Preprints aus dem Institut für Mathematik
http://edoc.hu-berlin.de/18452/68
2024-04-20T03:59:35ZOperator-Splitting Methods Respecting Eigenvalue Problems For Shallow Shelf Equations With Basal Drag
http://edoc.hu-berlin.de/18452/3493
Operator-Splitting Methods Respecting Eigenvalue Problems For Shallow Shelf Equations With Basal Drag
Calov, Reinhard; Geiser, Jürgen
http://dx.doi.org/10.18452/2841
We discuss different numerical methods for solving the shallow shelf equations with basal drag. The coupled equations are decomposed into operators for membranes stresses, basal shear stress and driving stress. Applying reasonable parameter values, we demonstrate that the operator of the membrane stresses is much stiffer than operator of the basal shear stress. Therefore, we propose a new splitting method, which alternates between the iteration on the membrane-stress operator and the basal-shear operator, with a stronger iteration on the operator of the membrane stress. We show that this splitting improves the computational performance of the numerical method, although the choice of the (standard) method to solve for all operators in one step speeds up the scheme too. (Based on the delicate and coupled equation we propose a new decomposition method to decouple into simpler solvable sub-equations. After a number of approximations we consider the error of the method and proposed a choice of the operators.)
2013-09-12T00:00:00ZReview of AdS/CFT Integrability, Chapter I.2
http://edoc.hu-berlin.de/18452/3492
Review of AdS/CFT Integrability, Chapter I.2
Sieg, Christoph
http://dx.doi.org/10.18452/2840
We review the constructions and tests of the dilatation operator and of the spectrum of composite operators in the flavour SU(2) subsector of N=4 SYM in the planar limit by explicit Feynman graph calculations with emphasis on analyses beyond one loop. From four loops on, the dilatation operator determines the spectrum only in the asymptotic regime, i.e. to a loop order which is strictly smaller than the number of elementary fields of the composite operators. We review also the calculations which take a first step beyond this limitation by including the leading wrapping corrections.
2011-10-20T00:00:00ZReview of AdS/CFT Integrability
http://edoc.hu-berlin.de/18452/3491
Review of AdS/CFT Integrability
Beisert, Niklas; Ahn, Changrim; Alday, Luis F.; Bajnok, Zoltán; Drummond, James M.; Freyhult, Lisa; Gromov, Nikolay; Janik, Romuald A.; Kazakov, Vladimir; Klose, Thomas; Korchemsky, Gregory P.; Kristjansen, Charlotte
http://dx.doi.org/10.18452/2839
This is the introductory chapter of a review collection on integrability in the context of the AdS/CFT correspondence. In the collection we present an overview of the achievements and the status of this subject as of the year 2010.
2011-10-20T00:00:00ZTwist operators in N=4 beta-deformed theory
http://edoc.hu-berlin.de/18452/3490
Twist operators in N=4 beta-deformed theory
Leeuw, Marius de; Lukowski, Tomasz
http://dx.doi.org/10.18452/2838
In this paper we derive both the leading order finite size corrections for twist-2 and twist-3 operators and the next-to-leading order finite-size correction for twist-2 operators in beta-deformed SYM theory. The obtained results respect the principle of maximum transcendentality as well as reciprocity. We also find that both wrapping corrections go to zero in the large spin limit. Moreover, for twist-2 operators we studied the pole structure and compared it against leading BFKL predictions.
2011-09-27T00:00:00ZReview of AdS/CFT Integrability, Chapter III.1
http://edoc.hu-berlin.de/18452/3489
Review of AdS/CFT Integrability, Chapter III.1
Staudacher, Matthias
http://dx.doi.org/10.18452/2837
The one-dimensional Heisenberg XXX spin chain appears in a special limit of the AdS/CFT integrable system. We review various ways of proving its integrability, and discuss the associated methods of solution. In particular, we outline the coordinate and the algebraic Bethe ansatz, giving reference to literature suitable for learning these techniques. Finally we speculate which of the methods might lift to the exact solution of the AdS/CFT system, and sketch a promising method for constructing the Baxter Q-operator of the XXX chain. It allows to find the spectrum of the model using certain algebraic techniques, while entirely avoiding Bethe's ansatz.
