# Variational approach in weighted Sobolev spaces to scattering by unbounded rough surface

*Authors*

- Chandler-Wilde, Simon
- Elschner, Johannes

*2010 Mathematics Subject Classification*

- 35J05 35J20 35J25 42B10 78A45

*Keywords*

- Non-smooth boundary, radiation condition, variational formulation, weighted Sobolev spaces, Helmholtz equation

*DOI*

*Appeared in*

- SIAM J. Math. Anal., 42 (2010) pp. 2554--2580.

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# Hamiltonian structure of propagation equations for ultrashort optical pulses

*Authors*

- Amiranashvili, Shalva

ORCID: 0000-0002-8132-882X - Demircan, Ayhan

*2008 Physics and Astronomy Classification Scheme*

- 42.65.-k, 42.81.Dp, 03.50.De, 45.20.Jj

*DOI*

*Abstract*

A Hamiltonian framework is developed for a sequence of ultrashort optical pulses propagating in a nonlinear dispersive medium. To this end a second-order nonlinear wave equation is first simplified using an unidirectional approximation. All non-resonant nonlinear terms are then rigorously eliminated using a suitable change of variables in the spirit of the canonical perturbation theory. The derived propagation equation operates with a properly defined complexification of the real electric field. It accounts for arbitrary dispersion, four-wave mixing processes, weak absorption, and arbitrary pulse duration. Thereafter the so called normal variables, i.e., classical fields corresponding to the quantum creation and annihilation operators, are introduced. Neglecting absorption we finally derive the Hamiltonian formulation. The latter yields the most essential integrals of motion for the pulse propagation. These integrals reflect the time-averaged fluxes of energy, momentum, and classical photon number transferred by the pulse. The conservation laws are further used to control the numerical solutions when calculating supercontinuum generation by an ultrashort optical pulse.

*Appeared in*

- Phys. Rev. A, 82 (2010) pp. 013812/1--013812/11.

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# Elastoplastic Timoshenko beams

*Authors*

- Krejčí, Pavel
- Sprekels, Jürgen
- Wu, Hao

*2010 Mathematics Subject Classification*

- 34C55 35Q72 47J40 74C05 74K10 74N30

*Keywords*

- elastoplasticity, Timoshenko beam, anisotropic hysteresis operators, Prandtl--Ishlinskii model, von Mises model

*DOI*

*Abstract*

A Timoshenko type elastoplastic beam equation is derived by dimensional reduction from a general 3D system with von Mises plasticity law. It consists of two second-order hyperbolic equations with an anisotropic vectorial Prandtl--Ishlinskii hysteresis operator. Existence and uniqueness of a strong solution for an initial-boundary value problem is proven via standard energy and monotonicity methods.

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# Semi-supervised novelty detection

*Authors*

- Blanchard, Gilles
- Lee, Gyemin
- Scott, Clayton

*2010 Mathematics Subject Classification*

- 62H30 62H15

*Keywords*

- Semi-supervised learning, novelty detection, Neyman-Pearson classification, learning reduction, two-sample problem, mutliple testing

*DOI*

*Abstract*

A common setting for novelty detection assumes that labeled examples from the nominal class are available, but that labeled examples of novelties are unavailable. The standard (inductive) approach is to declare novelties where the nominal density is low, which reduces the problem to density level set estimation. In this paper, we consider the setting where an unlabeled and possibly contaminated sample is also available at learning time. We argue that novelty detection in this semi-supervised setting is naturally solved by a general reduction to a binary classification problem. In particular, a detector with a desired false positive rate can be achieved through a reduction to Neyman-Pearson classification. Unlike the inductive approach, semi-supervised novelty detection (SSND) yields detectors that are optimal (e.g., statistically consistent) regardless of the distribution on novelties. Therefore, in novelty detection, unlabeled data have a substantial impact on the theoretical properties of the decision rule. We validate the practical utility of SSND with an extensive experimental study. We also show that SSND provides distribution-free, learning-theoretic solutions to two well known problems in hypothesis testing. First, our results provide a general solution to the general two-sample problem, that is, the problem of determining whether two random samples arise from the same distribution. Second, a specialization of SSND coincides with the standard $p$-value approach to multiple testing under the so-called random effects model. Unlike standard rejection regions based on thresholded $p$-values, the general SSND framework allows for adaptation to arbitrary alternative distributions.

*Appeared in*

- J. Mach. Learn. Res., 11 (2010) pp. 2973--3009.

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# From bell shapes to pyramids: A continuum model for self-assembled quantum dot growth

*Authors*

- Korzec, Maciek D.
- Evans, Peter L.

*2010 Mathematics Subject Classification*

- 74A50 41A60

*Keywords*

- Anisotropic surface energy, self-assembly of quantum dots, small-slope approximation, stationary solutions, coarsening, linear stability

*DOI*

*Abstract*

A continuum model for the growth of self-assembled quantum dots that incorporates surface diffusion, an elastically deformable substrate, wetting interactions and anisotropic surface energy is presented. Using a small slope approximation a thin film equation for the surface profile that describes facetted growth is derived. A linear stability analysis shows that anisotropy acts to destabilize the surface. It lowers the critical height of flat films and there exists an anisotropy strength above which all thicknesses are unstable. A numerical algorithm based on spectral differentiation is presented and simulation are carried out. These clearly show faceting of the growing islands and a logarithmically slow coarsening behavior.

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# Local existence, uniqueness, and smooth dependence for nonsmooth quasilinear parabolic problems

*Authors*

- Griepentrog, Jens André
- Recke, Lutz

*2010 Mathematics Subject Classification*

- 35K60 35R05 35K50 35K90

*Keywords*

- Sobolev-Morrey spaces, maximal regularity, Implicit Function Theorem, Lipschitz domains, mixed boundary conditions

*DOI*

*Abstract*

A general theory on local existence, uniqueness, regularity, and smooth dependence in Hölder spaces for a general class of quasilinear parabolic initial boundary value problems with nonsmooth data has been developed. As a result the gap between low smoothness of the data, which is typical for many applications, and high smoothness of the solutions, which is necessary for the applicability of differential calculus to the abstract formulations of the initial boundary value problems, has been closed. The main tools are new maximal regularity results of the first author in Sobolev-Morrey spaces, linearization techniques and the Implicit Function Theorem. Typical applications are transport processes of charged particles in semiconductor heterostructures, phase separation processes of nonlocally interacting particles, chemotactic aggregation in heterogeneous environments as well as optimal control by means of quasilinear elliptic and parabolic PDEs with nonsmooth data.

*Appeared in*

- J. Evol. Equ., 10 (2010) pp. 341--375.

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# Implementing exact absorbing boundary condition for the linear one-dimensional Schrödinger problem with variable potential by Titchmarsh--Weyl theory

*Authors*

- Ehrhardt, Matthias
- Zheng, Chunxiong

*2010 Mathematics Subject Classification*

- 65M99 81-08

*Keywords*

- absorbing boundary conditions, variable potential, Schrödinger equation, Titchmarsh-Weyl m-function, unbounded domain

*DOI*

*Abstract*

A new approach for simulating the solution of the time-dependent Schrödinger equation with a general variable potential will be proposed. The key idea is to approximate the Titchmarsh-Weyl m-function (exact Dirichlet-to-Neumann operator) by a rational function with respect to a suitable spectral parameter. With the proposed method we can overcome the usual high-frequency restriction for absorbing boundary conditions of general variable potential problems. We end up with a fast computational algorithm for absorbing boundary conditions that are accurate for the full frequency band.

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# Plasma induced pulse breaking in filamentary self-compression

*Authors*

- Brée, Carsten
- Demircan, Ayhan
- Skupin, Stefan
- Bergé, Luc
- Steinmeyer, Günter

*2010 Mathematics Subject Classification*

- 78A60 81V80 35Q55 37K40

*2008 Physics and Astronomy Classification Scheme*

- 42.65.Tg 42.65.-k 52.38.Hb 42.68.Ay

*Keywords*

- Nonlinear Schrödinger Equations, Optical self-focusing, Ultrashort pulse propagation

*DOI*

*Abstract*

A plasma induced temporal break-up in filamentary propagation has recently been identified as one of the key events in the temporal self-compression of femtosecond laser pulses. An analysis of the Nonlinear Schrödinger Equation coupled to a noninstantaneous plasma response yields a set of stationary states. This analysis clearly indicates that the emergence of double-hump, characteristically asymmetric temporal on-axis intensity profiles in regimes where plasma defocusing saturates the optical collapse caused by Kerr self-focusing is an inherent property of the underlying dynamical model.

*Appeared in*

- Laser Physics, 20 (2010) pp. 1107--1113.

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# Mean field diffusion models for precipitation in crystalline GaAs including surface tension and bulk stresses

*Authors*

- Dreyer, Wolfgang
- Kimmerle, Sven-Joachim

*2010 Mathematics Subject Classification*

- 74Q99 35B27 35Q74 35R35 74A15 74F99 74N20 65P99

*2008 Physics and Astronomy Classification Scheme*

- 02.30 Mv 02.30.Jr 61.72.uj 64.70.D- 66.30.-h

*Keywords*

- mean field approximation, homogenisation, semi-insulating GaAs, Ostwald ripening including mechanics

*DOI*

*Abstract*

Based on a thermodynamically consistent model for precipitation in gallium arsenide crystals including surface tension and bulk stresses by Dreyer and Duderstadt, we propose different mathematical models to describe the size evolution of liquid droplets in a crystalline solid. The first class of models treats the diffusion-controlled regime of interface motion, while the second class is concerned with the interface-controlled regime of interface motion. Our models take care of conservation of mass and substance. We consider homogenised models, where different length scales of the experimental situation have been exploited in order to simplify the equations. These homogenised models generalise the well-known Lifshitz-Slyozov-Wagner model for Ostwald ripening. Mean field models capture the main properties of our system and are well adapted for numerics and further analysis. Numerical evidence suggests in which case which one of the two regimes might be appropriate to the experimental situation.

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# On the construction of Dulac--Cherkas functions for generalized Liénard systems

*Authors*

- Cherkas, Leonid
- Grin, Alexander
- Schneider, Klaus R.

*2010 Mathematics Subject Classification*

- 34C07 34C05

*Keywords*

- number of limit cycles, generalized Liénard systems, Dulac-Cherkas functions

*DOI*

*Abstract*

Dulac-Cherkas functions can be used to derive an upper bound for the number of limit cycles of planar autonomous differential systems, at the same time they provide information about their stability. In this paper we present a method to construct such functions for generalized Liénard systems by means of linear differential equations. If the degree m of the polynomial is not greater than 3, then the described algorithm works generically. By means of an example we show that this approach can be applied also to polynomials with degree m larger than 3.

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# Polarization properties in the transition from below to above lasing threshold in broad-area vertical-cavity surface-emitting lasers

*Authors*

- Schulz-Ruhtenberg, Malte
- Babushkin, Ihar
- Loiko, Natalia Aleksandrovna
- Ackemann, Thorsten
- Huang, Kai

*2010 Mathematics Subject Classification*

- 78A60 35R60 37K50

*2008 Physics and Astronomy Classification Scheme*

- 42.55.Px, 42.60.Jf, 42.25.Ja, 05.40.-a

*Keywords*

- semiconductor lasers, light polarization, bifurcation, stochastic PDE

*DOI*

*Abstract*

For highly divergent emission of broad-area vertical-cavity surface-emitting lasers (VCSELs) a rotation of the polarization direction by up to 90 degrees occurs when the pump rate approaches the lasing threshold. Well below threshold the polarization is parallel to the direction of the transverse wave vector and is determined by the transmissive properties of the Bragg reflectors that form the cavity mirrors. In contrast, near-threshold and above-threshold emission is more affected by the reflective properties of the reflectors and is predominantly perpendicular to the direction of transverse wave vectors. Two qualitatively different types of polarization transition are demonstrated: an abrupt transition, where the light polarization vanishes at the point of the transition, and a smooth one, where it is significantly nonzero during the transition.

*Appeared in*

- Phys. Rev. A, 81 (2010) pp. 023819/1--023819/11.

