# Modelling of microstructure and its evolution in shape-memory-alloy single-crystals, in particular in CuAlNi

*Authors*

- Kružík, Martin
- Mielke, Alexander

ORCID: 0000-0002-4583-3888 - Roubíček, Tomáš

*2010 Mathematics Subject Classification*

- 35K85 49S05 74C15 74N15

*Keywords*

- Young measures, laminates, cubic-to-orthorhombic martensitic transformation, rate-independent evolution, energetic formulation, space-time discretization

*DOI*

*Abstract*

A continuum-mechanical description of the stored energy in shape-memory alloys is presented, with its multi-well character giving rise to a microstructure described, with a certain approximation, by special gradient Young measures. A rate-independent phenomenological dissipation is then considered to model a hysteretic response. Isothermal simulations with CuAlNi single crystal are presented.

*Appeared in*

- Meccanica, 40 (2005) pp. 389--418.

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# Pointwise asymptotic convergence of solutions for a phase separation model

*Authors*

- Krejčí, Pavel
- Zheng, Songmu

*2010 Mathematics Subject Classification*

- 80A22 35K50 35B40

*Keywords*

- Phase-field system, asymptotic phase separation, energy, entropy

*DOI*

*Abstract*

A new technique, combining the global energy and entropy balance equations with the local stability theory for dynamical systems, is used for proving that every solution to a non-smooth temperature-driven phase separation model with conserved energy converges pointwise in space to an equilibrium as time tends to infinity. Three main features are observed: the limit temperature is uniform in space, there exists a partition of the physical body into at most three constant limit phases, and the phase separation process has a hysteresis-like character.

*Appeared in*

- Discrete Contin. Dyn. Syst., 16 (2006) pp. 1-18.

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# Lubrication models with small to large slip lengths

*Authors*

- Münch, Andreas
- Wagner, Barbara
- Witelski, Thomas P.

*2010 Mathematics Subject Classification*

- 76A2 76D27

*2008 Physics and Astronomy Classification Scheme*

- 97.10.Gz 97.30.Qt

*Keywords*

- lubrication models, matched asymptotics, stability analysis, Navier slip condition, dewetting films

*DOI*

*Abstract*

A set of lubrication models for the thin film flow of incompressible fluids on solid substrates is derived and studied.The models are obtained as asymptotic limits of the Navier-Stokes equations with the Navier-slip boundary condition for different orders of magnitude for the slip-length parameter. Specifically, the influence of slip on the dewetting behavior of fluids on hydrophobic substrates is investigated here. Matched asymptotics are used to describe the dynamic profiles for dewetting films and comparison is given with computational simulations. The motion of the dewetting front shows transitions from being nearly linear in time for no-slip to $t^2/3$ as the slip is increased. For much larger slip lengths the front motion appears to become linear again. Correspondingly, the dewetting profiles undergo a transition from oscillatory to monotone decay into the uniform film layer for large slip.Increasing the slip further, to very large values, is associated with an increasing degree of asymmetry in the structure of the dewetting ridge profile.

*Appeared in*

- J. Engrg. Math., 53 (2005) pp. 359-383.

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# Electronic structure and optoelectronic properties of strained InAsSb/GaSb multi quantum wells

*Authors*

- Koprucki, Thomas

ORCID: 0000-0001-6235-9412 - Baro, Michael
- Bandelow, Uwe

ORCID: 0000-0003-3677-2347 - Tien, Tran Q.
- Weik, Fritz
- Tomm, Jens W.
- Grau, Markus
- Amann, Markus-Christian

*2010 Mathematics Subject Classification*

- 82D37 78A60

*Keywords*

- Optoelectronic devices, light emitting devices, Indium-Arsenide-Antimonide, strained multi quantum-wells, photoluminescence meassurements, absorption measurements, Stokes shift, kp simulations

*DOI*

*Abstract*

A study of the optical properties of a set of InAs_{x}Sb_{1-x}/Al_{0.15}In_{0.85}As_{0.77}Sb_{0.23}/GaSb multiple quantum-wells (for x between 0.82 and 0.92) with build-in strains in the -0.62% to +0.05%-range is presented. The energy of the lowest quantum-confined optical transition is calculated by kp perturbation theory and experimentally determined by absorption measurements. Stokes shift of photoluminescence, photocurrent and of the emission from light emitting devices against the absorption edge of the quantum-well are quantified. The impact of the decreasing carrier confinement in the InAs_{x}Sb_{1-x} quantum well system with increasing mole fraction is analyzed theoretically, and experimentally demonstrated by photoluminescence measurement. Our results allow for the improvement of optoelectronic devices, in particular for tailoring emission spectra of light emitting diodes.

*Appeared in*

- Appl. Phys. Lett. 87, 181911 (2005)

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# Induced gelation in a two-site spatial coagulation model

*Authors*

- Siegmund-Schultze, Rainer
- Wagner, Wolfgang

*2010 Mathematics Subject Classification*

- 60K40

*Keywords*

- Spatial coagulation model, induced gelation, stochastic particle systems

*DOI*

*Abstract*

A two-site spatial coagulation model is considered. Particles of masses $m$ and $n$ at the same site form a new particle of mass $m+n$ at rate $mn,.$ Independently, particles jump to the other site at a constant rate. The limit (for increasing particle numbers) of this model is expected to be non-deterministic after the gelation time, namely, one or two giant particles randomly jump between the two sites. Moreover, a new effect of induced gelation is observed - the gelation happening at the site with the larger initial number of monomers immediately induces gelation at the other site. Induced gelation is shown to be of logarithmic order. The limiting behaviour of the model is derived rigorously up to the gelation time, while the expected post-gelation behaviour is illustrated by a numerical simulation.

*Appeared in*

- Ann. Appl. Probab., 16 (2006) pp. 370-402.