2011-09-27T00:00:00ZFrom convergence principles to stability and optimality
http://edoc.hu-berlin.de/18452/3488
From convergence principles to stability and optimality
Klatte, Diethard; Kruger, Alexander; Kummer, Bernd
http://dx.doi.org/10.18452/2836
We show in a rather general setting that Hoelder and Lipschitz stability properties of solutions to variational problems can be characterized by convergence of more or less abstract iteration schemes. Depending on the principle of convergence, new and intrinsic stability conditions can be derived. Our most abstract models are (multi-) functions on complete metric spaces. The relevance of this approach is illustrated by deriving both classical and new results on existence and optimality conditions, stability of feasible and solution sets and convergence behavior of solution procedures.
2011-09-27T00:00:00ZIterative Operator Splitting Method for Coupled Problems
http://edoc.hu-berlin.de/18452/3487
Iterative Operator Splitting Method for Coupled Problems
Geiser, Jürgen; Küttel, Felix
http://dx.doi.org/10.18452/2835
In this article a new approach is considered for implementing operator splitting methods for transport problems, influenced by eletric fields. Our motivation came to model PE-CVD (plasma-enhanced chemical vapor deposition) processes, means the flow of species to a gas-phase, which are influenced by an electric field. We consider a convection-diffusion equation and a Lorence force in the electrostatic case. The iterative splitting schemes is given as an embedded coupling method and we apply such a scheme as a fast solver. The decomposition analysis is discussed for the nonlinear case. Numerical experiments are given with respect to explicit Adam-Bashforth schemes. We discuss the convergence behavior in time and space for the iterative schemes.
2011-09-27T00:00:00ZOperator-Splitting Methods Respecting Eigenvalue Problems For Shallow Shelf Equations With Basal Drag
http://edoc.hu-berlin.de/18452/3486
Operator-Splitting Methods Respecting Eigenvalue Problems For Shallow Shelf Equations With Basal Drag
Geiser, Jürgen; Calov, Reinhard; Recknagel, Thomas
http://dx.doi.org/10.18452/2834
We discuss different numerical methods for solving the shallow shelf equations with basal drag. The coupled equations are decomposed into operators for membranes stresses, basal shear stress and driving stress. Applying reasonable parameter values, we demonstrate that the operator of the membrane stresses is much stiffer than operator of the basal shear stress. Therefore, we propose a new splitting method, which alternates between the iteration on the membrane-stress operator and the basal-shear operator, with a stronger iteration on the operator of the membrane stress. We show that this splitting improves the computational performance of the numerical method, although the choice of the (standard) method to solve for all operators in one step speeds up the scheme too. (Based on the delicate and coupled equation we propose a new decomposition method to decouple into simpler solvable sub-equations. After a number of approximations we consider the error of the method and proposed a choice of the operators.)
2011-09-27T00:00:00ZSuperspace calculation of the four-loop spectrum in N=6 supersymmetricChern-Simons theory
http://edoc.hu-berlin.de/18452/3485
Superspace calculation of the four-loop spectrum in N=6 supersymmetricChern-Simons theory
Leoni, Matias; Mauri, Andrea; Minahan, Joseph A.; Sax, Olof Ohlsson; Santambrogio, Alberto; Sieg, Christoph; Tartaglino-Mazzucchelli, Gabriele
http://dx.doi.org/10.18452/2833
2011-09-27T00:00:00ZPorous Media Based Modeling of parallel Plate PE-CVD Apparatus
http://edoc.hu-berlin.de/18452/3484
Porous Media Based Modeling of parallel Plate PE-CVD Apparatus
Geiser, Jürgen; Arab, M.
http://dx.doi.org/10.18452/2832
The numerous technical applications in deposit metal plates with new materials like SiC and TiC has an advantage to overcome the leaking corrosive behavior and have additional a good electrical behavior. Here we present an application of a porous media to model a homogenized deposition with a parallel plate PE-CVD apparatus. Special geometries of parallel Anodes and cathodes helps to obtain at least a laminar flow field. By the way the delicate arrangement of the anode and cathode has to be simulated. The flux of the precursors are important to simulate to the porous media given as the plasma background. Here we can optimize the transport to the delicate geometry respecting the flux field in the permeable layers. To derive a mathematical model, we deal with a model for the transport and kinetics of the different species. Underlying physical experiments help to approximate the parameters of the numerical model. We introduce a multi regression method to approximate the physical to the mathematical parameters. We present results of some numerical simulations and help to foresee some effects to find on optimal deposition process.
