Numerical Studies of Recovery Chances for a Simplified EIT Problem

Author:    Christopher Hofmann, Bernd Hofmann, Roman. Unger

Published in: Bernd Hofmann, Antonio Leitao and Jorge P. Zubelli (Eds.). New Trends in Parameter Identification for Mathematical Models, Birkhäuser Basel 2018

This study investigates a simplified discretized EIT model with eight electrodes distributed equally spaced at the boundary of a disc covered with a small number of material stripes of varying conductivity. The goal of this paper is to evaluate the chances of identifying the conductivity values of each stripe from rotating measurements of potential differences. This setting comes from an engineering background, where the used EIT model is exploited for the detection of conductivities in carbon nanotubes (CNT) and carbon nanofibers (CNF). Connections between electrical conductivity and mechanical strain have been of major interest within the engineering community and has motivated the investigation of such a stripe structure. Up to five conductivity values can be recovered from noisy 8x8 data matrices in a stable manner by a least squares approach. Hence, this is a version of regularization by discretization and additional tools for stabilizing the recovery seem to be superfluous. To our astonishment, no local minima of the squared misfit functional were observed, which seems to indicate uniqueness of the recovery if the number of stripes is quite small.

Clustering by optimal subsets to describe environment interdependencies

Author:    J. Glänzel, R. Unger,

Published in the Preprint Series of Department of Mathematics as PREPRINT 2017-03 at the Technische Universität Chemnitz, 2017.

Download in pdf-format as Glaenzel-Unger--Clustering_by_optimal_subsets_to_describe_environment_interdependencies.pdf

The paper copes with the problem of finding an optimal subset of interpolation points out of a given large set of computed values, arising from a finite element simulation. This simulation computes environment data, which are on their part input data for finite element simulations of machine tools. For machine tool manufacturers it is still a seriously problem that the machine works imprecisely and products wastrel if environment values like temperature changes. The change of the environment boundary conditions contribute to the phenomenon through sunlight or cold draught owing to open doors of the machine hall or factory. Resulting thermo-elastic effects on the tool center point are one of the major reasons for positioning errors in machine tools. A genetic search algorithm for clustering relevant heat transfer coefficient values over the geometric surface through computational fluid dynamics (CFD) simulations will be described. These values are the input data for a developed thermo-elastic correction algorithm.

Extended bisection method for parameter identification of the transient heat conduction equation for thermo-elastic deformations during drilling

Author:    J. Glänzel, A. Meyer, R. Unger, M. Bräunig, V. Wittstock, S. Ihlenfeldt

Published in the The International Journal of Advanced Manufacturing Technology 88:1279-1288, 2017.

A new interdisciplinary approach is discribed to identifying unknown parameters using an extended version of the known interval bisection method. This developed method is based on the use of finite elements for calibrating the simulation calculation. The resulting thermo-elastic deformations which occur in drilling processes with impaired cooling lubrication are to be used as correction values for tool positioning in the NC control. Based on the strong impact on workpiece temperature of machining, a simulation approach is presented for calculating the temperature fields and their thermo-elastic consequences. In addition, methods are presented to correct these effects. This paper particularly deals with the temperature fields of drilling operations. Special attention is paid to the technique employed for iterative numerical determination of the unknown heat flux and heat transfer coefficient values.

Simulation-based correction approach for thermo-elastic workpiece deformations during milling processes.

Author:    J. Glänzel, R. Herzog, S. Ihlenfeldt, A. Meyer, R. Unger Published in the 7th HPC 2016 - CIRP Conference on High Performance Cutting, Procedia CIRP } 46:103-106, 2016

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Based on the method of adaptive finite elements (FEM) a correction approach has been considered to identify the influence of thermo-elastic workpiece deformations during the production process milling. The paper presents a simulation-based tolerance variation calculation of cutting paths, which is caused by the heat input of the machine tool. Therefore a mathematical method is developed to numerically depict the progress of the miller with different curves A(t), B(t) and C(t). These curves are used to map the state of the milling path during the production process as well as to compare the current workpiece contour and the target workpiece contour. The tool center point (TCP) correction results from mapping of time-dependent deformation fields from the FE simulation. The aim is, on the one hand, to make statements before the production about keeping the tolerance, and on the other hand, to derive other correction approaches for the adaption of the cutting path coordinates.