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# Comparison of the continuous, semi-discrete and fully-discrete transparent boundary conditions (TBC) for the parabolic wave equation 1. Theory

*Authors*

- Sumichrast, Lubomir
- Ehrhardt, Matthias

*2010 Mathematics Subject Classification*

- 65M12 35Q40

*2008 Physics and Astronomy Classification Scheme*

- 02.70.Bf 31.15.Fx

*Keywords*

- Transparent boundary conditions, beam propagation method, parabolic wave equation

*DOI*

*Abstract*

For the simulation of the propagation of optical waves in open wave guiding structures of integrated optics the parabolic approximation of the scalar wave equation is commonly used. This approach is commonly termed the beam propagation method (BPM). It is of paramount importance to have well-performing transparent boundary conditions applied on the boundaries of the finite computational window, to enable the superfluous portion of the propagating wave to radiate away from the wave guiding structure. Three different formulations (continuous, semi-discrete and fully-discrete) of the non-local transparent boundary conditions are described and compared here.

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# Executing large orders in a microscopic market model

*Authors*

- Weiss, Alexander

*2010 Mathematics Subject Classification*

- 91B24 62P05

*Keywords*

- market micro structure, illiquid markets, optimal trading strategies

*DOI*

*Abstract*

In a recent paper, Alfonsi, Schied and Schulz (ASS) propose a simple order book based model for the impact of large orders on stock prices. They use this model to derive optimal strategies for the execution of large orders. We test this model in the context of an agent based microscopic stochastic order book model that was recently proposed by Bovier, Černý and Hryniv. While the ASS model captures some features of real markets, some assumptions in the model contradict our simulation results. In particular, from our simulations the recovery speed of the market after a large order is clearly depended on the order size, whereas the ASS model assumes the speed to be given by a constant. For this reason, we propose a generalisation of the model of ASS that incorporates this dependency, and derive the optimal investment strategies. We show that within our artificial market, correct fitting of this parameter leads to optimal hedging strategies that reduce the trading costs, compared to the ones produced by ASS. Finally, we show that the costs of applying the optimal strategies of the improved ASS model to the artificial market still differ significantly from the model predictions, indicating that even the improved model does not capture all of the relevant details of a real market.

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# BV solutions and viscosity approximations of rate-independent systems

*Authors*

- Mielke, Alexander

ORCID: 0000-0002-4583-3888 - Rossi, Riccarda
- Savaré, Giuseppe

*2010 Mathematics Subject Classification*

- 49Q20 58E99

*Keywords*

- doubly nonlinear, differential inclusions, generalized gradient flows, viscous regularization, vanishing-viscosity limit, vanishing-viscosity contact potential, parametrized solutions

*DOI*

*Abstract*

In the nonconvex case solutions of rate-independent systems may develop jumps as a function of time. To model such jumps, we adopt the philosophy that rate independence should be considered as limit of systems with smaller and smaller viscosity. For the finite-dimensional case we study the vanishing-viscosity limit of doubly nonlinear equations given in terms of a differentiable energy functional and a dissipation potential which is a viscous regularization of a given rate-independent dissipation potential. The resulting definition of `BV solutions' involves, in a nontrivial way, both the rate-independent and the viscous dissipation potential, which play a crucial role in the description of the associated jump trajectories. We shall prove a general convergence result for the time-continuous and for the time-discretized viscous approximations and establish various properties of the limiting $BV$ solutions. In particular, we shall provide a careful description of the jumps and compare the new notion of solutions with the related concepts of energetic and local solutions to rate-independent systems.

*Appeared in*

- ESAIM Control Optim. Calc. Var., 18 (2012) pp. 36--80.

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# Collision in a cross-shaped domain --- A steady 2D Navier--Stokes example demonstrating the importance of mass conservation in CFD

*Authors*

- Linke, Alexander

ORCID: 0000-0002-0165-2698

*2010 Mathematics Subject Classification*

- 35Q30 76D07 76M10

*2008 Physics and Astronomy Classification Scheme*

- 47.10.ad

*Keywords*

- incompressible Navier-Stokes equations, mixed finite elements, poor mass conservation, numerical instability

*DOI*

*Abstract*

In the numerical simulation of the incompressible Navier-Stokes equations different numerical instabilities can occur. While instability in the discrete velocity due to dominant convection and instability in the discrete pressure due to a vanishing discrete LBB constant are well-known, instability in the discrete velocity due to a poor mass conservation at high Reynolds numbers sometimes seems to be underestimated. At least, when using conforming Galerkin mixed finite element methods like the Taylor-Hood element, the classical grad-div stabilization for enhancing discrete mass conservation is often neglected in practical computations. Though simple academic flow problems showing the importance of mass conservation are well-known, these examples differ from practically relevant ones, since specially designed force vectors are prescribed. Therefore we present a simple steady Navier-Stokes problem in two space dimensions at Reynolds number 1024, a colliding flow in a cross-shaped domain, where the instability of poor mass conservation is studied in detail and where no force vector is prescribed.

*Appeared in*

- Comput. Methods Appl. Mech. Engrg., 198 (2009) pp. 3278--3286.

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# Stable Crank--Nicolson discretisation for incompressible miscible displacement problems of low regularity

*Authors*

- Jensen, Max
- Müller, Rüdiger

ORCID: 0000-0003-2643-722X

*2010 Mathematics Subject Classification*

- 65M60 65M12 76S05

*Keywords*

- discontinuous Galerkin, low regularity, Crank-Nicolson

*DOI*

*Abstract*

In this article we study the numerical approximation of incompressible miscible displacement problems with a linearised Crank-Nicolson time discretisation, combined with a mixed finite element and discontinuous Galerkin method. At the heart of the analysis is the proof of convergence under low regularity requirements. Numerical experiments demonstrate that the proposed method exhibits second-order convergence for smooth and robustness for rough problems.

*Appeared in*

- Numerical Mathematics and Advanced Applications 2009, Part 2, G. Kreiss, P. Lötstedt, A. Målqvist, M. Neytcheva, eds., Springer, Heidelberg et al., pp. 469--477

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# Optimal control of robot-guided laser material treatment

*Authors*

- Hömberg, Dietmar
- Steinbrecher, Andreas
- Stykel, Tatjana

*2010 Mathematics Subject Classification*

- 35K05 49K15 49K20 49M05

*Keywords*

- Optimal control, coupled systems, heat equation, equations of motion, gradient based method

*DOI*

*Abstract*

In this article we will consider the optimal control of robot guided laser material treatments, where the discrete multibody system model of a robot is coupled with a PDE model of the laser treatment. We will present and discuss several optimization approaches of such optimal control problems and its properties in view of a robust and suitable numerical solution. We will illustrate the approaches in an application to the surface hardening of steel.

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# Global spatial regularity for a regularized elasto-plastic model

*Authors*

- Bumb, Andreas
- Knees, Dorothee

*2010 Mathematics Subject Classification*

- 35B65 49N60 74C10

*Keywords*

- global spatial regularity, nonsmooth domain, regularized elasto-viscoplastic model

*DOI*

*Abstract*

In this note the spatial regularity of weak solutions for a class of elasto-viscoplastic evolution models is studied for nonsmooth domains. The considered class comprises e.g. models which are obtained through a Yosida regularization from classical, rate-independent models. The corresponding evolution model consists of an elliptic PDE for the (generalized) displacements which is coupled with an ordinary differential equation with a Lipschitz continuous nonlinearity describing the evolution of the internal variable. It is shown that the global spatial regularity of the displacements and the inner variables is exactly determined through the mapping properties of the underlying elliptic operator.

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# Damage of nonlinearly elastic materials at small strain --- Existence and regularity results

*Authors*

- Thomas, Marita
- Mielke, Alexander

ORCID: 0000-0002-4583-3888

*2010 Mathematics Subject Classification*

- 35K85 49S05 74C15 74R20

*Keywords*

- Damage evolution with spatial regularization, partial damage, rate-independent systems, energetic formulation via energy functional and dissipation distance, energetic solutions, convexity of energy functional, temporal Lipschitz- and Hölder-continuity of solutions

*DOI*

*Abstract*

In this paper an existence result for energetic solutions of rate-independent damage processes is established and the temporal regularity of the solution is discussed. We consider a body consisting of a physically nonlinearly elastic material undergoing small deformations and partial damage. The present work is a generalization of [Mielke-Roubicek 2006] concerning the properties of the stored elastic energy density as well as the suitable Sobolev space for the damage variable: While previous work assumes that the damage variable z satisfies z ∈ W^1,r (Omega) with r>d for Omega ⊂ R^d, we can handle the case r>1 by a new technique for the construction of joint recovery sequences. Moreover, this work generalizes the temporal regularity results to physically nonlinearly elastic materials by analyzing Lipschitz- and Hölder-continuity of solutions with respect to time.

*Appeared in*

- ZAMM Z. Angew. Math. Mech., 90 (2010) pp. 88--112.

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# Analytical and numerical aspects of time-dependent models with internal variables

*Authors*

- Gruber, Peter
- Knees, Dorothee
- Nesenenko, Sergiy
- Thomas, Marita

*2010 Mathematics Subject Classification*

- 74C05 74C10 49N60 65M60

*Keywords*

- Elasto-plasticity, visco-plasticity, models of monotone type, existence of solutions, monotone operator method, spatial regularity, Slant Newton Method, energetic formulation of rate-independent processes, temporal regularity

*DOI*

*Abstract*

In this paper some analytical and numerical aspects of time-dependent models with internal variables are discussed. The focus lies on elasto/visco-plastic models of monotone type arising in the theory of inelastic behavior of materials. This class of problems includes the classical models of elasto-plasticity with hardening and viscous models of the Norton-Hoff type. We discuss the existence theory for different models of monotone type, give an overview on spatial regularity results for solutions to such models and illustrate a numerical solution algorithm at an example. Finally, the relation to the energetic formulation for rate-independent processes is explained and temporal regularity results based on different convexity assumptions are presented.

*Appeared in*

- ZAMM Z. Angew. Math. Mech., 90 (2010) pp. 861--902.

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# Finite rank perturbations, scattering matrices and inverse problems

*Authors*

- Behrndt, Jussi
- Malamud, Mark M.
- Neidhardt, Hagen

*2010 Mathematics Subject Classification*

- 47A40 81U40 47A55 47B44

*Keywords*

- Scattering system, scattering matrix, boundary triplet, Weyl function, dissipative operator, Lax-Phillips scattering

*DOI*

*Abstract*

In this paper the scattering matrix of a scattering system consisting of two selfadjoint operators with finite dimensional resolvent difference is expressed in terms of a matrix Nevanlinna function. The problem is embedded into an extension theoretic framework and the theory of boundary triplets and associated Weyl functions for (in general nondensely defined) symmetric operators is applied. The representation results are extended to dissipative scattering systems and an explicit solution of an inverse scattering problem for the Lax-Phillips scattering matrix is presented.

*Appeared in*

- Operator Theory in Krein Spaces and Spectral Analysis, J. Behrndt, K.-H. Förster, C. Trunk, H. Winkler, eds., vol. 198 of Operator Theory: Advances and Applications, Birkhäuser, Basel, 2009, pp. 61--85

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# Representations for optimal stopping under dynamic monetary utility functionals

*Authors*

- Krätschmer, Volker
- Schoenmakers, John G. M.

ORCID: 0000-0002-4389-8266

*2010 Mathematics Subject Classification*

- 49L20 60G40 91B16

*Keywords*

- monetary utility functionals, optimal stopping, duality, policy iteration

*DOI*

*Abstract*

In this paper we consider the optimal stopping problem for general dynamic monetary utility functionals. Sufficient conditions for the Bellman principle and the existence of optimal stopping times are provided. Particular attention is payed to representations which allow for a numerical treatment in real situations. To this aim, generalizations of standard evaluation methods like policy iteration, dual and consumption based approaches are developed in the context of general dynamic monetary utility functionals. As a result, it turns out that the possibility of a particular generalization depends on specific properties of the utility functional under consideration.

*Appeared in*

- SIAM J. Financial Math., 1 (2010) pp. 811--832.

