
Projects
Possible Dissertation Projects
Direct Numerical Simulation of TwoPhase Flow Through Porous Media
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Supervisors: Bastian (IWR Heidelberg), Roth (IUP Heidelberg)
Keywords:Multiscale, Multilevel, Parallel Computing
Background: Multiphase flow is an important process arising in porous
media with applications in the exploitation of natural resources or environmental
problems of contaminated site remediation. Simulation of multiphase flow on
large scales, as required by the applications above, is based on macroscopic
models (Richards equation, extended Darcy law) where the interaction of the
fluids with each other and the the porescale geometry is incorporated into
nonlinear algebraic couplings.
Research Goal: In this project we aim at computing macroscopic
parameters of incompressible twophase flow from direct numerical
simulation on the porescale in order to understand in more detail the
influence of the porescale on the macroscopic parameters. One outcome
would be a more rigorous understanding of the hysteresis of the
capillary pressure saturation relationship, [1].
Numerical simulation of twophase flow on the porescale has so far
been accomplished with network models and LatticeBoltzmann
methods. The former oversimplifies the complicated porescale geometry
of natural porous media while the latter method has so far not been
able to treat watergas flows with the true values for density and
viscosity. We therefore attempt to numerically solve the governing
PDE for twophase flow in a true porescale geometry. This is possible due
to recent advances in experimental techniques and high performance computing.
High resolution Xray tomography or the Nuclear Magnetic Resonance imaging
technique are able to acquire 3D images of the pore space with a
resolution of 1 µm while large scale parallel supercomputers
allow the simulation of nonlinear timedependent partial differential
equations with up to 1 billion degrees of freedom per time step. We
therefore think to be able to compute a volume containing 50^{3}...100^{3} grains.
The system of model equations is solved using a combination of level set and
volumeoffluid method. The complex geometry is tackled by using
discontinuous Galerkin finite elements, [2], [3], on a structured
threedimensional mesh intersected with the pore space. This avoids
the mesh generation issue and allows a massively parallel
implementation.
This project contributes to the main topics of research in the
Graduiertenkolleg as the upscaling of complex microscale process is
considered and stateoftheart mathematical techniques including multigrid
methods and parallel computing are developed further to promote the
application field.
Literature:
[1] 
H.J. Vogel and K. Roth 
Quantitative morphology and network representation of soil pore structure
Advances in Water Resources, 24, 233242 (2001). 
[2] 
P.Bastian 
Higher order discontinuous Galerkin methods for flow and transport in porous media
in E. Bänsch, ed., Challenges in Scientific Computing  CISC 2002, number 35 in LNCSE,
122, 2003. 
[3] 
P. Bastian and B. Riviùre 
Superconvergence and H(div)projection for discontinuous Galerkin methods
Int. J. Numer. Meth. Fluids., 42:10431057, 2003. 

Efficient Parallel Inverse Modeling of Multiphase Transport in Variably Saturated Porous Media
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Supervisors: Bastian (IWR Heidelberg), Schlöder (IWR Heidelberg)
Keywords: Allatonce Optimization, Multilevel Methods, Parallel Computing
Background: The determination of hydraulic parameters necessary for the quantitative modeling of flow and transport in porous media is often tedious and
time consuming. Direct measurements of the relative permeability/saturation function are hardly possible. For homogeneous porous media a combination of multistep outflow (MSO)
experiments and inverse modeling is often the method of choice, [1], [2]. However, natural porous media are often heterogeneous. Our preliminary investigations showed that parameter estimation might still be possible if the sample is composed of a limited number of materials and their structural arrangement is known, e.g. from Xray tomography, [2], [3].
Research Goal: Aim of this project is the development of efficient numerical inverse modeling methods for 3D multiphase problems with data from MSO experiments. Basis of the methods will be a suitable multiple shooting parametrization and a generalized
GaussNewton approach, [1], [4]. Essential will be the exploitation of parallelism on all levels, in particular in the generation of derivatives and the solution of the largescale linear systems of discretized PDE. Alternative realizations of internal numerical differentiation based on forward and adjoint modes will be developed in order to determine the derivative generation method with lowest computational complexity. The algorithms will be implemented in such a way that models realized in the Distributed and Unified Numerics Environment Dune can easily be used. This allows for parameter estimation of the whole class of applications realizable within that framework.
Literature:
[1] 
A. E. AltmannDieses, J. P. Schlöder, H. G. Bock, O. Richter: 
Optimal experimental design for parameter estimation
in column outflow experiments,
Water Resources Research, 38(10), 11861196, 2002. 
[2] 
S. I. Hwang and S. E. Powers. 
Estimating unique soil hydraulic parameters for sandy media from multistep outflow experiments,
Advanced Water Resources, 26(4), 445456, 2003., 
[3] 
O. Ippisch, A. Samouëlian, H.J. Vogel, P. Bastian. 
Estimation of hydraulic parameters for heterogeneous porous media,
EGS 1st General Assembly, Nice, France, 2530 April 2004. 
[4] 
S. Körkel, E. Kostina, H. G. Bock and J. P. Schlöder. 
Numerical methods for optimal control problems in design of robust optimal experiments for nonlinear dynamic processes,
Optimization Methods and Software, 19, 327338, 2004. 

Nonlinear Mixed Integer Optimal Control
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Supervisors: Bock (IWR Heidelberg), Reinelt (IfI Heidelberg)
Keywords: Mixed ContinuousDiscrete Optimization, BioChemistry
Background: The project aims at the development of numerical methods for optimization problems that involve nonlinear differential algebraic equation models and decision variables that are both reals and integers, and that may be time dependent controls or time constant design parameters. This problem class is very important in applications, but its efficient numerical solution still presents a major difficulty for real world problems. To name just two examples from process engineering, design and operation of coupled distillation columns involves integer design variables that determine which trays
are coupled, as well as continuous controls like flux rates. On the other hand, in the optimal scheduling and control of batch processes, time dependent integer controls arise together with continuous controls. The numerical solution of this type of problem is
challanging due to the fact that algorithmic elements from dynamic process optimization need to be combined with techniques from the field of nonlinear mixed integer optimization.
Research Goals: The aim of the project is the development of an optimal control
package for generic mixed integer control problems that is applicable to large scale nonlinear DAE models with continuous and integer decision variables. The algorithms shall not aim at finding global minima but shall be based on fast heuristics. These heuristics will be based on concepts from linear and nonlinear mixed integer optimization
(branchandbound, cutting planes, generalized Benders decomposition) as well as on ideas from optimal control (bang bang solutions, indirect approach, homotopy and penalty methods, efficient derivative generation).
Besides applications in optimization problems in engineering, the algorithms may also be applied to model identification problems from cell biology, where measurement data shall be optimally reproduced by differential equation models with unknown structure, depending on the existence or nonexistence of reaction pathways from a given set.
Literature:
[1] 
S. Engell, A. Märkert, G. Sand, R. Schultz and C. Schulz: 
Online scheduling of multiproduct batch plants under uncertainty,
In M. Grötschel, S. O. Krumke and J. Rambau, editors, Online Optimization of Large Scale Systems: State of the Art, pages 649678. Springer, 2001. 
[2] 
I. E. Grossmann: 
Review of nonlinear mixedinteger and disjunctive programming techniques,
Optimization and Engineering, 3:227252, 2002. 
[3] 
M. Jünger and G. Reinelt: 
Combinatorial optimization and integer programming,
In U. Derigs, editor, Encyclopedia of Life Support Systems EOLSS, 6.5: Optimization and Operations Research, pages 321327. Eolss Publishers, 2004. 
[4] 
S. Sager, M. Diehl, H. G. Bock and G. Reinelt: 
Numerical methods for optimal control with binary control functions,
Poster at 12th FrenchGermanSpanish Conference on Optimization, Avignon, September 2004. 
[5] 
O. Stein, J. Oldenburg and W. Marquardt: 
Continuous reformulations of discretecontinuous optimization problems, Comp. & Chem. Eng., 28(10):1951  1966, 2004. 