Nonparametric Copula Density Estimation Using a Petrov-Galerkin Projection

Author:    D. Uhlig, R. Unger. Published in the Innovations in Quantitative Risk Management. Springer Proceedings in Mathematics & Statistics 99:423-438, 2015

Nonparametrical copula density estimation is a meaningful tool for analyzing the dependence structure of a random vector from given samples. Usually kernel estimators or penalized maximum likelihood estimators are considered. We propose solving the Volterra integral equation to find the copula density c(u1,...,ud) of the given copula C. In the statistical framework, the copula C is not available and we replace it by the empirical copula of the pseudo samples, which converges to the unobservable copula C for large samples. Hence, we can treat the copula density estimation from given samples as an inverse problem and consider the instability of the inverse operator, which has an important impact if the input data of the operator equation are noisy. The well-known curse of high dimensions usually results in huge nonsparse linear equations after discretizing the operator equation. We present a Petrov–Galerkin projection for the numerical computation of the linear integral equation. A special choice of test and ansatz functions leads to a very special structure of the linear equations, such that we are able to estimate the copula density also in higher dimensions.

High Quality FEM-Postprocessing and Visualization Using a Gnuplot Based Toolchain

Author:    J. Glänzel. R. Unger.

Published in the Chemnitz Scientific Computing Preprints as csc14-03.pdf at the Technische Universität Chemnitz, 2014.

Download in pdf-format as Glaenzel-Unger--High Quality_FEM-Postprocessing_and_Visualization_Using_a_Gnuplot_Based_Toolchain.pdf

In the paper a toolchain for postprocessing and visualization of simulation datas, arising in finite element computations is described. This toolchain is based on gnuplot and free software. It is developed on the simulation problem of thermoelasticity, but it is not restricted on this problem class and adaptable on all other simulation problems, dealing with scalar- and vector fields. The most involved programs are gnuplot, perl, make and ffmpeg.

Some examples, files, scripts e.g. are downloadable at CSC14-03

The Petrov-Galerkin projection for copula density estimation isn't counting

Authors:    D. Uhlig,    R. Unger.

Published in the Preprint Series of Department of Mathematics as PREPRINT 2014-03 at the Technische Universität Chemnitz, 2014.

Download in pdf-format as Uhlig-Unger--The_Petrov_Galerkin_projection_for_copula_density_estimation_isnt_counting.pdf

Non-parametric copula density estimation in the d-dimensional case is a big challenge in particular if the dimension d of the problem increases. In Preprint 2013-07 we proposed to solve the d-dimensional Volterra integral equation \int_0^u c(s)ds = C(u) for a given copula C.
In the statistical framework the copula C is unobservable and hence we solved the linear integral equation for the empirical copula. For the numerical computation we used a Petrov-Galerkin projection for the approximated piecewise constant function c_h(u)=Sum_{i=1}^N c_i \phi_i(u).
Other than might be expected, the coefficient vector c does not count the number of samples in the elements of the discretized grid, even the approximated solution c_h is a piecewise constant function on the elements.
We will establish that solving the Volterra integral equation by a Petrov-Galerkin projection is not simple counting.

A Petrov Galerkin projection for copula density estimation

Authors:    D. Uhlig,    R. Unger.

Published in the Preprint Series of Department of Mathematics as PREPRINT 2013-07 at the Technische Universität Chemnitz, 2013.

Download in pdf-format as Uhlig-Unger--A_Petrov_Galerkin_projection_for_copula_density_estimation.pdf

The reconstruction of the dependence structure of two or more random variables (d > 1) is a big issue in finance and many other applications. Looking at samples of the random vector, neither the common distribution nor the copula itself are observable. So the identification of the copula C or the copula density can be treated as an inverse problem. In the statistical literature usually kernel estimators or penalized maximum likelihood estimators are considered for the non-parametric estimation of the copula density c from given samples of the random vector.
Even though the copula C itself is unobservable we can treat the empirical copula as a noisy representation, since it is well known that the empirical copula converges for large samples to the copula and solve the d-dimensional linear integral equation for determining the copula density c.
We present a Petrov-Galerkin projection for the numerical computation of the linear integral equation and discuss the assembling algorithm of the non-sparce matrices and vectors. Furthermore we analyze the stability of the discretized linear equation.

Population dispersal via diffusion-reaction equations

Authors:    A. Kandler,    R. Unger.

Published in the Preprint Series of Department of Mathematics as PREPRINT 2010-16 at the Technische Universität Chemnitz, 2010.