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# Asymptotic statistical equivalence for ergodic diffusions: The multidimensional case

*Authors*

- Dalalyan, Arnak
- Reiss, Markus

*2010 Mathematics Subject Classification*

- 62G20 62B15 62G05 62G07 62M05

*Keywords*

- Asymptotic equivalence, statistical experiment, Le Cam distance, ergodic diffusion, Gaussian shift, heteroskedasticregression

*DOI*

*Abstract*

Asymptotic local equivalence in the sense of Le Cam is established for inference on the drift in multidimensional ergodic diffusions and an accompanying sequence of Gaussian shift experiments. The nonparametric local neighbourhoods can be attained for any dimension, provided the regularity of the drift is sufficiently large. In addition, a heteroskedastic Gaussian regression experiment is given, which is also locally asymptotically equivalent and which does not depend on the centre of localisation. For one direction of the equivalence an explicit Markov kernel is constructed.

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# Optimal calibration for exponential Levy models

*Authors*

- Belomestny, Denis
- Reiss, Markus

*2010 Mathematics Subject Classification*

- 62G20 60G51 91B28

*Keywords*

- European option, jump diffusion, minimax rates, severely ill-posed, nonlinear inverse problem, spectral cut-off

*DOI*

*Abstract*

Based on options data at the market the problem of calibrating an exponential Lévy model for the underlying asset is investigated. It is shown that this statistical inverse problem is in general severely ill-posed and exact minimax rates of convergence are derived. The estimation procedure we propose is based on the explicit inversion of the option price formula in the spectral domain and a cut-off scheme for high frequencies as regularisation. Its performance is illustrated by numerical simulations.

*Appeared in*

- Finance Stoch., 10 (2006) pp. 449--474.

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# Simulationsbasierte Regelung der Laserhärtung von Stahl

*Authors*

- Alder, Holger
- Hömberg, Dietmar
- Weiss, Wolf

*2010 Mathematics Subject Classification*

- 80A20 93A30

*2008 Physics and Astronomy Classification Scheme*

- 44.05.+e

*Keywords*

- Heat treatment control phase transitions

*DOI*

*Abstract*

Bei der Oberflächenhärtung mit Hilfe von Laserstrahlen ist eine konstante Einhärtetiefe erwünscht, wobei gleichzeitig Anschmelzungen vermieden werden sollen. Um Anschmelzungen zu verhindern, kann die Temperatur im Auftreffpunkt des Lasers gemessen werden und die Laserleistung entsprechend geregelt werden. Eine konstante Temperatur führt bei geometrisch komplizierten Bauteilen jedoch nicht zu einer konstanten Einhärtetiefe. In dieser Arbeit wird ein Verfahren aufgezeigt, wobei durch numerische Simulationen eine nichtkonstante Oberflächentemperatur berechnet wird, die eine konstante Einhärtetiefe liefert. Die berechnete Oberflächentemperatur kann als Solltemperatur im realen Prozess benutzt werden.

*Appeared in*

- HTM Z. Werkst. Waermebeh. Fertigung, 61 (2006) pp. 103--108.

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# Conditional large deviations for a sequence of words

*Authors*

- Birkner, Matthias

*2010 Mathematics Subject Classification*

- 60F10 60G10 82D60

*Keywords*

- Conditional process level large deviations, quenched free energy

*DOI*

*Abstract*

Cut an i.i.d. sequence $(X_i)$ of ``letters'' into ``words'' according to an independent renewal process. Then one obtains an i.i.d. sequence of words, and thus the level three large deviation behaviour of this sequence of words is governed by the specific relative entropy. We consider the corresponding problem for the conditional empirical process of words, where one conditions on a typical underlying $(X_i)$. We find that if the tails of the word lengths decay super-exponentially, the large deviations under the conditional distribution are again governed by the specific relative entropy, but the set of attainable limits is restricted. We indicate potential applications of such a conditional LDP to the computation of the quenched free energy for directed polymer models with random disorder.

*Appeared in*

- Ann. Probab., 118 (2008) pp. 703--729.

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# Rate-independent damage processes in nonlinear elasticity

*Authors*

- Mielke, Alexander

ORCID: 0000-0002-4583-3888 - Roubíček, Tomáš

*2010 Mathematics Subject Classification*

- 35K85 49S05 74C15 74R20

*Keywords*

- Inelastic damage, large deformations, unilateral contact, variational inequality, energetic formulation.

*DOI*

*Abstract*

Damage of an elastic body undergoing large deformations by a ``hard-device'' loading possibly combined with an impact (modelled by a unilateral frictionless contact) of another, ideally rigid body is formulated as an activated, rate-independent process. The damage is assumed to absorb a specific and prescribed amount of energy. A solution is defined by energetic principles of stability and balance of stored and dissipated energies with the work of external loading, realized here through displacement on a part of the boundary. Rigorous analysis by time discretization is performed.

*Appeared in*

- Math. Models Methods Appl. Sci., 16 (2006) pp. 177--209.

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# Analysing fMRI experiments with structural adaptive smoothing procedures

*Authors*

- Tabelow, Karsten

ORCID: 0000-0003-1274-9951 - Polzehl, Jörg

ORCID: 0000-0001-7471-2658 - Spokoiny, Vladimir

ORCID: 0000-0002-2040-3427

*2010 Mathematics Subject Classification*

- 62P10 92C55 62G05 62G10

*2008 Physics and Astronomy Classification Scheme*

- 07.05.Pj

*Keywords*

- functional MRI, spatially adaptive smoothing, signal detection

*DOI*

*Abstract*

Data from functional magnetic resonance imaging (fMRI) consists of time series of brain images which are characterized by a high noise level and a low signal-to-noise ratio. We provide a complete procedure for fMRI analysis. In order to reduce noise and to improve signal detection the fMRI data is spatially smoothed. However, the common application of a Gaussian filter does this at the cost of loss of information on spatial extend and shape of the activation area. We suggest to use the propagation-separation procedures introduced by Polzehl and Spokoiny (2005) instead. We show that this significantly improves the information on the spatial extend and shape of the activation region with similar results for the noise reduction. Signal detection is based on locally varying thresholds defined by random field theory. Effects of adaptive and non adaptive smoothing are illustrated by artificial examples and an analysis of real data.

