TY - GEN A1 - Klapproth, Corinna A1 - Schiela, Anton A1 - Deuflhard, Peter T1 - Consistency Results for the Contact-Stabilized Newmark Method N2 - The paper considers the time integration of frictionless dynamical contact problems between viscoelastic bodies in the frame of the Signorini condition. Among the numerical integrators, interest focuses on the contact-stabilized Newmark method recently suggested by Deuflhard et al., which is compared to the classical Newmark method and an improved energy dissipative version due to Kane et al. In the absence of contact, any such variant is equivalent to the Störmer-Verlet scheme, which is well-known to have consistency order 2. In the presence of contact, however, the classical approach to discretization errors would not show consistency at all because of the discontinuity at the contact. Surprisingly, the question of consistency in the constrained situation has not been solved yet. The present paper fills this gap by means of a novel proof technique using specific norms based on earlier perturbation results due to the authors. The corresponding estimation of the local discretization error requires the bounded total variation of the solution. The results have consequences for the construction of an adaptive timestep control, which will be worked out subsequently in a forthcoming paper. T3 - ZIB-Report - 09-06 KW - Dynamical contact problems KW - viscoelasticity KW - Signorini condition KW - consistency KW - Newmark method Y1 - 2009 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11164 SN - 1438-0064 ER - TY - GEN A1 - Weiser, Martin T1 - On goal-oriented adaptivity for elliptic optimal control problems N2 - The paper proposes goal-oriented error estimation and mesh refinement for optimal control problems with elliptic PDE constraints using the value of the reduced cost functional as quantity of interest. Error representation, hierarchical error estimators, and greedy-style error indicators are derived and compared to their counterparts when using the all-at-once cost functional as quantity of interest. Finally, the efficiency of the error estimator and generated meshes are demonstrated on numerical examples. T3 - ZIB-Report - 09-08 KW - optimal control KW - error estimation KW - adaptive mesh refinement Y1 - 2009 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11192 SN - 1438-0064 ER - TY - GEN A1 - Schiela, Anton A1 - Weiser, Martin T1 - Barrier methods for a control problem from hyperthermia treatment planning N2 - We consider an optimal control problem from hyperthermia treatment planning and its barrier regularization. We derive basic results, which lay the groundwork for the computation of optimal solutions via an interior point path-following method. Further, we report on a numerical implementation of such a method and its performance at an example problem. T3 - ZIB-Report - 09-36 KW - hyperthermia treatment planning KW - optimal control KW - barrier methods Y1 - 2009 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11567 SN - 1438-0064 ER - TY - THES A1 - Zschiedrich, Lin T1 - Transparent boundary conditions for Maxwell's equations: Numerical concepts beyond the PML method N2 - Optical technologies are ubiquitously used in hi-tech devices. As a common feature of such devices one finds structures with dimensions in the order of the wavelength of the used light. To design and produce such devices, the wave nature of light must be taken into account. Accordingly, robust simulation tools are required which are based on rigorously solving Maxwell's equations, the governing equations of light propagation within macroscopic media. This thesis contributes to the modeling and the numerical computation of light scattering problems: Light scattering problems are typically posed on the entire space. The Perfectly-Matched -Layer method (PML) is widely used to restrict the simulation problem onto a bounded computational domain. We propose an adaptive PML method which exhibits a good convergence even for critical problems where standard PML implementations fail. Besides the computation of the near field, that is the electromagnetic field within the computational domain, it is of major interest to evaluate the electromagnetic field in the exterior domain and to compute the far field. So far, this was numerically only possible for simple geometries such as homogeneous exterior domains or layered media. To deal with more complicated devices, for example with waveguide inhomogeneities, we develop an evaluation formula based on the PML solution which allows for an exterior domain field evaluation in a half space above the device. Finally, we generalize the PML method to problems with multiply structured exterior domains. The term “multiply structured exterior domain” is defined in this thesis and means that the exterior domain exhibits several half-infinite structures. Mathematically, this gives rise to various complications. For example, no analytical solutions to Maxwell's equations for standard light sources are available in the exterior domain, which are needed to describe the incoming field in a light scattering problem. To tackle this we propose a new light scattering problem formulation which fits well into the PML method framework and which may be regarded as an extension of classical contributions by Sommerfeld, Wiener and Hopf. An exterior domain evaluation formula for multiply structured exterior domains with an extended illumination is derived as well. KW - transparent boundary conditions KW - electromagnetic scattering KW - Maxwell's equations KW - perfectly matched layers KW - PML KW - Wiener-Hopf techniques Y1 - 2009 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:188-fudissthesis000000013994-8 ER - TY - THES A1 - Röblitz, Susanna T1 - Statistical Error Estimation and Grid-free Hierarchical Refinement in Conformation Dynamics N2 - The understanding of geometric structures and dynamical properties of molecular conformations gives insight into molecular long-term behavior. The identification of metastable conformations together with their life times and transition patterns is the intention of conformation dynamics. Conformation dynamics is a multi-scale approach that leads to a reduced description of the dynamical system in terms of a stochastic transition probability matrix. The present thesis deals with the error analysis of computed matrices and the resulting matrix functions. Since conformational membership vectors, as they are computed by the Robust Perron Cluster Analysis (PCCA+), form an invariant subspace of the transition matrix, subspace-based error estimators are of particular interest. The decomposition of the state space into basis functions and the approximation of integrals by Monte-Carlo quadrature give rise to row-wise correlated random matrices, for which stochastic norms are computed. Together with an appropriate statistical model for the distribution of matrix rows, this allows for the calculation of error bounds and error distributions of the invariant subspace and other variables of interest. Equilibration of errors among the basis functions can be achieved by enhanced sampling in regions where the trajectories are mixing slowly. Hierarchical refinement of such basis functions systematically improves the clustering into metastable conformations by reducing the error in the corresponding invariant subspace. These techniques allow for an evaluation of simulation results and pave the way for the analysis of larger molecules. Moreover, the extension of PCCA+ to non-reversible Markov chains, verified by the corresponding perturbation theory, and the modification of the objective function for the case of soft membership vectors represent a further generalization of the clustering method, thus continuing the development from PCCA over PCCA+ to PCCA++. The methods developed in this thesis are useful for but not limited to conformation dynamics. In fact, they are applicable to a broader class of problems which combine domain decomposition with Monte-Carlo quadrature. Possible application areas may include the chemical master equation or quantum dynamical systems. N2 - Das Verständnis von geometrischen Strukturen und dynamischen Eigenschaften molekularer Konformationen ist essentiell für die Vorhersage des Langzeitverhaltens von Molekülen. Die Identifikation metastabiler Konformationen sowie die Bestimmung von Übergangswahrscheinlichkeiten und Haltezeiten sind Bestandteil der Konformationdynamik. Dabei handelt es sich um eine Mehrskalenmethode, die auf eine reduzierte Beschreibung des Systems mittels einer stochastischen Übergangsmatrix führt. In der vorliegenden Dissertation wurde untersucht, wie man die Genauigkeit der Matrizen sowie der daraus berechneten Größen quantifizieren kann. Im Mittelpunkt stehen dabei Fehlerschätzer für den invarianten Unterraum, da die rechten Eigenvektoren als Grundlage der Robusten Perron Cluster Analyse (PCCA+) zur Identifizierung der metastabilen Konformationen dienen. Die Zerlegung