(The full reports of the Department of Mathematics are available undefinedhere.)

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

The goal of this project is to develop, analyze and apply highly efficient optimization methods for optimal control problems with control- and state constraints governed by time dependent partial differential algebraic equations (PDAEs). We combine in a modular way the state of the art software package KARDOS with modern multilevel second order optimization methods. The essential first and second order derivative information is determined by the continuous adjoint approach. The resulting correlated PDAEs are solved with KARDOS using adaptive Rosenbrock methods for the time integration and adaptive multilevel finite elements in space. Appropriate error estimates are derived to additionally control the mesh refinement according to the optimization advancement such that most of the optimization can be carried out on comparably cheap grids. The results are applied to the real engineering optimal control problem of glass cooling elaborately modeled by radiative heat transfer.

[1] D. Clever, J. Lang: Optimal Control of Radiative Heat Transfer in Glass Cooling with Restrictions on the Temperature Gradient, Preprint-Nr.: SPP1253-20-01,  2008

Contact: D. Clever, J. Lang
Support: DFG-Schwerpunktprogramm 1253, joint project with S. Ulbrich

Adaptive Two-Step Peer Methods for Incompressible Navier-Stokes Equations

The aim of this project is to analyze the potential of adaptive linearly implicit time integrators of higher order in CFD computations. We consider classical one-step Rosenbrock methods of order three and four as well as modern two-step methods of Peer-type up to order five. These new peer methods show good stability properties, but the main advantage over one-step methods lies in the fact that even in the application to PDEs no order reduction is observed. Therefore, they are good candidates for CFD computations that demand for high resolution.
We compare the accuracy and efficiency of the new methods equipped with variable time steps to second-order methods and investigate whether the higher order of convergence of the two-step peer methods pays off in practically relevant CFD computations. This will help rating the potential of linearly implicit methods as time discretizations in LES to solve practically relevant fluid flow problems.

Contact: B. Peth , J. Lang
Support: DFG-Graduiertenkolleg 1344

Anisotropic Mesh Adaptation Based upon A Posteriori Error Estimates

Error estimates are frequently used for mesh adaptation, especially for refinement strategies as well as in connection with the moving finite element method, but up to now they have been used mostly for isotropic mesh adaptation.
The objective of this project is to study the use of a posteriori error estimates in anisotropic mesh adaptation and compare it with commonly used Hessian-recovery-based adaptation methods. A key idea in the new approach is the use of the global hierarchical error estimator for reliable directional information of the solution.

[1] W. Huang, L. Kamenski, and J. Lang: Anisotropic mesh adaptation based upon a posteriori error estimates, Preprint Nr. 2570, 2008.

Contact: L. Kamenski, J. Lang
Support: German Research Foundation (DFG) under grants SFB568/3 and SPP1276 (MetStroem);
National Science Foundation (USA) under grants DMS-0410545 and DMS-0712935.


Autumn School Mathematical Modelling for Pupils

In this project 8 different groups of 5 pupils and 2 teachers each are working every autumn since 2000, on 8 different problems of different real life applications, under guidance of universitary assistance.
Within a week the problems have to be formulated in mathematical language, a solution strategy has to be developed and to be implemented. The results are presented in final talks and documented in a written paper. The project material is then available for use in education at schools or universities.

[1] M. Kiehl: Mathematisches Modellieren für die Sekundarstufe II, Cornelsen Verlag Scriptor, 2006

[2] Mathematische Modellierung mit Schülern, M. Kiehl (Ed), Jahrgang 2000-2003

Contact: M. Kiehl
Support: Volkswagenstiftung; Zentrum für Mathematik, Bensheim

Discrete-Continuous Optimization of Complex Dynamic Water Supply and Wastewater Networks

Today's demands in the management of water supply and wastewater networks require the close collaboration between industry and scientists from engineering and applied mathematics. The aim of this project is to tackle recent problems in water management with state-of-the-art numerical methods and to develop new adapted algorithms. The underlying tasks include the solution of hyperbolic partial differential equations on networks as well as discrete-continuous optimal control problems.
In this projects, scientists from various mathematical and engineering disciplines work in cooperation with our industry partners.