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# Regression methods for stochastic control problems and their convergence analysis

*Authors*

- Belomestny, Denis
- Kolodko, Anastasia
- Schoenmakers, John G. M.

ORCID: 0000-0002-4389-8266

*2010 Mathematics Subject Classification*

- 90C40 90-08 91B28

*Keywords*

- optimal control, dynamic programming, regression estimator, Monte Carlo simulation

*DOI*

*Abstract*

In this paper we develop several regression algorithms for solving general stochastic optimal control problems via Monte Carlo. This type of algorithms is particulary useful for problems with a high-dimensional state space and complex dependence structure of the underlying Markov process with respect to some control. The main idea behind the algorithms is to simulate a set of trajectories under some reference measure and to use the Bellman principle combined with fast methods for approximating conditional expectations and functional optimization. Theoretical properties of the presented algorithms are investigated and the convergence to the optimal solution is proved under mild assumptions. Finally, we present numerical results for the problem of pricing a high-dimensional Bermudan basket option under transaction costs in a financial market with a large investor.

*Appeared in*

- SIAM J. Control Optim., 48 (2010) pp. 3562--3588.

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# A model of an electrochemical flow cell with porous layer

*Authors*

- Ehrhardt, Matthias
- Fuhrmann, Jürgen

ORCID: 0000-0003-4432-2434 - Linke, Alexander

ORCID: 0000-0002-0165-2698

*2010 Mathematics Subject Classification*

- 65N99 35K20

*Keywords*

- limiting current, finite volume method, boundary layer, fluid-porous interface problem

*DOI*

*Abstract*

In this paper we discuss three different mathematical models for fluid-porous interfaces in a simple channel geometry that appears e.g. in thin-layer channel flow cells. Here the difficulties arise from the possibly different orders of the corresponding differential operators in the different domains. A finite volume discretization of this model allows to calculate the limiting current of the H_2 oxidation in a porous electrode with platinum catalyst particles.

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# Parameter choice methods using minimization schemes

*Authors*

- Bauer, Frank
- Mathé, Peter

ORCID: 0000-0002-1208-1421

*2010 Mathematics Subject Classification*

- 65J22 65F22 62H12

*Keywords*

- Inverse Problems, Heuristic parameter choice, Minimization schemes

*DOI*

*Abstract*

In this paper we establish a generalized framework, which allows to prove convergenence and optimality of parameter choice schemes for inverse problems based on minimization in a generic way. We show that the well known quasi-optimality criterion falls in this class. Furthermore we present a new parameter choice method and prove its convergence by using this newly established tool.

*Appeared in*

- J. Complexity, 27 (2011) pp. 68--85.

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# The real multiple dual

*Authors*

- Schoenmakers, John G. M.

ORCID: 0000-0002-4389-8266

*2010 Mathematics Subject Classification*

- 60G40 62L15 91B28

*Keywords*

- Optimal stopping, Dual representations, Multiple callable derivatives

*DOI*

*Abstract*

In this paper we present a dual representation for the multiple stopping problem, hence multiple exercise options. As such it is a natural generalization of the method in Rogers (2002) and Haugh and Kogan (2004) for the standard stopping problem for American options. We consider this representation as the real dual as it is solely expressed in terms of an infimum over martingales rather than an infimum over martingales and stopping times as in Meinshausen and Hambly (2004). For the multiple dual representation we present three Monte Carlo simulation algorithms which require only one degree of nesting.

*Appeared in*

- Finance and Stochastics, 16 (2012) pp. 319-334 as: A pure martingale dual for multiple stopping

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# Asymptotic behavior of a hydrodynamic system in the nematic liquid crystal flows

*Authors*

- Liu, Chun
- Wu, Hao
- Xu, Xiang

*2010 Mathematics Subject Classification*

- 35B40 35B41 35Q35 76D05

*Keywords*

- Liquid crystal flows, Navier--Stokes equation, kinematic transport, uniqueness of asymptotic limit, Łojasiewicz--Simon inequality

*DOI*

*Abstract*

In this paper we study the long time behavior of the classical solutions to a hydrodynamical system modeling the flow of nematic liquid crystals. This system consists of a coupled system of Navier--Stokes equations and kinematic transport equations for the molecular orientations. By using a suitable Łojasiewicz--Simon type inequality, we prove the convergence of global solutions to single steady states as time tends to infinity. Moreover, we provide estimates for the convergence rate.

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# Strong stationary solutions to equilibrium problems with equilibrium constraints with applications to an electricity spot market model

*Authors*

- Henrion, René
- Outrata, Jiří
- Surowiec, Thomas

ORCID: 0000-0003-2473-4984

*2010 Mathematics Subject Classification*

- 90C30 49J53

*Keywords*

- Equilibrium problems with equilibrium constraints, EPEC, strong stationary solutions, electricity spot market

*DOI*

*Abstract*

In this paper, we consider the characterization of strong stationary solutions to equilibrium problems with equilibrium constraints (EPECs). Assuming that the underlying generalized equation satisfies strong regularity in the sense of Robinson, an explicit multiplier-based stationarity condition can be derived. This is applied then to an equilibrium model arising from ISO-regulated electricity spot markets.

*Appeared in*

- ESAIM Control Optim. Calc. Var., 18 (2012) pp. 295--317 under the new title "Analysis of M-stationary points to an EPEC modeling oligopolistic competition in an electricity spot market''.

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# Trotter--Kato product formula for unitary groups

*Authors*

- Exner, Pavel
- Neidhardt, Hagen

*2010 Mathematics Subject Classification*

- 47A55, 47D03, 81Q30, 32A40 47B25

*Keywords*

- Trotter product formula, Trotter-Kato product formula, unitary groups, Feynman path integrals, holomorphic Kato functions

*DOI*

*Abstract*

Let $A$ and $B$ be non-negative self-adjoint operators in a separable Hilbert space such that its form sum $C$ is densely defined. It is shown that the Trotter product formula holds for imaginary times in the $L^2$-norm. The result remains true for the Trotter-Kato product formula for so-called holomorphic Kato functions; we also derive a canonical representation for any function of this class.

*Appeared in*

- Integral Equations Operator Theory, 69 (2011) pp. 451--478.

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# An iterative method for the multipliers of periodic delay-differential equations and the analysis of a PDE milling model

*Authors*

- Rott, Oliver
- Jarlebring, Elias

*2010 Mathematics Subject Classification*

- 47J10 39A30

*Keywords*

- Time-periodic delay-differential equations, stability, nonlinear eigenvalue problems

*DOI*

*Abstract*

Locally convergent iterative schemes have turned out to be very useful in the analysis of the characteristic roots of delay-differential equations (DDEs) with constant coefficients. In this work we present a locally convergent iterative scheme for the characteristic multipliers of periodic-coefficient DDEs. The method is an adaption of an iterative method called residual inverse iteration. The possibility to use this method stems from an observation that the characteristic matrix can be expressed with the fundamental solution of a differential equation. We apply the method to a coupled milling model containing a partial and an ordinary differential equation. The conclusion of the numerical results is that the stability diagram of the coupled model differs significantly from the combined stability diagrams for each subsystem.

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# Solving conical diffraction with integral equations

*Authors*

- Goray, Leonid I.
- Schmidt, Gunther

*2010 Mathematics Subject Classification*

- 78A45 78M15 65N38 35J05 35Q60 45F15

*Keywords*

- Diffraction, periodic structure, integral equation method, oblique incidence, energy conservation, numerical tests

*DOI*

*Abstract*

Off-plane scattering of time-harmonic plane waves by a diffraction grating with arbitrary conductivity and general border profile is considered in a rigorous electromagnetic formulation. The integral equations for conical diffraction were obtained using the boundary integrals of the single and double layer potentials including the tangential derivative of single layer potentials interpreted as singular integrals. We derive an important formula for the calculation of the absorption in conical diffraction. Some rules which are expedient for the numerical implementation of the theory are presented. The efficiencies and polarization angles compared with those obtained by Lifeng Li for transmission and reflection gratings are in a good agreement. The code developed and tested is found to be accurate and efficient for solving off-plane diffraction problems including high-conductive surfaces, borders with edges, real border profiles, and gratings working at short wavelengths.

*Appeared in*

- J. Opt. Soc. Amer. A, 27 (2010) pp. 585--597 under new title: Solving conical diffraction grating problems with integral equations

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# Bifurcations in a model of monolithic passively mode-locked semiconductor laser

*Authors*

- Vladimirov, Andrei G.
- Pimenov, Alexander
- Rachinskii, Dmitry

*2010 Mathematics Subject Classification*

- 78A60 34C23

*2008 Physics and Astronomy Classification Scheme*

- 42.60.Fc 42.55.Px 42.60.Mi 42.65.Pc

*Keywords*

- semiconductor laser, quantum dots, bifurcations, mode-locking

*DOI*

*Abstract*

Operation regimes of a two section monolithic quantum dot mode-locked laser are studied theoretically using a model that takes into account carrier exchange between quantum dots and wetting layer. It is shown that when the absorber section length is large enough the laser exhibits bistability between laser off state and different mode-locking regimes. Q-switching instability leading to slow modulation of the mode-locked pulse peak intensity is completely eliminated in this case.

*Appeared in*

- IEEE J. Quantum Electron., 45 (2009) pp. 462--468.

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# Optimal and robust a posteriori error estimates in $L^infty(L^2)$ for the approximation of Allen--Cahn equations past singularities

*Authors*

- Bartels, Sören
- Müller, Rüdiger

ORCID: 0000-0003-2643-722X

*2010 Mathematics Subject Classification*

- 65M60 65M15 35K55

*Keywords*

- Allen-Cahn equation, mean curvature flow, finite element method, error analysis, adaptive methods

*DOI*

*Abstract*

Optimal a posteriori error estimates in $L^infty(0,T;L^2(O))$ are derived for the finite element approximation of Allen-Cahn equations. The estimates depend on the inverse of a small parameter only in a low order polynomial and are valid past topological changes of the evolving interface. The error analysis employs an elliptic reconstruction of the approximate solution and applies to a large class of conforming, nonconforming, mixed, and discontinuous Galerkin methods. Numerical experiments illustrate the theoretical results.

*Appeared in*

- Math. Comp., 80 (2011) pp. 761--780.

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# Uniform exponential decay of the free energy for Voronoi finite volume discretized reaction-diffusion systems

*Authors*

- Glitzky, Annegret

*2010 Mathematics Subject Classification*

- 35B40 35K57 35R05 46E39 65M12

*Keywords*

- Reaction-diffusion systems, energy estimates, thermodynamic equilibria, asymptotic behaviour, time and space discretization, boundary conforming Delaunay grid, Voronoi finite volume scheme, discrete Sobolev-Poincaré inequality

*DOI*

*Abstract*

Our focus are energy estimates for discretized reaction-diffusion systems for a finite number of species. We introduce a discretization scheme (Voronoi finite volume in space and fully implicit in time) which has the special property that it preserves the main features of the continuous systems, namely positivity, dissipativity and flux conservation. For a class of Voronoi finite volume meshes we investigate thermodynamic equilibria and prove for solutions to the evolution system the monotone and exponential decay of the discrete free energy to its equilibrium value with a unified rate of decay for this class of discretizations. The essential idea is an estimate of the free energy by the dissipation rate which is proved indirectly by taking into account sequences of Voronoi finite volume meshes. Essential ingredient in that proof is a discrete Sobolev-Poincaré inequality.

*Appeared in*

- Math. Nachr., 284 (2011) pp. 2159--2174.

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# Padé approximant for refractive index and nonlocal envelope equations

*Authors*

- Amiranashvili, Shalva

ORCID: 0000-0002-8132-882X - Bandelow, Uwe

ORCID: 0000-0003-3677-2347 - Mielke, Alexander

ORCID: 0000-0002-4583-3888

*2010 Mathematics Subject Classification*

- 78A60

*2008 Physics and Astronomy Classification Scheme*

- 42.65.-k, 42.65.Re, 31.15.-p, 02.30.Mv

*Keywords*

- Short optical pulses, Envelope equation, Padé approximant

*DOI*

*Abstract*

Padé approximant is superior to Taylor expansion when functions contain poles. This is especially important for response functions in complex frequency domain, where singularities are present and intimately related to resonances and absorption. Therefore we introduce a diagonal Padé approximant for the complex refractive index and apply it to the description of short optical pulses. This yields a new nonlocal envelope equation for pulse propagation. The model offers a global representation of arbitrary medium dispersion and absorption, e.g., the fulfillment of the Kramers-Kronig relation can be established. In practice, the model yields an adequate description of spectrally broad pulses for which the polynomial dispersion operator diverges and can induce huge errors.