des Zustandsraumes in Basisfunktionen sowie die Approximation der Matrixeinträge mittels Monte-Carlo-Quadratur führen zu zeilenweise korrelierten Zufallsmatrizen. Mit Hilfe einer stochastischen Norm sowie einem geeigneten statistischen Modell für die Verteilung der Matrixzeilen können u.a. Fehlerschranken und -verteilungen für den invarianten Unterraum brechnet werden. Eine Equilibrierung des Fehlers zwischen den Basisfunktionen kann durch erweitertes Sampling in solchen Regionen erreicht werden, in denen die Trajektorien nur langsam mischen.Eine hierarchische Zerlegung dieser Basisfunktionen verbessert systematisch die Zerlegung in metastabile Konformationen, indem sie den Fehler im invarianten Unterraum reduziert. Diese Techniken gestatten eine Evaluierung der Simulationsergebnisse und ebnen den Weg zur Behandlung komplexerer Moleküle. Desweiteren wurden Verallgemeinerungen der PCCA+ untersucht. Die Erweiterung der PCCA+ auf nicht-reversible Markov-Ketten sowie die Modifizierung der Zielfunktion für den Fall der weichen Clusterung setzen die Entwicklung von der PCCA über PCCA+ zu PCCA++ fort. Somit können neue Anwendungsfelder für dieses Cluster-Verfahren erschlossen werden. Die Methoden wurden zwar in Rahmen der Konformationsdynamik entwickelt, jedoch lassen sie sich auf eine weite Problemklasse anwenden, in der Gebietszerlegungsverfahren mit Monte-Carlo-Quadratur kombiniert werden. Mögliche Anwendungsgebiete umfassen die chemische Master-Gleichung oder quantenchemische Systeme. KW - Konformationsdynamik KW - Metastabilität KW - gitterfreie Methoden KW - Monte-Carlo-Quadratur KW - Perron-Cluster-Analyse KW - conformation dynamics KW - metastability KW - mesh-free methods KW - Monte Carlo quadrature KW - Perron cluster analysis Y1 - 2009 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:188-fudissthesis000000008079-9 ER - TY - GEN A1 - Kilian, Susanne A1 - Münch, Matthias T1 - A new generalized domain decomposition strategy for the efficient parallel solution of the FDS-pressure equation. Part I: Theory, Concept and Implementation N2 - A world-wide used program for the simulation of fire-induced flows is the Fire Dynamics Simulator (FDS) which originally was developed for a purely serial execution on single-processor computing systems. Due to steadily increasing problem sizes and accuracy requirements as well as restrictions in storage capacity and computing power on single-processor systems, the efficient simulation of the considered fire scenarios can only be achieved on modern high-performance systems based on multi-processor architectures. The transition to those systems requires the elaborate parallelization of the underlying numerical methods which must guarantee the same result for a given problem as the corresponding serial execution. Unfortunately, one fundamental serial serial solver of FDS, the pressure solver, only possesses a low degree of inherent parallelizm. Its current parallelization may cause additional numerical errors, casually leading to significant losses of accuracy or even numerical instabilities. In order to ensure that the parallelization errors are limited by the leading error of the numerical scheme such that second order convergence for the whole method can be acchieved, optimized parallelization concepts must be designed. With respect to these considerations this articles gives an overview of the current parallel pressure solver as well as the problems related to it and presents an alternative method, SCARC, to overcome the existing complicacies. Part I explains the theory, concept and implementation of this new strategy, whereas Part II describes a series of validation and verification tests to proof its correctness. T3 - ZIB-Report - 09-19 KW - Fire Dynamics Simulator (FDS) KW - Zero-Mach number KW - domain decomposition KW - pressure-solver KW - fire safety engineering Y1 - 2009 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11332 SN - 1438-0064 ER - TY - GEN A1 - Münch, Matthias A1 - Kilian, Susanne T1 - A new generalized domain decomposition strategy for the efficient parallel solution of the FDS-pressure equation. Part II: Verification and Validation N2 - Because CFD programs, like FDS, generally consist of a large number of different components representing the variety of participating numerical algorithms and chemical / physical processes, it is nearly impossible to verify such codes in their entirety, for example with comparisons of fire tests. Instead, a careful verification and validation with respect to the underlying mathematical conditions and applied numerical schemes is indispensable. In particular, error cancelations between single program components can only be detected by such detailed component-level tests. In part I of this article series a conceptual deficiency of the FDS program package with regard to multi-mesh computations was illustrated and an alternative domain decomposition strategy FDS-ScaRC was introduced. In this second part we will present the structure of a comprehensive test concept and the needs for a more mathematically and numerically orientated test procedure that is much more suited for a reliable evaluation than only a simple visual comparison of the numerical results with experimental fire tests. After a general introduction of our test concept we will demonstrate the high potential of the new FDS-\scarc{} technique compared to the FDS-FFT technique which is used in the FDS program package as yet. Based on this concept, we will present a comprehensive set of analytical and numerical test results. T3 - ZIB-Report - 09-20 KW - Fire Dynamics Simulator (FDS) KW - domain decomposition KW - Verification and Validation KW - fire safety engineering Y1 - 2009 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11340 SN - 1438-0064 ER - TY - GEN A1 - Ruprecht, Daniel A1 - Klein, Rupert A1 - Majda, Andrew J. T1 - Moisture-Gravity Wave Interactions in a Multiscale Environment N2 - Starting from the conservation laws for mass, momentum and energy together with a three species, bulk microphysic model, a model for the interaction of internal gravity waves and deep convective hot towers is derived by using multiscale asymptotic techniques. From the resulting leading order equations, a closed model is obtained by applying weighted averages to the smallscale hot towers without requiring further closure approximations. The resulting model is an extension of the linear, anelastic equations, into which moisture enters as the area fraction of saturated regions on the microscale with two way coupling between the large and small scale. Moisture reduces the effective stability in the model and defines a potential temperature sourceterm related to the net effect of latent heat release or consumption by microscale up- and downdrafts. The dispersion relation and group velocity of the system is analyzed and moisture is found to have several effects: It reduces energy transport by waves, increases the vertical wavenumber but decreases the slope at which wave packets travel and it introduces a lower horizontal cutoff wavenumber, below which modes turn into evanescent. Further, moisture can cause critical layers. Numerical examples for steadystate and timedependent mountain waves are shown and the effects of moisture on these waves are investigated. T3 - ZIB-Report - 09-21 Y1 - 2009 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11358 SN - 1438-0064 ER - TY - THES A1 - Massing, André T1 - 1D-Reduktion thermal signifikanter Aderstränge in der Hyperthermie-Modellierung T1 - 1D-Reduction of Thermal Significant Blood Vessels in the Context of Hyperthermia-Modelling N2 - In dieser Arbeit wird ein neuer Ansatz zur Modellierung von thermal signifikanten Gefäßsträngen im Hyperthermie-Kontext betrachtet. Ausgehend von einer Konvektions-Diffusions-Gleichung wird durch Reskalierung des Massenflussterms eine Reduktion des Adergebietes auf eine 1D-Struktur erreicht. Nach numerischen Vorbetrachtungen wird die Grenzgleichung innerhalb einer verallgemeinerten Sobolev-Algebra formuliert. Die Untersuchung der Lösungsfamilie in klassischen Funktionenräumen zeigt, dass deren schwacher Grenzwert die Lösung der korrespondierenden Diffusions-Gleichung ist. Die Diskretisierung einer formalen Grenzgleichung mit Linienstromanteil stellt jedoch eine gute Approximation an die Diskretisierung des ursprünglichen Problems dar, wenn man die lokale Maschenweite an die Gefäßradien koppelt und bei erhöhtem Genauigkeitsbedarf auf ein vollständiges 3D-Modell umschaltet. KW - Finite Elemente Methode KW - Mehrskalen-Modellierung KW - Bio-Heat-Transer-Equation KW - Asymptotische Analysis KW - Finite Element Method KW - Multiscale-Modelling KW - Bio-Heat-Transer-Equation KW - Asymptotic Analysis Y1 - 2009 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11407 ER - TY - GEN A1 - Schmidt, Heiko A1 - Oevermann, Michael A1 - Bastiaans, Rob J.M. A1 - Kerstein, Alan R. T1 - A Priori Tabulation of Turbulent Flame Speeds via a Combination of a Stochastic Mixing Model and