Continuous Optimization and Numerical DAE (Prof. Dr. Dr. h.c. Hans Georg Bock, Dr. Johannes P. Schlöder)

Analysis, development and implementation of efficient parallel numerical methods for parameter estimation and optimum experimental design, optimal control and feedback control of large-scale processes are central research topics. Process models are nonlinear differential equations, typically instationary partial differential algebraic equations, subject to boundary conditions and further technical constraints. Optimization variables comprise continuous parameters and control functions but also discrete variables resulting in hybrid optimization problems.

Future work concentrates on

• Direct multiple shooting methods for optimal control in PDE
• Adaptivity and error control in optimization methods for PDE
• Computation of feedback controls for large-scale nonlinear processes
• Robust methods for experimental design and control under uncertainties
• Non-standard optimal control problems based on derivatives of the process model
• Nonlinear dynamic mixed-integer optimization

The optimization boundary value problem approach developed in the group, in particular the direct multiple shooting method for DAEs, will be extended to instationary PDEs in higher dimensions. For the treatment of the constrained initial boundary value problems on each multiple shooting element adaptive discretization methods for the differential equations and the parametrization of the controls will be developed in cooperation with the Rannacher group. Research topics include suitable error criteria, domain decomposition techniques and fast iterative solvers for discretized PDE as well as efficient generation of derivatives for the optimization solver, e.g. by adjoint methods.

Advanced control strategies use feedback to compensate for disturbances and to stabilise nonlinear processes. Challenges in this field are fast computation of feedback controls in real-time for complex large-scale processes. Our emphasis will be on inexact Newton-type methods based on the gradient of the Lagrangian for equality and inequality constrained PDEs parametrized by multiple shooting. For efficient gradient evaluation methods based on internal numerical differentiation techniques and the reverse mode of automatic differentiation will be developed. In cooperation with the Gradon group at ICM, Warsaw, applications in chemical engineering will be treated.

For robust design or operation of processes new algorithms for an efficient treatment of min-max-problems based on a model linearization have been developed and successfully applied to nonlinear optimum experimental design. This approach will be extended to optimal data evaluation for parameter and state estimation in real-time in the framework of NMPC. Aim is the computation of controls that allow a reliable parameter and state estimation and simultaneously operate the process optimally. Based on experience in fast optimization methods for NMPC, suitable formulations of this problem and the corresponding algorithms will be developed and practically tested in chemical engineering applications in cooperation with the Gradon group.

For periodic processes robustness to disturbances can qualitatively be described by the spectral radius of the monodromy matrix. Stability optimization thus leads to optimal control problems with intricate non-differentiable objective functions. Stability optimization methods applicable to mechanical DAEs with discontinuities have been developed in the group. These methods will be enhanced for the treatment of problems with many optimization variables. Applications include multilegged locomotion of detailed (bio)mechanical systems.

Modeling complex multi-scale processes often leads to different model components for the different scales. They have to be coupled in a suitable way in order to represent a given process adequately. The optimal selection of components and the estimation of parameters leads to mixed-inter parameter identification and optimum experimental design problems. Together with the Reinelt group heuristics based on a combination of direct multiple shooting and branch-and-bound methods will be developed using practical instances from cell biology (e.g. metabolic pathways in cooperation with Leysing, Warsaw) and chemical engineering.

The theoretical analysis of the methods and the development of algorithms is driven by applications in industry and academia including

• Chemical engineering problems like
  • Coupled distillation columns
  • Simulated moving bed chromatography
• Environmental problems
  • Global aerosol distribution and climate feedback
  • Inverse modeling of soil processes
• Legged locomotion in biomechanics and robotics
• Modeling of cellular processes
  • Metabolic pathways, signaling processes.

From these applications benchmark problems will be selected to demonstrate the performance of the newly developed methods. Furthermore it is planned to include experimental investigations, in particular with the colleagues from Warsaw. All structures of problem classes (e.g. hierarchy in optimization problems, bifurcation structures, structure of problem parametrization) will be exploited for efficiency and parallelization.