Optimal Control of Time Dependent Partial Differential Algebraic Equation (PDAE) Models
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Supervisors: Bock (IWR Heidelberg), Rannacher (IAM Heidelberg)
Keywords: Allatonce Optimization, Adaptivity
Background: Constrained optimal control problems in nonlinear time dependent partial differential algebraic equations (PDAE) pose important challenges for the development of numerical algorithms. To ensure a reliable and efficient solution to inequality constrained problems, algorithms are needed which employ a direct simultaneous approach that combines adaptive grid refinement techniques such as developed in [1] in time and space, inexact linear algebra solvers, and modern nonlinear optimization methods, with stateoftheart techniques for efficient derivative generation. The combination of all these techniques has not been accomplished yet for realistic problems.
Research Goal: The algorithms to be developed will use a direct multiple shooting parameterization that divides the time horizon of interest into subintervals, on each of which adaptive grid refinement in time and space and important computations can be performed independently. In contrast to direct multiple shooting for differential algebraic equations, where the parameterized optimization problems are finite dimensional, adaptivity in space introduces the need to formulate the optimization problems in a function space setting, and continuity conditions need to be imposed in a weak formulation, [2]. The numerical solution of the resulting optimization problems in function space shall be achieved by tailored variants of sequential
quadratic programming (SQP) with inexact Jacobians and inexact linear system solvers, [3]. For this aim it is of crucial importance that the Lagrange gradient can accurately and efficiently be computed. This shall be achieved by suitable generalization of adjoint derivative calculation methods similar to the reverse mode of automatic differentiation. A major focus will be the treatment of inequalities in the presence of inexact derivatives.
An important aim of the project is the development of a software package that allows to treat generic time dependent optimal control problems in PDAE and can be easily coupled to existing modelling environments that are used in industry. Furthermore, the algorithms shall be designed in such a way that an extension to realtime optimization and nonlinear model predictive control is possible.
Literature:
[1] 
R. Becker, H. Kapp, and R. Rannacher: 
Adaptive finite element methods for optimal control of partial differential equations: Basic concepts,
SIAM J. Control Optimization, 39:113132, 2000. 

[2] 
H. G. Bock, M. Diehl, E. A. Kostina and J. P. Schlöder: 
Constrained optimal feedback control of systems governed by large differential algebraic equations,
In RealTime and Online PDEConstrained Optimization (L. Biegler, et al., eds.), 2004, submitted. 
[3] 
H. G. Bock, W. Egartner, W. Kappis and V. Schulz: 
Practical shape optimization for turbine and compressor blades by the use of PRSQP methods,
Optimization and Engineering, 3(4):395414, 2002. 

Numerical Solution of Large Eigenvalue Problems for Modeling Intermolecular Coulombic Decay
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Supervisors: Cederbaum (PCI Heidelberg), Rannacher (IAM Heidelberg)
Keywords: Multilevel, Adaptivity, Parallel Computing, Quantum Dynamics
Background: Several years ago we have predicted theoretically a new
fundamental decay mechanism [1,2] termed Interatomic Coulombic Decay (ICD).
This novel decay mechanism becomes the dominant decay channel once the excited
atom is placed in an environment. Very recently, the ICD has been verified
experimentally [3]. Currently, several groups are measuring the ICD with great success. To model the ICD, large complex symmetric matrices (n>10^{5}) have to be constructed and diagonalized. The classical Krylovspace approach is by far too slow for calculations on realistic systems. Other methods should be developed and utilized. The numerical solution of the large eigenvalue problem will provide the input data for the dynamic modelling of the total ICD process.
Research Goal: The classical Krylovspace methods require particularly for the computation of eigenvalue clusters of complex matrices such enormous CPUtimes that parameter studies are hardly possible. This barrier can only be broken by the combination of proven LanczosArnoldi methods, [4], with multilevel techniques. A further improvement in efficiency is reached by 'shiftandinvert' strategies for the computation of interior eigenpairs. First experiences with such a combined approach are reported in [5], where it is shown that largescale eigenvalue problems with up to millions of unknowns can be efficiently treated on normal workstations. This method has since been further developed for computing groups of eigenvalues. By exploiting the inherent parallel structures of this algorithm the size of the manageable problems can be further increased.
Another performance enhancement could be reached by goaloriented model reduction by adaptive choice of the Galerkin ansatz in the discretization. This approach is commonly used in the Galerkin finite element method for solving partial differential equations but has not been applied yet to spectraltype Galerkin methods. This approach could also be useful in other quantum dynamical simulations of large configurations which are presently not possible by standard methods.
Literature:
[1] 
L. S. Cederbaum, J. Zobeley and F. Tarantelli: 
Giant intermolecular decay and fragmentation of clusters, Phys. Rev. Lett. 79, 4778 (1997). 
[2] 
R. Santra and L. S. Cederbaum: 
Coulombic energy transfer and triple ionization in clusters, Phys. Rev. Lett. 90, 153401 (2003). 
[3] 
S. Marburger, O. Kugeler, U. Hergenhahn and T. Moeller: 
Experimental evidence for interatomic Coulombic decay in Ne clusters, Phys. Rev. Lett. (2003). 
[4] 
V. Heuveline and M. Sadkane: 
ArnoldiFaber method for large nonhermitian eigenvalue problems, ETNA 5 (1997). 
[5] 
V. Heuveline and C. Bertsch: 
On multigrid methods for the eigenvalue computation of non selfadjoint elliptic operators, EastWest J. Numer. Math. 8, 275297(2000). 