Download in pdf-format as Kandler-Unger--Population_dispersal_via_diffusion-reaction_equations.pdf

Diffusion-reaction systems are well-established in different life-science disciplines. When applied to 'human questions' they are used to estimate the demographic processes involved in major human (or animal) dispersal episodes and to estimate the general spread pattern of new ideas or technologies through cultures.
This manuscript gives an introduction to diffusion-reaction systems for a non-mathematical audience. We focus on describing dispersal processes and start with modelling and analysing the spread dynamic of a single population under different dispersal and growth hypotheses. Further, we focus on the impacts of population interactions on spread behaviour of a particular population.
Lastly we introduce an open software package 'CultDiff' which provides a solution tool for diffusion reaction systems.

Language shift, bilingualism and the future of britains's celtic languages

Authors:    A. Kandler,    J. Steele,    R. Unger.

Published in the Philosophical Transactions of the Royal Socienty B: Biological Sciences (B 2010 365, 3855-3864)

Language shift is the process whereby members of a community in which more than one language is spoken abandon their original vernacular language in favour of another. The historical shifts to English by Celtic language speakers of Britain and Ireland are particularly well-studied examples, for which good census data exist for the most recent 100-120 years in many areas where Celtic languages were once the prevailing vernaculars. We model the dynamics of language shift as a competition process in which the numbers of speakers of each language (both monolingual and bilingual) vary as a function both of internal recruitment (as the net outcome of birth, death, immigration and emigration rates of native speakers), and of gains and losses due to language shift. We examine two models: a basic model in which bilingualism is simply the transitional state for households moving between alternative monolingual states, and a diglossia model in which there is an additional demand for the endangered language as the preferred medium of communication in some restricted sociolinguistic domain, superimposed on the basic shift dynamic. Fitting our models to census data we successfully reproduce the demographic trajectories of both languages over the past century. We estimate the rates of recruitment of new Scottish Gaelic speakers that would be required each year (for instance, through school education) to counteract the 'natural wastage' as households with one or more Gaelic speaker fail to transmit the language to the next generation informally, for different rates of loss during informal intergenerational transmission.

A Reaction-Diffusion Model of language shift with a bilingual transition state

Authors:    A. Kandler,    J. Steele,    R. Unger.

to appear

High velocity human range expansion: numerical models

Authors:    L. Hazelwood,    A. Kandler,    J. Steele,    J. Steele,    R. Unger,    T. Sluckin.

to appear

Obstacle Description with Radial Basis Functions for Contact Problems in Elasticity

Author:    R. Unger.

Published in the Chemnitz Scientific Computing Preprints as csc09-01.pdf at the Technische Universität Chemnitz, 2009.

Download in pdf-format as Unger--Obstacle_Description_with_Radial_Basis_Functions_for_Contact_Problems_in_Elasticity.pdf

In this paper the obstacle description with Radial Basis Functions for contact problems in three dimensional elasticity will be done.
A short Introduction of the idea of Radial Basis Functions will be followed by the usage of Radial Basis Functions for approximation of isosurfaces. Then this isosurfaces are used for the obstacle-description in three dimensional elasticity contact problems. In the last part some computational examples will be shown.

A New Methodology for Modeling, Analysis, Synthesis, and simulation of Time-Optimal Train Traffic in Large Networks

Authors:    Y.Bavafa-Toosi, Ch.Blendinger, V.Mehrmann, A.Steinbrecher, R.Unger.

Published in IEEE Transactions on Automation Science and Engineering Vol. 5 Number 1 01/2008.

Based on my Diploma Thesis Numerische Simulation von Zugfahrten unter Realbedingungen
respectively the preprint Numerical simulation of train traffic in large networks via time-optimal control

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From a system-theoretic standpoint, a constrained state-space model for train traffic in a large railway network is developed. The novelty of the work is the transformation or rather reduction of the directed graph of the network to some parallel lists. Mathematization of this sophisticated problem is thus circumvented. All the aspects of a real network (such as that of the German Rail) are completely captured by this model. Some degrees of freedom, as well as some robustness can be injected into the operation of the system. The problem of time-optimal train traffic in large networks is then defined and solved using the maximum principle. The solution is obtained by reducing the boundary value problem arising from the time-optimality criterion to an initial value problem for an ordinary differential equation. A taxonomy of all possible switching points of the control actions is presented. The proposed approach is expected to result in faster-than-real-time simulation of time-optimal traffic in large networks and, thus, facilitation of real-time control of the network by dispatchers. This expectation is quantitatively justified by analysis of simulation results of some small parts of the German Rail Network.