Flamelet Generated Manifolds, Extended to Incorporate Strain Effects N2 - In this paper we revisit the a priori turbulent flame speed tabulation (TFST) technique for a given parameter space within the region of flamelet combustion-regimes. It can be used as a subgrid-scale (SGS) model in Large Eddy Simulation (LES). In a first step, stationary laminar flamelets are computed and stored over the progress variable following the ideas of flamelet generated manifolds (FGM). In a second step, the incompressible one-dimensional Navier-Stokes equations supplemented by the equation for the progress variable are solved on a grid that resolves all turbulent scales. Additionally, turbulent transport is implemented via the linear eddy model (LEM). The turbulent flame structures are solved until a statistically stationary mean value of the turbulent flame speed has been reached. The results are stored in a table that could be used by large scale premixed combustion models, e.g. front tracking schemes. First results are compared quantitatively with direct numerical simulations (DNS) taken from literature. Then it is illustrated in one example how the new method could help to fix constants in algebraic models for turbulent flame speeds. Further it is shown how the technique can be extended to incorporate turbulent strain effects. Finally we investigate the effect of the use of detailed and tabulated chemistry under unsteady conditions. T3 - ZIB-Report - 09-09 KW - Turbulente Verbrennung KW - Flammenstrukturen KW - Linear Eddy Model KW - FGM KW - Turbulente Brenngeschwindigkeit KW - Streckungseffekte KW - turbulent premixed combustion KW - flame structures KW - linear eddy model KW - flamelet generated manifolds KW - turbulent flame speed tabulation KW - strain effects Y1 - 2009 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11214 SN - 1438-0064 ER - TY - GEN A1 - Weber, Marcus T1 - A Subspace Approach to Molecular Markov State Models via an Infinitesimal Generator N2 - Supercomputers can simulate complex molecular systems. However, there is a very large gap between the fastest oscillations of covalent bonds of a molecule and the time-scale of the dominant processes. In order to extract the dominant time-scales and to identify the dominant processes, a clustering of information is needed. This thesis shows that only the subspace-based Robust Perron Cluster Analysis (PCCA+) can solve this problem correctly by the construction of a Markov State Model. PCCA+ allows for time-extrapolation in molecular kinetics. This thesis shows the difference between molecular dynamics and molecular kinetics. Only in the molecular kinetics framework a definition of transition rates is possible. In this context, the existence of an infinitesimal generator of the dynamical processes is discussed. If the existence is assumed, the Theorem of Gauß can be applied in order to compute transition rates efficiently. Molecular dynamics, however, is not able to provide a suitable statistical basis for the determination of the transition pattern. T3 - ZIB-Report - 09-27 KW - Robuste Perron Cluster Analyse KW - Molekülkinetik KW - Übergangsraten KW - Robust Perron cluster analysis KW - molecular kinetics KW - transition rates Y1 - 2009 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11432 SN - 1438-0064 ER - TY - GEN A1 - Lasser, Caroline A1 - Röblitz, Susanna T1 - Computing expectation values for molecular quantum dynamics N2 - We compute expectation values for the solution of the nuclear Schrödinger equation. The proposed particle method consists of three steps: sampling of the initial Wigner function, classical transport of the sampling points, weighted phase space summation for the final computation of the expectation values. The Egorov theorem guarantees that the algorithm is second order accurate with respect to the semiclassical parameter. We present numerical experiments for a two-dimensional torsional potential with three different sets of initial data and for a six-dimensional Henon-Heiles potential. By construction, the computing times scale linearly with the number of initial sampling points and range between three seconds and one hour. T3 - ZIB-Report - 09-30 KW - time dependent Schrödinger equation KW - molecular quantum dynamics KW - expectation values KW - Wigner functions Y1 - 2009 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11461 SN - 1438-0064 ER - TY - GEN A1 - Schmidt, Heiko A1 - Jimenéz, Carmen T1 - Numerical study of the direct pressure effect of acoustic waves in planar premixed flames N2 - Recently the unsteady response of 1-D premixed flames to acoustic pressure waves for the range of frequencies below and above the inverse of the flame transit time was investigated experimentally by Wangher et al. using OH chemiluminescence. They compared the frequency dependence of the measured response to the prediction of an analytical model proposed by Clavin et al., derived from the standard flame model (one-step Arrhenius kinetics). Discrepancies between the experimental results and the model led to the conclusion that the standard model does not provide an adequate description of the unsteady response of real flames and that it is necessary to investigate more realistic chemical models. Here we follow exactly this suggestion and perform numerical studies of the reponse of lean methane flames using different reaction mechanisms. We find that the global flame response obtained with both detailed chemistry (GRI3.0) and a reduced multi-step model by Peters lies slightly above the predictions of the analytical model, but is close to experimental results. We additionally used an irreversible one-step reaction model which yields good results at least for frequencies close to the inverse flame transit time. T3 - ZIB-Report - 09-31 KW - Druckeffekt KW - Akustik KW - Flammenantwort KW - inkompressibel KW - Direct pressure effect KW - acoustics KW - flame response KW - incompressible Y1 - 2009 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11476 SN - 1438-0064 ER - TY - GEN A1 - Weiser, Martin T1 - Optimization and Identification in Regional Hyperthermia N2 - Regional hyperthermia is a cancer therapy aiming at heating tumors using phased array applicators. This article provides an overview over current mathematical challenges of delivering individually optimal treatments. The focus is on therapy planning and identification of technical as well as physiological quantities from MR thermometry measurements. T3 - ZIB-Report - 08-40 KW - Hyperthermie-Therapieplanung KW - Identifizierung KW - MR-Thermometrie KW - Innere-Punkte-Methoden KW - hyperthermia treatment planning KW - identification KW - MR thermometry KW - interior point methods Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-10916 SN - 1438-0064 ER - TY - GEN A1 - Weiser, Martin A1 - Erdmann, Bodo A1 - Deuflhard, Peter T1 - On Efficiency and Accuracy in Cardioelectric Simulation N2 - Reasons for the failure of adaptive methods to deliver improved efficiency when integrating monodomain models for myocardiac excitation are discussed. Two closely related techniques for reducing the computational complexity of linearly implicit integrators, deliberate sparsing and splitting, are investigated with respect to their impact on computing time and accuracy. T3 - ZIB-Report - 08-41 KW - elektrische Herzanregung KW - Diskretisierungsfehler KW - splitting KW - mass lumping KW - cardioelectric excitation KW - discretization error KW - splitting KW - mass lumping Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-10934 SN - 1438-0064 ER - TY - GEN A1 - Haack, Fiete T1 - Properties of Reduced Reversible Markov Chains N2 - The enormous time lag between fast atomic motion and complex pro- tein folding events makes it almost impossible to compute molecular dy- namics on a high resolution. A common way to tackle this problem is to model the system dynamics as a Markov process. Yet for large molec- ular systems the resulting Markov chains can hardly be handled due to the curse of dimensionality. Coarse graining methods can be used to re- duce the dimension of a Markov chain, but it is still unclear how far the coarse grained Markov chain resembles the original system. In order to answer this question, two different coarse-graining methods were analysed and compared: a classical set-based reduction method and an alternative subspace-based approach, which is based on membership vectors instead of sets. On the basis of a small toy system, it could be shown, that in con- trast to the subset-based approach, the subspace-based reduction method preserves the Markov property as well as the essential dynamics of the original system. T3 - ZIB-Report - 08-42 KW - Markov Chain KW - PCCA+ KW - Conformation dynamics Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-10941 SN - 1438-0064 ER - TY - GEN A1 - Carqué, Gunter A1 - Schmidt, Heiko A1 - Stevens, Bjorn A1 - Klein, Rupert T1 - Plausibility Check of an Asymptotic Column Model for Deep Convective Clouds N2 - By use of