*Appeared in*

- NeuroImage, 33 (2006) pp. 55--62.

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# Block operator matrices, optical potentials, trace class perturbations and scattering

*Authors*

- Behrndt, Jussi
- Neidhardt, Hagen
- Rehberg, Joachim

*2010 Mathematics Subject Classification*

- 47A40 47A55 47B44

*Keywords*

- Feshbach decomposition, optical potential, Lax-Phillips scattering theory, dissipative scatteringtheory, scattering matrix, characteristicfunction, dissipative operators

*DOI*

*Abstract*

For an operator-valued block-matrix model, which is called in quantum physics a Feshbach decomposition, a scattering theory is considered. Under trace class perturbation the channel scattering matrices are calculated. Using Feshbach's optical potential it is shown that for a given spectral parameter the channel scattering matrices can be recovered either from a dissipative or from a Lax-Phillips scattering theory.

*Appeared in*

- Operator Theory in Inner Product Spaces, K.-H. Förster, P. Jonas, H. Langer, C. Trunk, eds., Vol. 175 of Operator Theory: Advances and Applications, Birkhäuser, Basel, 2007, 33--49

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# Adaptive simulation algorithms for pricing American and Bermudan options by local analysis of the financial market

*Authors*

- Belomestny, Denis
- Milstein, Grigori

*2010 Mathematics Subject Classification*

- 60H30 65C05 91B28

*Keywords*

- American and Bermudan options, Lower and Upper bounds, Monte Carlo simulations, Local analysis, Consumption

*DOI*

*Abstract*

Here we develop an approach for efficient pricing discrete-time American and Bermudan options which employs the fact that such options are equivalent to the European ones with a consumption, combined with analysis of the market model over a small number of steps ahead. This approach allows constructing both upper and low bounds for the true price by Monte Carlo simulations. An adaptive choice of local low bounds and use of the kernel interpolation technique enhance efficiency of the whole procedure, which is supported by numerical experiments.

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# Proof of a counterexample to the finiteness conjecture in the spirit of the theory of dynamical systems

*Authors*

- Kozyakin, Victor

*2010 Mathematics Subject Classification*

- 15A60 26E25 37E10 37E45 39A11

*Keywords*

- Infinite matrix products; Generalized spectral radius; Joint spectral radius; Extremal norms; Irreducibility; Discontinuous circle maps; Rotation number

*DOI*

*Abstract*

In 1995 J.C. Lagarias and Y. Wang conjectured that the generalized spectral radius of a finite set of square matrices can be attained on a finite product of matrices. The first counterexample to this Finiteness Conjecture was given in 2002 by T. Bousch and J. Mairesse and their proof was based on measure-theoretical ideas. In 2003 V.D. Blondel, J. Theys and A.A. Vladimirov proposed another proof of a counterexample to the Finiteness Conjecture which extensively exploited combinatorial properties of permutations of products of positive matrices. In this paper, it is proposed one more proof of a counterexample of the Finiteness Conjecture fulfilled in a rather traditional manner of the theory of dynamical systems. It is presented description of the structure of trajectories with the maximal growing rate in terms of extremal norms and associated with them so called extremal trajectories. The construction of the counterexample is based on a detailed analysis of properties of extremal norms of two-dimensional positive matrices in which the technique of the Gram symbols is essentially used. At last, notions and properties of the rotation number for discontinuous orientation preserving circle maps play significant role in the proof.

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# Enhanced policy iteration for American options via scenario selection

*Authors*

- Bender, Christian
- Kolodko, Anastasia
- Schoenmakers, John G. M.

ORCID: 0000-0002-4389-8266

*2010 Mathematics Subject Classification*

- 60G40 62L15 91B28

*Keywords*

- American options, Monte Carlo simulation, optimal stopping, policy improvement

*DOI*

*Abstract*

In Kolodko & Schoenmakers (2004) and Bender & Schoenmakers (2004) a policy iteration was introduced which allows to achieve tight lower approximations of the price for early exercise options via a nested Monte-Carlo simulation in a Markovian setting. In this paper we enhance the algorithm by a scenario selection method. It is demonstrated by numerical examples that the scenario selection can significantly reduce the number of actually performed inner simulations, and thus can heavily speed up the method (up to factor 10 in some examples). Moreover, it is shown that the modified algorithm retains the desirable properties of the original one such as the monotone improvement property, termination after a finite number of iteration steps, and numerical stability.

*Appeared in*

- Quantitative Finance, Vol. 8, Number 2, pp. 135-146

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# Compact gradient tracking in shape optimization

*Authors*

- Eppler, Karsten
- Harbrecht, Helmut

*2010 Mathematics Subject Classification*

- 49Q10 49M15 65N38 65K10 49K20 65T60

*Keywords*

- shape calculus, boundary integral equations, multiscale methods, sufficient second order conditions, ill-posed problem

*DOI*

*Abstract*

In the present paper we consider the minimization of gradient tracking functionals defined on a compact and fixed subdomain of the domain of interest. The underlying state is assumed to satisfy a Poisson equation with Dirichlet boundary conditions. We prove that, in contrast to other type of objectives, defined on the whole domain, the shape Hessian is not strictly $H^1/2$-coercive at the optimal domain which implies ill-posedness of the shape problem under consideration. Shape functional and gradient require only knowledge of the cauchy data of the state and its adjoint on the boundaries of the domain and the subdomain. These data can be computed in terms of boundary integral equations when reformulating the underlying differential equations as transmission problems. Thanks to fast boundary element techniques, we derive an efficient and accurate computation of the ingredients for optimization. Consequently, difficulties in the solution are related to the ill-posedness of the problem under consideration.