2011-09-27T00:00:00ZIterative Operator Splitting Methods For Differential Equations
http://edoc.hu-berlin.de/18452/3483
Iterative Operator Splitting Methods For Differential Equations
Geiser, Jürgen
http://dx.doi.org/10.18452/2831
In this paper we describe an iterative operator-splitting method for bounded operators. Our contribution is a novel iterative method that can be applied as a splitting method to ordinary and partial differential equations. A simple relation between the number of iterative steps and order of the splitting scheme makes this an alternative method to a time decomposition method. The iterative splitting scheme can be applied to a physical problem, but the original problem is not divided as in standard splitting schemes. We present error bounds for iterative splitting methods in the presence of bounded operators. We discuss efficient algorithms for computing the integral formulation of the splitting scheme. In experiments, we consider the benefits of the novel splitting method in terms of the number of iterations and time steps. Ordinary differential equations and convection-diffusion-reaction equations are presented in the numerical results.
2011-09-27T00:00:00ZOscillator Construction of su(n|m) Q-Operators
http://edoc.hu-berlin.de/18452/3482
Oscillator Construction of su(n|m) Q-Operators
Frassek, Rouven; Lukowski, Tomasz; Meneghelli, Carlo; Staudacher, Matthias
http://dx.doi.org/10.18452/2830
We generalize our recent explicit construction of the full hierarchy of Baxter Q-operators of compact spin chains with su(n) symmetry to the supersymmetric case su(n|m). The method is based on novel degenerate solutions of the graded Yang-Baxter equation, leading to an amalgam of bosonic and fermionic oscillator algebras. Our approach is fully algebraic, and leads to the exact solution of the associated compact spin chains while avoiding Bethe ansatz techniques. It furthermore elucidates the algebraic and combinatorial structures underlying the system of nested Bethe equations. Finally, our construction naturally reproduces the representation, due to Z. Tsuboi, of the hierarchy of Baxter Q-operators in terms of hypercubic Hasse diagrams.
2011-09-27T00:00:00ZOn the existence of weak solutions to a coupled system of two turbulent flows
http://edoc.hu-berlin.de/18452/3481
On the existence of weak solutions to a coupled system of two turbulent flows
Neumann, Joachim; Wolf, Jörg
http://dx.doi.org/10.18452/2829
2011-09-27T00:00:00ZAlgorithms of intrinsic complexity for point searching in real singular hypersurfaces
http://edoc.hu-berlin.de/18452/3480
Algorithms of intrinsic complexity for point searching in real singular hypersurfaces
Bank, Bernd; Giusti, Marc; Heintz, Joos; Lehmann, Lutz; Pardo, Luis-Miguel
http://dx.doi.org/10.18452/2828
2011-09-27T00:00:00ZMagnus integrator and successive approximation for solving time-dependent problems
http://edoc.hu-berlin.de/18452/3479
Magnus integrator and successive approximation for solving time-dependent problems
Geiser, Jürgen
http://dx.doi.org/10.18452/2827
Magnus integrator and successive approximation for solving time-dependent problems. The Magnus expansion has been intensely studied and widely applied for solving explicitly time-dependent problems. Due to its exponential character, it is rather difficult to derive practical algorithms beyond the sixth-order. An alternative method is based on successive approximation methods, that taken into account the temporally inhomogeneous equation (method of Tanabe and Sobolevski). In this work, we show that the recently derived ideas of the successive approximation method in a splitting method. Examples are discussed.
2011-09-27T00:00:00ZA Shortcut to the Q-Operator
http://edoc.hu-berlin.de/18452/3478
A Shortcut to the Q-Operator
Bazhanov, Vladimir V.; Lukowski, Tomasz; Meneghelli, Carlo; Staudacher, Matthias
http://dx.doi.org/10.18452/2826
Baxter's Q-operator is generally believed to be the most powerful tool for the exact diagonalization of integrable models. Curiously, it has hitherto not yet been properly constructed in the simplest such system, the compact spin-1/2 Heisenberg-Bethe XXX spin chain. Here we attempt to fill this gap and show how two linearly independent operatorial solutions to Baxter's TQ equation may be constructed as commuting transfer matrices if a twist field is present. The latter are obtained by tracing over infinitely many oscillator states living in the auxiliary channel of an associated monodromy matrix. We furthermore compare and differentiate our approach to earlier articles addressing the problem of the construction of the Q-operator for the XXX chain. Finally we speculate on the importance of Q-operators for the physical interpretation of recent proposals for the Y-system of AdS/CFT.