asymptotic analysis Carqué et al. [ZIB-Report 08-03] derived an asymptotic column model for deep convective clouds based on the three dimensional compressible flow equations and a bulk microphysics parameterization. In the present study we check the plausibility of the reduced model equations by comparing implications of the model for the scaling of various terms in the governing equations with those extracted from large eddy simulation data based on the computational model UCLA-LES1.1. This code solves an anelastic system of equations with complete droplet based microphysics and LES closures. We observe that the simulation data corroborate the basic assumptions of the asymptotic analysis and the main conclusions implied by the asymptotically reduced model. The code output reflects the scales of space and time: The deep convective clouds show an anisotropic structure where the horizontal scale is considerably narrower than the vertical scale; with a period of about 20 min, from emergence to breakup, the life cycle of one particular deep convective cloud corresponds exactly to the reference time of the reduced model. The characteristic properties of dynamics as predicted by the reduced model are also reflected in the simulation data: The horizontal flow is controlled by the pressure field; the vertical velocity develops freely independent of pressure over the depth of the convective column; the vertical velocity is directly determined by the buoyancy induced by the potential temperature deviation relative to the background stratification. With respect to grid resolution we observe that refining the spatial step size of the equidistant computational grid from 125 m to 62.5 m does not influence the results: Even with the coarser grid the relevant physical phenomena are sufficiently resolved. Somewhat surprisingly, the Coriolis term involving vertical velocity and acting on the horizontal (east-west) velocity component appears at leading order in the asymptotics. Accordingly, we expected to find a nontrivial impact of this Coriolis effect on the horizontal flow velocity components within columns of updrafts. However, switching the term on and off in subsequent simulations did not sizeably affect the results. T3 - ZIB-Report - 08-44 KW - atmospheric moist convection KW - deep convection KW - precipitating clouds KW - asymptotic analysis KW - Large Eddy Simulation Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-10967 SN - 1438-0064 ER - TY - GEN A1 - Weiser, Martin A1 - Röllig, Mathias A1 - Arndt, Ralf A1 - Erdmann, Bodo T1 - Development and test of a numerical model for pulse thermography in civil engineering N2 - Pulse thermography of concrete structures is used in civil engineering for detecting voids, honeycombing and delamination. The physical situation is readily modeled by Fourier's law. Despite the simplicity of the PDE structure, quantitatively realistic numerical 3D simulation faces two major obstacles. First, the short heating pulse induces a thin boundary layer at the heated surface which encapsulates all information and therefore has to be resolved faithfully. Even with adaptive mesh refinement techniques, obtaining useful accuracies requires an unsatisfactorily fine discretization. Second, bulk material parameters and boundary conditions are barely known exactly. We address both issues by a semi-analytic reformulation of the heat transport problem and by parameter identification. Numerical results are compared with measurements of test specimens. T3 - ZIB-Report - 08-45 KW - finite element model KW - KARDOS KW - pulse thermography KW - civil engineering KW - inverse solution Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-10980 SN - 1438-0064 ER - TY - GEN A1 - Schiela, Anton A1 - Günther, Andreas T1 - Interior Point Methods in Function Space for State Constraints - Inexact Newton and Adaptivity N2 - We consider an interior point method in function space for PDE constrained optimal control problems with state constraints. Our emphasis is on the construction and analysis of an algorithm that integrates a Newton path-following method with adaptive grid refinement. This is done in the framework of inexact Newton methods in function space, where the discretization error of each Newton step is controlled by adaptive grid refinement in the innermost loop. This allows to perform most of the required Newton steps on coarse grids, such that the overall computational time is dominated by the last few steps. For this purpose we propose an a-posteriori error estimator for a problem suited norm. T3 - ZIB-Report - 09-01 KW - interior point methods KW - function space KW - adaptivity KW - state constraints Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11007 SN - 1438-0064 ER - TY - GEN A1 - Prüfert, Uwe A1 - Schiela, Anton T1 - The minimization of an L^{\infty}-functional subject to an elliptic PDE and state constraints N2 - We study the optimal control of a maximum-norm objective functional subject to an elliptic-type PDE and pointwise state constraints. The problem is transformed into a problem where the non-differentiable L^{\infty}-norm in the functional will be replaced by a scalar variable and additional state constraints. This problem is solved by barrier methods. We will show the existence and convergence of the central path for a class of barrier functions. Numerical experiments complete the presentation. T3 - ZIB-Report - 08-17 Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-10714 SN - 1438-0064 ER - TY - GEN A1 - Deuflhard, Peter A1 - Dössel, Olaf A1 - Louis, Alfred A1 - Zachow, Stefan T1 - Mehr Mathematik wagen in der Medizin N2 - Der Artikel gibt einen Einblick in das reiche Feld der Zusammenarbeit zwischen Mathematik und Medizin. Beispielhaft werden drei Erfolgsmodelle dargestellt: Medizinische Bildgebung, mathematische Modellierung und Biosignalverarbeitung im Bereich der Dynamik des Herzens sowie mathematische Modellierung und Simulation in der Krebstherapie Hyperthermie und der Mund-Kiefer-Gesichts-Chirurgie. In allen Fällen existiert ein Gleichklang der Interessen von Medizin und Mathematik: Beide Disziplinen wollen die Resultate schnell und zuverlässig. Für die Klinik heißt das, dass notwendige Rechnungen in möglichst kurzer Zeit, und zwar auf dem PC, ablaufen müssen und dass die Resultate so genau und belastbar sein müssen, dass medizinische Entscheidungen darauf aufbauen können. Für die Mathematik folgt daraus, dass höchste Anforderungen an die Effizienz der verwendeten Algorithmen und die darauf aufbauende Software in Numerik und Visualisierung zu stellen sind. Jedes Kapitel endet mit einer Darstellung der Perspektive des jeweiligen Gebietes. Abschließend werden mögliche Handlungsoptionen für Politik und Wirtschaft diskutiert. T3 - ZIB-Report - 08-25 Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-10776 SN - 1438-0064 ER - TY - GEN A1 - Klapproth, Corinna A1 - Deuflhard, Peter A1 - Schiela, Anton T1 - A Perturbation Result for Dynamical Contact Problems N2 - This paper is intended to be a first step towards the continuous dependence of dynamical contact problems on the initial data as well as the uniqueness of a solution. Moreover, it provides the basis for a proof of the convergence of popular time integration schemes as the Newmark method. We study a frictionless dynamical contact problem between both linearly elastic and viscoelastic bodies which is formulated via the Signorini contact conditions. For viscoelastic materials fulfilling the Kelvin-Voigt constitutive law, we find a characterization of the class of problems which satisfy a perturbation result in a non-trivial mix of norms in function space. This characterization is given in the form of a stability condition on the contact stresses at the contact boundaries. Furthermore, we present perturbation results for two well-established approximations of the classical Signorini condition: The Signorini condition formulated in velocities and the model of normal compliance, both satisfying even a sharper version of our stability condition. T3 - ZIB-Report - 08-27 KW - Dynamical contact problems KW - stability KW - (visco-)elasticity KW - Signorini condition KW - Newmark method Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-10793 SN - 1438-0064 ER - TY - GEN A1 - Weber, Marcus T1 - An efficient analysis of rare events in canonical ensemble dynamics N2 - For an analysis of a molecular system from a computational statistical thermodynamics point of view, extensive molecular dynamics simulations are very inefficient. During this procedure, at lot of redundant data is generated. Whereas the algorithms spend most of the computing time for a sampling of configurations within the basins of the potential energy landscape of the molecular system, the important information about the long-time behaviour of the molecules is given by transition regions and barriers between the basins, which are sampled rarely only. Thinking of molecular dynamics trajectories, researchers try to figure out which kind of