MultiDimensional Quantum Dynamics: Development of Efficient Algorithms, Parallelization
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Supervisors: Meyer (IWR Heidelberg), Bastian (IWR Heidelberg)
Keywords: Parallel Computing, Molecular Dynamics
Background: The numerical treatment of quantum dynamics of molecules with four or more atoms is a very difficult and numerically demanding problem.
A very efficient method [1] for solving the timedependent Schrödinger equation has been developed in the Theoretical Chemistry group. This multiconfiguration timedependent Hartree (MCTDH) method shall be further developed and improved. This will allow us to solve the dynamics of larger molecules. The numericalmathematical methods, e.g. integration schemes and representation techniques, used in MCTDH are to be improved.
Research Goals: Due to the high dimensionality of the problem the MCTDH algorithm is computationally very intensive and all possibilities of decreasing its runtime should be exploited. A very attractive and not yet explored way of improvement is parallelization. The IWR group has extensive
experience in the parallelization of algorithms for the numerical solution of partial differential equations including adaptive parallel multigrid methods [2,3]. In [3] we demonstrated the solution of very large systems on the order of 1 billion equations on 400 processors of the IWR LINUX cluster. In a detailled analysis the inherent parallelism of the MCTDH algorithm will be explored and possible ways of improving parallelizability will be investigated. The aim of the project is a dataparallel implementation of the MCTDH algorithm for message passing architectures. Thus it will be portable
to any type of parallel computer from cluster of PCs to supercomputers.
Literature:
[1] 
M. H. Beck, A. Jäckle, G. A. Worth and H.D. Meyer: 
The multiconfiguration timedependent Hartree method: A highly efficient algorithm for propagating wavepackets, Phys. Rep. 324(2000), 1105. 
[2] 
P. Bastian, R. Helmig: 
Efficient fullycoupled solution techniques for twophase flow in porous media. Parallel multigrid solution and large scale computations. Advances in Water Resources, 23, 199216, 1999. 
[3] 
P. Bastian, M. Droske, C. Engwer, R. Klöfkorn, T. Neubauer, M. Ohlberger, and M. Rumpf: 
Towards a unified framework for scientific computing. Proc. 15th Conf. Domain Decomposition Methods (R. Kornhuber, et al.), LNCSE. SpringerVerlag, 2004. 

Multiscale Modeling for Processes on Surfaces and in Channels of Membranes
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Supervisors: Jäger (IAM Heidelberg), Bastian (IWR Heidelberg)
Keywords: Multiscale Modeling, Biochemical Processes
Background: The transport of ions through channels of membranes is modeled mainly by diffusiontransport systems coupled by transition conditions, adapted to experimental data, [1]. Since detailed information on the processes on the surfaces and the channels of the membranes is available,
macroscopic description should include this information. Multiscale modeling is providing the tools for this task. Assuming that the transported ions are changing distributions of charged molecules on the channel wall, W. Jäger and M. NeussRadu derived with new two scale techniques transmission conditions similar to those used by Choi and coauthors. There are no chemical reactions and processes on the surfaces considered, as it is necessary e.g. for the transport through nuclear pores of a cell. Special transporter molecules are used to move molecules through the membranes. In this project new mathematical methods have to be developed to derive a priori estimates for the strongly nonlinear differential equation on the microscopic level and to perform the homogenization limit,[2].
Research Goals: In this project new mathematical methods have to be developed to derive a priori estimates for the strongly nonlinear differential equation on the microscopic level and to perform the homogenization limit.
The theoretical results will be incorporated into a numerical simulation code. The discontinuous Galerkin method [3] is especially suited to incorporate transmission conditions at interfaces in an efficient way because it computes fluxes at interelement boundaries (local mass conservation) and is able to represent discontinuous functions. A method currently being developed in the Bastian group is able to handle geometrically complex interfaces between
subdomains. The aim of the numerical part of this project is to extend this method by nonlinear transmission conditions. The resulting numerical code can then be applied to a variety of models in computational biology, e.g. cell nucleus interchange, which then also enables comparison with experimental data.
Literature:
[1] 
Y. S. Choi, D. Resasco, J. Schaff, B. Slepchenko: 
Electrodiffusion of ions inside living cells, IMA Journal of Applied Mathematics, 62, 227244, (1999) 
[2] 
H. Gajewski, I.V. Skrypnik: 
Existence and uniqueness results for reactiondiffusion processes of electrically charged particles, Preprint 904, WIAS Berlin 2004 
[3] 
P. Bastian: 
Higher order discontinuous Galerkin methods for flow and transport in
porous media. Challenges in Scientific Computing  CISC 2002 (E. Bänsch, ed.), number 35 in LNCSE, 122, 2003. 

Optimization with Partial Differential Equations by Adaptive Discretization and SQPMultigrid Methods
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Supervisors: Rannacher (IAM Heidelberg), Bock (IWR Heidelberg)
Keywords: Allatonce Optimization, Adaptivity, Multilevel Methods
Background: As a result of the increasing demand for quantitatively correct description of technical processes the underlying mathematical models involve more and more complex partial differential equations. The optimization with such models necessarily requires the discretization of the state equation which limits the achievable accuracy. The control of this approximation can be accomplished by global `duality techniques' similar to those used in the optimization process itself. The resulting large discrete optimization problems have to be solved by efficient methods such as SQPmultigrid methods which largely exploit the structure of the problem inherited from the underlying continuous model.
Research goal: In this project the emphasize is on the economical discretization of PDEconstrained optimization as a crucial step towards problemoriented model reduction. Starting point is the reformulation of the optimal control problem as a system of coupled PDE using the allatonce
approach. The coupling of fast optimization algorithms (reduced SQPtechniques) with a posteriori controlled discretization and multigrid iteration is to be developed for certain classes of optimization problems. Prototypical applications are sought in the optimal design of chemical flow reactors and heat exchangers. The main difficulty in these problems are the very anisotropic geometries and the strong nonlinearities due to the chemistry which require a fully implicit solution process. However, this is very expensive on fine discretization meshes and model reduction by systematic mesh adaptition is highly dersirable. A coupling of optimization and multilevel solution has been achieved in [1], while in [2] a general concept for combining optimization and mesh adaption has been developed. This adaptive discretization in optimization has been used in [3] for problems in optimal flow control such as drag minimization in pipe flow and the identification of viscosity parameters. The goaloriented mesh adaptation results in an accurate discretization which is coarse enough to allow for the use of highly efficient optimization algorithms. First steps have also been done in the direction towards optimal control of timedependent models using socalled `check pointing' techniques.
Literature:
[1] 
T. Dreyer, H. G. Bock, V. H. Schulz and T. Speer: 
Optimum Shape Design of Turbine Blades: Operations Research Proceedings 1995 (P. Kleinschmidt et al., eds), pp. 190195, Springer, Heidelberg, 1996. 
[2] 
R. Becker, H. Kapp and R. Rannacher: 
Adaptive finite element methods for optimal control of partial differential equations: basic concepts, SIAM J. Optim.Control. 39, 113132 (2000). 
[3] 
R. Becker: 
Mesh adaptation for stationary flow control, J. Math. Fluid Mech. 3, 317341 (2001). 