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# Coupling of FEM and BEM in shape optimization

*Authors*

- Eppler, Karsten
- Harbrecht, Helmut

*2010 Mathematics Subject Classification*

- 49Q10 49M15 65N38 65K10 49K20 65T60

*Keywords*

- shape calculus, Newton method, boundary integral equations, finite element method, multiscale methods, sufficient second order conditions

*DOI*

*Abstract*

In the present paper we consider the numerical solution of shape optimization problems which arise from shape functionals of integral type over a compact region of the unknown domain, especially $L^2$-tracking type functionals. The underlying state equation is assumed to satisfy a Poisson equation with Dirichlet boundary conditions. We proof that the shape Hessian is not strictly $H^1/2$-coercive at the optimal domain which implies ill-posedness of the optimization problem under consideration. Since the adjoint state depends directly on the state, we propose a coupling of finite element methods (FEM) and boundary element methods (BEM) to realize an efficient first order shape optimization algorithm. FEM is applied in the compact region while the rest is treated by BEM. The coupling of FEM and BEM essentially retains all the structural and computational advantages of treating the free boundary by boundary integral equations.

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# Stochastic weighted particle method -- Theory and numerical examples

*Authors*

- Rjasanow, Sergej
- Wagner, Wolfgang

*2010 Mathematics Subject Classification*

- 65C05 82C80

*Keywords*

- Boltzmann equation, Stochastic numerics, Variance reduction

*DOI*

*Abstract*

In the present paper we give a theoretical background of the Stochastic Weighted Particle Method (SWPM) for the classical Boltzmann equation. This numerical method was developed for problems with big deviation in magnitude of values of interest. We describe the corresponding algorithms, give a brief summary of the convergence theory and illustrate the new possibilities by numerical tests.

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# Nonparametric risk management with generalized hyperbolic distributions

*Authors*

- Chen, Ying
- Härdle, Wolfgang
- Jeong, Seok-Oh

*2010 Mathematics Subject Classification*

- 62H12 62G05 62G07 62G08

*Keywords*

- adaptive volatility estimation, generalized hyperbolic distribution, value at risk, risk management

*DOI*

*Abstract*

In this paper we propose the GHADA risk management model that is based on the generalized hyperbolic (GH) distribution and on a nonparametric adaptive methodology. Compared to the normal distribution, the GH distribution possesses semi-heavy tails and represents the financial risk factors more appropriately. Nonparametric adaptive methodology has the desirable property of being able to estimate homogeneous volatility over a short time interval and reflects a sudden change in the volatility process. For DEM/USD exchange rate and German bank portfolio data, the proposed GHADA model provides more accurate Value at Risk calculations than the models with assumptions of the normal and t distributions. All calculations and simulations are done with XploRe.

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# The critical Galton--Watson process without further power moments

*Authors*

- Nagaev, Sergei
- Wachtel, Vitali

*2010 Mathematics Subject Classification*

- 60J80 60F05

*Keywords*

- critical Galton-Watson process, conditional limit theorem, slowly varying function, functional normalization

*DOI*

*Abstract*

In this paper we prove a conditional limit theorem for a critical Galton-Watson branching process with offspring generating function s+(1-s)L((1-s)^-1), where L(x) is slowly varying. In contrast to a well-known theorem of Slack (1968, 1972) we use a functional normalization, which gives an exponential limit. We give also an alternative proof of Sze's (1976) result on the asymptotic behavior of the nonextinction probability.

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# Sobolev-Morrey spaces associated with evolution equations

*Authors*

- Griepentrog, Jens André

*2010 Mathematics Subject Classification*

- 35D10 35R05 35K90

*Keywords*

- Evolution equations, monotone operators, second order parabolic bondary value problems, instationary drift-diffusion problems, nonsmooth coefficients, mixed boundary conditions, LIPSCHITZ domains, LIPSCHITZ hypersurfaces, regular sets, MORREY-CAMPANATO spaces, SOBOLEV-MORREY spaces, POINCARE inequalities

*DOI*

*Abstract*

In this text we introduce new classes of SOBOLEV-MORREY spaces being adequate for the regularity theory of second order parabolic boundary value problems on LIPSCHITZ domains of space dimension greater or equal than three with nonsmooth coefficients and mixed boundary conditions. We prove embedding and trace theorems as well as invariance properties of these spaces with respect to localization, LIPSCHITZ transformation, and reflection. In the second part of our presentation we show that the class of second order parabolic systems with diagonal principal part generates isomorphisms between the above mentioned SOBOLEV-MORREY spaces of solutions and right hand sides.

*Appeared in*

- Adv. Differential Equations, 12 (2007) pp. 781--840.

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# $W^1,q$ regularity results for elliptic transmission problems on heterogeneous polyhedra

*Authors*

- Elschner, Johannes
- Kaiser, Hans-Christoph
- Rehberg, Joachim
- Schmidt, Gunther

*2010 Mathematics Subject Classification*

- 35B65 35J25 35Q40 35R05

*Keywords*

- Elliptic transmission problems, polyhedral domains, $W^1q$ regularity

*DOI*

*Abstract*

Let $Upsilon$ be a three-dimensional Lipschitz polyhedron, and assume that the matrix function $mu$ is piecewise constant on a polyhedral partition of $Upsilon$. Based on regularity results for solutions to two-dimensional anisotropic transmission problems near corner points we obtain conditions on $mu$ and the intersection angles between interfaces and $partial Upsilon$ ensuring that the operator $-nabla cdot mu nabla$ maps the Sobolev space $W^1,q_0(Upsilon)$ isomorphically onto $W^-1,q(Upsilon)$ for some $q > 3$.