dynamical model is suitable for an efficient simulation. This article suggests to change the point of view from extensive simulation of molecular dynamics trajectories to more efficient sampling strategies of the conformation dynamics approach. T3 - ZIB-Report - 08-36 KW - Moleküldynamik KW - Kanonische Gesamtheit KW - Metastabilität KW - molecular dynamics KW - canonical ensemble KW - metastability Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-10871 SN - 1438-0064 ER - TY - GEN A1 - Schiela, Anton A1 - Wollner, Winnifried T1 - Barrier Methods for Optimal Control Problems with Convex Nonlinear Gradient Constraints N2 - In this paper we are concerned with the application of interior point methods in function space to gradient constrained optimal control problems, governed by partial differential equations. We will derive existence of solutions together with first order optimality conditions. Afterwards we show continuity of the central path, together with convergence rates depending on the interior point parameter. T3 - ZIB-Report - 08-47 KW - interior point method KW - necessary optimality conditions KW - convergence of the central path KW - gradient constrained optimization Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11014 SN - 1438-0064 ER - TY - GEN A1 - Mellado, Juan Pedro A1 - Stevens, Bjorn A1 - Schmidt, Heiko A1 - Peters, Norbert T1 - Buoyancy-reversal in cloud-top mixing layers N2 - A theoretical and numerical small-scale study of the evaporative cooling phenomenon that might appear in the stratocumulus-topped boundary layers is presented. An ideal configuration of a cloud-top mixing layer is considered as defined by two nonturbulent horizontal layers, stably stratified and with buoyancy reversal within a certain range of mixture fractions due to the evaporative cooling. Linear stability analysis of the shear-free configuration is employed to provide a new interpretation of the buoyancy reversal parameter, namely, in terms of a time-scale ratio between the unstable and the stable modes of the system. An incompressible high-order numerical algorithm to perform direct numerical simulation of the configuration is described and two-dimensional simulations of single-mode perturbations are discussed. These simulations confirm the role of the different parameters identified in the linear stability analysis and show that convoluted flow patterns can be generated by the evaporative cooling even for the low levels of buoyancy reversal found in stratocumulus clouds. They also show that there is no enhancement of entrainment of upper layer fluid in the shear-free configuration, and mixing enhancement by the evaporative cooling is restricted to the lower layer. T3 - ZIB-Report - 08-50 KW - Stratocumulus Wolken KW - freie Konvektion KW - turbulente Strömungen KW - stratocumulus KW - free convection KW - free turbulent flows Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11046 SN - 1438-0064 ER - TY - THES A1 - Ruprecht, Daniel T1 - Analysis of a multi-scale asymptotic model for internal gravity waves in a moist atmosphere N2 - The thesis presents the analysis of a reduced model for modulation of internal gravity waves by deep convective clouds. The starting point for the derivation are conservation laws for mass, momentum and energy coupled with a bulk micro-physics model describing the evolution of mixing ratios of water vapor, cloud water and rain water. A reduced model for the identified scales of the regime is derived, using multi-scale asymptotics. The closure of the model employs conditional averaging over the horizontal scale of the convective clouds. The resulting reduced model is an extension of the anelastic equations, linearized around a constant background state, which are well-known from meteorology. The closure of the model is achieved purely by analytical means and involves no additional physically motivated assumptions. The essential new parameter arising from the coupling to a micro-physics model is the area fraction of saturated regions on the horizontal scale of the convective clouds. It turns out that this parameter is constant on the employed short timescale. Hence the clouds constitute a constant background, modulating the characteristics of propagation of internal waves. The model is then investigated by analytical as well as numerical means. Important results are, among others, that in the model moisture (i) inhibits propagation of internal waves by reducing the modulus of the group velocity, (ii) reduces the angle between the propagation direction of a wave-packet and the horizontal, (iii) causes critical layers and (iv) introduces a maximum horizontal wavelength beyond which waves are no longer propagating but become evanescent. The investigated examples of orographically generated gravity waves also feature a significant reduction of vertical momentum flux by moisture. The model is extended by assuming systematically small under-saturation, that is saturation at leading order. The closure is similar to the original case but requires additional assumptions. The saturated area fraction in the obtained model is no longer constant but now depends nonlinearly on vertical displacement and thus on vertical velocity. KW - gravity-waves KW - moisture KW - multiscale KW - reduced model KW - deep convective clouds Y1 - 2010 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:188-fudissthesis000000018355-1 ER - TY - GEN A1 - Götschel, Sebastian A1 - Weiser, Martin A1 - Schiela, Anton T1 - Solving Optimal Control Problems with the Kaskade 7 Finite Element Toolbox N2 - This paper presents concepts and implementation of the finite element toolbox Kaskade 7, a flexible C++ code for solving elliptic and parabolic PDE systems. Issues such as problem formulation, assembly and adaptivity are discussed at the example of optimal control problems. Trajectory compression for parabolic optimization problems is considered as a case study. T3 - ZIB-Report - 10-25 KW - partial differential equations KW - optimal control KW - finite elements KW - generic programming KW - adaptive methods Y1 - 2010 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11909 SN - 1438-0064 ER - TY - GEN A1 - Schiela, Anton A1 - Wachsmuth, Daniel T1 - Convergence Analysis of Smoothing Methods for Optimal Control of Stationary Variational Inequalities N2 - In the article an optimal control problem subject to a stationary variational inequality is investigated. The optimal control problem is complemented with pointwise control constraints. The convergence of a smoothing scheme is analyzed. There, the variational inequality is replaced by a semilinear elliptic equation. It is shown that solutions of the regularized optimal control problem converge to solutions of the original one. Passing to the limit in the optimality system of the regularized problem allows to prove C-stationarity of local solutions of the original problem. Moreover, convergence rates with respect to the regularization parameter for the error in the control are obtained. These rates coincide with rates obtained by numerical experiments, which are included in the paper. T3 - ZIB-Report - 11-23 KW - Variational inequalities KW - optimal control KW - control constraints KW - path-following KW - C-stationarity Y1 - 2011 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-13125 ER - TY - GEN A1 - Fackeldey, Konstantin A1 - Röblitz, Susanna A1 - Scharkoi, Olga A1 - Weber, Marcus T1 - Soft Versus Hard Metastable Conformations in Molecular Simulations N2 - Particle methods have become indispensible in conformation dynamics to compute transition rates in protein folding, binding processes and molecular design, to mention a few. Conformation dynamics requires at a decomposition of a molecule's position space into metastable conformations. In this paper, we show how this decomposition can be obtained via the design of either ``soft'' or ``hard'' molecular conformations. We show, that the soft approach results in a larger metastabilitiy of the decomposition and is thus more advantegous. This is illustrated by a simulation of Alanine Dipeptide. T3 - ZIB-Report - 11-27 KW - Proteins, Conformation Space, Meshfree Methods Y1 - 2011 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-13189 ER - TY - THES A1 - Weber, Marcus T1 - A Subspace Approach to Molecular Markov State Models via a New Infinitesimal Generator N2 - Supercomputers can simulate complex molecular systems. However, there is a very large gap between the fastest oscillations of covalent bonds of a molecule and the time-scale of the dominant processes. In order to extract the dominant time-scales and to identify the dominant processes, a clustering of information is needed. This thesis shows that only the subspace-based Robust Perron Cluster Analysis (PCCA+) can solve this problem correctly by the construction of a Markov State Model. PCCA+ allows for time-extrapolation in molecular kinetics. This thesis shows the difference between molecular dynamics and molecular kinetics. Only