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Theoretical Chemistry (Prof. Dr. Lorenz S. Cederbaum, PD Dr. Hans-Dieter Meyer)

The Theoretical Chemistry group possesses broad expertise in describing various electronic processes in molecules and clusters and in molecular quantum dynamics. More recently, a research activity on non-linear quantum mechanics has been started with the objective of modeling Bose-Einstein condensates.

The work of the group includes the development of the theory and methods needed to describe the complex electronic processes including the nuclear quantum dynamics accompanying these processes. The main goal is to apply the methods to simulate realistic systems. All the applications involve large scale numerical calculations. The numerical methods and computational techniques as well as the codes are extensive and must be further developed and where possible, new and more efficient ones introduced. Here, we strongly need the collaboration with the numerical mathematicians within the Graduiertenkolleg.

An interesting complex process we would like to study is ICD. A few years ago our group predicted a new and very efficient energy transfer process called Interatomic Coulombic decay (ICD). This process is extremely sensitive to the environment of an atom (or molecule). The isolated atom does not decay electronically following the ionization of an inner-valence electron. In an environment, e.g., in a cluster, there is an ultrafast decay. The energy is transferred to the neighboring atoms and an electron is emitted from them. After the emission of this ICD electron the thus created ions repel each other strongly and a Coulomb explosion takes place. Recently, our predictions have been confirmed experimentally. In the meantime several German and international experimental groups have started working on the subject and results are already available and more is to be expected.

To model the ICD, large eigenvalue problems of complex, non-hermitian matrices must be solved numerically. Often the non-hermicity originates from the introduction of a complex absorbing potential (CAP), which turns a scattering problem into a bound state one. It should be emphasized that one is seeking for internal states of a dense spectrum. For this project we plan to collaborate with the group of R. Rannacher which has worked on related problems. A combination of the Arnoldi method and multi level approaches are promising.

The study of quantum molecular dynamics becomes very elaborate if more than three degrees of freedom have to be treated. A new algorithm for solving the time-dependent Schrödinger equation, Multi-Configuration Time-Dependent Hartree (MCTDH), was therefore developed in Heidelberg. MCTDH has been applied to a wide range of applications, the most successful of which was the study of femto-second quantum dynamics on conical intersections. A full quantum mechanical treatment is needed here.

The MCTDH approach is a reduction of complexity approach, where the reduction of complexity is accomplished by the introduction of the so called single-particle functions. The integration of the non-linear equations of motion is performed by multiple scale techniques (so called constant mean field integrator.

The most important step for the future development of MCTDH will be the parallelization of the code. As MCTDH is a very intricate algorithm, this will be a difficult task. Here the help from the experts (P. Bastian) will be essential. On the other hand the numerical-mathematical methods used in MCTDH, like integration schemes and representation techniques, must be further improved. Here again we will profit from the expertise and knowledge present in the Graduiertenkolleg (H. G. Bock).

Bose-Einstein condensates are currently of high actuality. The condensates are usually modeled by a mean-field ansatz (non-linear Schrödinger equation). This ansatz has been successfully applied in many cases, but fails to describe several experimentally relevant phenomena like fragmentation. We have succeeded to formulate a general mean-field which can describe fragmentation. To model the non-linear quantum dynamics with these equations, powerful numerical codes are necessary. Here, we plan to collaborate with the group of B.~Lesyng from Warsaw which has experience with parallel and vector-parallel architectures and is also studying condensates.

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Multiphase Flows and Combustion (Prof. Dr. Eva Gutheil)

The research in the group Multiphase Flows and Combustion within the Graduate College concerns the link between basic research in the area of spray vaporization, ignition, and combustion and application as well as improved mathematical and numerical methods for the solution of the governing equations of the system. The Graduate College enables the interaction with scientific groups both in the area of technical applications such as the groups of Prof. Wolfrum at the IWR and Prof. Schulz at Duisburg University whose groups experimentally investigate spray systems. Moreover, joint research with Prof. Bałdyga (Warsaw University of Technology) whose strength is the investigation of droplet interaction with both experimental and numerical methods and their application to both energy and biological systems is extremely beneficial for the testing and implementation of the developed models and methods.