[1] O. Kolb, P. Bales, J. Lang: Moving Penalty Functions for Optimal Control with PDEs on Networks; Technische Universität Darmstadt, Fachbereich Mathematik, Preprint Nr. 2562 (2008)

Contact: O. Kolb, J. Lang
Support: Federal Ministry of Education and Research (BMBF) under the grant 03MAPAK1

Global Error Control for Parabolic PDEs

In this project we investigate efficiency and reliability questions for finite difference approximations of parabolic problems. Existing popular codes focus on efficiency by adaptively optimizing time grids in accordance to local error control. The reliability question, that is, how large are the global errors, has received much less attention. Therefore, we are developing strategies for global error estimation, based on the solution of linearized error transport equations, and global error control, based on the property of tolerance proportionality.

[1] K. Debrabant, J. Lang: On global error estimation and control for parabolic equations; Technische Universität Darmstadt, Fachbereich Mathematik, Preprint Nr. 2512 (2007)

[2] J. Lang, J. Verwer: On global error estimation and control for initial value problems; SIAM Journal on Scientific Computing 29 (2007), pp. 1460-1475

Contact: K. Debrabant, J. Lang
Partner: J. Verwer (CWI)

Hierarchical Modelling and Model Adaptivity for Gas Flow on Networks

Recent demands for gas transmission companies are to satisfy the customers requirements at designated times. Therefore, one needs to react fast and flexibly to short-term changes in the requested quantity and quality of gas. To meet the demands, reliable mathematical models as a basis for decisions on changing the configuration of the network are needed. Realistic problems in practice necessitate the consideration of thousands of pipes which makes global optimization with high resolution impossible. We develop a strategy to automatically employ different models in different regions of the network according to actual measurements of the gas flow using adjoint techniques.

[1] P. Domschke, O. Kolb, J. Lang: An adaptive model switching and discretization algorithm for gas flow on networks; Procedia Computer Science, Volume 1, Issue 1; ICCS 2010, pp. 1325-1334, DOI: 10.1016/j.procs.2010.04.148, May 2010.

[2] P. Bales, O. Kolb, and J. Lang: Hierarchical Modelling and Model Adaptivity for Gas Flow on Networks; ICCS 2009, Part I, LNCS 5544, pp. 337–346, Springer-Verlag (2009); (Preprint)

Contact: P. Domschke, J. Lang
Support: German Research Foundation (DFG) under the grant LA1372/5-1.

Large-Eddy Simulation with Adaptive Grids for Meteorological Applications

The aim of this project is to design elementary mathematical and numerical methods of dynamical grid adaption for Large Eddy Simulations (LES). Therefore, we apply LES combined with the moving mesh PDE (MMPDE) approach to problems with meteorological background, like the turbulent flow over periodic hills, using the finite-element software KARDOS. The MMPDE method continuously redistributes a fixed number of grid points due to a grid refinement criterion. The main advantage of this method is that during the integration process the mesh topology is preserved and no new degrees of freedom are added and therefore the data structures are preserved as well. In this project we are mainly concerned with developing such refinement criteria particularly for LES.

[1] S. Löbig, A. Dörnbrack, J. Fröhlich, C. Hertel, C. Kühnlein and J. Lang: Towards large eddy simulation on moving grids; Proceedings in Applied Mathematics and Mechanics 2009 (submitted for review)

Contact: S. Löbig, J. Lang
Support: DFG Schwerpunktprogramm (SPP) 1276 MetStröm: Skalenübergreifende Modellierung in der Strömungsmechanik und Meteorologie

Model Reduction for Large-Eddy Simulation

The project aims at model reduction for problems in computational fluid dynamics. In flow optimization often a Navier-Stokes model has to be solved consecutively using slightly different settings. The numerical solution of the incompressible Navier-Stokes equations is usually the computationally most demanding part of such an optimization process. Using the proper orthogonal decomposition (POD) or the centroidal Voronoi tessellation (CVT) it is possible to create a low-dimensional reduced model from snapshots of the solution of the original flow model. With this technique only a single run of the full model is required to create the snapshots, the other solutions can be obtained efficiently with the reduced model. Research is done on model reduction in the context of large-eddy simulation (LES) of turbulent flows with the finite-element software KARDOS.