in the molecular kinetics framework a definition of transition rates is possible. In this context, the existence of an infinitesimal generator of the dynamical processes is discussed. If the existence is assumed, the Theorem of Gauß can be applied in order to compute transition rates efficiently. Molecular dynamics, however, is not able to provide a suitable statistical basis for the determination of the transition pattern. KW - Conformation Dynamics KW - Molecular Kinetics KW - Transition Rates KW - Markov State Models Y1 - 2011 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-14025 ER - TY - GEN A1 - Schiela, Anton A1 - Hintermüller, Michael T1 - On the Length of the Primal-Dual Path in Moreau-Yosida-based Path-following for State Constrained Optimal Control: Analysis and Numerics N2 - We derive a-priori estimates on the length of the primal-dual path that results from a Moreau-Yosida approximation of the feasible set for state constrained optimal control problems. These bounds depend on the regularity of the state and the dimension of the problem. Comparison with numerical results indicates that these bounds are sharp and are attained for the case of a single active point. T3 - ZIB-Report - 11-37 KW - state constraints KW - PDE constrained optimization KW - path-following Y1 - 2011 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-14071 SN - 1438-0064 ER - TY - GEN A1 - Weiser, Martin A1 - Götschel, Sebastian T1 - State Trajectory Compression for Optimal Control with Parabolic PDEs N2 - In optimal control problems with nonlinear time-dependent 3D PDEs, full 4D discretizations are usually prohibitive due to the storage requirement. For this reason gradient and quasi-Newton methods working on the reduced functional are often employed. The computation of the reduced gradient requires one solve of the state equation forward in time, and one backward solve of the adjoint equation. The state enters into the adjoint equation, again requiring the storage of a full 4D data set. We propose a lossy compression algorithm using an inexact but cheap predictor for the state data, with additional entropy coding of prediction errors. As the data is used inside a discretized, iterative algorithm, lossy coding maintaining an error bound is sufficient. T3 - ZIB-Report - 10-05 KW - optimal control KW - adjoint gradient computation KW - trajectory storage Y1 - 2010 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11676 SN - 1438-0064 ER - TY - GEN A1 - Boer, H. Marike T. A1 - Stötzel, Claudia A1 - Röblitz, Susanna A1 - Deuflhard, Peter A1 - Veerkamp, Roel F. A1 - Woelders, Henri T1 - A simple mathematical model of the bovine estrous cycle: follicle development and endocrine interactions N2 - Bovine fertility is the subject of extensive research in animal sciences, especially because fertility of dairy cows has declined during the last decades. The regulation of estrus is controlled by the complex interplay of various organs and hormones. Mathematical modeling of the bovine estrous cycle could help in understanding the dynamics of this complex biological system. In this paper we present a mathematical model of the bovine estrous cycle that includes the processes of follicle and corpus luteum development and the key hormones that interact to control these processes. Focus in this paper is on development of the model, but also some simulation results are presented, showing that a set of equations and parameters is obtained that describes the system consistent with empirical knowledge. Even though the majority of the mechanisms that are included are only known qualitatively as stimulatory or inhibitory effects, the model surprisingly well features quantitative observations made in reality. This model of the bovine estrous cycle could be used as a basis for more elaborate models with the ability to study effects of external manipulations and genetic differences. T3 - ZIB-Report - 10-06 KW - cow KW - reproduction KW - hormone patterns KW - ordinary differential equations KW - systems biology Y1 - 2010 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11682 SN - 1438-0064 ER - TY - GEN A1 - Dierkes, Thomas A1 - Wade, Moritz A1 - Nowak, Ulrich A1 - Röblitz, Susanna T1 - BioPARKIN - Biology-related Parameter Identification in Large Kinetic Networks N2 - Modelling, parameter identification, and simulation play an important rôle in Systems Biology. In recent years, various software packages have been established for scientific use in both licencing types, open source as well as commercial. Many of these codes are based on inefficient and mathematically outdated algorithms. By introducing the package BioPARKIN recently developed at ZIB, we want to improve this situation significantly. The development of the software BioPARKIN involves long standing mathematical ideas that, however, have not yet entered the field of Systems Biology, as well as new ideas and tools that are particularly important for the analysis of the dynamics of biological networks. BioPARKIN originates from the package PARKIN, written by P.Deuflhard and U.Nowak, that has been applied successfully for parameter identification in physical chemistry for many years. T3 - ZIB-Report - 11-15 KW - systems biology KW - ordinary differential equations KW - sensitivity analysis KW - parameter identification KW - affine invariant Gauss-Newton algorithm KW - numerical library KW - graphical user interface Y1 - 2011 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-12706 ER - TY - GEN A1 - Röblitz, Susanna A1 - Stötzel, Claudia A1 - Deuflhard, Peter A1 - Jones, Hannah M. A1 - Azulay, David-Olivier A1 - van der Graaf, Piet A1 - Martin, Steven T1 - A mathematical model of the human menstrual cycle for the administration of GnRH analogues N2 - This study presents a differential equation model for the feedback mechanisms between Gonadotropin-releasing Hormone (GnRH), Follicle-Stimulating Hormone (FSH), Luteinizing Hormone (LH), development of follicles and corpus luteum, and the production of estradiol (E2), progesterone (P4), inhibin A (IhA), and inhibin B (IhB) during the female menstrual cycle. In contrast to other models, this model does not involve delay differential equations and is based on deterministic modelling of the GnRH pulse pattern, which allows for faster simulation times and efficient parameter identification. These steps were essential to tackle the task of developing a mathematical model for the administration of GnRH analogues. The focus of this paper is on model development for GnRH receptor binding and the integration of a pharmacokinetic/pharmacodynamic model for the GnRH agonist Nafarelin and the GnRH antagonist Cetrorelix into the menstrual cycle model. The final mathematical model describes the hormone profiles (LH, FSH, P4, E2) throughout the menstrual cycle in 12 healthy women. Moreover, it correctly predicts the changes in the cycle following single and multiple dose administration of Nafarelin or Cetrorelix at different stages in the cycle. T3 - ZIB-Report - 11-16 KW - human menstrual cycle KW - systems biology KW - mathematical modelling KW - GnRH analogues KW - differential equations Y1 - 2011 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-12732 ER - TY - GEN A1 - Stötzel, Claudia A1 - Plöntzke, Julia A1 - Röblitz, Susanna T1 - Advances in modelling of the bovine estrous cycle: Administration of PGF2alpha N2 - Our model of the bovine estrous cycle is a set of ordinary differential equations which generates hormone profiles of successive estrous cycles with several follicular waves per cycle. It describes the growth and decay of the follicles and the corpus luteum, as well as the change of the key substances over time. In this work we describe recent improvements of this model, including the introduction of new components, and elimination of time delays. We validate our model by showing that the simulations agree with observations from synchronization studies and with measured progesterone data after a single dose administration of synthetic prostaglandin F2alpha. T3 - ZIB-Report - 11-17 KW - cow KW - reproduction KW - hormone patterns KW - differential equations KW - systems biology Y1 - 2011 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-12740 ER - TY - GEN A1 - Günther, Andreas A1 - Lamecker, Hans A1 - Weiser, Martin T1 - Direct LDDMM of Discrete Currents with Adaptive Finite Elements N2 - We consider Large Deformation Diffeomorphic Metric Mapping of general $m$-currents. After stating an optimization algorithm in the function space of admissable morph generating velocity fields, two innovative aspects in this framework are presented and numerically investigated: First, we spatially discretize the velocity field with conforming adaptive finite elements and discuss advantages of this new approach. Second, we directly compute the temporal evolution of discrete $m$-current attributes. T3 - ZIB-Report - 11-22 KW - Large Deformation KW - Diffeomorphic Registration KW - Matching KW - Currents KW - Adaptive