Both laminar and turbulent spray combustion with improved models for the interaction of spray dynamics and the gas phase comprising chemical reactions are investigated. In particular, focus is given to the formation of pollutants through use of a flamelet model for turbulent spray systems. Here, laminar spray flamelets are computed that account for the energy consumption of the gas phase due to droplet vaporization. This is important in particular in areas where both the chemical reactions and the vaporization occur. In situations where unsteadiness becomes important (e.g. flame lift--off, flame extinction and re-ignition), unsteady flamelets will be computed. There are various studies that address this problem, but it has not been resolved. The counterflow configuration with its well-defined boundary and initial conditions is suitable to study the interaction of dilute sprays where previous investigations on the ignition behavior of various fuel sprays in stagnant air are extremely beneficial for the success of the model.

A computer code will be developed to account for the unsteadiness of flamelets. The resulting laminar spray flame structures will be implemented into commercial codes such as the KIVA code to study ignition in turbulent internal engine combustion.

Moreover, the planned activities with the groups of Prof. Wittum at the IWR and Prof. Feistauer in Prague within the Graduiertenkolleg with a joint proposed project that focuses on improved numerical methods for use in combustions systems, will be extremely worthwhile for combustion systems (either gas or spray) that require the use of more complex geometries and/or the implementation of adaptive grids in areas where chemical reactions or vaporization take place. These regimes usually do not coincide since the vaporization of the liquid is a process that needs to be initiated substantially before ignition and combustion may take place in particular in combustion systems where no pre-vaporized fuel is added. Therefore, this issue considerably affects the consumed computer time and improvements are essential.

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Applied Analysis, Multiscale Problems ( Prof. Dr. Dr. h.c. mult. Willi Jäger)

Modeling and simulation of multiscale systems will be in the center of research of the graduate program also in the coming years. This team is going to continue the program of the last period, however, putting more attention to processes in microscopic scales. Due to an improvement of experimental research and new technologies e.g. in microscopy, more qualitative and quantitative information about these processes is available. The mathematical modeling and the analytic and algorithmic method are not developed to the extend needed to structure and to understand the data obtained by the experimental research. Discrete, mixed discrete and continuous modeling, stochastic processes and geometries are getting more important. As far as the areas of applications are concerned, problems in biosciences and biotechnology will play a more important role. Here, the cooperation with the partners at ICM is already working very well and will be increased in the future.

Despite the importance of the specific test applications, the development of analytic and numerical methods of general relevance will be central. Techniques for dynamical systems, partial differential equations, asymptotic analysis, stochastic processes and random media are used to formulate model equations, to study their dependence on data, with special emphasis on their characteristic scales. The structure, stability and asymptotic behaviour of solutions are analyzed. Furthermore, methods to discretize the solutions and to estimate the quality of the approximations are developed. Advanced methods in statistics and optimization will be used to determine data which are necessary for the simulation of the models.

The following main challenges arise

• mathematical modeling in nano- and micro-scales:
  

  the classical descriptions will no longer be valid, new types of models will have to be derived using available information; classic transport and flow models like Navier-Stokes will have to be replaced.

• transition between different scales:
  

  understanding of this topic will help not only to describe the systems better, but also to improve numerical computations by using the multi-scale information in the algorithms; the information about a process in a micro scale and the macroscopic description have to be quantitatively correlated. Special techniques have to be developed to switch between scales also in the numerical simulations and to choose according to the needs the level of scale for computation.

• stochastic processes in random media:
  

  in real nature we are forced to take the random feature in time and in space into account; here is a crucial gap, which has to be filled. Each single cell, the tissues, the roots in a plant, the vessel system in an organ are biological random media with specific structures and in their geometric properties similar to structures to be seen in soil physics.

