Analysis, development and implementation of efficient parallel numerical methods
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
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.
Group of Prof. Dr. Dr. h.c. Hans Georg Bock
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
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
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
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
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
Group of Prof. Dr. Lorenz S. Cederbaum
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.
Group of Prof. Dr. Eva Gutheil
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
Group of Prof. Dr. Dr. h.c. mult. Willi Jäger
- 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.
Numerical Methods for Partial Differential Equations (
Prof. Dr. Rolf Rannacher)
Mathematical models in science and engineering usually contain
systems of differential equations. Their numerical solution requires accurate
and economical discretization by adaptive methods, efficient solution by
techniques and the use of parallel computers in the case of very large
This is the subject of the current work of the Numerical Methods Group of
The development of new methods is oriented at the needs of practical
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
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
A Galerkin finite element discretization of the Navier-Stokes
has been developed using either equal-order or Hood-Taylor-type shape
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
in the estimation of parameters from experimental data. In a planned PhD
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
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
Group of Prof. Dr. Rolf Rannacher
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
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.
Group of Prof. Dr. Gerhard Reinelt
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.
Group of Prof. Dr. Jeremy C. Smith
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
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:
Group of Prof. Dr. Jörg Langowski
- 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.
The group of Prof. Dr. Angela Stevens starts operating as of April 1st. Details about the research topics will be provided soon.
Group of Prof. Dr. Angela Stevens
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.
Group of Prof. Dr. Dr. h.c. Jürgen Warnatz
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
Group of Prof. Dr. Gabriel Wittum
- 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
M t = A t, t ∈ T with a "testing space" T,
yielding a linear iteration of the type
xnew = xold M-1 (A xold - 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
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.
- 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.
- 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".
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.
Group of Prof. Dr. Jürgen Wolfrum