Finite Elements Y1 - 2011 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-13090 ER - TY - GEN A1 - Meyer, Christian A1 - Panizzi, Lucia A1 - Schiela, Anton T1 - Uniqueness criteria for solutions of the adjoint equation in state-constrained optimal control N2 - The paper considers linear elliptic equations with regular Borel measures as inhomogeneity. Such equations frequently appear in state-constrained optimal control problems. By a counter-example of Serrin, it is known that, in the presence of non-smooth data, a standard weak formulation does not ensure uniqueness for such equations. Therefore several notions of solution have been developed that guarantee uniqueness. In this note, we compare different definitions of solutions, namely the ones of Stampacchia and the two notions of solutions of Casas and Alibert-Raymond, and show that they are the same. As side results, we reformulate the solution in the sense of Stampacchia, and prove the existence and uniqueness of solutions in in case of mixed boundary conditions. T3 - ZIB-Report - 10-28 KW - optimal control KW - elliptic partial differential equations KW - state constraints KW - measure right hand sides Y1 - 2010 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11933 ER - TY - GEN A1 - Berg, Mascha A1 - Ritter, Julia A1 - Röblitz, Susanna T1 - A mathematical model for the Zinc response of young pigs N2 - In a number of studies the feeding of pharmacological levels of Zinc to weanling pigs was examined. The publications agree in the fact that the achieved performance increase of the pigs depends on the amount of Zinc as well as on the type of mineral complex used as feed additive. The results are based on measurements of the animals' body weight and the Zinc concentration in the blood within the first two weeks of the experiments. From the measurement values, conclusions are drawn about adequate Zinc doses, but a deeper understanding of the physiological processes that influence the Zinc distribution in the organism is still missing. In this report, we want to close this gap by using a systems biology approach. We present a small mathematical model that describes the organic feedback mechanisms with the help of differential equations. The simulation results not only match with measurement values, but also give insight into non-measurable processes, which will hopefully contribute to animal experiments being organized in a more selective and reduced way. T3 - ZIB-Report - 11-05 KW - systems biology KW - ordinary differential equations KW - feeding study KW - weanling pigs Y1 - 2011 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-12191 ER - TY - GEN A1 - Deuflhard, Peter A1 - Schiela, Anton A1 - Weiser, Martin T1 - Mathematical Cancer Therapy Planning in Deep Regional Hyperthermia N2 - This paper surveys the required mathematics for a typical challenging problem from computational medicine, the cancer therapy planning in deep regional hyperthermia. In the course of many years of close cooperation with clinics, the medical problem gave rise to quite a number of subtle mathematical problems, part of which had been unsolved when the common project started. Efficiency of numerical algorithms, i.e. computational speed and monitored reliability, play a decisive role for the medical treatment. Off-the-shelf software had turned out to be not sufficient to meet the requirements of medicine. Rather, new mathematical theory as well as new numerical algorithms had to be developed. In order to make our algorithms useful in the clinical environment, new visualization software, a virtual lab, including 3D geometry processing of individual virtual patients had to be designed and implemented. Moreover, before the problems could be attacked by numerical algorithms, careful mathematical modelling had to be done. Finally, parameter identification and constrained optimization for the PDEs had to be newly analyzed and realized over the individual patient's geometry. Our new techniques had an impact on the specificity of the individual patients' treatment and on the construction of an improved hyperthermia applicator. T3 - ZIB-Report - 11-39 KW - hyperthermia therapy planning KW - mathematical modelling KW - Maxwell equations KW - state constraints KW - parameter identification Y1 - 2011 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-14204 SN - 1438-0064 ER - TY - GEN A1 - Sander, Oliver A1 - Klapproth, Corinna A1 - Youett, Jonathan A1 - Kornhuber, Ralf A1 - Deuflhard, Peter T1 - Towards an Efficient Numerical Simulation of Complex 3D Knee Joint Motion N2 - We present a time-dependent finite element model of the human knee joint of full 3D geometric complexity. Its efficient numerical simulation requires advanced numerical algorithms that have been developed just recently. Up to now, the model comprises bones, cartilage, and the major ligaments (patella and menisci are still missing). Bones (femur, tibia, and fibula) are modelled by linear elastic materials, cartilage by viscoelastic materials, ligaments by one-dimensional so-called Cosserat rods. In order to capture the dynamical contact problems correctly, we solve the full PDEs of elasticity in the presence of strict contact inequalities. For the total spatio-temporal discretization we apply a method of layers approach (first time, then space discretization). For the time discretization of the elastic and viscoelastic parts, we apply a new contact-stabilized Newmark method, while for the Cosserat rods we choose an energy-momentum method. For the space discretization, we use linear finite elements for the elastic and viscoelastic parts and novel geodesic finite elements for the Cosserat rods. The coupled system is solved by a Dirichlet-Neumann method, and the arising large algebraic systems are solved by a recent fast multigrid solver, the truncated non-smooth Newton multigrid method. T3 - ZIB-Report - 12-06 KW - contact problem KW - contact-stabilized Newmark method KW - domain decomposition KW - energy-momentum method KW - geodesic finite elements KW - knee model Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-14512 SN - 1438-0064 ER - TY - GEN A1 - Lubkoll, Lars A1 - Schiela, Anton A1 - Weiser, Martin T1 - An optimal control problem in polyconvex hyperelasticity N2 - We consider a shape implant design problem that arises in the context of facial surgery. We introduce a reformulation as an optimal control problem, where the control acts as a boundary force. The state is modelled as a minimizer of a polyconvex hyperelastic energy functional. We show existence of optimal solutions and derive - on a formal level - first order optimality conditions. Finally, preliminary numerical results are presented. T3 - ZIB-Report - 12-08 KW - polyconvex elasticity KW - implant design KW - optimal control Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-14745 SN - 1438-0064 ER - TY - GEN A1 - Weber, Marcus T1 - The funnel trap paradox N2 - In this article, an illustrative example is given for the coarse-graining of a Markov process which leads to a shift in the statistical weights of a two-states-system. The example is based on a 2D-funnel trap. The funnel trap is constructed in such a way, that the area inside and outside of the trap is identical. However, observing the flight of the insect as a Markov process, the probability for being “in the trap” is higher. This example can be transferred to several kinds of processes (like receptor-ligandbinding processes in chemistry) and describes the influence of “re-entering events”. T3 - ZIB-Report - 12-12 KW - Markov process KW - Robust Perron Cluster Analysis KW - Conformation Dynamics Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-14765 SN - 1438-0064 ER - TY - GEN A1 - Klapproth, Corinna A1 - Schiela, Anton A1 - Deuflhard, Peter T1 - Adaptive Timestep Control for the Contact-Stabilized Newmark Method N2 - The aim of this paper is to devise an adaptive timestep control in the contact--stabilized Newmark method (CONTACX) for dynamical contact problems between two viscoelastic bodies in the framework of Signorini's condition. In order to construct a comparative scheme of higher order accuracy, we extend extrapolation techniques. This approach demands a subtle theoretical investigation of an asymptotic error expansion of the contact--stabilized Newmark scheme. On the basis of theoretical insight and numerical observations, we suggest an error estimator and a timestep selection which also cover the presence of contact. Finally, we give a numerical example. T3 - ZIB-Report - 10-09 KW - dynamical contact problems KW - contact--stabilized Newmark method KW - extrapolation methods KW - adaptivity KW - timestep control Y1 - 2010 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11714 SN - 1438-0064 ER - TY - GEN A1 - Fackeldey, Konstantin T1 - Coupling Meshbased and Meshfree Methods by a Transfer Operator Approach N2 - In contrast to the well known meshbased methods like the finite element method, meshfree methods do not rely on a mesh. However