*Appeared in*

- Math. Models Methods Appl. Sci., 17 (2007) pp. 593--615.

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# Moments and distribution of the local time of a random walk on $Z^2$

*Authors*

- Černý, Jiri

*2010 Mathematics Subject Classification*

- 60F15 60G50

*Keywords*

- random walk, local time

*DOI*

*Abstract*

Let l(n,x) be the local time of a random walk on Z^2. We prove a strong law of large numbers for the quantity L_n(a)=sum_xin Z^2 l(n,x)^a $ for all a>0. We use this result to describe the distribution of the local time of a typical point in the range of the random walk.

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# Metastability: A potential theoretic approach

*Authors*

- Bovier, Anton

*2010 Mathematics Subject Classification*

- 60J45 82C26

*Keywords*

- Metastability, Markov processes, potential theory, capacity, spectral theory

*DOI*

*Abstract*

Metastability is an ubiquitous phenomenon of the dynamical behaviour of complex systems. In this talk, I describe recent attempts towards a model-independent approach to metastability in the context of reversible Markov processes. I will present an outline of a general theory, based on careful use of potential theoretic ideas and indicate a number of concrete examples where this theory was used very successfully. I will also indicate some challenges for future work.

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# Numerical experiments on the modulation theory for the nonlinear atomic chain

*Authors*

- Dreyer, Wolfgang
- Herrmann, Michael

*2010 Mathematics Subject Classification*

- 34K60 35L65 70F10 74A25 82C21

*Keywords*

- atomic chain, modulation theory, traveling waves, thermodynamic limit

*DOI*

*Abstract*

Modulation theory with periodic traveling waves is a powerful, but not rigorous tool to derive a thermodynamic description for the atomic chain. We investigate the validity of this theory by means of several numerical experiments.

*Appeared in*

- Phys. D, 237 (2008) pp. 255-282.

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# Exponential bounds for the probability deviations of sums of random fields

*Authors*

- Kurbanmuradov, Orazgeldy
- Sabelfeld, Karl

*2010 Mathematics Subject Classification*

- 65C05 60F10 65C50

*Keywords*

- Moderately large deviations, Bernstein's inequality, sums of random fields, deviation probability, optimal asymptotics, sample continuity modulus

*DOI*

*Abstract*

Non-asymptotic exponential upper bounds for the deviation probability for a sum of independent random fields are obtained under Bernstein's condition and assumptions formulated in terms of Kolmogorov's metric entropy. These estimations are constructive in the sense that all the constants involved are given explicitly. In the case of moderately large deviations, the upper bounds have optimal log-asymptotices. The exponential estimations are extended to the local and global continuity modulus for sums of independent samples of a random field.

*Appeared in*

- Monte Carlo Methods Appl., 12 (2006) pp. 211--229.

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# Dynamics of a surface-gradient-driven liquid film rising from a reservoir onto a substrate

*Authors*

- Evans, Peter
- Münch, Andreas

*2010 Mathematics Subject Classification*

- 76D08 37N10 76D45 76B45 76A20 34E10 34B60 76D27

*2008 Physics and Astronomy Classification Scheme*

- 68.15.+e 47.20.Ky 47.15.Gf 68.03.Kn 68.03.Cd

*Keywords*

- Lubrication theory, Gravity and surface tension driven liquid flows, undercompressive waves, Landau-Levich drag-out problem, coating flows, capillary meniscus, rarefaction fans

*DOI*

*Abstract*

On a tilted heated substrate, surface tension gradients can draw liquid up out of a reservoir. The resulting film thickness profile is controlled by the tilt of the substrate, the imposed temperature gradient, and the thickness of a postulated thin precursor layer. We study the evolution of this film in time, using a lubrication model. A number of distinct behaviours are possible as the substrate tilt angle and other parameters are varied. We use recent results for the multiple stationary profiles possible near the meniscus and examine how these can interact with the advancing front. We show that it is in fact possible to systematically determine the evolution of the entire film profile from the meniscus to the apparent contact line. This allows a categorisation of the range of behaviours for a transversely-uniform profile, in a two-dimensional parameter space. In addition to combinations of meniscus profiles involving capillary fronts and double shock structures, we describe a new combination of a Type I meniscus with a rarefaction fan, and either undercompressive or classical waves for the advancing front, that arises for certain ranges of large substrate tilt and of precursor thickness.

*Appeared in*

- SIAM J. Appl. Math., 66 (2006) pp. 1610-1631.

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# Structural adaptive smoothing by propagation-separation methods

*Authors*

- Polzehl, Jörg

ORCID: 0000-0001-7471-2658 - Spokoiny, Vladimir

ORCID: 0000-0002-2040-3427

*2010 Mathematics Subject Classification*

- 62G05

*2008 Physics and Astronomy Classification Scheme*

- 07.05.Pj

*Keywords*

- adaptive weights; local structure; propagation; separation; image processing

*DOI*

*Abstract*

Propagation-Separation stands for the main properties of a new class of adaptive smoothing methods. An assumption that a prespecified type of models allows for a good local approximation within homogeneous regions in the design (structural assumption), is utilized to both recover homogeneous regions and to efficiently estimate the regression function. Locality is defined by pairwise weights. Propagation stands for the unrestricted expansion of weights within homogeneous regions. Separations characterizes the restriction of positive weights to homogeneous regions with respect to the specified model. The procedures have remarkable properties like preservation of edges and contrast, and (in some sense) optimal reduction of noise. They are fully adaptive and dimension free. We here provide a short introduction into Propagation-Separation procedures in the context of image processing. Properties are illustrated by a series of examples.