2011-09-27T00:00:00ZBaxter Q-Operators and Representations of Yangians
http://edoc.hu-berlin.de/18452/3477
Baxter Q-Operators and Representations of Yangians
Bazhanov, Vladimir V.; Frassek, Rouven; Lukowski, Tomasz; Meneghelli, Carlo; Staudacher, Matthias
http://dx.doi.org/10.18452/2825
We develop a new approach to Baxter Q-operators by relating them to the theory of Yangians, which are the simplest examples for quantum groups. Here we open up a new chapter in this theory and study certain degenerate solutions of the Yang-Baxter equation connected with harmonic oscillator algebras. These infinite-state solutions of the Yang-Baxter equation serve as elementary, "partonic" building blocks for other solutions via the standard fusion procedure. As a first example of the method we consider sl(n) compact spin chains and derive the full hierarchy of operatorial functional equations for all related commuting transfer matrices and Q-operators. This leads to a systematic and transparent solution of these chains, where the nested Bethe equations are derived in an entirely algebraic fashion, without any reference to the traditional Bethe ansatz techniques.
2011-09-27T00:00:00ZOn the construction of a class of Dulac-Cherkas functions for generalized Liénard systems
http://edoc.hu-berlin.de/18452/3476
On the construction of a class of Dulac-Cherkas functions for generalized Liénard systems
Cherkas, Leonid; Grin, Alexander; Schneider, Klaus
http://dx.doi.org/10.18452/2824
2011-09-27T00:00:00ZMonte Carlo simulations concerning modeling DC and high power pulsed magnetron sputtering for Ti₃SiC₂ including high pressures and ion deposition probabilities
http://edoc.hu-berlin.de/18452/3475
Monte Carlo simulations concerning modeling DC and high power pulsed magnetron sputtering for Ti₃SiC₂ including high pressures and ion deposition probabilities
Geiser, Jürgen
http://dx.doi.org/10.18452/2823
We motivate our study by simulating the particle transport of a thin film deposition process done by PVD (physical vapor deposition) processes. In this paper we present a new model taken into account a higher pressure regimes in a sputter process. We propose a collision models for projectile and target collisions in order to compute the mean free path and include the virial coefficients that considered interacting gas particles. A detailed description of collision models of the Monte Carlo Simulations is discussed for high power impulse magnetron sputtering (HIPIMS) and DC sputtering in lower pressure regimes. We derive an equation for the mean free path for arbitrary interactions (cross sections) which (most important) includes the relative velocity between the projectiles and targets based on physical first principles and extend with higher order Virial terms . At the substrate we simulate the implantation of the particles with the help of TRIM, based on result of the energy that are computed with the Monte Carlo methods. We apply our results to three interaction models: hard sphere interaction, Screened Coulomb interaction and a mixture of the last mentioned interactions. The deposition to realistic geometries, which have sharp angles included, are presented. Because of the strong convective process of a HIPIMS method, the low diffusion process allows not to deposit into delicate geometries, see [Christ2005]. This can be improved by rotating the target to a more or less perpendicular angle.
2011-09-27T00:00:00ZModel of PE-CVD apparatus
http://edoc.hu-berlin.de/18452/3474
Model of PE-CVD apparatus
Geiser, Jürgen; Buck, V.; Arab, M.
http://dx.doi.org/10.18452/2822
In this paper we present the simulation of a chemical vapor deposition for metallic bipolar plates. For chemical vapor deposition, the delicate optimization between temperature, pressure and plasma power is important to obtain a homogeneous depositio. The aim is to reduce real-life experiments of a given CVD plasma reactor, based on a large physical parameter space we have a hugh amount of experiments. A detail study of the physical experiments on a CVD plasma reactor allows to reduce to an approximated mathematical model, which is the underlying transport-reaction model. Significant region of the CVD apparatus are approximated and physical parameters are transferred to the mathematical parameters. Such approximation reduced the mathematical parameter space to a realistic amount of numerical experiments. Based on interpolation and regression functions we fit to the physical parameter space and can give first prediction to deposition rates with the simulation model. Here numerical experiments help to understand the deposition process and the control the positions of the sources for the deposition and precursor gases. For the simulations we apply analytical as well as numerical methods to obtain results to predict the growth of thin layers. The results are discussed with physical experiments to give a valid model for the assumed growth. Here an important transfer of engineering research on modelling real-life processes to achieve a simulatable mathematical model. Such a model can be solved by numerical solvers and discretisation schemes. The results can be used to obtain a new understanding of the technical processes in engineering research.