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Numerical Methods for Partial Differential Equations ( Prof. Dr. Rolf Rannacher)

Mathematical models in science and engineering usually contain complex systems of differential equations. Their numerical solution requires accurate and economical discretization by adaptive methods, efficient solution by multi-level techniques and the use of parallel computers in the case of very large problems. This is the subject of the current work of the Numerical Methods Group of the IAM. The development of new methods is oriented at the needs of practical applications. It employs most advanced numerical algorithms and software techniques and is supplemented by theoretical analysis and systematic benchmark testing. A large part of the developed software is available via internet and is permanently developed further:

• Flow-solver package HiFlow
• Flow-solver package Gascoigne
• FEM package deal.II

A general approach to error control and mesh adaptation has been developed for finite element discretizations of variational problems. Within a feed-back process residual-based a posteriori error estimates are obtained by solving global dual problems. High accuracy can be obtained in computing quantities of physical interest. This method is currently applied to a variety of problems some of them being closely related to the research program of the GK, e.g.,

• Drag and lift of bodies in laminar flows ;
• Sedimentation of rigid bodies in viscous fluids;
• Chemical flow reactors and combustion;
• Optimal control and parameter estimation in reactive flows;
• Eigenvalue computation;

Most of this research is carried out in close cooperation with experts from various fields from outside Mathematics such as Physical Chemistry, and Fluid Mechanics. The synergy gained by this cooperation is essential for producing results which are of real practical value for applications.

A Galerkin finite element discretization of the Navier-Stokes equations has been developed using either equal-order or Hood-Taylor-type shape functions for velocity and pressure. For this method our concept of goal-oriented mesh adaptivity has been applied to 2D/3D stationary flows. Currently it is extended to include nonstationary flows and hp-adaptivity. This adaptive discretization is also used in the optimal control of flow properties as well as in the estimation of parameters from experimental data. In a planned PhD project of this GK this method will be extended for the simulation and optimization of flows in micro-reactors. This requires close cooperation with the Optimization Group at IWR (Bock) and data input by experimental measurements done in the Physical Chemistry Group at PCI (Wolfrum).

The new approach to adaptive discretization has also been applied to symmetric as well as nonsymmetrical eigenvalue problems like those occurring in hydrodynamic stability theory. This method combines systematic mesh adaptation for model reduction with efficient solution of the discrete eigenvalue problems by multigrid methods. The experience gained in this work will be utilized in a PhD project dealing with large eigenvalue problems occurring in molecular dynamics. Here, the cooperation with the Theoretical Chemistry Group at PCI (Cederbaum) is essential for exploiting the particular structure of the problem and for evaluating the results. The cooperation with the colleagues at the ICM Warsaw opens the possibility to learn new analytical techniques and to attack new areas of applications not so well represented at Heidelberg. We are mainly interested in the cooperation with the Analysis Group (Niezgódka) and the Medical Science Group at ICM (Waniewski).

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Discrete Optimization ( Prof. Dr. Gerhard Reinelt)

Development of optimization methods for nonlinear mixed-integer programing problems

In cooperation with the group "Continuous Optimization and Numerical DAE" of Prof. Bock we plan to continue research on the solution of hybrid optimization problems, i.e., problems which have both continuous and discrete model components. Such problems integrate mixed-integer and nonlinear optimization problems. Algorithmic approaches will be of branch-and-bound type. Tasks to be accomplished in such frameworks are the suitable partitioning of problems into subproblems, the development of appropriate relaxations which are computationally tractable and provide good bounds, and the development of effective primal heuristics for obtaining feasible solutions. Special techniques to be exploited are e.g. outer approximation, Benders decomposition, cutting plane generation and disjunctive programing.

As application areas for the methods to be developed we aim at process engineering (e.g. synthesis of heat exchange networks, optimization of reactor networks, moving bed chromotography processes), scheduling, investment models, cell biology (e.g. control of calcium concentration in hepatocytes as a second messenger), design problems in mechanics and eletronics.

The work will concentrate on the development of innovative methods and algorithms combining and mutually exploiting the different approaches from discrete and continous optimization. Progress shall be achieved as well in theory as in practical implementations for application problems.

Choice models and games on graphs

In this project we want to consider choice models and games on graphs. The research shall be conducted in cooperation with the group of Prof. Wislicki from ICM Warsaw.

Sets of communication paths between agents, described in terms of directed graphs, are in general dynamic (i.e., evolving in time) and non-passive (i.e., being an active party of a game) networks. An abstract communication network with reinforcement learning formation mechanism represents a good example of such a system, being at the same time an efficient tool to model various application areas, as energy routing on networks, data routing on computational grids, airline transportation or interpersonal communication in organizations. In case of an active network operator, one deals with a two-party game rather than a choice model.