Numerical Simulation of Flows in MicroChannels
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Supervisors: Rannacher (IAM Heidelberg), Wolfrum (PCI
Heidelberg)
Keywords: Multiscale, Adaptivity
Background: Microtechnology belongs to the most promising new
technologies. Its potential relies on the better efficiency and
controllability of mechanical
and chemical processes on small spatial scales.
After the first development, one is now in the phase of exploring new
applications and systematic system optimization, [1]. This has to be
supported by numerical simulation and experimental validation. In this project
special `diagnostic chips' and `DNA chips' are considered. In these systems
fluid mechanical and chemical processes happen on the µm scale and
nonstandard multiscale effects such as capillary forces and
slipboundary conditions become important. Their simulation
and experimental diagnostics for system optimization possess high
requirements which are not met by standard commercial software. Hence, new
specially adapted numerical methods are to be developed.
Research goal: Goal of the numerical part of the project is the
development of simulation tools for the design of microchips in which the
flow of liquid and particle transport is driven by capillary or
electromagnetic forces. The software
basis is provided by the flowsolver package `Gascoigne'
(http://www.gascoigne.unihd.de/) which has been
developed in the
Rannacher group. For the simulation and optimization efficient methods as
described in [2] will be used. These methods
employ higherorder spatial and time discretization by finite elements and
automatic mesh and timestep adaptativity. For handling the new
physical
effects such as travelling capillary
fronts and slip boundary conditions, these methods will be combined with front
tracking/level set techniques. In order to capture smallscale wall effects
the combination with particleoriented methods such as the LatticeBoltzmann
method may be necessary. For a validated simulation and optimization of
microchips, quantitative data about wall properties and flow behavior are
necessary. These will be provided by measurements of real configurations
within a cooperation with the Wolfrum group, [3].
Literature:
[1] 
H. Herwig: 
Flow and heat transfer in micro systems: Is everything
different or just smaller?,
Z. Angew. Math. Mech. 82, 579586 (2002). 
[2] 
R. Becker, M. Braack, R. Rannacher: 
Numerical simulation of chemical reacting flows,
Proc. Workshop on Interdisciplinary Design and Simulation Tools for
Micro and Biomedical Fluid Applications, IMM,
http://www.imumainz.de/content.html,
2000. 
[3] 
M. Sauer, J. Wolfrum, et al. : 
Single molecule DNA sequencing in submicrometer
channels:State of the art and future prospects,
J. Biotechnology 86, 181201 (2001). 

Adaptive Transition Networks for Molecular Systems
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Supervisors: Reinelt (IfI Heidelberg), Smith (IWR
Heidelberg)
Keywords: Adaptivity, Multigrid, Molecular Dynamics
Background: he generation of transition networks for molecular
systems is computationally expensive since the addition of each node and
edge requires an optimization of a function in thousands of variables. It
is thus vital to keep the transition network as small as possible while
still obtaining sufficient information on the desired macroscopic
properties. For the computation of effective rates, it is sufficient to
have a good resolution at the highest separating energy barrier. This
highest barrier corresponds to a minimum cut in a network where the edge
weights correspond to the rates of microtransitions.
Research Goals:
In this PhD project we aim at computing adaptive transition networks for small
molecular systems with relatively low energy barriers (as e.g., short
peptides). A very efficient method to generate transition networks, in the
spirit of adaptive multigrid methods, is to generate nodes and edges at
different levels of resolution, until the highest barrier of the network has
converged. The required methodology will also be developed in this
project.
For systems considered here, the macroscopic properties which are
available
with transition networks can also be obtained from a direct molecular dynamics
simulation. The validity of the generated transition networks can therefore
be tested by comparing the values of these properties.
The project is based on the complementary expertise of the two
participating
groups. The Smith group has expert knowledge in the analysis of protein
structures (see e.g., [1]) and the Reinelt group provides the necessary
algorithmic approaches from graph theory and combinatorial optimization (see
e.g., [2]). Current work of the Phd student No\'e has already exhibited
promising perspectives of integrating combinatorial optimization methods
into analysis tools in molecular dynamics.
Literature:
[1] 
F. Noe, S.M. Schwarzl, S. Fischer and J.C.
Smith: 
Computational tools for analyzing structural changes
in proteins in solution, Applied Bioinformatics. 2(3): 1117
(2003) 
[2] 
M. Jünger and G. Reinelt 
Combinatorial Optimization and Integer
Programming, In: 6.5: Optimization and Operations Research, U. Derigs
(ed.), Encyclopedia of Life Support Systems, EOLSS publishers, 321327
(2004) 

Exploring Protein Energy Landscapes: Optimization of Protein
Conformational Pathways
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Supervisors: Smith (IWR Heidelberg), Bock (IWR Heidelberg)
Keywords: Direct Multiple Shooting, Molecular Dynamics
Background:
Protein landscapes are highly complex hypersurfaces of many thousands of
dimensions. Their exploration in particular involves finding optimal
pathways for conformal transitions and thus gives insight into
the kinetics of proteins, [1].
Determining lowest energy pathways between two conformations can be
regarded as an optimal control problem. The high dimension of the
underlying differential equation, its strong nonlinearity and the
high number of controls provide hard challenges for the
numerical methods.
Project Goals:
This dissertation project will first investigate numerically appropriate
formulations of the problem (selection of performance index, choice and
parametrization of controls, boundary conditions, path and control
constraints, etc) and consider direct optimal control methods for their
numerical treatment.
Based on the direct multiple shooting method, [2,3], novel
simultaneous optimization methods will be developed that take advantage
of the structure of the problem, in particular of the largescale
differential equation system.
The new methods will be compared to currently used tools for
the analysis of structural changes in proteins [1] and applied
to problems investigated in the Smith group including the mechanism of
muscle contraction and Gprotein molecular switches.
Literature:
[1] 
F. Noe, S. M. Schwarzl, S. Fischer and J. C. Smith
(2003): 
Computational tools for analysing structural changes
in proteins in solution}, in: Appl Bioinformatics, vol 2, pp S11S17. 
[2] 
D. B. Leineweber, I. Bauer, H. G. Bock
and J. P. Schlöder (2003) : 
An efficient multiple shooting based reduced SQP
strategy for largescale dynamic process optimization  part I:
theoretical aspects, Comput. Chem. Engng. 27, pp. 157166. 
[3] 
A. Schäfer (2004): 
Effiziente reduzierte Newtonähnliche Verfahren
zur Behandlung hochdimensionaler strukturierter Optimierungsprobleme mit
Anwendung bei biologischen und chemischen Prozessen, Dissertation, Uni
Heidelberg. 