Contact: S. Ullmann, J. Lang
Support: Sonderforschungsbereich 568

Monotone approximation schemes for fully non-linear diffusion equations of Bellman-Isaacs type

The aim of this project is to develop monotone approximation schemes for linear and fully non-linear diffusion equations of Bellman-Isaacs type which converge also for degenerate diffusion equations having general non-diagonal dominant coefficient matrices and are easy to implement and analyze.

[1] K. Debrabant, E. R. Jakobsen. Semi-Lagrangian schemes for linear and fully non-linear diffusion equations. Technische Universität Darmstadt, Fachbereich Mathematik, Preprint No. 2593 (2009).

Contact: K. Debrabant
Partner: Espen R. Jakobsen (NTNU Trondheim)

Multiscale structure-functional modeling of musculoskeletal mineralized tissues

Musculoskeletal mineralized tissues (MMTs) are natural examples ofachieving a unique combination and variability of stiffness andstrength. One of the striking features of MMTs is that this diversityof elastic function is achieved by only one common building unit,i.e. the mineralized collagen fibril, but variable structuralarrangements at several levels of hierarchical organization. A profound understanding of the structure-function relations in MMTsrequires both experimental assessment of heterogeneous elastic andstructural parameters and theoretical modeling of the elasticdeformation behavior. Multi-scale and multi-modal assessment of MMTswill be used to probe not only the microarchitecture, but alsoanisotropic linear elastic properties from the nanoscale to themacroscale. By combining experimental data obtained from MMTs atvarious length scales with numerical homogenization approaches incontinuum mechanics, we hypothesize to gain new insight into selfassembly mechanisms, construction rules and physiological boundaryconditions of MMTs.

Within this joint project with the group of K. Raum (Charité -Universitätsmedizin Berlin) we focus in Darmstadt on the developmentas well as efficient and reliable implementation of numericalhomogenisation techniques. Together with the group in Berlin we devisenew mathematical models in order to aid the understanding of MMTs. Theexperimental assessment of MMTs is performed in Berlin and withexternal cooperation partners.

Contact: S. Tiburtius, A. Gerisch
Support: DFG Schwerpunktprogramm (SPP) 1420 Biomimetic Materials Research: Functionality by Hierarchical Structuring of  Materials (joint project with K. Raum)

Numerical Solution of Stochastic Ordinary Differential Equations

In many applications, e.g. in epidemiology or mathematical finance, including stochastic effects in the modeling of continuous time dynamical systems leads to stochastic differential equations. Often one is not interested in approximating the solution pathwise (strong convergence), but in the expectation of functionals of the solution, which leads to weak convergence.
Together with Dr. A. Rößler (TU Darmstadt) we developed explicit and implicit 3-stage stochastic Runge-Kutta methods of weak stochastic order 2 and deterministic order 2 or 3 and classified these schemes in the explicit case completely. We also developed continuous stochastic Runge-Kutta methods.
For stochastic differential equations with (multidimensional) additive noise we introduced a class of weak third order Runge-Kutta methods.
For implicit stochastic Runge-Kutta methods, the equations for the stage values are solved by iterative schemes like simple iteration, modified Newton iteration or full Newton iteration. Together with Prof. A. Kværnø (NTNU Trondheim) we employed a unifying approach for the construction of stochastic B-series which is valid both for Itô- and Stratonovich-SDEs and applicable both for weak and strong convergence to analyze the order of the iterated Runge-Kutta method.  We also examined the relations between this approach and Wagner-Platen-expansions.
Further, we extended our general B-series approach to stochastic Taylor methods, giving for the first time order conditions of (implicit) Taylor methods in terms of rooted trees. As an example, we applied these order conditions to derive in a simple manner a family of implicit strong order 1.5 Taylor methods applicable to Itô SDEs. Finally, we determined the order of implicit Taylor methods in dependence on the numerical method used to approximate the solution of the implicit equation and on the number of iterations.