Unterraum-CG-Techniken zur Bearbeitung von Kontaktproblemen

Autor:    R. Unger.

Dissertationsschrift, veröffentlicht im MONARCH der TU-Chemnitz unter

Download als PDF-Version: dissertation_roman_unger.pdf

Der Gegenstand dieser Arbeit ist die Untersuchung spezieller Lösungsmethoden zum Problem des Kontaktes eines elastischen Körpers mit einem festen Hindernis sowie des Kontaktes zweier elastischer Körper miteinander. Grundlage der Betrachtungen ist dabei ein Lösungsverfahren, das auf Unterraum-CG-Techniken beruht.

Die zu Grunde liegende partielle Differentialgleichung zur Modellierung der Verformung eines elastischen Körpers ist die Lame-Gleichung. Aufbauend auf dieser Gleichung wird das Problem des Kontaktes in einer neuen Formulierung, die auch große Verformungen zuläßt, betrachtet.

Um diese Probleme mit Hilfe der Finiten-Elemente-Methode numerisch lösen zu können, erfolgt die Betrachtung der üblichen Variationsformulierung mit Hilfe von Variationsungleichungen sowie die Angabe einer alternativen Formulierung, die auf einer Variationsgleichung beruht.

Zur Konstruktion eines effektiven Lösungsalgorithmus werden die Problematiken der a-posteriori Fehlerschätzung, Voraussetzungen an Vernetzungen sowie moderner Lösungsmethoden zum Auflösen des Finiten-Elemente-Gleichungssystems betrachtet.

Um die aus dem Kontaktproblem resultierenden Restriktionen zu erfüllen, wird die Klasse der Unterraum-CG-Verfahren einführend betrachtet und es wird die Anpassung dieser Verfahren auf die betrachteten Probleme vorgestellt. Die für derartige Lösungsmethoden verwendeten Projektoren werden formuliert und es werden verschiedene Formulierungen dieser Projektoren in Bezug auf Effektivität der Implementierung sowie Speicheraufwand miteinander verglichen.

Es wird auf einige verschiedene Möglichkeiten der Beschreibung von Hindernissen sowie des Kontaktproblems zweier elastischer Körper miteinander eingegangen und es werden Referenzimplementierungen zu diesen Problemen angegeben.

Zu den implementierten Projektoren werden Beispielrechnungen am Ende der jeweiligen Abschnitte vorgestellt sowie die Rechenzeiten und Konvergenzverhalten restringierter und unrestringierter Elastizitätsprobleme verglichen. Es zeigt sich dabei der Vorteil der entwickelten Verfahren in einem vergleichbaren numerischen Aufwand zwischen restringierten und unrestringierten Problemen bei einer übersichtlichen Implementierbarkeit und guter Stabilität.

Die Problemklasse von Restriktionen im Inneren des betrachteten Gebietes wird anhand des Clinch-Problems formuliert, und die zur Lösung derartiger Probleme verwendeten Projektoren betrachtet.

Die Referenzimplementierung aller vorgestellen Algorithmen und Projektoren erfolgt dabei in einem adaptiven 2D-FEM-Programm, welches innerhalb des DFG-Sonderforschungsbereichs 393 "Parallele Numerische Simulation für Physik und Kontinuumsmechanik" entstanden ist.

Subspace-cg-techniques for clinch-problems

Authors:    A. Meyer, R. Unger.

Published in the preprint series of the SFB393 "Parallele Numerische Simulation für Physik und Kontinuumsmechanik" at the Technische Universität Chemnitz, 2005.

Download as Postscript or PDF Version.

Subspace-cg-techniques with projection methods are useful for an easy extension of an arbitrary finite element code with error estimation and adaptive strategies to an algorithm for solving contact problems with additional restrictions such as contact problems.

In this paper we use the method to apply restrictions in the interior of the domain of an elastic body, not on the boundary.

Projection methods for contact problems in elasticity

Authors:    A. Meyer, R. Unger.

Published in the preprint series of the SFB393 "Parallele Numerische Simulation für Physik und Kontinuumsmechanik" at the Technische Universität Chemnitz, 2004.

Download as Postscript or PDF Version.