Transition networks for proteins

In this project we want to develop, improve and apply the methodology of transition networks for proteins and other molecular systems. The research is conducted in cooperation with the group ''Computational Molecular Biophysics'' of Prof. Smith.

Determining transition states for conformational changes in proteins is often infeasible in experiment. Consequently, simulation methods are used to gain insight into the mechanism of such transitions. Molecular dynamics can simulate the natural dynamics of any molecule, but computational limitations currently do not allow to observe transitions which occur on a timescale longer than nanoseconds.

The methodology of transition networks is a promising new approach, where the macroscopic transition of a high-dimensional system, e.g. protein, is described in terms of a network of microscopic transitions. Such a network is a weighted graph and can therefore be analyzed using graph-theoretic approaches. Macroscopic properties which are directly comparable to experimental measurements (e.g. effective rate constants, equilibrium distributions) can thus be computed effectively. As an advantage over experimental measurements, the full microscopic information (e.g. molecular structures) is available in transition networks.

In their cooperation the two groups are planning to develop and improve efficient methods of generating the nodes (low-energy conformations) and edges (transition states) of the network. Furthermore, the applications should be extended to different molecular and nonmolecular systems.

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Computational Molecular Biophysics ( Prof. Dr. Jeremy C. Smith)

An important part of our activity in the next 5 years will concern the development of methods for efficiently exploring protein energy landscapes. Protein energy landscapes are complex hypersurfaces of many thousands of dimensions. Understanding and optimizing pathways over these surfaces are critical to biology and of immense potential use in medicine.

Of particular importance in the near future will be the determination of methods for finding pathways of minimum energy over the protein energy hypersurfaces. This optimization problem will involve a combination of model reduction with multilevel techniques. This will involve close cooperation with the Simulation/Optimization group (Bock) and the Discrete Optimization Group (Reinelt) within the Graduiertenkolleg .

Also to be pursued are multilevel techniques and adaptive methods for modeling protein electrostatics (Wittum) with a view to enabling efficient calculation of forces in biomolecular systems and corresponding extension of accessible timescales and lengthscales in simulations of biomolecules. This research area will be of particular interest when combined with Brownian dynamics multiscaling methods for biomoleular dynamics via the Langevin equation (Langowski).

Our pathway-finding saddle-point optimization methods will also be combined with the model-reduction methodology of the Theoretical Chemistry group here (Cederbaum) in excited-state optimizations of biomolecules. Efficient parallelization of the molecular dynamics algorithms currently in use in the group will also be effected. Applications will include large-scale and entropic functional conformational transitions in motor proteins and proton transfer mechanisms in enzymes.

Finally, our group has a particularly strong cooperation with the group of Prof. Bogdan Lesyng at the ICM, Warsaw. Prof. Lesyng's group works in almost exactly the same field as we do. In the context of the Graduiertenkolleg, several cooperative projects will be undertaken with the Lesyng group. For example, we wish to combine our expertise in saddle point optimization with the knowledge of the Lesyng group in quantum chemical methodology for reaction dynamics in order to investigate enzyme reactions.

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Biophysics of Macromolecules (Prof. Dr. Jörg Langowski)

Our group studies the influence of protein- or sequence-induced structural transitions on the global conformation of large DNA domains, the role of DNA conformation in transcriptional regulation at a distance, chromatin structure and the functional organization of the interphase nucleus. As model systems we investigate the structure and dynamics of superhelical DNA, of complexes between DNA and regulatory proteins, of oligonucleosomes, and of interphase chromosomes in living cells. Modeling of such large-scale systems requires methods that allow describing structural transitions in large biomolecules on a millisecond to second time. For this purpose our group is very active in the development of Monte-Carlo and Brownian dynamics simulation techniques. The modeling work is backed up by experimental biophysical techniques such as single molecule fluorescence, light and neutron scattering and scanning force microscopy.