*Appeared in*

- Opt. Commun., 283 (2010) pp. 480--485.

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# Fast numerical methods for waves in periodic media

*Authors*

- Ehrhardt, Matthias
- Zheng, Chunxiong

*2010 Mathematics Subject Classification*

- 65M99 35B27 35Q60 35J05 81-08

*Keywords*

- artificial boundary conditions, periodic potential, Schrödinger equation, Helmholtz equation, hyperbolic equation, unbounded domain, Dirichlet-to-Neumann maps, Robin-to-Robin maps, band structure, Floquet-Bloch theory, high-order finite elements

*DOI*

*Abstract*

Periodic media problems widely exist in many modern application areas like semiconductor nanostructures (e.g. quantum dots and nanocrystals), semi-conductor superlattices, photonic crystals (PC) structures, meta materials or Bragg gratings of surface plasmon polariton (SPP) waveguides, etc. Often these application problems are modeled by partial differential equations with periodic coefficients and/or periodic geometries.

In order to numerically solve these periodic structure problems efficiently one usually confines the spatial domain to a bounded computational domain (i.e. in a neighborhood of the region of physical interest). Hereby, the usual strategy is to introduce so-called emphartificial boundaries and impose suitable boundary conditions. For wave-like equations, the ideal boundary conditions should not only lead to well-posed problems, but also mimic the perfect absorption of waves traveling out of the computational domain through the artificial boundaries.

In the first part of this chapter we present a novel analytical impedance expression for general second order ODE problems with periodic coefficients. This new expression for the kernel of the Dirichlet-to-Neumann mapping of the artificial boundary conditions is then used for computing the bound states of the Schrödinger operator with periodic potentials at infinity. Other potential applications are associated with the exact artificial boundary conditions for some time-dependent problems with periodic structures. As an example, a two-dimensional hyperbolic equation modeling the TM polarization of the electromagnetic field with a periodic dielectric permittivity is considered.

In the second part of this chapter we present a new numerical technique for solving periodic structure problems. This novel approach possesses several advantages. First, it allows for a fast evaluation of the Sommerfeld-to-Sommerfeld operator for periodic array problems. Secondly, this computational method can also be used for bi-periodic structure problems with local defects. In the sequel we consider several problems, such as the exterior elliptic problems with strong coercivity, the time-dependent Schrödinger equation and the Helmholtz equation with damping.

Finally, in the third part we consider periodic arrays that are structures consisting of geometrically identical subdomains, usually called periodic cells. We use the Helmholtz equation as a model equation and consider the definition and evaluation of the exact boundary mappings for general semi-infinite arrays that are periodic in one direction for any real wavenumber. The well-posedness of the Helmholtz equation is established via the emphlimiting absorption principle (LABP).

An algorithm based on the doubling procedure of the second part of this chapter and an extrapolation method is proposed to construct the exact Sommerfeld-to-Sommerfeld boundary mapping. This new algorithm benefits from its robustness and the simplicity of implementation. But it also suffers from the high computational cost and the resonance wave numbers. To overcome these shortcomings, we propose another algorithm based on a conjecture about the asymptotic behaviour of limiting absorption principle solutions. The price we have to pay is the resolution of some generalized eigenvalue problem, but still the overall computational cost is significantly reduced. Numerical evidences show that this algorithm presents theoretically the same results as the first algorithm. Moreover, some quantitative comparisons between these two algorithms are given.

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# Simulations of 3D/4D precipitation processes in a turbulent flow field

*Authors*

- John, Volker

ORCID: 0000-0002-2711-4409 - Roland, Michael

*2010 Mathematics Subject Classification*

- 76F65 65M06 65M60

*Keywords*

- population balance system, precipitation process, incompressible Navier--Stokes equations, transport equations, FEM--FCT scheme, upwind finite difference method

*DOI*

*Abstract*

Precipitation processes are modeled by population balance systems. A very expensive part of the simulation of population balance systems is the solution of the equation for the particle size distribution (PSD) since this equation is defined in a higher dimensional domain than the other equations in the system. This paper studies different approaches for the solution of this equation: two finite difference upwind schemes and a linear finite element flux--corrected transport method. It is shown that the different schemes lead to qualitatively different solutions for an output of interest.

*Appeared in*

- Numerical Mathematics and Advanced Applications 2009, G. Kreiss, P. Lötstedt, A. Målqvist, M. Neytcheva, eds., Springer, Heidelberg et al., 2010, pp. 479-- 487

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# Random and deterministic fragmentation models

*Authors*

- Wagner, Wolfgang

*2010 Mathematics Subject Classification*

- 60J75

*Keywords*

- fragmentation models, kinetic equations, Markov jump processes, explosion property

*DOI*

*Abstract*

Random and deterministic fragmentation models are considered. Their relationship is studied by deriving different forms of the kinetic fragmentation equation from the corresponding stochastic models. Results related to the problem of non-conservation of mass (phase transition into dust) are discussed. Illustrative examples are given and some open problems are mentioned.

*Appeared in*

- Monte Carlo Methods Appl., 16 (2010) pp. 399--420.

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# Profile reconstruction in EUV scatterometry: Modeling and uncertainty estimates

*Authors*

- Gross, Hermann
- Rathsfeld, Andreas
- Scholze, Frank
- Bär, Markus

*2010 Mathematics Subject Classification*

- 35R30 74J20 35J05

*2008 Physics and Astronomy Classification Scheme*

- 42.30.Tz

*Keywords*

- EUV scatterometry, inverse scattering, lithography masks, uncertainty estimates

*DOI*

*Abstract*

Scatterometry as a non-imaging indirect optical method in wafer metrology is also relevant to lithography masks designed for Extreme Ultraviolet Lithography, where light with wavelengths in the range of 13 nm is applied. The solution of the inverse problem, i.e. the determination of periodic surface structures regarding critical dimensions (CD) and other profile properties from light diffraction patterns, is incomplete without knowledge of the uncertainties associated with the reconstructed parameters. With decreasing feature sizes of lithography masks, increasing demands on metrology techniques and their uncertainties arise. The numerical simulation of the diffraction process for periodic 2D structures can be realized by the finite element solution of the two-dimensional Helmholtz equation. For typical EUV masks the ratio period over wave length is so large, that a generalized finite element method has to be used to ensure reliable results with reasonable computational costs. The inverse problem can be formulated as a non-linear operator equation in Euclidean spaces. The operator maps the sought mask parameters to the efficiencies of diffracted plane wave modes. We employ a Gauß-Newton type iterative method to solve this operator equation and end up minimizing the deviation of the measured efficiency or phase shift values from the calculated ones. We apply our reconstruction algorithm for the measurement of a typical EUV mask composed of TaN absorber lines of about 80 nm height, a period of 420 nm resp. 720 nm, and with an underlying MoSi-multilayer stack of 300 nm thickness. Clearly, the uncertainties of the reconstructed geometric parameters essentially depend on the uncertainties of the input data and can be estimated by various methods. We apply a Monte Carlo procedure and an approximative covariance method to evaluate the reconstruction algorithm. Finally, we analyze the influence of uncertainties in the widths of the multilayer stack by the Monte Carlo method.

*Appeared in*

- Meas. Sci. Technol., 20 (2009) pp. 105102 (11 pp).

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# Stripe-array diode-laser in an off-axis external cavity: Theory and experiment

*Authors*

- Jechow, Andreas
- Lichtner, Mark
- Menzel, Ralf
- Radziunas, Mindaugas
- Skoczowsky, Danilo
- Vladimirov, Andrei G.

*2010 Mathematics Subject Classification*

- 78A60 37M05 78A45

*2008 Physics and Astronomy Classification Scheme*

- 42.55.Px 42.60.Mi 42.60.Da 42.65.Sf

*Keywords*

- semiconductor laser array, broad area laser, synchronization, supermode

*DOI*

*Abstract*

Stripe-array diode lasers naturally operate in an anti-phase supermode. This produces a sharp double lobe far field at angles $pm alpha$ depending on the period of the array. In this paper a 40 emitter gain guided stripe-array laterally coupled by off-axis filtered feedback is investigated experimentally and numerically. We predict theoretically and confirm experimentally that at doubled feedback angle $2 alpha$ a stable higher order supermode exists with twice the number of emitters per array period. The theoretical model is based on time domain traveling wave equations for optical fields coupled to the carrier density equation taking into account diffusion of carriers. Feedback from the external reflector is modeled using Fresnel integration.

*Appeared in*

- Optics Express, 17 (2009) pp. 19599-19604.

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# Structural adaptive smoothing: Principles and applications in imaging

*Authors*

- Polzehl, Jörg

ORCID: 0000-0001-7471-2658 - Tabelow, Karsten

ORCID: 0000-0003-1274-9951

*2010 Mathematics Subject Classification*

- 68U10 92C55 62G08

*Keywords*

- Image Enhancement, Functional Magnetic Resonance Imaging, Diffusion Tensor Imaging, Structural Adaptive Smoothing

*DOI*

*Abstract*

Structural adaptive smoothing provides a new concept of edge-preserving non-parametric smoothing methods. In imaging it employs qualitative assumption on the underlying homogeneity structure of the image. The chapter describes the main principles of the approach and discusses applications ranging from image denoising to the analysis of functional and diffusion weighted Magnetic Resonance experiments.

*Appeared in*

- P. Jörg, T. Karsten, Structural adaptive smoothing: principles and applications in imaging, F. L., D. R., J. G., L. M. - C. van, D. L., eds., vol. 41 of Computational imaging and vision, Springer, London, 2012, pp. 65--81

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# A microeconomic explanation of the EPK paradox

*Authors*

- Härdle, Wolfgang Karl
- Krätschmer, Volker
- Moro, Rouslan

*2010 Mathematics Subject Classification*

- 15A29 62G07 62G35

*Keywords*

- Pricing kernel, representative agent, empirical pricing kernel, epk paradox, state dependent utilities, switching points

*DOI*

*Abstract*

Supported by some recent investigations the empirical pricing kernel paradox might be viewed as a stylized fact. In ChabiYo et al. (2008) simulation studies have been presented which suggest that this paradox might be caused by regime switching of stock prices in financial markets. Alternatively, we want to emphasize a microeconomic view. Based on an economic model with state dependent utilities for the financial investors we succeed in explaining the paradox by changes of risk attitudes. Theoretically, the change behaviour is compressed in the pricing kernels. As a starting point for empirical insights we shall develop and investigate inverse problems in terms of data fits for estimated basic values of the pricing kernel.

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# On the unitary equivalence of absolutely continuous parts of self-adjoint extensions

*Authors*

- Malamud, Mark M.
- Neidhardt, Hagen

*2010 Mathematics Subject Classification*

- 47A57 47B25, 47A55

*Keywords*

- Symmetric operators, self-adjoint extensions, boundary triplets, Weyl functions, spectral multiplicity, unitary equivalence, direct sums of symmetric operators, Sturm-Liouville operators with operator potentials

*DOI*

*Abstract*

The classical Weyl-von Neumann theorem states that for any self-adjoint operator $A$ in a separable Hilbert space $gotH$ there exists a (non-unique) Hilbert-Schmidt operator $C = C^*$ such that the perturbed operator $A+C$ has purely point spectrum. We are interesting whether this result remains valid for non-additive perturbations by considering self-adjoint extensions of a given densely defined symmetric operator $A$ in $mathfrak H$ and fixing an extension $A_0 = A_0^*$. We show that for a wide class of symmetric operators the absolutely continuous parts of extensions $widetilde A = widetilde A^*$ and $A_0$ are unitarily equivalent provided that their resolvent difference is a compact operator. Namely, we show that this is true whenever the Weyl function $M(cdot)$ of a pair $A,A_0$ admits bounded limits $M(t) := wlim_yto+0M(t+iy)$ for a.e. $t in mathbbR$. This result is applied to direct sums of symmetric operators and Sturm-Liouville operators with operator potentials.