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# Stochastic spectral and Fourier-wavelet methods for vector Gaussian random fields

*Authors*

- Kurbanmuradov, Orazgeldy
- Sabelfeld, Karl

*2010 Mathematics Subject Classification*

- 65C05 60F05 60F10

*Keywords*

- Randomized Spectral models, Fourier-Wavelet method, plane wave decomposition

*DOI*

*Abstract*

Randomized Spectral Models (RSM) and Randomized Fourier-Wavelet Models (FWM) for simulation of homogeneous Gaussian random fields based on spectral representations and plane wave decomposition of random fields are developed. Extensions of FWM to vector random processes are constructed. Convergence of the constructed Fourier-Wavelet models (in the sense of finite-dimensional distributions) under some general conditions on the spectral tensor is given. A comparative analysis of RSM and FWM is made by calculating Eulerian and Lagrangian statistical characteristics of a 3D isotropic incompressible random field through an ensemble and space averaging

*Appeared in*

- Monte Carlo Methods Appl., 12 (2006) pp. 395--446.

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# A tomography of the GREM: Beyond the REM conjecture

*Authors*

- Bovier, Anton
- Kurkova, Irina

*2010 Mathematics Subject Classification*

- 60G70 82B45

*Keywords*

- level statistics, random energy model, generalized random energy model, extreme value theory, disordered systems, spin glasses

*DOI*

*Abstract*

Recently, Bauke and Mertens conjectured that the local statistics of energies in random spin systems with discrete spin space should in most circumstances be the same as in the random energy model. This was proven in a large class of models for energies that do not grow too fast with the system size. Considering the example of the generalized random energy model, we show that the conjecture breaks down for energies proportional to the volume of the system, and describe the far more complex behavior that then sets in.

*Appeared in*

- Comm. Math. Phys., 263 (2006) pp. 535-552.

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# Local energy statistics in disordered systems: A proof of the local REM conjecture

*Authors*

- Bovier, Anton
- Kurkova, Irina

*2010 Mathematics Subject Classification*

- 60G70 82B45

*Keywords*

- universality, level statistics, random energy model, extreme value theory, disordered systems, spin glasses

*DOI*

*Abstract*

Recently, Bauke and Mertens conjectured that the local statistics of energies in random spin systems with discrete spin space should in most circumstances be the same as in the random energy model. Here we give necessary conditions for this hypothesis to be true, which we show to hold in wide classes of examples: short range spin glasses and mean field spin glasses of the SK type. We also show that, under certain conditions, the conjecture holds even if energy levels that grow moderately with the volume of the system are considered.

*Appeared in*

- Comm. Math. Phys., 263 (2006) pp. 513-533.

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# Renormalization analysis of catalytic Wright--Fisher diffusions

*Authors*

- Fleischmann, Klaus
- Swart, Jan M.

*2010 Mathematics Subject Classification*

- 82C28 82C22 60J60 60J80

*Keywords*

- Renormalization, catalytic Wright-Fisher diffusion, rescaled iterates, catalyzing function, renormalization branching process, embedded particle system, extinction, unbounded growth, interacting diffusions, duality, coupling, Poisson-cluster, fixed point equation, universality class

*DOI*

*Abstract*

Recently, several authors have studied maps where a function, describing the local diffusion matrix of a diffusion process with a linear drift towards an attraction point, is mapped into the average of that function with respect to the unique invariant measure of the diffusion process, as a function of the attraction point. Such mappings arise in the analysis of infinite systems of diffusions indexed by the hierarchical group, with a linear attractive interaction between the components. In this context, the mappings are called renormalization transformations. We consider such maps for catalytic Wright-Fisher diffusions. These are diffusions on the unit square where the first component (the catalyst) performs an autonomous Wright-Fisher diffusion, while the second component (the reactant) performs a Wright-Fisher diffusion with a rate depending on the first component through a catalyzing function. We determine the limit of rescaled iterates of renormalization transformations acting on the diffusion matrices of such catalytic Wright-Fisher diffusions.

*Appeared in*

- Electron. J. Probab., 11 (2006) pp. 585-654.

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# GHICA --- Risk analysis with GH distributions and independent components

*Authors*

- Chen, Ying
- Härdle, Wolfgang
- Spokoiny, Vladimir

ORCID: 0000-0002-2040-3427

*2010 Mathematics Subject Classification*

- 62G05 62H12 62H10

*Keywords*

- independent component analysis, generalized hyperbolic distribution, adaptive volatility, Value-at-Risk

*DOI*

*Abstract*

Risk management technology applied to high dimensional portfolios needs simple and fast methods for calculation of Value-at-Risk (VaR). The multivariate normal framework provides a simple off-the-shelf methodology but lacks the heavy tailed distributional properties that are observed in data. A principle component based method (tied closely to the elliptical structure of the distribution) is therefore expected to be unsatisfactory. Here we propose and analyze a technology that is based on 1) performing an Independent Component (IC) search and 2) adaptively fitting the resulting independent marginals by Generalized Hyperbolic (GH) distributions. We study the proposed GHICA methodology in an extensive simulation study. We then apply GHICA to exchange rate portfolios with different trading strategies and a high-dimensional German stocks portfolio. Our analysis with GHICA yields very accurate VaRs.

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# Spectral analysis of Sinai's walk for small eigenvalues

*Authors*

- Bovier, Anton
- Faggionato, Alessandra

ORCID: 0000-0002-6168-3517

*2010 Mathematics Subject Classification*

- 60K37 82B41 82B44

*Keywords*

- disordered systems, random dynamics, trap models, ageing, spectral properties

*DOI*

*Abstract*

Sinai's walk can be thought of as a random walk on $ZZ$ with random potential $V$, with $V$ weakly converging under diffusive rescaling to a two-sided Brownian motion. We consider here the generator $LL _N$ of Sinai's walk on $[-N,N]cap ZZ$ with Dirichlet conditions on $-N,N$. By means of potential theory, for each $h>0$ we show the relation between the spectral properties of $LL_N$ for eigenvalues of order $oleft(expleft(-h sqrtNright)right)$ and the distribution of the $h$-extrema of the rescaled potential $V_N(x)equiv V(Nx)/sqrtN$ defined on $[-1,1]$. Information about the $h$-extrema of $V_N$ is derived from a result of Neveu and Pitman concerning the statistics of $h$-extrema of Brownian motion. As first application of our results, we give a proof of a refined version of Sinai's localization theorem.