Multiscale Modeling of Chromatin
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Supervisors: Langowski (DKFZ Heidelberg), Smith (IWR
Heidelberg)
Keywords: Model Reduction, Parallel Computing, Molecular
Dynamics
Background: The goal of this project is to build a
"coarsegrained" model of interaction between
DNA and the histone octamer. Molecular dynamics simulation on
the atomic scale will be used to identify relevant degrees of freedom of
motion in the nucleosome, and to build realistic largescale interaction
potentials between its components. These parameters will be used in Monte
Carlo and Brownian dynamics simulations to reproduce in silico various
experiments revealing structure and dynamics of mono and
oligonucleosomes. The simulations will be important for understanding of
chromatin organization and functioning; furthermore, the methodological
development of connecting MD with higherlevel simulations will be a
major advance in biomolecular simulation techniques.
From the biological side, we plan to contribute some answers to the
following questions:
 What is the geometry of the linker DNA in mono and
oligonucleosomes as a function of buffer composition, presence/absence of
H1 and degree of histone acetylation?
 What parameters are necessary to describe the known
structural properties of the chromatin fiber and their variations due to
histone modifications and external conditions?
Since so far only little reliable data on the physical behavior of
chromatin exists, the results gained here are expected to contribute
significantly to our understanding of chromatin function in its different
states in the cell.
Research Goals:
In the framework of the IGK, this project will combine multiple methods,
multiple time and length scales by transferring information of
atomiclevel molecular dynamics to largerscale Brownian dynamics and
MonteCarlo simulations.
For this, strategies of homogenization and model reduction need
to be developed. We shall also need to make extensive use of parallelization in
our software. Our interaction partners within the GK will be
Prof. Jeremy Smith (IWR) and Prof. Antosiewicz in Warsaw. In addition,
Dr. Rebecca Wade (EML) will participate with her expertise in
atomicscale Brownian dynamics and electrostatics.
Literature:
[1] 
G. Wedemann, J. Langowski 2002 : 
Computer simulation of the 30nanometer chromatin
fiber, Biophys J 82(6):284759. 
[2] 
K. Tóth, N. Brun, J. Langowski 2001: 
Trajectory of nucleosomal linker DNA studied by
fluorescence resonance energy transfer, Biochemistry
40(23):69216928. 

Optimization of Brownian Dynamics Simulation of Entangled Polymer
Chains, Application to Chromatin in the Cell Nucleus
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Supervisors: Langowski (DKFZ Heidelberg), Smith (IWR
Heidelberg)
Keywords: Model Reduction, Parallel Computing, Brownian
Dynamics
Background: Here we plan to develop methods for fast
simulation of the distribution of
flexible polymer chains in a confined space. This problem is important for our
understanding of the distribution of chromosomes in the cell nucleus;
in particular the question, whether and how chromosomes keep their relative
arrangement after cell division and how this changes through the cell cycle
is not understood at all at the moment.
The ensemble of chromosomes in the human cell nucleus constitutes a
very
complex system, comprising 46 chromosomes with a total of 6*10^{9}
base pairs
of DNA, which are compacted at several levels of folding hierarchies.
In previous work, we developed a parallelized hybrid MonteCarlo model that
allows simulation of properties of interphase chromosomes, using a chain of
about 1.5*10^{6} polymer segments that are interconnected in a
topology given
by the known biological and structural properties [1, 2].
Research Projects:
In this project, we will develop a new Brownian dynamics model that allows
to simulate 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. The fractal nature of the interphase chromatin folding makes
this system an obvious target for model reduction.
Literature:
[1] 
C. Münkel and J. Langowski (1998): 
Chromosome structure described by a polymer model,
Physical Review E 57, 58885896. 
[2] 
C. Münkel, R. Eils, S. Dietzel, D. Zink,
C. Mehring, G. Wedemann, T. Cremer and J. Langowski (1999): 
Compartmentalization of interphase chromosomes
observed in simulation and experiment, Journal of Molecular Biology
285, 10531065. 

Modeling and Simulation of Chemistry and Transport in SOFC
Porous Electrodes
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Supervisors: Warnatz (IWR Heidelberg), Bastian (IWR
Heidelberg)
Keywords: Parallel Computing, Power Cells
Background:
Solid oxide fuel cell (SOFC) technology offers a unique potential for
efficient energy conversion without pollutant emission in stationary
applications. Since SOFCs can be operated with hydrocarbon fuels
(e.g. natural gas), they are expected to play an important midterm
role in sustainable energy supply. The core of an SOFC consists of an
assembly of anode, electrolyte and cathode. Our investigations focus
on the anode, which is typically a porous structure consisting of a
metallic and a ceramic component.
Research Goals:
The goal of this project is the development of detailed models of the
electrochemical fuel oxidation reaction and multiscale mass transport
within the anode. The electrochemical reaction takes place at
the threephase boundary of gas phase and the two
solid phases of electrode and electrolyte. Its mechanism, however,
is subject of widely differing hypotheses [1].
Besides surface
chemistry, effects of surface diffusion and surface phase transitions
will be implemented in the model, as well as interactions with the gas
phase, all of which are taking place on different spatial and temporal
scales [2].
A large body of empirical
evidence shows the large influence of porous electrode microstructure on fuel
cell performance. However, these effects cannot be resolved with
currentlyapplied homogenized transport models. In
collaboration with the Bastian group, simulations of porous diffusion
will be performed on a microscopic scale. This approach is based on the
numerical tools and parallel computing
capabilities of the Bastian group. The governing partial
differential equations will be discretized with the discontinuous
Galerkin method [3] which is especially suited to tackle
the complex pore scale geometry. The parallel implementation will be
based on the software framework DUNE [4].
Literature:
[1] 
W. G. Bessler and J. Warnatz : 
Calculation of impedance by transient numerical
simulation of SOFC elementary electrochemistry, Proc. of the 6th
European SOFC Forum (Lucerne, Switzerland, 2004), 754763. 
[2] 
F. Behrendt, O. Deutschmann, B. Ruf, R. Schmidt and J.
Warnatz: 
Simulation of Heterogeneous Reaction Systems, In:
Gas phase chemical reaction systems: experiment and models, 100 years
after Max Bodenstein, J. Wolfrum, et al., Editors (Springer, 1996),
pp. 265278. 
[3] 
P. Bastian: 
Higher order discontinuous Galerkin methods for flow
and transport in porous media}, in E. Bänsch, editor, Challenges
in Scientific Computing  CISC 2002}, number 35 in LNCSE, pages 122,
2003. 
[4] 
P. Bastian, M. Droske, C. Engwer, R. Klöfkorn,
T. Neubauer, M. Ohlberger and M. Rumpf: 
Towards a unified framework for scientific
computing, Proc. 15th Conf. Domain Decomposition Methods
(R. Kornhuber, et al., eds.), LNCSE. Springer, 2004. 