*Appeared in*

- J. Funct. Anal., 360 (2011) pp. 613--638.

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# The heat treatment of steel --- A mathematical control problem

*Authors*

- Hömberg, Dietmar
- Kern, Daniela

*2010 Mathematics Subject Classification*

- 74P10 80A20 93C20

*Keywords*

- Laser surface hardening, optimal control, pyrometer control

*DOI*

*Abstract*

The goal of this paper is to show how the heat treatment of steel can be modelled in terms of a mathematical optimal control problem. The approach is applied to laser surface hardening and the cooling of a steel slab including mechanical effects. Finally, it is shown how the results can be utilized in industrial practice by a coupling with machine-based control.

*Appeared in*

- Materialwiss. Werkstofftech., 40 (2009) pp. 438--442.

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# Optimal control of 3D state-constrained induction heating problems with nonlocal radiation effects

*Authors*

- Druet, Pierre-Étienne

ORCID: 0000-0001-5303-0500 - Klein, Olaf

ORCID: 0000-0002-4142-3603 - Sprekels, Jürgen
- Tröltzsch, Fredi
- Yousept, Irwin

*2010 Mathematics Subject Classification*

- 49K20 35D10 35J60 78M50 80M50

*Keywords*

- State-constrained optimization, Maxwell equations, nonlocal radiation boundary conditions, induction heating, crystal growth

*DOI*

*Abstract*

The paper is concerned with a class of optimal heating problems in semiconductor single crystal growth processes. To model the heating process, time-harmonic Maxwell equations are considered in the system of the state. Due to the high temperatures characterizing crystal growth, it is necessary to include nonlocal radiation boundary conditions and a temperature-dependent heat conductivity in the description of the heat transfer process. The first goal of this paper is to prove the existence and uniqueness of the solution to the state equation. The regularity analysis associated with the time harmonic Maxwell equations is also studied. In the second part of the paper, the existence and uniqueness of the solution to the corresponding linearized equation is shown. With this result at hand, the differentiability of the control-to-state mapping operator associated with the state equation is derived. Finally, based on the theoretical results, first oder necessary optimality conditions for an associated optimal control problem are established.

*Appeared in*

- SIAM J. Control Optim., 49 (2011) pp. 1707--1736.

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# Tensor product approximations of high dimensional potentials

*Authors*

- Lanzara, Flavia
- Maz'ya, Vladimir
- Schmidt, Gunther

*2010 Mathematics Subject Classification*

- 41A30 65D15 41A63 41A25

*Keywords*

- Cubature of integral operators, multivariate approximation, tensor product approximation

*DOI*

*Abstract*

The paper is devoted to the efficient computation of high-order cubature formulas for volume potentials obtained within the framework of approximate approximations. We combine this approach with modern methods of structured tensor product approximations. Instead of performing high-dimensional discrete convolutions the cubature of the potentials can be reduced to a certain number of one-dimensional convolutions leading to a considerable reduction of computing resources. We propose one-dimensional integral representions of high-order cubature formulas for n-dimensional harmonic and Yukawa potentials, which allow low rank tensor product approximations.

*Appeared in*

- Math. Comp., 80 (2011) pp. 887--904 under the new title Ön the fast computation of high dimensional volume potentials".

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# Integral methods for conical diffraction

*Authors*

- Schmidt, Gunther

*2010 Mathematics Subject Classification*

- 31A10 35J05 35Q60 35K50 45F15 78A45

*Keywords*

- Diffraction, periodic structure, integral equation method, oblique incidence, system of singular integral equations

*DOI*

*Abstract*

The paper is devoted to the scattering of a plane wave obliquely illuminating a periodic surface. Integral equation methods lead to a system of singular integral equations over the profile. Using boundary integral techniques we study the equivalence of these equations to the electromagnetic formulation, the existence and uniqueness of solutions under general assumptions on the permittivity and permeability of the materials. In particular, new results for materials with negative permittivity or permeability are established.

*Appeared in*

- Around the Research of Vladimir Maz'ya II. Partial Differential Equations, A. Laptev, ed., 12 of International Mathematical Series, Springer Science+Business Media, New York [et al.], 2010, pp. 337--363 under the new title "Boundary integral methods for periodic scattering problems''.

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# Parameter estimation in time series analysis

*Authors*

- Spokoiny, Vladimir

ORCID: 0000-0002-2040-3427

*2010 Mathematics Subject Classification*

- 62F10 62J12 62F25 62H12

*Keywords*

- Exponential risk bounds, rate function, quasi maximum likelihood autoregression, generalized linear models, quantile regression

*DOI*

*Abstract*

The paper offers a novel unified approach to studying the accuracy of parameter estimation for a time series. Important features of the approach are: (1) The underlying model is not assumed to be parametric. (2) The imposed conditions on the model are very mild and can be easily checked in specific applications. (3) The considered time series need not to be ergodic or stationary. The approach is equally applicable to ergodic, unit root and explosive cases. (4) The parameter set can be unbounded and non-compact. (5) No conditions on parameter identifiability are required. (6) The established risk bounds are nonasymptotic and valid for large, moderate and small samples. (7) The results describe confidence and concentration sets rather than the accuracy of point estimation. The whole approach can be viewed as complementary to the classical one based on the asymptotic expansion of the log-likelihood. In particular, it claims a consistency of the considered estimate in a rather general sense, which usually is assumed to be fulfilled in the asymptotic analysis. In standard situations under ergodicity conditions, the usual rate results can be easily obtained as corollaries from the established risk bounds. The approach and the results are illustrated on a number of popular time series models including autoregressive, Generalized Linear time series, ARCH and GARCH models and meadian/quantile regression.

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# Hysteresis in the context of hydrogen storage and lithium-ion batteries

*Authors*

- Dreyer, Wolfgang
- Guhlke, Clemens
- Huth, Robert

*2010 Mathematics Subject Classification*

- 74N30 74A15 74F25 74G65

*2008 Physics and Astronomy Classification Scheme*

- 82.47.Aa 82.60.Hc 82.60.Qr

*Keywords*

- thermodynamics, phase transitions, hysteresis, chemical potentials, elasticity, hydrogen, lithium-ion batteries

*DOI*

*Abstract*

The processes of reversible storage of hydrogen in a metal by loading and unloading and of charging and discharging of lithium-ion batteries have many things in common. The both processes are accompanied by a phase transition and loading and unloading run along different paths, so that hysteretic behavior is observed. For hydrogen storage we consider a fine powder of magnesium (Mg) particles and lithium storage is studied for iron phosphate (FePO$_4$) particles forming the cathode of a lithium-ion battery. The mathematical models that are established in citeDGJ08 and citeDGH09a, describe phase transitions and hysteresis exclusively in a single particle and on that basis they can predict the observed hysteretic plots with almost horizontal plateaus. Interestingly the models predict that the coexistence of a 2-phase system in an individual particle disappears, if its size is below a critical value. However, measurements reveal that this is qualitatively not reflected by the mentioned hysteretic plots of loading and unloading. In other words: The behavior of a storage system consisting of many particles is qualitatively independent of the fact whether the individual particles itself develop a 2-phase system or if they remain in a single phase state. This apparent paradoxical observation will be resolved in this article. It will be shown that if each of the individual particles homogeneously distributes the supplied matter, nevertheless the many particle ensemble exhibits phase transition and hysteresis, because one of the two phases is realized in some part of the particles while the remaining part is in the other phase.

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# Complete-damage evolution based on energies and stresses

*Authors*

- Mielke, Alexander

ORCID: 0000-0002-4583-3888

*2010 Mathematics Subject Classification*

- 35K65 35K85 49S05 74C05 74R05

*Keywords*

- Energetic solution, rate-independent system, complete damage, parametrized Gamma convergence

*DOI*

*Abstract*

The rate-independent damage model recently developed in Bouchitté, Mielke, Roubíček ``A complete-damage problem at small strains" allows for complete damage, such that the deformation is no longer well-defined. The evolution can be described in terms of energy densities and stresses. Using concepts of parametrized Gamma convergence, we generalize the theory to convex, but non-quadratic elastic energies by providing Gamma convergence of energetic solutions from partial to complete damage under rather general conditions.

*Appeared in*

- Discrete Contin. Dyn. Syst. Ser. S, 4 (2011) pp. 423--439.

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# Variational approach to scattering of plane elastic waves by diffraction gratings

*Authors*

- Elschner, Johannes
- Hu, Guanghui

*2010 Mathematics Subject Classification*

- 74J20 74B05 35J55 35Q72

*Keywords*

- Elastic waves, diffraction gratings, Navier equation, variational formulation

*DOI*

*Abstract*

The scattering of a time-harmonic plane elastic wave by a two-dimensional periodic structure is studied. The grating profile is given by a Lipschitz curve on which the displacement vanishes. Using a variational formulation in a bounded periodic cell involving a nonlocal boundary operator, existence of solutions in quasi-periodic Sobolev spaces is investigated by establishing the Fredholmness of the operator generated by the corresponding sesquilinear form. Moreover, by a Rellich identity, uniqueness is proved under the assumption that the grating profile is given by a Lipschitz graph. The direct scattering problem for transmission gratings is also investigated. In this case, uniqueness is proved except for a discrete set of frequencies.

*Appeared in*

- Math. Methods Appl. Sci., 33 (2010) pp. 1924--1941.

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# A priori error analysis for state constrained boundary control problems. Part I: Control discretization

*Authors*

- Krumbiegel, Klaus
- Meyer, Christian
- Rösch, Arnd

*2010 Mathematics Subject Classification*

- 49K20 49M25 49M29

*Keywords*

- Optimal control, state constraints, boundary control, regularization, virtual control, numerical approximation, finite elements

*DOI*

*Abstract*

This is the first of two papers concerned with a state-constrained optimal control problems with boundary control, where the state constraints are only imposed in an interior subdomain. We apply the virtual control concept introduced in [20] to regularize the problem. The arising regularized optimal control problem is discretized by finite elements and linear and continuous ansatz functions for the boundary control. In the first part of the work, we investigate the errors induced by the regularization and the discretization of the boundary control. The second part deals with the error arising from discretization of the PDE. Since the state constraints only appear in an inner subdomain, the obtained order of convergence exceeds the known results in the field of a priori analysis for state-constrained problems.

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# A priori error analysis for state constrained boundary control problems. Part II: Full discretization

*Authors*

- Krumbiegel, Klaus
- Meyer, Christian
- Rösch, Arnd

*2010 Mathematics Subject Classification*

- 49K20 49M25 49M29

*Keywords*

- Optimal control, state constraints, boundary control, regularization, virtual control, numerical approximation, finite elements

*DOI*

*Abstract*

This is the second of two papers concerned with a state-constrained optimal control problems with boundary control, where the state constraints are only imposed in an interior subdomain. We apply the virtual control concept introduced in [26] to regularize the problem. The arising regularized optimal control problem is discretized by finite elements and linear and continuous ansatz functions for the boundary control. In the first part of the work, we investigate the errors induced by the regularization and the discretization of the boundary control. The second part deals with the error arising from discretization of the PDE. Since the state constraints only appear in an inner subdomain, the obtained order of convergence exceeds the known results in the field of a priori analysis for state-constrained problems. The theoretical results are illustrated by numerical computations.