The aim of the paper is showing, how projection methods can be used for computing contact-problems in elasticity for different classes of obstacles.
Starting with the projection idea for handling hanging nodes in finite element discretizations the extension of the method for handling penetrated nodes in contact problems will be described for some obstacle classes.

Lösung parabolischer Differentialgleichungen mit zufälligen Randbedingungen mittels FEM

Authors:    A. Kandler, J. vom Scheidt, R. Unger.

Veröffentlicht im Tagungsband zum Workshop Stochastische Analysis vom 29.09.2003 - 01.10.2003.

In dieser Arbeit werden stochastische Charakteristiken der Lösung parabolischer Differentialgleichungen mit zufälligen Neumann-Randbedingungen mit Hilfe der Finite-Elemente-Methode angegeben.
Dabei wird der Berechnung der Korrelations-bzw. Varianzfunktion besondere Bedeutung beigemessen.
Das stochastische Randanfangswertproblem wird durch Anwendung von FEM-Techniken durch ein System gewöhnlicher Differentialgleichungen mit stochastischen inhomogenen Termen approximiert.
Die Modellierung der stochastischen Eingangsparameter durch epsilon-korrelierte Felder gestattet Entwicklungen der Lösungscharakteristiken nach der Korrelationslänge.
Numerische Beispiele enthalten den Vergleich zwischen analytischen Ergebnissen und Simulationsresultaten.

Numerical simulation of train traffic in large networks via time-optimal control

Ch.Blendinger, V.Mehrmann, A.Steinbrecher, R.Unger.

Published in the preprint series at the Institute of Mathematics, Technische Universität Berlin, 2001. Technical Report 722-2001

Download as Postscript or PDF Version.

We discuss the mathematical modelling of schedule based rail traffic. The model is used to develop efficient numerical simulation methods for the time optimal control of a large number of interacting trains in a large network. The time optimal control is used to model the realistic behaviour of driving in a network like that of Deutsche Bahn. It is used to allow a real time simulation with incomplete information on real train velocities. We present numerical examples that demonstrate the efficiency of the model and the simulation method.

Nachfolge - Versionen:

Y.Bavafa-Toosi, Ch.Blendinger, V.Mehrmann, H.Ohmori, A.Steinbrecher and R.Unger.
Time-Optimal Train Traffic in Large Networks based on a New Model.
Preprints of the 10th IFAC/IFORS/IMACS/IFIP Symposium on Large Scale Systems: Theory and Applications, Osaka, Japan, 2004, pages 729-734, 2004.

Y.Bavafa-Toosi, Ch.Blendinger, V.Mehrmann, A.Steinbrecher and R.Unger.
Modeling, analysis, synthesis, and simulation of time-optimal train traffic in large networks.
Technical Report 27-2006 Institute of Mathematics, Technische Universit\E4t Berlin, 2006. Download as PDF Version.

Y.Bavafa-Toosi, Ch.Blendinger, V.Mehrmann, A.Steinbrecher and R.Unger.
A New Methodology for Modeling, Analysis, Synthesis, and simulation of Time-Optimal Train Traffic in Large Networks.
IEEE Trans. Autom. Sci. Eng., 2007 (to appear)

Numerische Simulation von Zugfahrten unter Realbedingungen

Autor:    R. Unger.


Download als PDF-Version: diplomarbeit_roman_unger.pdf

Die in dieser Arbeit entwickelten Algorithmen sollen als schnelles Simulationswerkzeug zur Fahrzeitrechnung zum Einsatz kommen. Hauptzielgruppe sind die Betriebszentralen der Deutschen Bahn. Ziel war es, Dispositionshilfen für die Bearbeitung eines einzelnen Zuges und eines Systems aus mehreren, sich gegenseitig beeinflussenden Zügen zu schaffen. Schwerpunkt der Entwicklung der Algorithmen war neben der obligatorischen sinnvollen Genauigkeit der Rechnung eine schnelle Laufzeit der Programme, um Testreihen für Optimierungsstrategien in effektiven Rechenzeiten absolvieren zu können. Es wurden sowohl die theoretischen Grundlagen der Simulation sowie der optimalen Steuerung als auch die algorithmische Umsetzung in ein zeitgemäßes C++ Programm betrachtet. Auf hohe Portabilität und ISO-Konformität des Quellcodes wurde besonderer Wert gelegt, da die spätere Einsatzplattform noch nicht feststeht.