besides their great applicability, meshfree methods are rather time consuming. Thus, it seems favorable to combine both methods, by using meshfree methods only in a small part of the domain, where a mesh is disadvantageous, and a meshbased method for the rest of the domain. We motivate, that this coupling between the two simulation techniques can be considered as saddle point problem and show the stability of this coupling. Thereby a novel transfer operator is introduced, which interacts in the transition zone, where both methods coexist. T3 - ZIB-Report - 10-12 KW - Gitterlose Methoden KW - inf-sup-Bedingung KW - Kopplung KW - meshfree KW - meshbased KW - inf-sup-Condition Y1 - 2010 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11755 SN - 1438-0064 ER - TY - GEN A1 - Schädle, Achim A1 - Zschiedrich, Lin T1 - Additive Schwarz method for scattering problems using the PML method at interfaces N2 - Scattering problems in integrated optics can be modeled in simple cases by the Helmholtz equation. The computational domain is truncated by a non-reflecting boundary condition. We investigate Schwarz algorithms with a sort of DtN operator, realized by the PML-method, at the interfaces of the sub-domains as an iterative solver. T3 - ZIB-Report - 05-27 KW - perfectly matched layer KW - Helmholtz equation KW - scattering KW - domain decomposition Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8613 ER - TY - GEN A1 - Franzone, Piero Colli A1 - Deuflhard, Peter A1 - Erdmann, Bodo A1 - Lang, Jens A1 - Pavarino, Luca Franco T1 - Adaptivity in Space and Time for Reaction-Diffusion Systems in Electrocardiology N2 - Adaptive numerical methods in space and time are introduced and studied for multiscale cardiac reaction-diffusion models in three dimensions. The evolution of a complete heartbeat, from the excitation to the recovery phase, is simulated with both the anisotropic Bidomain and Monodomain models, coupled with either a variant of the simple FitzHugh-Nagumo model or the more complex phase-I Luo-Rudy ionic model. The simulations are performed with the {\sc kardos} library, that employs adaptive finite elements in space and adaptive linearly implicit methods in time. The numerical results show that this adaptive method successfully solves these complex cardiac reaction-diffusion models on three-dimensional domains of moderate sizes. By automatically adapting the spatial meshes and time steps to the proper scales in each phase of the heartbeat, the method accurately resolves the evolution of the intra- and extra-cellular potentials, gating variables and ion concentrations during the excitation, plateau and recovery phases. T3 - ZIB-Report - 05-30 KW - reaction-diffusion equations KW - cardiac Bidomain and Monodomain models KW - adaptive finite elements KW - adaptive time integration Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8649 ER - TY - THES A1 - Friese, Tilmann T1 - Eine Mehrgitter-Methode zur Lösung des Eigenwertproblems der komplexen Helmholtzgleichung Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:188-1999000133 ER - TY - GEN A1 - Prüfert, Uwe A1 - Tröltzsch, Fredi A1 - Weiser, Martin T1 - The convergence of an interior point method for an elliptic control problem with mixed control-state constraints N2 - The paper addresses primal interior point method for state constrained PDE optimal control problems. By a Lavrentiev regularization, the state constraint is transformed to a mixed control-state constraint with bounded Lagrange multiplier. Existence and convergence of the central path are established, and linear convergence of a short-step pathfollowing method is shown. The behaviour of the regularizations are demonstrated by numerical examples. T3 - ZIB-Report - 04-47 KW - interior point methods in function space KW - optimal control KW - mixed control-state constraints KW - Lavrentiev regularization Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8223 ER - TY - GEN A1 - Lopez-Fernandez, Maria A1 - Lubich, Christian A1 - Schädle, Achim T1 - Adaptive, Fast and Oblivious Convolution in Evolution Equations with Memory N2 - To approximate convolutions which occur in evolution equations with memory terms, a variable-stepsize algorithm is presented for which advancing $N$ steps requires only $O(N\log N)$ operations and $O(\log N)$ active memory, in place of $O(N^2)$ operations and $O(N)$ memory for a direct implementation. A basic feature of the fast algorithm is the reduction, via contour integral representations, to differential equations which are solved numerically with adaptive step sizes. Rather than the kernel itself, its Laplace transform is used in the algorithm. The algorithm is illustrated on three examples: a blow-up example originating from a Schrödinger equation with concentrated nonlinearity, chemical reactions with inhibited diffusion, and viscoelasticity with a fractional order constitutive law. T3 - ZIB-Report - 06-45 KW - convolution quadrature KW - adaptivity KW - Volterra integral equations KW - numerical inverse Laplace transform KW - anomalous diffusion KW - fractional order visco Y1 - 2006 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-9399 ER - TY - GEN A1 - Weiser, Martin A1 - Deuflhard, Peter A1 - Erdmann, Bodo T1 - Affine conjugate adaptive Newton methods for nonlinear elastomechanics N2 - The paper extends affine conjugate Newton methods from convex to nonconvex minimization, with particular emphasis on PDE problems originating from compressible hyperelasticity. Based on well-known schemes from finite dimensional nonlinear optimization, three different algorithmic variants are worked out in a function space setting, which permits an adaptive multilevel finite element implementation. These algorithms are tested on two well-known 3D test problems and a real-life example from surgical operation planning. T3 - ZIB-Report - 04-01 KW - affine conjugate Newton methods KW - nonconvex minimization KW - nonlinear elastomechnics KW - cranio-maxillofacial surgery KW - soft tissue simulation KW - multilev Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-7768 ER - TY - GEN A1 - Weiser, Martin A1 - Gänzler, Tobias A1 - Schiela, Anton T1 - A Control Reduced Primal Interior Point Method for PDE Constrained Optimization N2 - A primal interior point method for control constrained optimal control problems with PDE constraints is considered. Pointwise elimination of the control leads to a homotopy in the remaining state and dual variables, which is addressed by a short step pathfollowing method. The algorithm is applied to the continuous, infinite dimensional problem, where discretization is performed only in the innermost loop when solving linear equations. The a priori elimination of the least regular control permits to obtain the required accuracy with comparable coarse meshes. Convergence of the method and discretization errors are studied, and the method is illustrated at two numerical examples. T3 - ZIB-Report - 04-38 KW - interior point methods in function space KW - optimal control KW - finite elements KW - discretization error Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8138 ER - TY - GEN A1 - Deuflhard, Peter A1 - Huisinga, Wilhelm A1 - Jahnke, Tobias A1 - Wulkow, Michael T1 - Adaptive Discrete Galerkin Methods Applied to the Chemical Master Equation N2 - \begin{abstract} In systems biology, the stochastic description of biochemical reaction kinetics is increasingly being employed to model gene regulatory networks and signalling pathways. Mathematically speaking, such models require the numerical solution of the underlying evolution equat ion, also known as the chemical master equation (CME). Up to now, the CME has almost exclusively been treated by Monte-Carlo techniques, the most prominent of which is the simulation algorithm suggest ed by Gillespie in 1976. Since this algorithm requires an update for each single reaction event, realizations can be computationally very costly. As an alternative, we here propose a novel approach, which focuses on the discrete partial differential equation (PDE) structure of the CME and thus allows to adopt ideas from adaptive discrete Galerkin methods (as designed by two of the present authors in 1989), which have proven to be highly efficient in the mathematical modelling of polyreaction kinetics. Among the two different options of discretizing the CME as a discrete PDE, the method of lines approach (first space, then time) and the Rothe method (first time, then space), we select the latter one for clear theoretical and algorithmic reasons. First numeric al experiments at a challenging model problem illustrate the promising features of the proposed method and, at the same time, indicate lines of necessary further research. \end{abstract} T3 - ZIB-Report - 07-04 KW - adaptive discrete Galerkin methods KW - adaptive Rothe method KW - discrete Chebyshev polynomials KW - stochastic reactions kinetics KW - chemical master equation Y1 - 2007 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-9471 ER -