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# Random walk on fixed spheres method for electro- and elastostatics problems

*Authors*

- Sabelfeld, Karl
- Levykin, Alexander I.
- Shalimova, Irina

*2010 Mathematics Subject Classification*

- 65C05 65F10 65Z05

*Keywords*

- Random Walk methods, stochastic iterative pprocedures, Lame equation, systems of spherical integral equations

*DOI*

*Abstract*

Stochastic algorithms for solving Dirichlet boundary value problems for the Laplace and Lame equations governing 2D elasticity problems are developed. The approach presented is based on the Poisson integral formula written for each disc of a domain consisting of a family of overlapping discs. The original differential boundary value problem is reformulated in the form of equivalent system of integral equations defined on the intersection surfaces, i.e., arcs in 2D. A Random Walk algorithm can be applied then directly to the obtained system of integral equations where the random walks are living on the intersecting surfaces. We develop also a discrete random walk technique for solving the system of linear equations approximating the system of integral equations. We construct a randomized version of the successive over relaxation (SOR) method. In [6] we have demonstrated that in the case of classical potential theory our method considerably improves the convergence rate of the standard Random Walk on Spheres method. In this paper we extend the algorithm to the system of Lame equations which cannot be solved by the conventional Random Walk on Spheres method. Illustrating computations for 2D Laplace and Lame equations, and comparative analysis of different stochastic algorithms are presented.

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# Stochastic flow simulation in 3D porous media

*Authors*

- Kolyukhin, Dmitry R.
- Sabelfeld, Karl

*2010 Mathematics Subject Classification*

- 65C05 76N20

*Keywords*

- Stochastic hydraulic conductivity, Lognormal random field, Darcy law, Randomized spectral representation, Lagrangian statistics, Mean squared separation

*DOI*

*Abstract*

Stochastic models and Monte Carlo algorithms for simulation of flow through porous media beyond the small hydraulic conductivity fluctuation assumptions are developed. The hydraulic conductivity is modelled as an isotropic random field with a lognormal distribution and prescribed correlation or spectral functions. It is sampled by a Monte Carlo method based on a randomized spectral representation. The Darcy and continuity equations with the random hydraulic conductivity are solved numerically, using the successive over relaxation method in order to extract statistical characteristics of the flow. Hybrid averaging is used: we combine spatial and ensemble avergaing to get efficient numerical procedure. We provide some conceptual and numerical comparison of various stochastic simulation techniques, and focus on the prediction of applicability of the randomized spectral models derived under the assumption of small hydraulic conductivity fluctuations.

*Appeared in*

- Monte Carlo Methods Appl., 11 (2005) pp. 15--37.

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# Approximate approximations from scattered data

*Authors*

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

*2010 Mathematics Subject Classification*

- 41A30 65D15 41A63 41A25

*Keywords*

- scattered data quasi-interpolation, cubature of integral operators, multivariate approximation, error estimates

*DOI*

*Abstract*

The aim of this paper is to extend the approximate quasi-interpolation on a uniform grid by dilated shifts of a smooth and rapidly decaying function on a uniform grid to scattered data quasi-interpolation. It is shown that high order approximation of smooth functions up to some prescribed accuracy is possible, if the basis functions, which are centered at the scattered nodes, are multiplied by suitable polynomials such that their sum is an approximate partition of unity. For Gaussian functions we propose a method to construct the approximate partition of unity and describe the application of the new quasi-interpolation approach to the cubature of multi-dimensional integral operators.

*Appeared in*

- J. Approx. Theory, 145 (2007), pp. 141--170

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# Thin film dynamics on vertically rotating disks

*Authors*

- Afanasiev, Konstantin
- Münch, Andreas
- Wagner, Barbara

*2010 Mathematics Subject Classification*

- 76A20 76D27 65M60 41A60

*2008 Physics and Astronomy Classification Scheme*

- 47.85.mf 47.32.Ef

*Keywords*

- Lubrication theory, FEM, matched asymptotic expansions, rotating flow, free boundary flow

*DOI*

*Abstract*

The axisymmetric flow of a thin liquid film subject to surface tension, gravity and centrifugal forces is considered for the problem of a vertically rotating disk that is partially immersed in a liquid bath. This problem constitutes a generalization of the classic Landau-Levich drag-out problem to axisymmetric flow. A generalized lubrication model that includes the meniscus region connecting the thin film to the bath is derived. The resulting nonlinear fourth-order partial differential equation is solved numerically using a finite element scheme. For a range of parameters steady states are found. While the solutions for the height profile of the film near the drag-out region show excellent agreement with the asymptotic solutions to the corresponding classic Landau-Levich problem, they show novel patterns away from the meniscus region. The implications for possible industrial applications are discussed.

*Appeared in*

- Appl. Math. Modelling, 32 (2008) pp. 1894-1911.

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# Distance to uncontrollability for convex processes

*Authors*

- Henrion, René
- Lewis, Adrian
- Seeger, Alberto

*2010 Mathematics Subject Classification*

- 93B05 34A60

*Keywords*

- convex processes, controllability, rank condition, uncontrollable modes, adjoint processes, cone-constrained controls, distance to uncontrollability

*DOI*

*Abstract*

The classical study of controllability of linear systems assumes unconstrained control inputs. The 'distance to uncontrollability' measures the size of the smallest perturbation to the matrix description of the system rendering it uncontrollable, and is a key measure of system robustness. We extend the standard theory of this measure of controllability to the case where the control input must satisfy given linear inequalities. Specifically, we consider the control of differential inclusions, concentrating on the particular case where the control input takes values in a given convex cone.