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# Polyhedral systems in finite and infinite dimensions with applications to robust stability of variational inequalities

*Authors*

- Henrion, René
- Mordukhovich, Boris
- Mau Nam, Nguyen

*2010 Mathematics Subject Classification*

- 49J52 49K40 58C20

*Keywords*

- Variational analysis and optimization, reflexive Banach spaces, polyhedral sets, parametric variational inequalities, robust stability, generalized differentiation, coderivatives and second-order subdifferentials

*DOI*

*Abstract*

This paper concerns second-order analysis for a remarkable class of variational systems in finite-dimensional and infinite-dimensional spaces, which is particularly important for the study of optimization and equilibrium problems with equilibrium constraints. Systems of this type are described via variational inequalities over polyhedral convex sets and allow us to provide a comprehensive local analysis by using appropriate generalized differentiation of the normal cone mappings for such sets. In this paper we efficiently compute the required coderivatives of the normal cone mappings exclusively via the initial data of polyhedral sets in reflexive Banach spaces. This provides the main tools of second-order variational analysis allowing us, in particular, to derive necessary and sufficient conditions for robust Lipschitzian stability of solution maps to parameterized variational inequalities with evaluating the exact bound of the corresponding Lipschitzian moduli. The efficient coderivative calculations and characterizations of robust stability obtained in this paper are the first results in the literature for the problems under consideration in infinite-dimensional spaces. Most of them are also new in finite dimensions.

*Appeared in*

- SIAM J. Optim., 20 (2010) pp. 2199--2227 under the new title "Second-order analysis of polyhedral systems in finite and infinite dimensions with applications to robust stability of variational inequalities''.

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# Absorbing boundary conditions for hyperbolic systems

*Authors*

- Ehrhardt, Matthias

*2010 Mathematics Subject Classification*

- 65M06, 35L50

*DOI*

*Abstract*

This paper deals with absorbing boundary conditions for hyperbolic systems in one and two space dimensions. We prove the strict well-posedness of the resulting initial boundary value problem in 1D. Afterwards we establish the GKS-stability of the corresponding Lax-Wendroff-type finite difference scheme. Hereby, we have to extend the classical proofs, since the (discretized) absorbing boundary conditions do not fit the standard form of boundary conditions for hyperbolic systems.

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# Absorbing boundary conditions for hyperbolic systems

*Authors*

- Ehrhardt, Matthias

*2010 Mathematics Subject Classification*

- 65M06, 35L50

*Keywords*

- Absorbing boundary conditions, hyperbolic system, Engquist and Majda approach, strict well-posedness, GKS-stability

*DOI*

*Abstract*

This paper deals with absorbing boundary conditions for hyperbolic systems in one and two space dimensions. We prove the strict well-posedness of the resulting initial boundary value problem in 1D. Afterwards we establish the GKS-stability of the corresponding Lax-Wendroff-type finite difference scheme. Hereby, we have to extend the classical proofs, since the (discretized) absorbing boundary conditions do not fit the standard form of boundary conditions for hyperbolic systems.

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# Error estimates for space-time discretizations of a rate-independent variational inequality

*Authors*

- Mielke, Alexander

ORCID: 0000-0002-4583-3888 - Paoli, Laetitia
- Petrov, Adrien
- Stefanelli, Ulisse

*2010 Mathematics Subject Classification*

- 49J40, 74C05, 74N30

*Keywords*

- shape-memory materials, evolutionary variational inequalities, rate-independent processes, space-time discretization, error estimates

*DOI*

*Abstract*

This paper deals with error estimates for space-time discretizations in the context of evolutionary variational inequalities of rate-independent type. After introducing a general abstract evolution problem, we address a fully-discrete approximation and provide a priori error estimates. The application of the abstract theory to a semilinear case is detailed. In particular, we provide explicit space-time convergence rates for the isothermal Souza-Auricchio model for shape-memory alloys.

*Appeared in*

- SIAM J. Numer. Anal., 48 (2010) pp. 1625--1646.

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# Thin film rupture for large slip

*Authors*

- Peschka, Dirk

ORCID: 0000-0002-3047-1140 - Münch, Andreas
- Niethammer, Barbara

*2010 Mathematics Subject Classification*

- 76A20 35B40 76D08 60G18

*Keywords*

- Asymptotic behavior of solutions, Thin fluid films, Lubrication theory, Self-similar processes

*DOI*

*Abstract*

This paper studies the rupture of thin liquid films on hydrophobic substrates, assuming large slip at the liquidsolid interface. Using a recently developed em strong slip lubrication model, it is shown that the rupture passes through up to three self-similar regimes with different dominant balances and different scaling exponents. For one of these regimes the similarity is of second kind, and the similarity exponent is determined by solving a boundary value problem for a nonlinear ODE. For this regime we also prove finite-time rupture.

*Appeared in*

- J. Engrg. Math., 66 (2010) pp. 33--51.

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# Homogenization in gradient plasticity

*Authors*

- Hanke, Hauke

*2010 Mathematics Subject Classification*

- 35B27 74C05 74Q15

*Keywords*

- two-scale convergence, folding and unfolding, elasto-plasticity, gradient plasticity, Γ-convergence

*DOI*

*Abstract*

This paper yields a two-scale homogenization result for a rate-independent elastoplastic system. The presented model is a generalization of the classical model of linearized elastoplacticity with hardening, which is extended by a gradient term of the plastic variables. The associated stored elastic energy density has periodically oscillating coefficients, where the period is scaled by ε > 0 . The additional gradient term of the plastic variables z is contained in the elastic energy with a prefactor ε^{γ} (γ ≥ 0) . We derive different limiting models for ε → 0 in dependence of &gamma ;. For γ > 1 the limiting model is the two-scale model derived in [MielkeTimofte07], where no gradient term was present. For γ = 1 the gradient term of the plastic variable survives on the microscopic cell poblem, while for γ ∈ [0,1) the limit model is defined in terms of a plastic variable without microscopic fluctuation. The latter model can be simplified to a purely macroscopic elastoplasticity model by homogenisation of the elastic part.

*Appeared in*

- Math. Models Methods Appl. Sci., 21 (2011) pp. 1651--1684.

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# Heuristic parameter selection based on functional minimization: Optimality and model function approach

*Authors*

- Lu, Shuai
- Mathé, Peter

ORCID: 0000-0002-1208-1421

*2010 Mathematics Subject Classification*

- 65J20 47A52

*Keywords*

- parameter choice, L-curve, model function

*DOI*

*Abstract*

We analyze some parameter choice strategies in regularization of inverse problems, in particular the (modified) L-curve method and a variant of the Hanke-Raus rule. These are heuristic rules, free of the noise level, and they are based on minimization of some functional. We analyze these functionals, and we prove some optimality results under general smoothness conditions. We also devise some numerical approach for finding the minimizers, which uses model functions. Numerical experiments indicate that this is an efficient numerical procedure.

*Appeared in*

- Math. Comp., 82 (2013) pp. 1609--1630.

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# Direct computation of elliptic singularities across anisotropic, multi-material edges

*Authors*

- Haller-Dintelmann, Robert
- Kaiser, Hans-Christoph
- Rehberg, Joachim

*2010 Mathematics Subject Classification*

- 35B65 35J25 35R05

*Keywords*

- Elliptic transmission problems, mixed boundary problems, $W^1,p$ regularity

*DOI*

*Abstract*

We characterise the singularities of elliptic div-grad operators at points or edges where several materials meet on a Dirichlet or Neumann part of the boundary of a two- or three-dimensional domain. Special emphasis is put on anisotropic coefficient matrices. The singularities can be computed as roots of a characteristic transcendental equation. We establish uniform bounds for the singular values for several classes of three- and four-material edges. These bounds can be used to prove optimal regularity results for elliptic div-grad operators on three-dimensional, heterogeneous, polyhedral domains with mixed boundary conditions. We demonstrate this for the benchmark L--shape problem.

*Appeared in*

- J. Math. Sci. (N. Y.), 172 (2011) pp. 589--622.

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# On phase change of a vapor bubble in liquid water

*Authors*

- Dreyer, Wolfgang
- Duderstadt, Frank
- Hantke, Maren
- Warnecke, Gerald

*2010 Mathematics Subject Classification*

- 35Q53 35G25 76E15

*Keywords*

- Conservation laws, phase change, bubbles

*DOI*

*Abstract*

We consider a bubble of vapor and inert gas surrounded by the corresponding liquid phase. We study the behavior of the bubble due to phase change, i.e. condensation and evaporation, at the interface. Special attention is given to the effects of surface tension and heat production on the bubble dynamics as well as the propagation of acoustic elastic waves by including slight compressibility of the liquid phase. Separately we study the influence of the three phenomena heat conduction, elastic waves, and phase transition on the evolution of the bubble. The objective is to derive relations including the mass, momentum, and energy transfer between the phases. We find ordinary differential equations, in the cases of heat transfer and the emission of acoustic waves partial differential equations, that describe the bubble dynamics. From numerical evidence we deduce that the effect of phase transition and heat transfer on the behavior of the radius of the bubble is negligible. It turns out that the elastic waves in the liquid are of greatest importance to the dynamics of the bubble radius. The phase transition has a strong influence on the evolution of the temperature, in particular at the interface. Furthermore the phase transition leads to a drastic change of the water content in the bubble, so that a rebounding bubble is only possible, if it contains in addition an inert gas. In a forthcoming paper the equations derived are sought in order to close equations for multi-phase mixture balance laws for dispersed bubbles in liquids involving phase change. Also the model is used to make comparisons with experimental data on the oscillation of a laser induced bubble. For this case it was necessary to include the effect of an inert gas in the thermodynamic modeling of the phase transition.

*Appeared in*

- Contin. Mech. Thermodyn., 24 (2012) pp. 461--483.

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# Numerical algorithms for Schrödinger equation with artificial boundary conditions

*Authors*

- Čiegis, Raimondas
- Laukaitytė, Inga
- Radziunas, Mindaugas

*2010 Mathematics Subject Classification*

- 65M06 65M12 35Q40

*Keywords*

- finite difference method, Schrödinger problem, absorbing boundary conditions, transparent boundary conditions, numerical experiments

*DOI*

*Abstract*

We consider a one-dimensional linear Schrödinger problem defined on an infinite domain and approximated by the Crank-Nicolson type finite difference scheme. To solve this problem numerically we restrict the computational domain by introducing the reflective, absorbing or transparent artificial boundary conditions. We investigate the conservativity of the discrete scheme with respect to the mass and energy of the solution. Results of computational experiments are presented and the efficiency of different artificial boundary conditions is discussed.

*Appeared in*

- Numer. Funct. Anal. Optim., 30 (2009) pp. 903--923.

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# Global region of attraction of a periodic solution to a singularly perturbed parabolic problem in case of exchange of stability

*Authors*

- Butuzov, Valentin F.
- Nefedov, Nikolai N.
- Recke, Lutz
- Schneider, Klaus R.

*2010 Mathematics Subject Classification*

- 35B25 35B10 35K20 35K57

*Keywords*

- singularly perturbed reaction diffusion equation, exchange of stability, asymptotically stable periodic solution, global region of attraction

*DOI*

*Abstract*

We consider a singularly perturbed parabolic differential equation in case that the degenerate equation has two intersecting roots. In a previous paper we presented conditions under which there exists an asymptotically stable periodic solution satisfying no-flux boundary conditions. In this note we characterize a set of initial functions belonging to the global region of attraction of that periodic solution.

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# Analysis of M-stationary points to an EPEC modeling oligopolistic competition in an electricity spot market

*Authors*

- Henrion, René
- Outrata, Jiří
- Surowiec, Thomas

ORCID: 0000-0003-2473-4984

*2010 Mathematics Subject Classification*

- 90C30 49J53

*Keywords*

- Equilibrium problems with equilibrium constraints, EPEC, $M$-stationary solutions, electricity spot market, calmness

*DOI*

*Abstract*

We consider an equilibrium problem with equilibrium constraints (EPEC) as it arises from modeling competition in an electricity spot market (under ISO regulation). For a characterization of equilibrium solutions, so-called $M$-stationarity conditions are derived. This requires a structural analysis of the problem first (constraint qualifications, strong regularity). Second, the calmness property of a certain multifunction has to be verified in order to justify $M$-stationarity. Third, for stating the stationarity conditions, the co-derivative of a normal cone mapping has to be calculated. Finally, the obtained necessary conditions are made fully explicit in terms of the problem data for one typical constellation. A simple two-settlements example serves as an illustration.

*Appeared in*

- ESAIM Control Optim. Calc. Var., 18 (2012) pp. 295--317.