2011-09-27T00:00:00ZA new algorithm for the index determination in DAEs by Taylor series using Algorithmic Differentiation
http://edoc.hu-berlin.de/18452/3473
A new algorithm for the index determination in DAEs by Taylor series using Algorithmic Differentiation
Lamour, René; Monett-Diaz, Dagmar
http://dx.doi.org/10.18452/2821
We present an approach for determining the tractability index using truncated polynomial arithmetic. In particular, computing the index this way generates a sequence of matrices that contains itself derivatives. We realize the time differentiations using Algorithmic Differentiation techniques, specially by using the standard ADOL-C package with which calculating the derivatives becomes a simple shift and scaling of coefficients. We present the theory supporting the procedure we propose, as well as the implementation issues behind it to provide a convenient interface to the standard ADOL-C functionality. We give also examples of academic and practical problems and report several experimental results we have obtained with them.
2011-09-27T00:00:00ZIterative operator-splitting method
http://edoc.hu-berlin.de/18452/3472
Iterative operator-splitting method
Geiser, Jürgen
http://dx.doi.org/10.18452/2820
In this paper we describe an iterative operator-splitting method for bounded operators. The contribution is a novel iterative method, that can be applied as a splitting method for ordinary and partial differential equations. A simple relation between the number of iterative steps and the order of the splitting scheme, can made it as an alternative method to consider it as a time decomposition method. The iterative splitting scheme is interested on physical problem, while the original problem is not divided as in standard splitting schemes. We present error bounds for the iterative splitting methods in the presence of bounded operators. We discuss efficient algorithms to compute the integral formulation of the splitting scheme. In experiments, we consider the benefits of the novel splitting method with respect to their number of iterations and time steps. Ordinary differential equations and convection-diffusion-reaction equations are presented in the numerical results.
2011-09-27T00:00:00ZSplitting Method of Convection-Diffusion Methods with Disentanglement methods
http://edoc.hu-berlin.de/18452/3471
Splitting Method of Convection-Diffusion Methods with Disentanglement methods
Geiser, Jürgen; Elbiomy, Mahmoud
http://dx.doi.org/10.18452/2819
In this paper, we discuss higher-order operator-splitting methods done by disentanglement methods. The idea is based on computing best fitted exponents to an exponential splitting scheme with more than two operators. We introduce the underlying splitting methods and the special scheme to compute the disentanglement method. First applications are done to consider finite difference methods to the spatial operators and derive their underlying Lie algebras. Based on the Lie algebra it is simple to compute the corresponding Lie group with $\exp$ functions. Such results help to derive the disentanglement of the operator splitting method. The verification of our improved splitting methods are done with first numerical examples.
2011-09-27T00:00:00ZKinetic processes and Phase-transition of CVD processes for Ti₃SiC₂
http://edoc.hu-berlin.de/18452/3470
Kinetic processes and Phase-transition of CVD processes for Ti₃SiC₂
Geiser, Jürgen; Roehle, Robert
http://dx.doi.org/10.18452/2818
In this paper we present a kinetic model based on numerical simulations of a chemical vapor deposition (CVD) process. We discuss a model that is based on kinetics of the deposition rates to the material. Such a simple model can explain the experimental results. Based on experiments with Ti3SiC2 we verify our model. Here different processes of ionized Ti+, Ti++ and C are important to achieve our stoichiometry. The numerical methods are based on iterative schemes to solve coupled and nonlinear differential equations. The results are discussed with physical experiments to give a valid model for the assumed growth of thin layers.
2011-09-20T00:00:00ZA combined BDF-semismooth Newton approach for time-dependent Bingham flow
http://edoc.hu-berlin.de/18452/3469
A combined BDF-semismooth Newton approach for time-dependent Bingham flow
Reyes, Juan Carlos De Los; Andrade, Sergio González
http://dx.doi.org/10.18452/2817
This paper is devoted to the numerical simulation of time-dependent convective Bingham flow in cavities. Motivated by a primal-dual regularization of the stationary model, a family of regularized time-dependent problems is introduced. Well posedness of the regularized problems is proved and convergence of the regularized solutions to a solution of the original multiplier system is verified. For the numerical solution of each regularized multiplier system, a fully-discrete approach is studied. A stable finite element approximation in space, together with a second order backward differentiation formula for the time discretization are proposed. The discretization scheme yields a system of Newton differentiable nonlinear equations in each time step, for which a semismooth Newton algorithm is utilized. We present two numerical experiments to verify the main properties of the proposed approach.
2011-09-20T00:00:00Z