Based upon a polymer chain model for the description of large DNAs, we developed an object oriented, parallelized simulation program for the structure of the 30 nm chromatin fiber. The model successfully predicted a new structure for the chromatin filament that has recently been confirmed by crystallographic studies. It is used for systematic studies of the effect of DNA structure and protein binding on the structure of the chromatin fiber, as well as for computing its nanomechanical properties. Necessary parameters such as elasticity and local curvature of the DNA are obtained from comparison with experimental data and molecular modeling of DNA-oligonucleotides.

In the context of this Graduiertenkolleg, we will concentrate on developing new multiscale modeling techniques that allow connecting information on the atomic scale - as obtained by molecular dynamics simulations - to larger-scale modeling techniques such as Brownian dynamics.

The particular activities that are planned within this program are:

• Atomic-scale simulations of the electrostatics around nucleosomes in the chromatin chain will be used for modeling the internucleosomal interactions. For this purpose, we need close cooperation with the Smith group (molecular dynamics modeling) as well as with the Wittum group for developing multilevel techniques. In a collaboration with Prof.~Antosiewicz in Warsaw and Dr. Rebecca Wade (EML Heidelberg, external to the Graduiertenkolleg), we will develop effective potentials of DNA binding to the histone core. By transfering the information from the atomic scale to the large-scale models, we will arrive at a detailed understanding of the forces that stabilize the chromatin fiber, enable the formation of alternate structures e.g. at telomers, and render the structure independent of variations in nucleosome spacing.
• Brownian dynamics simulation of the behavior of the 30 nm chromatin filament in the restricted space of the cell nucleus and under topological restriction such as attachment to the nuclear membrane and other organizing centers. The results of these simulations will be compared to experimental data on the in vivo mobility of chromosomes in yeast and higher eukaryotes. For this project, we need to develop modeling strategies for the dynamics of entangled polymer chains; this work will require cooperation with the Simulation/Optimization group of Prof. Bock.
• Monte-Carlo simulations of the distribution of interphase chromosomes in the nucleus and Brownian dynamics modeling of the diffusion of proteins in the resulting chromatin network. These studies are aimed at an understanding of protein and RNA transport inside the nucleus and will be complemented by single molecule experiments in our own group.

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Applied Mathematics for the Life-Sciences (Prof. Dr. Angela Stevens)

The group of Prof. Dr. Angela Stevens starts operating as of April 1st. Details about the research topics will be provided soon.

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Multi-Scale Modeling and Simulation of Reaction-Transport Processes (Prof. Dr. Dr. h.c. Jürgen Warnatz, Dr. Wolfgang Bessler, Dr. Dirk Lebiedz)

Modeling and simulation of multi-scale processes in physical chemistry and biochemistry is at the base of the research activities of Prof. Warnatz's Reactive Flows group at IWR. A particular focus is the incorporation of detailed chemical reaction mechanisms into accurate models for complex reaction-transport processes. The investigations proposed within the framework of the Graduiertenkolleg include two major topics at the interface between efficient mathematical and numerical treatment of multi-scale phenomena, and modeling and simulation of chemical-engineering related problems:

• Heterogeneous catalysis: Modeling and simulation of the gas-phase/surface processes in solid oxide fuel cells and automotive catalytic converters
• Model reduction for large-scale ordinary and partial differential equation systems: automatic reduction of complex reaction mechanisms
• Optimal control of self-organizing chemical and biochemical reaction systems

Heterogeneous catalysis in flow reactors is an outstanding example for a multi-scale process in chemistry. Microscopic aspects of surface reactivity couple to macroscopic fluid flow in the gas phase, and time scales involved cover the range from nanoseconds to minutes. Both accurate models based on detailed physical and chemical insight and appropriate numerical methods bridging the scales are required for efficient simulation of these processes and meaningful interpretation of the results. Two applications will be investigated: Solid oxide fuel cells, where microscopic diffusive transport in the porous structure will be treated in collaboration with Prof. Bastian (IWR Heidelberg), and automotive catalytic converters, where the models will be validated with experimental work from the group of Prof. Wolfrum (PCI Heidelberg).