Parallel Adaptive Multigrid Methods for Protein Folding and
Dynamics
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Supervisors: Wittum (IWR Heidelberg), Smith (IWR Heidelberg)
Keywods: Parallel Computing, Multilevel
Background: In atomicdetail computer simulations of biological
macromolecules in aqeous solution electrostatic interactions are usually
represented via Coulomb interactions between partial charges positioned on
the atoms of the macromolecule and its surrounding explicit solvent molecules.
Thus, a large amount of computation time is wasted on solvent dynamics. To avoid
this, methods have been developed that represent the solvent as a polarisable
continuum. Such implicitsolvent techniques rely on the numerical solution of
the PoissonBoltzmann equation (PBE) and have been shown to yield accurate
results at reasonable computational expense in the context of ligand binding
studies, [1]. However, current implementations to solve the PBE are
not optimized and thus slower than the state of the art would allow.
An optimized numerical protocol and implementation to be used with the
macromolecular modeling package CHARMM [2] would therefore be highly
desirable.
Research Goals: As computers become faster it becomes more and more
possible to tackle the question how enzymes achieve their remarkable
efficiency in speeding up chemical reactions by up to
10^{21}fold.
Such simulations require
the highlycomplex combination of quantum mechanical (QM) and molecular
mechanical (MM) simulation techniques. An implicitsolvent treatment of
longrange electrostatic effects to be used with QM/MM calculations has
to date not been mathematically formulated. The development and implementation
of such a formalism would enhance the capability to understand the functioning
of biological macromolecules at an atomic and subatomic level of detail.
An important issue in all the above as well as several of the following
projects is modeling the effects of solvent/water molecules.
To attack this problem, we plan to continue our development of continuum
methods using the PoissonBoltzmann
representation in combination with the UG multigrid method being
developed
in the Wittum group. The simulation system UG provides parallel
adaptive multigrid methods for complex application problems, [3].
In the framework of a diploma thesis, we already started a cooperation
on this topic. In this project we want to extend this work to more
realistic configurations.
Literature:
[1] 
S. M. Schwarzl, T. B. Tschopp, J. C. Smith,
S. Fischer: 
Can the Calculation of Ligand Binding Free Energies
Be Improved with Continuum Solvent Electrostatics and an IdelGas Entropy Correction?, J. Comput. Chem. 2002, 23, 11431149 
[2] 
B. R. Brooks, R. E. Bruccoleri, B. D. Olafson, D. J. States, S. Swaminathan, M. Karplus: 
CHARMM: A program for macromolecular energy,
minimization and dynamics calculations, J. Comp. Chem. 1983, 4,
187217. 
[3] 
P. Bastian, K. Birken, K. Johannsen, S. Lang,
V. Reichenberger, C. Wieners, G. Wittum, C. Wrobel: 
A parallel softwareplatform for solving problems of
partial differential equations using unstructured grids and adaptive
multigrid methods, In High Performance Computing in Science and
Engineering (W. Jäger and E. Krause, eds.), p 326339 Springer,
2000. 

Numerical Modeling of Nonlinear Diffusion on Complex Domains
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Supervisors: Bastian (IWR Heidelberg), Niezgódka (ICM
Warsaw)
Keywords: Parallel Computing, Multiscale
Background:
Nonlinear diffusion over domains of complex geometry represents a
challenge equally to analysts and numerical analysts. To gain a
qualitative knowledge on local effects, in particular for large time
intervals, in subdomains of strongly different geometric and
topological characteristics is of high interest for many realworld
applications such as those arising from the modeling of chemical or
biological systems. Of high applied relevance is the formation of
spatial structures due to diffusiondriven developments in
heterogeneous populations, [1].
Research Goals:
Within the proposed project, of special interest will be to develop a
systematic classification of models according to the
geometric/topological characteristics of the domains involved.
This will be possible through the combination of the modeling
techniques of the ICM group with the computational techniques
developed in the IWR group. In particular in the
numerical part of this project we propose to:
 Extend the discontinuous Galerkin method to systems of
nonlinear diffusionreaction equations and implement different
stabilization schemes such as the symmetric and nonsymmetric
interior penalty Galerkin method, the OdenBabuskaBaumann
method and the local discontinuous Galerkin method.
 Compare the different schemes with respect to accuracy and
robustness for applications including complex geometries
and highly varying diffusion coefficients. So far, thorough comparisons
are only available for simple model problems. In particular, some
methods have a stabilization parameter and the sensitivity of the
accuracy with respect to this parameter in real applications shall
be investigated.
 Develop fast, parallel and robust solvers for the arising linear
systems which allow us to solve very large problems on large scale
distributed memory parallel computers, [2].
One possibility here is to extend
aggregation based algebraic multigrid methods. These methods are
very robust for systems arising from standard discretizations and
are highly parallelizable, [3].
Literature:
[1] 
N. Kenmochi, M. Niezgódka, M. Otani, eds.: 
Mathematical Aspects of Modelling Structure Formation
Phenomena, Math. Sciences and Applications, Vol. 17, Gakkotosho,
Tokyo, 2001. 
[2] 
P. Bastian, M. Droske, C. Engwer, R. Klöfkorn,
T. Neubauer, M. Ohlberger, and M. Rumpf: 
Towards a unified framework for scientific
computing, Proc. 15th Conf. on Domain Decomposition
Methods(R. Kornhuber et al., eds.), LNCSE. SpringerVerlag, 2004, accepted
for publication. 
[3] 
M. Blatt: 
Parallel aggregation based algebraic multigrid,
Master's thesis, Faculty of Mathematics and Computer Science, University
Heidelberg, 2004. 