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# Differential, energetic, and metric formulations for rate-independent processes

*Authors*

- Mielke, Alexander

ORCID: 0000-0002-4583-3888

*2010 Mathematics Subject Classification*

- 49Q20 58E99

*Keywords*

- Energetic solutions, BV solutions, parametrized solutions, metric evolutionary systems, metric velocity, metric slope, vanishing-viscosity limit, vanishing-viscosity contact potential, doubly nonlinear equations, Gamma convergence

*DOI*

*Abstract*

We consider different solution concepts for rate-independent systems. This includes energetic solutions in the topological setting and differentiable, local, parametrized and BV solutions in the Banach-space setting. The latter two solution concepts rely on the method of vanishing viscosity, in which solutions of the rate-independent system are defined as limits of solutions of systems with small viscosity. Finally, we also show how the theory of metric evolutionary systems can be used to define parametrized and BV solutions in metric spaces.

*Appeared in*

- A. Mielke, Chapter: Differential, Energetic, and Metric Formulations for Rate-Independent Processes, in: Nonlinear PDE's and Applications, C.I.M.E. Summer School, Cetraro, Italy 2008, L. Ambrosio, G. Savaré, eds., vol. 2028 of Lecture Notes in Mathematics, Springer, Berlin Heidelberg, 2011, pp. 87--167

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# Self-similar rupture of viscous thin films in the strong-slip regime

*Authors*

- Peschka, Dirk

ORCID: 0000-0002-3047-1140 - Münch, Andreas
- Niethammer, Barbara

*2010 Mathematics Subject Classification*

- 76A20 35B40 76D08 60G18

*Keywords*

- Asymptotic behavior of solutions, Thin fluid films, Lubrication theory, Self-similar processes

*DOI*

*Abstract*

We consider rupture of thin viscous films in the strong-slip regime with small Reynolds numbers. Numerical simulations indicate that near the rupture point viscosity and van-der-Waals forces are dominant and that there are self-similar solutions of the second kind. For a corresponding simplified model we rigorously analyse self-similar behaviour. There exists a one-parameter family of self-similar solutions and we establish necessary and sufficient conditions for convergence to any self-similar solution in a certain parameter regime. We also present a conjecture on the domains of attraction of all self-similar solutions which is supported by numerical simulations.

*Appeared in*

- Nonlinearity, 23 (2010) pp. 409--427.

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# Eigenvalue fluctuations for lattice Anderson Hamiltonians

*Authors*

- Biskup, Marek
- Fukushima, Ryoki
- König, Wolfgang

ORCID: 0000-0002-4212-0065

*2010 Mathematics Subject Classification*

- 60F05 82B44 60H25

*Keywords*

- Anderson model, spectra of random operators, central limit theorem

*DOI*

*Abstract*

We consider the random Schrödinger operator on a large box in the lattice with a large prefactor in front of the Laplacian part of the operator, which is proportional to the square of the diameter of the box. The random potential is assumed to be independent and bounded; its expectation function and variance function is given in terms of continuous bounded functions on the rescaled box. Our main result is a multivariate central limit theorem for all the simple eigenvalues of this operator, after centering and rescaling. The limiting covariances are expressed in terms of the limiting homogenized eigenvalue problem; more precisely, they are equal to the integral of the product of the squares of the eigenfunctions of that problem times the variance function.

*Appeared in*

- SIAM J. Math. Anal., 48 (2016), pp. 2674--2700.

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# An anisotropic, inhomogeneous, elastically modified Gibbs--Thomson law as singular limit of a diffuse interface model

*Authors*

- Garcke, Harald
- Kraus, Christiane

*2010 Mathematics Subject Classification*

- 49Q20, 35B25 82B26 58B20

*Keywords*

- Van der Waals-Cahn-Hilliard energy, singular perturbations, anisotropic and inhomogeneous interfacial energy, elasticity, Gibbs-Thomson law

*DOI*

*Abstract*

We consider the sharp interface limit of a diffuse phase field model with prescribed total mass taking into account a spatially inhomogeneous anisotropic interfacial energy and an elastic energy. The main aim is the derivation of a weak formulation of an anisotropic, inhomogeneous, elastically modified Gibbs-Thomson law in the sharp interface limit. To this end we show that one can pass to the limit in the weak formulation of the Euler-Lagrange equation of the diffuse phase field energy.

*Appeared in*

- Adv. Math. Sci. Appl., 20 (2010) pp. 511--545.

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# Shifted linear systems in electromagnetics. Part I: Systems with identical right-hand sides

*Authors*

- Schlundt, Rainer

ORCID: 0000-0002-4424-4301 - Schmückle, Franz-Josef
- Heinrich, Wolfgang

*2010 Mathematics Subject Classification*

- 35Q60 65F10 65F15 65N22 78M25

*Keywords*

- Microwave device, Maxwell's equations, Scattering matrix, Boundary value problem, PML boundary condition, Eigenvalue problem, Linear algebraic equations, Multiple shifts, Krylov subspace method, Polynomial preconditioning

*DOI*

*Abstract*

We consider the solution of multiply shifted linear systems for a single right-hand side. The coefficient matrix is symmetric, complex, and indefinite. The matrix is shifted by different multiples of the identity. Such problems arise in a number of applications, including the electromagnetic simulation in the development of microwave and mm-wave circuits and modules. The properties of microwave circuits can be described in terms of their scattering matrix which is extracted from the orthogonal decomposition of the electric field. We discretize the Maxwell's equations with orthogonal grids using the Finite Integration Technique (FIT). Some Krylov subspace methods have been used to solve multiply shifted systems for about the cost of solving just one system. We use the QMR method based on coupled two-term recurrences with polynomial preconditioning.

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# Uniqueness in inverse elastic scattering with finitely many incident waves

*Authors*

- Elschner, Johannes
- Yamamoto, Masahiro

*2010 Mathematics Subject Classification*

- 35R30 35B60

*Keywords*

- Inverse scattering problem, uniqueness, elastic waves, polyhedral obstacle

*DOI*

*Abstract*

We consider the third and fourth exterior boundary value problems of linear isotropic elasticity and present uniqueness results for the corresponding inverse scattering problems with polyhedral-type obstacles and a finite number of incident plane elastic waves. Our approach is based on a reflection principle for the Navier equation.

*Appeared in*

- Inverse Problems, 26 (2010) pp. 045005/1--045005/8.

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# Random walks conditioned to stay in Weyl chambers of type C and D

*Authors*

- König, Wolfgang

ORCID: 0000-0002-4212-0065 - Schmid, Patrick

*2010 Mathematics Subject Classification*

- 60G50 60F17

*Keywords*

- random walks, Weyl chamber for type C and D, Dyson's Brownian motion, conditioning

*DOI*

*Abstract*

We construct the conditional versions of a multidimensional random walk given that it does not leave the Weyl chambers of type C and of type D, respectively, in terms of a Doob $h$-transform. Furthermore, we prove functional limit theorems for the rescaled random walks. This is an extension of recent work by Eichelsbacher and König who studied the analogous conditioning for the Weyl chamber of type A. Our proof follows recent work by Denisov and Wachtel who used martingale properties and a strong approximation of random walks by Brownian motion. Therefore, we are able to keep minimal moment assumptions. Finally, we present an alternate function that is amenable to an $h$-transform in the Weyl chamber of type C.

*Appeared in*

- Electron. Comm. Probab., (2010) pp. 286--295.

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# Dispersive stability of infinite dimensional Hamiltonian systems on lattices

*Authors*

- Mielke, Alexander

ORCID: 0000-0002-4583-3888 - Patz, Carsten

*2010 Mathematics Subject Classification*

- 37K60 34Dxx 41A60

*Keywords*

- Dispersive decay, oscillatory integrals, discrete Klein--Gordon equation, FPU chain

*DOI*

*Abstract*

We derive dispersive stability results for oscillator chains like the FPU chain or the discrete Klein-Gordon chain. If the nonlinearity is of degree higher than 4, then small localized initial data decay like in the linear case. For this, we provide sharp decay estimates for the linearized problem using oscillatory integrals and avoiding the nonoptimal interpolation between different $ell^p$ spaces.

*Appeared in*

- Appl. Anal., 89 (2010) pp. 1493--1512.

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# Global analytic expansion of solution for a class of linear parabolic systems with coupling of first order derivatives terms

*Authors*

- Kampen, Jörg

*2010 Mathematics Subject Classification*

- 35K40

*Keywords*

- parabolic systems of second order, global analytic expansions

*DOI*

*Abstract*

We derive global analytic representations of fundamental solutions for a class of linear parabolic systems with full coupling of first order derivative terms where coefficients may depend on space and time. Pointwise convergence of the global analytic expansion is proved. This leads to analytic representations of solutions of initial-boundary problems of first and second type in terms of convolution integrals or convolution integrals and linear integral equations. The results have both analytical and numerical impact. Analytically, our representations of fundamental solutions of coupled parabolic systems may be used to define generalized stochastic processes. Moreover, some classical analytical results based on a priori estimates of elliptic equations are a simple corollary of our main result. Numerically, accurate, stable and efficient schemes for computation and error estimates in strong norms can be obtained for a considerable class of Cauchy- and initial-boundary problems of parabolic type. Furthermore, there are obvious and less obvious applications to finance and physics.

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# Weighted energy-dissipation functionals for gradient flows

*Authors*

- Mielke, Alexander

ORCID: 0000-0002-4583-3888 - Stefanelli, Ulisse

*2010 Mathematics Subject Classification*

- 35K55

*Keywords*

- Variational principle, gradient flow, convergence

*DOI*

*Abstract*

We investigate a global-in-time variational approach to abstract evolution by means of the weighted energy-dissipation functionals proposed by Mielke & Ortiz in ``A class of minimum principles for characterizing the trajectories of dissipative systems''. In particular, we focus on gradient flows in Hilbert spaces. The main result is the convergence of minimizers and approximate minimizers of these functionals to the unique solution of the gradient flow. Sharp convergence rates are provided and the convergence analysis is combined with time-discretization. Applications of the theory to various classes of parabolic PDE problems are presented. In particular, we focus on two examples of microstructure evolution from S. Conti and M. Ortiz ``Minimum principles for the trajectories of systems governed by rate problems''.

*Appeared in*

- ESAIM Control Optim. Calc. Var., 17 (2011) pp. 52--85.

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# High-frequency averaging in semi-classical Hartree-type equations

*Authors*

- Giannoulis, Johannes
- Mielke, Alexander

ORCID: 0000-0002-4583-3888 - Sparber, Christof

*2010 Mathematics Subject Classification*

- 35B40 35C20 81Q20

*Keywords*

- Nonlinear Schrödinger equation, Hartree-type nonlinearity, Wiener space, propagation of pulses, justification of amplitude equations, high-frequency asymptotics, WKB approximation

*DOI*

*Abstract*

We investigate the asymptotic behavior of solutions to semi-classical Schröodinger equations with nonlinearities of Hartree type. For a weakly nonlinear scaling, we show the validity of an asymptotic superposition principle for slowly modulated highly oscillatory pulses. The result is based on a high-frequency averaging effect due to the nonlocal nature of the Hartree potential, which inhibits the creation of new resonant waves. In the proof we make use of the framework of Wiener algebras.

*Appeared in*

- Asymptot. Anal., 70 (2010) pp. 87--100.

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# R-matrix formalism for electron scattering in two dimensions

*Authors*

- Racec, Paul N.
- Racec, Roxana
- Neidhardt, Hagen

*2010 Mathematics Subject Classification*

- 35Q40 35P25 35B34

*2008 Physics and Astronomy Classification Scheme*

- 72.20.Dp, 73.40.-c, 73.63.-b

*Keywords*

- scattering, quantum transport, resonances, quantum dot, nanowire

*DOI*

*Abstract*

We investigate the scattering phenomena in two dimensions produced by a general finite-range nonseparable potential. This situation can appear either in a Cartesian geometry or in a heterostructure with cylindrical symmetry. Increasing the dimensionality of the scattering problem new processes as the scattering between conducting channels and the scattering from conducting to evanescent channels are allowed. For certain values of the energy, called resonance energy, the transmission through the scattering region changes dramatically in comparison with an one-dimensional problem. If the potential has an attractive character even the evanescent channels can be seen as dips of the total transmission. The multi-channel current scattering matrix is determined using its representation in terms of the R-matrix. The resonant transmission peaks are characterized quantitatively through the poles of the current scattering matrix. Detailed maps of the localization probability density sustain the physical interpretation of the resonances. Our formalism is applied to a quantum dot in a two dimensional electron gas and a conical quantum dot embedded inside a nanowire.