The computationally expensive evaluation of large chemical reaction mechanisms often requires complexity reduction, for example when coupled to transient flow simulations. Automatic model reduction techniques based on a firm mathematical treatment of the separation of multiple time scales in stiff chemical reaction systems are developed in close collaboration with the numerical mathematics groups of Prof.~Bock and Prof.~Rannacher (IWR Heidelberg). Applications cover both combustion and biochemical reaction networks. In the latter case, the method is closely related to complexity reduction of biological systems for analysis and efficient simulation and plays a crucial role in modern systems biology. Biochemical applications are treated in collaboration with Dr. Kummer (EML Heidelberg).

External control and optimization of self-organizing and pattern-forming spatiotemporal systems under non-equilibrium conditions is an important issue for both technical and future biomedical applications. From a numerical point of view, this is particularly challenging since optimal control problems for nonlinear PDEs involving highly unstable dynamics have to be solved. Here, a synergy between numerical developments in simultaneous nonlinear optimization and PDE numerics on the one hand and application related modeling aspects on the other hand is crucial. The collaboration with the groups of Prof. Bock and Prof. Rannacher (IWR Heidelberg) and with Prof. Gorecki (ICM Warsaw) aims at addressing model based control and manipulation of spatiotemporal chemical and biochemical reaction systems. The collaboration with Prof. Gorecki (ICM Warsaw) who is an expert in modeling and simulation of excitable chemical systems in direct relation to experimental investigations and their potential role in information processing of similar biochemical systems is promising.

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Simulation in Technology ( Prof. Dr. Gabriel Wittum)

Simulation in Technology aims at developing and applying methods and tools for the simulation of processes from Science and Technology. In particular fast solvers for partial differential equations (i.e. PDE) are developed like robust, parallel, and adaptive multigrid methods. These methods allow the solution of very large problems. In the framework of the Graduiertenkolleg we want to investigate the following topics:

1. Fast Solvers for large systems of equations: Filtering Algebraic Multigrid for Systems of PDE.

   The development of fast solvers for large systems of equations is our core project. Starting with robust multi-grid methods for systems of pde, we develop a lot of different methods nowadays.

   Filtering is another technique for solving linear systems A x = b. It is based on the filtering property: Find an approximate inverse M, such that:

   
   M t = A t, t ∈ T with a "testing space" T,
   yielding a linear iteration of the type
   x_new = x_old M^-1 (A x_old - b).

   Depending on T, the resulting method can be tailored to be a smoother or a corrector. It is possible to derive the filtering space T adaptively during the iteration by Adaptive Filtering. The efficiency of the filtering approach was demonstrated in these papers. However, only so-called filtering deocmpositions were available to construct filters. Those are based on block incomplete decompositions and thus are limited to structured grids only. We continued this development with the two-frequency decompositions. To generalize this method to unstructured grids, Wagner developed Filtering Algebraic Multigrid (FAMG). This method uses a filtering property for the construction of the grid transfers in the AMG method. Furthermore, the construction of the coarse grid correction takes the smoother into account, thus granting a coarse-grid correction optimized w.r.t the smoother. The method has been used to compute a strategy of bioremediation of an aquifer and in several other applications. The method has been parallelized by Wrobel, yielding a robust and at the same time efficient overall solver. Currently FAMG is generalised for systems of pde. FAMG is one of the most advanced AMG solvers available.

   Based on FAMG we want to investigate the adaptive filtering approach further. In the framework of transforming iterations, which we developed years ago, these methods can be generalized to systems of pde. the overall aim is a general solver for large classes of matrices.

2. Biotechnology and Process Engineering: Simulating Population Dynamics in flows.
   

   The main feature of population dynamics model is the high dimensionality: 2 or 3 spatial dimensions for flow and transport are extended by additional structural co-ordinates for the population dynamics. In case of e.g. a crystallizer we take the crystal length as structural co-ordinate. A practically relevant description of complicated behaviour of the crystallization process requires a fine numerical resolution in this structural co-ordinate. Furthermore, the population balance equations include integral terms that need special numerical treatment. For the integral terms we developed special adaptive and complexity reducing integration techniques. The high problem dimension leads to a high computational complexity of the simulations that can only be achieved by using parallel architectures and adaptive techniques.