[1] K. Debrabant, A. Kværnø. B-series analysis of iterated Taylor methods. Report 1-2010, Martin-Luther-Universität Halle-Wittenberg, Institut für Mathematik (2010).

[2] K. Debrabant, A. Kværnø. Composition of stochastic B-series with applications to implicit Taylor methods. Preprint Numerics 01/2010, Norwegian University of Science  and Technology, Trondheim (2010).

[3] K. Debrabant. Runge-Kutta methods for third order weak approximation of SDEs with multidimensional additive noise. Technische Universität Darmstadt, Fachbereich Mathematik, Preprint Nr. 2589 (2009)

[4] K. Debrabant, A. Kværnø. Stochastic Taylor Expansions: Weight functions of B-series expressed as multiple integrals. Stochastic Analysis and Applications 28 (2010), No. 2, pp. 293 - 302.

[5] K. Debrabant, A. Rößler. Diagonally drift-implicit Runge-Kutta methods of weak order one and two for Itô SDEs and stability analysis. Applied Numerical Mathematics 59 (2009), pp. 595-607.

[6] K. Debrabant, A. Rößler. Families of efficient second order Runge-Kutta methods for the weak approximation of Itô stochastic differential equations. Applied Numerical Mathematics 59 (2009), pp. 582-594.

[7] K. Debrabant, A. Kværnø. B-series analysis of stochastic Runge-Kutta methods that use an iterative scheme to compute their internal stage values. SIAM Journal on Numerical Analysis 47 (2008), No. 1, pp. 181-203.

[8] K. Debrabant, A. Rößler. Continuous Runge-Kutta methods for Stratonovich stochastic differential equations. Monte Carlo and Quasi-Monte Carlo Methods 2006, pp. 237-250, Springer-Verlag (2008).

[9] K. Debrabant, A. Rößler. Classification of Stochastic Runge-Kutta Methods for the Weak Approximation of Stochastic Differential Equations. Mathematics and Computers in Simulation 77 (2008), No. 4, pp. 408-420.  

[10] K. Debrabant, A. Rößler. Continuous Weak Approximation for Stochastic Differential Equations. Journal of Computational and Applied Mathematics 214 (2008), No. 1, pp. 259-273.

Contact: K. Debrabant

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Unsteady Adaptive Stochastic Collocation Methods on Sparse Grids

Numerical simulations become more reliable if random effects are taken into account. To this end, the describing parameters can be expressed by correlated random fields, which leads to partial differential equations (PDEs) with random parameters. Common numerical methods to solve such problems are spectral methods of Galerkin type and stochastic collocation on sparse grids. We focus on stochastic collocation, because it decouples the random PDE into a set of deterministic equations that can be solved in parallel. By mean of interpolation, the procedure provides a functional dependency between random input parameters and response of the system.

The aim of this work is to derive error estimates for adaptive strategies in order to gain efficiency for the more and more complex problems arising in Computational Engineering. We want to combine stochastic collocation with an adjoint approach in order to estimate and control the error of some stochastic quantity, such as the mean or variance of a solution functional.

Contact: B. Schieche, J. Lang
Support: German Research Foundation (DFG): Graduate School of Computational Engineering, TU Darmstadt

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AG Numerik
FB Mathematik
TU Darmstadt
Dolivostraße 15
64293 Darmstadt

Elke Dehnert

Tel.: +49 (0)6151 16-23163
Fax: +49 (0)6151 16-23164




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