*Appeared in*

- Trends in Nanophysics, A. Aldea, V. Bârsan, eds., Engineering Materials, Springer, Berlin/Heidelberg, 2010, pp. 149--174

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# Discrete Sobolev--Poincaré inequalities for Voronoi finite volume approximations

*Authors*

- Glitzky, Annegret
- Griepentrog, Jens André

*2010 Mathematics Subject Classification*

- 46E35 46E39 31B10

*Keywords*

- Discrete Sobolev inequality, Sobolev integral representation, Voronoi finite volume mesh

*DOI*

*Abstract*

We prove a discrete Sobolev-Poincare inequality for functions with arbitrary boundary values on Voronoi finite volume meshes. We use Sobolev's integral representation and estimate weakly singular integrals in the context of finite volumes. We establish the result for star shaped polyhedral domains and generalize it to the finite union of overlapping star shaped domains. In the appendix we prove a discrete Poincare inequality for space dimensions greater or equal to two.

*Appeared in*

- SIAM J. Numer. Anal., 48 (2010) pp. 372--391.

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# Improving the stability of distributed-feedback tapered master-oscillator power-amplifiers

*Authors*

- Tronciu, Vasile Z.
- Lichtner, Mark
- Radziunas, Mindaugas
- Bandelow, Uwe

ORCID: 0000-0003-3677-2347 - Wenzel, Hans

*2010 Mathematics Subject Classification*

- 35Q60 35B30

*2008 Physics and Astronomy Classification Scheme*

- 42.55.Px, 42.60.Pk, 42.60.Mi

*Keywords*

- high power lasers, DFB MOPA, coupling coefficient, continuous wave

*DOI*

*Abstract*

We report theoretical results on the wavelength stabilization in distributed-feedback master-oscillator power-amplifiers which are compact semiconductor laser devices capable of emitting a high brilliance beam at an optical power of several Watts. Based on a traveling wave equation model we calculate emitted optical power and spectral maps in dependence on the pump of the power amplifier. We show that a proper choice of the Bragg grating type and coupling coefficient allows to optimize the laser operation, such that for a wide range of injection currents the laser emits a high intensity continuous wave beam.

*Appeared in*

- Opt. Quantum Electron., 41 (2010) pp. 531--537.

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# Phase transition and hysteresis in a rechargeable lithium battery revisited

*Authors*

- Dreyer, Wolfgang
- Gaberšček, Miran
- Guhlke, Clemens
- Huth, Robert
- Jamnik, Janko

*2010 Mathematics Subject Classification*

- 74N20 74A15 74N05 74B99

*Keywords*

- lithium-ion-battery, FePO4, thermodynamics phase transitions, hysteresis, chemical potentials, surface stress, deviatoric stress, elasticity

*DOI*

*Abstract*

We revisit a model which describes the evolution of a phase transition that occurs in the cathode of a rechargeable lithium battery during the process of charging/discharging. The model is capable to simulate hysteretic behavior of the voltage - charge characteristics with two voltage plateaus. The cathode consists of small crystalline storage particles. During discharging of the battery, the interstitial lattice sites of the particles are filled up with lithium atoms and these are released again during charging. We show within the context of a sharp interface model for a single particle of core-shell type that two mechanical phenomena go along with the phase transition during supply and removal of lithium. The lithium atoms need more space than is available by the interstitial lattice sites, which leads to a maximal relative change of the crystal volume of about 6%. Furthermore there is an interface between two adjacent phases that has very large curvature of the order of magnitude 10^{8} m^{-1}, which evoke here a discontinuity of the normal component of the stress. In order to simulate the dynamics within a single storage particle we establish a new initial and boundary value problem for a nonlinear PDE system that can be reduced in some limiting case to an ODE system. Furthermore we verify the common assumption of phase nucleation at the external boundary of the particle. In case of quasi static loading inner hysteresis loops in the voltage-charge plots are not contained within the setting of a core-shell model for a single storage particle. The origin of this fact is discussed in detail.

*Appeared in*

- European J. Appl. Math., 22 (2011) pp. 267--290.

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# Maximal parabolic regularity for divergence operators on distribution spaces

*Authors*

- Haller-Dintelmann, Robert
- Rehberg, Joachim

*2010 Mathematics Subject Classification*

- 35A05 35B6 35K15 35K20

*Keywords*

- Maximal parabolic regularity, quasilinear parabolic equations, mixed Dirichlet--Neumann conditions

*DOI*

*Abstract*

We show that elliptic second order operators A of divergence type fulfill maximal parabolic regularity on distribution spaces, even if the underlying domain is highly non-smooth, the coefficients of A are discontinuous and A is complemented with mixed boundary conditions. Applications to quasilinear parabolic equations with non-smooth data are presented.

*Appeared in*

- Parabolic Problems: The Herbert Amann Festschrift, J. Escher, P. Guidotti, M. Hieber, P. Mucha, J.W. Pruess, Y. Shibata, G. Simonett, CH. Walker, W. Zajaczkowski, eds., vol. 80 of Progress in Nonlinear Differential Equations and Their Applications, Springer, Basel, 2011, pp. 313--342

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# Scar-like structures and their localization in a perfectly square optical billiard

*Authors*

- Babushkin, Ihar

*2010 Mathematics Subject Classification*

- 78A60 65F15 15A52

*2008 Physics and Astronomy Classification Scheme*

- 05.45.Mt 42.60.Jf 42.25.Ja 42.55.Px

*Keywords*

- Optical quantum billiards, Semiconductor lasers, Light polarization

*DOI*

*Abstract*

We show that scar-like structures (SLS) in a wide aperture vertical cavity surface emitting laser (VCSEL) can be formed even in a perfectly square geometry due to interaction of polarization and spatial degrees of freedom of light. We show also that dissipation in the system induces an order among the cavity modes, so that SLS become preferred at lasing threshold. More generally, modes which are more localized both in coordinate and momentum space have in average lower losses.

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# The spectrum of delay differential equations with large delay

*Authors*

- Lichtner, Mark
- Wolfrum, Matthias
- Yanchuk, Serhiy

*2010 Mathematics Subject Classification*

- 34K06 34K20 34K26

*Keywords*

- Spectrum, Delay Differential Equation, Large Delay

*DOI*

*Abstract*

We show that the spectrum of linear delay differential equations with large delay splits into two different parts. One part, called the strong spectrum, converges to isolated points when the delay parameter tends to infinity. The other part, called the pseudocontinuous spectrum, accumulates near criticality and converges after rescaling to a set of spectral curves, called the asymptotic continuous spectrum. We show that the spectral curves and strong spectral points provide a complete description of the spectrum for sufficiently large delay and can be comparatively easily calculated by approximating expressions.

*Appeared in*

- SIAM J. Math. Anal., 43 (2011) pp. 788-802.

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# Coercivity for elliptic operators and positivity of solutions on Lipschitz domains

*Authors*

- Haller-Dintelmann, Robert
- Rehberg, Joachim

*2010 Mathematics Subject Classification*

- 35B05 35J20 35R05

*Keywords*

- Coercivity, mixed boundary problems, positivity of solutions

*DOI*

*Abstract*

We show that usual second order operators in divergence form satisfy coercivity on Lipschitz domains if they are either complemented with homogeneous Dirichlet boundary conditions on a set of non-zero boundary measure or if a suitable Robin boundary condition is posed. Moreover, we prove the positivity of solutions in a general, abstract setting, provided that the right hand side is a positive functional. Finally, positive elements from $W^-1,2$ are identified as positive measures.

*Appeared in*

- Arch. Math. (Basel), 95 (2010) pp. 457--468.

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# Non-Poissonian statistics in an optical analog of quantum billiard with perfectly square boundaries

*Authors*

- Babushkin, Ihar

*2010 Mathematics Subject Classification*

- 78A60 65F15 15A52

*2008 Physics and Astronomy Classification Scheme*

- 05.45.Mt 42.60.Jf 42.25.Ja 42.55.Px

*Keywords*

- Optical quantum billiards, Semiconductor lasers, Light polarization

*DOI*

*Abstract*

We study deviation from the Poissonian statistics of the frequency spacing distribution, appearing due to coupling of polarizational and transverse degrees of freedom in a perfectly square vertical cavity surface emitting laser. The deviation can be controlled by strength of the intracavity anisotropy and its alignment to the device boundaries.

*Appeared in*

- Phys. Lett. A, 374 (2010) pp. 896-900.

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# The behavior of a many particle cathode in a lithium-ion battery

*Authors*

- Dreyer, Wolfgang
- Guhlke, Clemens
- Huth, Robert

*2010 Mathematics Subject Classification*

- 74N30 74A15 74F20 74G65

*2008 Physics and Astronomy Classification Scheme*

- 82.47.Aa 82.60.Hc 82.60.Qr 83.10.Tv

*Keywords*

- lithium-ion batteries, thermodynamics, phase transistion, hysteresis, many particle system, chemical potential, elasticity

*DOI*

*Abstract*

We study the almost reversible storage process of charging and discharging of lithium-ion batteries. That process is accompanied by a phase transition and charging and discharging run along different paths, so that hysteretic behavior is observed. We are interested in the storage problem of the cathode of a lithium-ion battery consisting of a system of many iron phosphate (FePO4) particles. There are mathematical models, see [DGJ08], [DGGHJ09] and [DG09], that describe phase transitions and hysteresis exclusively in a single storage particle and they can describe the observed hysteretic voltage-charge plots with almost horizontal plateaus. Interestingly the models predict that the coexistence of a 2-phase system in an individual particle disappears, if its size is below a critical value. The disappearance of the phase transition in the single particle model implies the disappearance of the hysteresis. However, in the experiment hysteretic behavior survives. In other words: The behavior of a storage system consisting of many particles is qualitatively independent of the fact whether the individual particles itself develop a 2-phase system or if they remain in a single phase state. This apparent paradoxical observation will be resolved in this article by a many particle model. It will be shown that if each of the individual particles is in a homogeneous state, nevertheless the many particle ensemble exhibits phase transition and hysteresis, because one of the two phases is realized in some part of the particles while the remaining particles are in the other phase. Mathematically speaking this phenomenon is due to the non-monotonicity of the relation between the chemical potential and the lithium mole fraction. The pressure-radius relation of a spherical elastic rubber balloon also exhibits non-monotone behavior. In fact, a system of many interconnected balloons behaves correspondingly to a cathode consisting of many storage particles. This analogy between the two systems is important, because the predictions of the many particle model can easier be tested with rubber balloons of macroscopic size than with an ensemble of microscopically small (FePO4) particles.

*Appeared in*

- Phys. D, 240 (2011) pp. 1008--1019.

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# Global spatial regularity for time dependent elasto-plasticity and related problems

*Authors*

- Knees, Dorothee

*2010 Mathematics Subject Classification*

- 35B65 49N60 74C05 74C10

*Keywords*

- elasto-plasticity, visco-plasticity, global regularity, reflection argument

*DOI*

*Abstract*

We study the global spatial regularity of solutions of generalized elasto-plastic models with linear hardening on smooth domains. Under natural smoothness assumptions on the data and the boundary we obtain that the displacements belong to L^∞((0,T);H^(3/2-δ)(Ω)) whereas the internal variables belong to L^∞((0,T);H^(1/2-δ)(Ω)). The key step in the proof is a reflection argument which gives the regularity result in directions normal to the boundary on the basis of tangential regularity results.

*Appeared in*

- Math. Models Methods Appl. Sci., vol. 20, no. 10 (2010) pp. 1823--1858 under the new title ``On global spatial regularity and convergence rates for time-dependent elasto-plasticity".

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