   In a first step, we developed a simulation strategy for these models based on our simulation system uG. There, the geometric space is covered by an unstructured grid, which is used to discretize the coupled flow and transport equations. The arising linear systems are handled by robust multigrid solvers. The unstructured grid is adaptively refined during time stepping according to an error indicator. In every node of this grid, another mesh for the discretization of the population balance equations is attached. This allowed solving the full problem for 3d (2d flow and 1d property space) and 4d (3d flow and 1d property) problems for the first time ever. In particular, space and time dependance of the numbser density function is able to explain observed fluctuations in crystal size in running crystal processors.

   To handle systems with even higher dimensions, we want to base these computations on sparse-grid approximations. We already have very good experiences with such approximations for Black Scholes Models from Financial Mathematics. There we were able to compute problems up to dimension 8. For dimensions 4-6 the methods perform greatly. Thus, we want to generalize our current approach to sparse grids. This is one of the main focuses of the next period.

3. Parallel adaptive Computations in Bioscience
   

   In cooperation with J. Smith, we will generalize our approach for the Poisson-Boltzmann equations to compute molecular forces as decribed in the separate doctoral project "Parallel adaptive Multigrid methods for Protein Folding and Dynamics".

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In-Situ Studies of Catalytic Reaction Processes at Surfaces ( Prof. Dr. Jürgen Wolfrum, PD Dr. Hans-Robert Volpp)

Heterogeneous catalysis is one of the least understood processes in chemical engineering. This is related to its physicochemical complexity, as the system consists of multiple phases (gas-phase, surface, bulk phase), where both chemical reactions and transport processes take place on widely varying temporal and spatial scales. Because heterogeneous reactions sensitively depend on the surface concentrations of reactants and products that are connected with adsorption and desorption equilibria and with gas-phase transport processes depending on operating conditions, different partial processes can become rate-determining. This has a large influences on the global reaction. As a consequence, the development of appropriate mathematical models for the simulation of surface reactions and their coupling to the surrounding gas phase is essential to an understanding of heterogeneous catalysis under technically relevant conditions.

The subject of investigations within the framework of the Graduiertenkolleg are catalysts for pollutant emission control processes such as the three-way catalyst (TWC) used for the after-treatment of automobile exhaust gases. In the TWC, a catalyst containing Pt/Pd/Rh converts the two reducing pollutants, CO and unburned hydrocarbons (HC), as well as the oxidizing pollutant, NO, to the stable products H₂O, CO₂, and N₂. Up to now, surface reaction mechanisms for this system have been derived mainly from studies of elementary surface reaction steps carried out under ultra-high vacuum (UHV) conditions (10^-12 - 10^-7 mbar) and on well-defined single-crystal surfaces. In contrast, however, technical processes usually take place at high pressure ("pressure gap") and, for example, on polycrystalline catalyst material ("materials gap"). This discrepancy can have a large impact on modeling and simulation results and thus emphasizes the importance of applying in situ diagnostics techniques that can be used under practical pressure and temperature conditions and on realistic catalysts. Optical diagnostic methods that probe interface vibrational resonances such as infrared-visible (IR-VIS) sum-frequency generation (SFG) offer significant advantages over conventional surface spectroscopic methods in which, for example, beams of charged particles are used as a probe. As we demonstrated in a close collaboration with the group of Prof.~Warnatz (IWR), in which IR-VIS SFG was employed for in situ CO coverage measurements during CO adsorption and heterogeneous CO oxidation on realistic polycrystalline Pt catalyst for intermediate CO and O2 partial pressures, as typically present in the exhaust gas of spark-ignited engines, IR-VIS SFG allows surface vibrational spectroscopic measurements with submonolayer sensitivity with the help of a tunable IR laser under realistic operation conditions. These investigations represent important steps toward developing validated adsorption/desorption and surface reaction mechanisms for catalysts with higher structural complexity. Such studies can be regarded as intermediate between the UHV surface science studies on well-defined single crystals and investigations of supported nanoparticles as used in practical automotive catalytic converters.

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