Efficient Techniques for Global Optimization
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Supervisors: Bock (IWR Heidelberg), Piela (ICM, Warsaw)
Keywords: Global Optimization, Multiscale
Background: Functions with numerous local minima are ubiquitous in many branches of natural
sciences and one is interested in finding the global minimum (GM). In chemistry, e.g., the GM conformation may tell us about the function an enzyme performs. But for flexible molecules (e.g. enzymes) the number of minima (conformations) may easily attain 10^{100}. Any systematic search of the conformational space based on local approaches seems to be hopeless.
There are many different approaches to global optimization in the literature, e.g. [4], differing from incomplete methods as genetic algorithms over asymptotically complete ones to rigorous methods that usually make use of outer approximation of the feasible set, underestimation of the objective function and interval arithmetic in a BranchandBound scheme. While these methods work well on a large set of problems, they seem unfit to treat the very highdimensional problem class under investigation, simply due to the exponential worstcase behaviour of BranchandBound algorithms. To overcome these problems a promising heuristics have been proposed in [2]. There a nonconvolutional way of smoothing of the objective function is considered. Imagine a wheelbarrow (of the wheel radius r) rolling under the target function f(x) and at any instance tangent to f(x). Then, the wheelbarrow axis trajectory F(x;r) for sufficiently large r has as the only minima those at the exact positions of the GM of the original function f(x).
Research Goals: The first part of the proposed thesis could be a study of 1D cases. It might be to `highlight' or `stress' the GM by a recursive transformation,
F(n,x,a)=a{exp[F(n1,x,a)/a]+1} for n=1,2,3, and a>0,
with f(x)= F(0,x,a). Then, a method of rolling the wheel has to be designed. Next, the method has to be generalized to the socalled global optimization standard test functions of higher dimension [1]. As benchmarks for the method the well known GM for the LennardJones atomic clusters may be taken. The mathematical properties of the new heuristics will be investigated and compared to established exact methods of global optimization [3]. In particular the complexity of the new techniques will be investigated and comparisons to exact methods will be performed.
Literature
[1] 
L. Piela: 
Handbook of Global Optimization, P.M.Pardalos, H.E.Romeijn, eds., Vol.2, 461488, Kluwer Acad.Publishers, 2002. 
[2] 
J. Pillardy, L. Piela: 
J. Comput. Chem., 18, 20402049, 1997. 
[3] 
C.A. Floudas and P.M. Pardalos: 
Optimization in Computational Chemistry and Molecular Biology: Local and Global Approaches, Kluwer, 2000. 
[4] 
A. Neumaier: 
Complete Search in Continuous Global Optimization and Constraint Satisfaction, pp. 271369 in: Acta Numerica 2004 (A. Iserles, ed.), Cambridge University Press, 2004. 

Modeling BoseEinstein Condensates by NonLinear Quantum Dynamics
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Supervisors:Cederbaum (PCI Heidelberg), Lesyng (ICM Warsaw)
Keywords: Parallel Computing, Nonlinear Quantum Dynamics
Background: Nonlinear Schrödinger equations are widely used in many areas of physics
and chemistry. A particularly interesting field of high current actuality,
where the nonlinear Schrödinger equation (also called the GrossPitaevskii
(GP) equation) is often used, refers to dilute gas BoseEinstein
condensates. The GP equation is a meanfield equation which has been
successfully applied in many cases, but fails to describe several phenomena
of interest, like fragmentation of condensates. Recently, a general
meanfield has been introduced and shown to be able to handle fragmentation
[1], [2].
Research Goals: This best meanfield leads to a set of n (n>1) coupled nonlinear
equations and the objective of the project is to develop numerical methods
to solve them efficiently in one, two and three dimensions of space and for
several n. For a given problem, the number n of coupled equations is
determined by the variational principle.
The solution of the new nonlinear Schrödinger equation provides a
wavefunction describing the condensate. This wavefunction is
discretized using discrete variable representations (DVR). The
resulting nonlinear working equations are solved iteratively until
selfconsistency is obtained.
The group of Lesyng (ICM Warsaw) has experience in parallel and vectorparallel
architectures, [3], and has already started working with the GP equation
[4]. In addition, their vast experience with efficient implementations of
dynamics of multidimensional quantum subsystems embedded in large classical
environments will be most helpful for treating also the dynamics of
condensates efficiently.
The doctoral student is expected to develop the code following the above lines
and, in particular, to parallelize the algorithm. Simulations of
realistic condensates like those experimentally prepared in Heidelberg
in the group of J. Schmiedmayer and M. Oberthaler (Physics Department)
should be carried out.
Literature:
[1] 
L. S. Cederbaum and A. I. Streltsov: 
Best meanfield for condensates, Phys. Lett. A 318, 564 (2003). 
[2] 
L. S. Cederbaum and A. I. Streltsov: 
Selfconsistent fragmented excited states of trapped condensates, Phys. Rev. A 70, 023610 (2004). 
[3] 
P. Bala, T. Clark, P. Grochowski, B. Lesyng, J. A. McCammon: 
Parallel version of the combined quantum classical molecular dynamics code for complex molecular and biomolecular systems, in "Recent Advances in Parallel Virtual Machine and Message Passing Interface", pp. 409416, SpringerVerlag (1997) 
[4] 
A. Gorecki, P. Bala and B. Lesyng: 
Parallelization of the quantum dynamics code for cluster architecture and its application to the GrossPitaevskii equation, Int. J. Quant. Chem. (submitted). 

Ignition of Liquid Fuel Sprays in Air in the Counterflow Configuration
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Supervisors: Gutheil (IAM Heidelberg), Rychter (ICM Warsaw)
Keywords: Multiscale, Combustion Processes
Background: The ignition of sprays has not been successfully studied to the extend that the ignition behavior of interacting droplets can be implemented into computer codes for technical spray applications.
The counterflow configuration has been identified as a powerful tool to study steady spray flames with both experimental and numerical methods.
Research Goals: An existing computer code for a steady system [1] will be extended to unsteady conditions to investigate the ignition behavior of laminar sprays with parameter studies for the initial droplet size and velocity as well as the dispersity of the spray. The parametric ignition times obtained will be implemented into the KIVA 3V (Release 2) code for turbulent spray ignition and combustion [2] where methods such as the PDF (probability density function) method [3] will be used to account for the turbulent character of the flow. The application is a diesel engine where both injection and ignition are major issues in the determination of the entire combustion process. The combined expertise of knowledge on heterogeneous ignition by the first supervisor and modeling of injection characteristics in connection with the use of the KIVA 3V code of the second supervisor will be extremely beneficial for the success of the proposed project.
Literature:
[1] 
E. Gutheil: 
Structure and Extinction of Laminar Ethanol/Air Spray Flames, Combustion Theory and Modeling 5: 115 (2001). 
[2] 
R. Jarnicki, T. J. Rychter: 
Theoretical Study of Combustion Intensification in SI Piston Engines, SAE Paper No. 2002011742, 2002; Also in the book: Engine Modeling Techniques: SI and Diesel., SP1711. 
[3] 
H.W. Ge, E. Gutheil: 
PDF Modeling of Turbulent Spray Flows, Atomization and Sprays, 2004, submitted. 

