimization method for all variables involved. This will potentially lead to optimization methods which require a numerical effort equivalent to only a few simulation runs regardless of the resolution of the discretizations. Because elastic effects play an important role in wing design, aspects of multi-disciplinary design optimization will be addressed as well.

 Harbrecht, Basel 
 Eppler, Dresden

Development of Efficient Numerical Methods for the Solution of Shape Optimization Problems with Elliptic State Equation

Image:Harbrecht.gif

Throughout the last 25-30 years, optimal shape design has become more and more important in engineering applications. Many problems that arise in structural mechanics, fluid dynamics and electro-magnetics lead to the minimization of functionals defined over a class of admissible domains, governed by the solutions of boundary value problems or initial boundary value problems. Our methodology is based on a mathematical shape sensitivity analysis and exploits the analytic computation of the shape gradient and, if available, even higher derivatives. Afterwords, the shape (and thus, the shape gradient) as well as the underlying state equation are discretized, which is in contrast to fully discrete shape optimization methods. The scope of the present project is the development of efficient algorithms for the solution of general shape optimization problems

 Haußer, Berlin 
 Voigt, Bonn

Control of Nanostructures through Electric Fields

This project is motivated by theoretical and experimental observations, that an external (macroscopic) electric field has large influence on the shape evolution of nanostructures on crystalline surfaces.

As a first step to solve a control problem for the evolution of a complicated surface consisting of terraces/islands separated by atomic height steps, we will consider a single island given as a closed curve. The aim is to drive the island by means of an external electric field to a predefined shape. Mathematically this leads to the optimal control of a free boundary problem.

We will investigate this optimal control problem analytically and provide efficient numerical methods. The free boundary problem will be formulated as a degenerate phase-field model where the island boundary is given as the 1/2 level set of a phase-field variable whose dynamic is derived from a Ginzburg-Landau free energy functional.

A key point in the numerical treatment will be the extensive use of adaptive mesh refinement and coarsening. Descretizing the state and adjoint variables on independently adapted meshes will significantly reduce the computational cost.


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 Heuveline, Karlsruhe 
 S. Ulbrich, Darmstadt 
 M. Ulbrich, München

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 as well as control of the wave generation at the water surface. This involves optimization problems including very large and highly coupled system of PDE constraints. For the solution process the theoretical part of this project will focus on

  • multilevel methods based on inexact trust-region SQP-techniques
  • semismooth Newton and interior point methods in the context of inequality constraints
  • 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.

In that framework, the following optimization problems are considered for the optimization of the hull-keel-winglet configuration:

  • shape optimization within constraints of weight, structural strength and lift
  • tracking type problems for the control of the wave generation at the water surface.

The developed techniques can be applied directly for the Shosholoza boat competing in the America's Cup for the next two trophys and for which the first applicant of this proposal is leading the Scientific Advisory Team.

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.
 Heuveline, Karlsruhe 
 Walther, Dresden

Automatic Differentiation for Large Scale Flow Control with Application to Non-Newtonian Flows

This project focuses on the development, analysis, and implementation of efficient numerical optimization algorithms using Automatic Differentiation techniques in the context of flow control problems including highly nonlinear PDE constraints. The developed PDE-constrained optimization algorithms will be applied to the stabilization of flows involving non-Newtonian fluids as for example blood flows and sedimentation problems. The considered models include e.g. memory effects of the fluid which lead to complex and highly nonlinear state equations. These problems have in common that the determination of the linearized state equation needed for the adjoints or for the sensitivities would be an extremely tedious task if possible at all. Our goal is to apply in a systematic way techniques of automatic differentiation in that context. A special emphasis is given to goal oriented adaptivity, optimal experimental design toward model calibration, and parallel processing in that framework.

 Hoppe, Augsburg/Houston 
 Wixforth, Augsburg

Multilevel Based All-At-Once Methods in PDE Constrained Optimization with Applications to Shape Optimization of Active Microfluidic Biochips

This project within the area of PDE constrained optimization focuses on the development, analysis and implementation of optimization algorithms that combine efficient solution techniques from the numerics of PDEs, namely multilevel iterative solvers, and state-of-the-art optimization approaches, the so-called `all-at-once' optimization methods. It is well-known that multilevel techniques provide efficient PDE solvers of optimal algorithmic complexity. On the other hand, optimization methods within the all-at-once approach, such as sequential quadratic programming (SQP) methods and primal-dual Newton interior-point methods, have the appealing feature that in contrast to more traditional approaches, the numerical solution of the state equations is an integral part of the optimization routine. This is realized by incorporating the PDEs as constraints into the optimization routine. These strategies allow to save a considerable amount of computational work compared to methods that treat the PDE solution as an implicit function of the control/design variables. Moreover, the proper combination of multilevel techniques and optimization algorithms makes it possible to extract essential structural information from the originally infinite dimensional optimization problem.

 Klar, Kaiserslautern 
 Leugering, Erlangen

Optimal Nodal Control Of Networked Systems Of Conservation Laws

The objective of this proposal is the investigation of controllability and optimality issues for networks of gas and water flow. The gained insight is then applied to derive adapted numerical methods for simulation and optimization of real-world gas and water networks.

 Pinnau, Kaiserslautern 
 Siedow, Kaiserslautern

Optimal Control and Inverse Problems in Radiative Heat Transfer

The main objective of this project is the derivation and investigation of efficient mathematical methods for the solution of optimal control and identification problems for radiation dominant processes, which are described by a nonlinear integro–differential system or diffusive type approximations. These processes are for example relevant in glass production or in the layout of gas turbine combustion chambers. Since already their simulation is very complex and time consuming due to the radiation, the main focus of the project will be on the investigation of optimization algorithms based on the adjoint variables, which will be applied to the full radiative heat transfer system as well as to diffusive type approximations. In contrast to black–box optimization algorithms, this allows to keep a balance between the optimization and simulation effort for arbitrary fine discretizations. Special focus will be on the investigation of different numerical methods which allow to solve three dimensional problems at reasonable computational costs. In addition to the optimization we will also study new approaches to the reconstruction of the initial temperature from boundary measurements, since its precise knowledge is mandatory for any satisfactory simulation. Especially, we will develop a fast, derivative–free method for the solution of the inverse problem, such that we can use many different models for the simulation of the radiative process.

 Tröltzsch, Berlin

Numerical Analysis of State-Constrained Optimal Control Problems for PDEs

The project is a contribution to the optimal control of nonlinear systems of PDEs with pointwise state-constraints. The work is focussed on two aspects of associated numerical methods and their analysis. In a first topic, regularization techniques of Lavrentiev type will be studied to solve state-constrained problems. Exemplarily, special emphasis is placed on semilinear parabolic equations with boundary control and state constraints in the domain. A second part of the project is devoted to the case where the controls are given by a linear combination of finitely many ansatz functions, where the coefficients are constant or may depend on time. This situation is characteristic for many applications in practice and leads to semi-infinite optimization problems.



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