Proton Transfer and KetoEnol Tautomerization in
Malonaldehyde
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Supervisors: Meyer (IWR Heidelberg), Dodziuk (ICM Warsaw)
Keywords: Adaptive Methods, Molecular Dynamis
Background: Malonaldehyde (C_{3}H_{4}O_{2}) exists in the Keto and Enol form. The Enol form is cyclic with the two Oxygen atoms sharing one Hydrogen. By proton transfer this Hydrogen exchanges the Oxygen to which it is bound. These dynamical processes, i.e. the tautomerization and the proton transfer, shall be studied theoretically with the multiconfiguration timedependent Hartree(MCTDH) method [1]. MCTDH has been developed in Heidelberg over the last decade. A program package (250,000 lines, including the documentation) has been written, which allows to propagate a multidimensional (3 to 30 degrees of freedom, say) wavepacket very efficiently, as well as to analyse this propagated wavepacket to obtain crosssections, spectra, etc. Problems which were not feasible before, have become feasible through the advent of MCTDH. MCTDH is a novel, powerful, and convenient tool to attack problems of molecular quantum dynamics. MCTDH has proven its power in several applications (see the MCTDH website http://www.pci.uniheidelberg.de/tc/usr/mctdh/index.html for a list of applications).
Research Goals: In parallel Malonaldehyde shall be studied experimentally by Dodziuk (ICM Warsaw) using IR, UV, and NMR spectroscopic techniques [2]. This will lead to a deeper understanding of the complicated dynamics of this molecule [3]. Experimentally, the probes will consist of a mixture of isomers. The calculation of NMR parameters [4] is hence important for interpreting the spectra.
Literature:
[1] 
M. H. Beck, A. Jäckle, G. A. Worth and H.D. Meyer: 
The multiconfiguration timedependent Hartree method: A highly efficient algorithm for propagating wavepackets, Phys. Rep. 324 (2000), 1105. 
[2] 
H. Dodziuk: 
Rigidity versus flexibility. A review of experimental and theoretical studies pertaining to the cyclodextrin nonrigidity, J. Mol. Struct. 614 (2002), 33. 
[3] 
S. Yamabe, N. Tsuchida and K. Miyajima: 
Reaction path of ketoenol tautomerization of Þdiketones, J. Phys. Chem. A 108 (2004), 27502757. 
[4] 
M. Jaszunski, G. Dolgonos, and H. Dodziuk: 
An ab initio study of NMR chemical shifts and spinspin coupling constants of Bicyclobutane, Theor. Chem. Acc. 108 (2002). 240245. 

Numerical Modeling of the Dynamics of Structured Biopopulations:
DiffusionDriven Structural Transformation
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Supervisors Rannacher (IAM Heidelberg), Niezgódka (ICM Warsaw)
Keywords: Multiscale, Adaptivity
Background: Diffusive phase separation is a phenomenon exhibiting multiple temporal scales. In multicomponent systems, possibly with intrinsic multiphase nature, this process contributes to formation of spatial structures subject to thermodynamic nonequilibrium forcing. The class of mathematical models to be considered in their diffusion part involves a rather subtle interplay of two driving phenomena: thermodynamically destabilizing nonmonotonicity of bulk constitutive relations, in the simplest case resembling van der Waals functions, [2], and the stabilizing contribution due to surface forcing, in form of Ginzburg terms, [3].
Those effects heavily differ in their magnitude, up to several
orders, often getting close to floatingpoint accuracy of real
implementations. An additional challenge concerns the shape of geometric
domain, with a special interest in the case of few large subdomains connected by thin segments. It would be also highly interesting to tackle the case of dynamic couplings of diffusion to convection and diffusion to heat transfer. The latter effect plays a special role for large times, where due to experimental evidence thermal fluctuations have strong impact on the developments.
Research goal: For dealing with the problems desribed above, we need to develop numerical approaches capable of reproducing localized in space largetime behavior of the system. The presence of extremely fast temporal modes versus rather slow ones, manifesting only for very long time intervals and, even then, on quite low level of articulation, contributes to severe difficulties with designing efficient computational approaches. This requires the use of adaptive methods such as described in [1]. The contribution to the error by the local residuals at the various scales, i.e. their sensitivities, is captured by numerically solving approporiate dual problems. Here, the crucial question is whether dual information obtained on an intermediate scale can actually describe the effect by the finest scales on the coarse scale phenomena. This requires intensive test calculations with
nonlinear multiscale problems. This is related to the even more challenging question of turbulence simulation by adaptive LES (Large Eddy Simulation) approaches as recently proposed in [4]. The techniques used in this field should also be tried on partially chaotic biological models.
Literature:
[1] 
R. Becker, V. Heuveline, R. Rannacher: 
An optimal control approach to adaptivity in computational fluid mechanics, Int. J. Numer. Meth. Fluids. 40, 105120 (2002). 
[2] 
J. Cahn, A. NovickCohen: 
Evolution equations for phase separation and modelling in binary alloys, J. Stat. Phys. 76, 877909 (1994). 
[3] 
N. Kenmochi, M. Niezgódka, I. Pawlow: 
Subdifferential approach to the ChanHilliard equation with constraint, J. Diff. Equations 117, 320356 (1995). 
[4] 
T. J. R Hughes, G. R. Feijoo, L. Mazzei, and J.B. Quincy: 
The variational multiscale method  a paradigm for computational mechanics, Comput. Methods Appl. Mech. Eng., 166:324, 1998. 

Calculation of Thermodynamic Potentials and Response Functions for
LargeScale Networks
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Supervisors: Reinelt (IfI Heidelberg), Wislicki (ICM Warsaw)
Keywords: Graph Algorithms, Statistical Thermodynamics
Background: Finding equilibria for a largescale choicemodel network needs an extensive simulation and integration over the state space whose size depends factorially on the network size. Computational shortcuts are certainly needed for problems of the size comparable to real life problems in transportation, energy routing or communication in large groups, where the numbers of nodes are of the order of thousands and the number of agents exceeds the number of nodes by one or more orders of magnitudes. For simulating the system thermodynamic partition functions are to be computed. These are sums over all states of the system where the number of states not only depends on the size of
the network but also depends on the size of the population of agents distributed on such a network. This is a very hard computational task.
Simulations so far have been performed for rather small systems only. In this collatboration numerical simulations will be extended to larger
systems. This task requires finding algorithmic shortcuts in order to simplify the problem. For example, one can suppress a large number of the
largestutility paths as negligibly contributing to the partition
function. Effective combinatorial algorithms on graphs are needed and a
detailed analysis of an estimator's bias is necessary.
Research Goals: In a first PhD project in this area we want to consider a population of communicating agents, represented by a directed graph with a nonfixed, emergent topology with reinforcement learning as a topology formation formalism and to attempt to define a realistic utility function for such a system. The system's behaviour with respect to special network features as node transmittivity, link transmittivity or connectivity of nodes will be investigated.
For research work in this area the expertises of the two participating groups (statistical thermodynamics, see e.g., [1], [2],
and efficient graph algorithms, see e.g., [3], [4]) ideally complement one another.
Literature:
[1] 
A. Majka and W. Wislicki: 
Uniformity of the phase space and fluctuations in thermal equilibrium, Physica A, 322C, 313 (2003) 
[2] 
A. Majka and W. Wislicki: 
Statistical thermodynamics for choice models on graphs, Physica A 337, 645  663 (2004) 
[3] 
D. Ahr and G. Reinelt: 
A Tabu Search Algorithm for the MinMax kChinese Postman Problem, To appear in: Computers and Operations Research, Special Issue on Arc Routing, (2004) 
[4] 
G. Reinelt: 
The Traveling Salesman: Computational Solutions for TSP Applications, Lecture Notes in Computer Science 840, Springer (1994) 