*Appeared in*

- SIAM J. Optim., 45 (2006) pp. 26--50.

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# Iterative operator-splitting methods for linear problems

*Authors*

- Farago, Istvan
- Geiser, Jürgen

*2010 Mathematics Subject Classification*

- 80A20 80M25 74S10 76R50 35J60 35J65 65M99 65Z05

*2008 Physics and Astronomy Classification Scheme*

- 02.60.Cb 44.05.+e

*Keywords*

- Numerical simulation, Operator-Splitting Methods, Iterative Operator-Splitting Method, Ordinary Differential Equations, Consistency Analysis

*DOI*

*Abstract*

The operator-splitting methods base on splitting of the complex problem into the sequence of the simpler tasks. A useful method is the iterative splitting method which ensures a consistent approximation in each step. In our paper, we suggest a new method which is based on the combination of splitting the time interval and the traditional iterative operator splitting. We analyze the local splitting error of the method. Numerical examples are given in order to demonstrate the method.

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# Stationary energy models for semiconductor devices with incompletely ionized impurities

*Authors*

- Glitzky, Annegret
- Hünlich, Rolf

*2010 Mathematics Subject Classification*

- 80A20

*Keywords*

- Energy models, mass, charge and energy transport in heterostructures, strongly coupled elliptic systems, mixed boundary conditions, Implicit FunctionTheorem, existence, uniqueness, regularity

*DOI*

*Abstract*

The paper deals with two-dimensional stationary energy models for semiconductor devices, which contain incompletely ionized impurities. We reduce the problem to a strongly coupled nonlinear system of four equations, which is elliptic in nondegenerated states. Heterostructures as well as mixed boundary conditions have to be taken into account. For boundary data which are compatible with thermodynamic equilibrium there exists a thermodynamic equilibrium. Using regularity results for systems of strongly coupled linear elliptic differential equations with mixed boundary conditions and nonsmooth data and applying the Implicit Function Theorem we prove that in a suitable neighbourhood of such a thermodynamic equilibrium there exists a unique stationary solution, too.

*Appeared in*

- Z. angew. Math. Mech. 85(11) (2005) 778-792

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# Scattering matrix, phase shift, spectral shift and trace formula for one-dimensional Schrödinger-type operators

*Authors*

- Neidhardt, Hagen
- Rehberg, Joachim

*2010 Mathematics Subject Classification*

- 47A20 47B44 47A40

*Keywords*

- dissipative Schroedinger-type operators, Sturm-Liouville operators, self-adjoint dilation, characteristic function, Lax-Phillips scattering theory, scattering matrix, phase shift, spectral shift, trace formula, Birman-Krein formula

*DOI*

*Abstract*

The paper is devoted to Schroedinger operators on bounded intervals of the real axis with dissipative boundary conditions. In the framework of the Lax-Phillips scattering theory the asymptotic behaviour of the phase shift is investigated in detail and its relation to the spectral shift is discussed, in particular, trace formula and Birman-Krein formula are verified directly. The results are used for dissipative Schroedinger-Poisson systems.

*Appeared in*

- Integral Equations Operator Theory, 58 (2007) pp. 407--431.

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# Uniqueness for dissipative Schrödinger--Poisson systems

*Authors*

- Neidhardt, Hagen
- Rehberg, Joachim

*2010 Mathematics Subject Classification*

- 47B44 47E05 35J05

*Keywords*

- dissipative Schroedinger-type operators, dissipative Schroedinger-Poisson systems, carrier and current densities, density matrices, smooth operators, scattering theory

*DOI*

*Abstract*

The paper is devoted to the dissipative Schroedinger-Poisson system. We indicate conditions in terms of the Schroedinger-Poisson data which guarantee the uniqueness of the solution. Moreover, it is shown that if the system is sufficiently small shrunken, then it always admits a unique solution.

*Appeared in*

- J. Math. Phys., 46 (2005) pp. 113513/1--113513/28.

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# Simulation of microwave and semiconductor laser structures including PML: Computation of the eigen mode problem, the boundary value problem, and the scattering matrix

*Authors*

- Hebermehl, Georg
- Schefter, Juergen
- Schlundt, Rainer

ORCID: 0000-0002-4424-4301 - Tischler, Thorsten
- Zscheile, Horst
- Heinrich, Wolfgang

*2010 Mathematics Subject Classification*

- 35Q60 65N22 65F15 65F10 78M25

*Keywords*

- Microwave device, Semiconductor laser, Simulation, Maxwell's equations, Scattering matrix, Boundary value problem, PML boundary condition, Eigenvalue problem, Linear algebraic equations, Rectangular grids, Tetrahedral nets

*DOI*

*Abstract*

The properties of microwave circuits and optical structures can be described in terms of their scattering matrix which is extracted from the orthogonal decomposition of the electric field. We discretize the Maxwell' equations with ortogonal grids using the Finite Integration Technique (FIT). Maxwellian grid equations are formulated for staggered nonequidistant rectangular grids and for tetrahedral nets with corresponding dual Voronoi cells. The surface of the computation domain is assumed to be an electric or magnetic wall, open-region problems require uniaxial Perfectly Matched Layer (PML) absorbing boundary conditions. Calculating the excitations at the ports, one obtains eigenvalue problems and then large-scale systems of linear algebraic equations. This paper is a revised version of the preprint no. 987.

*Appeared in*

- Proc. 5th International Workshop Scientific Computing in Electrical Engineering (SCEE), Capo D'Orlando, Italy, September 5--9, 2004, A. Anile, G. Ali, G. Mascali, eds., Scientific Computing in Electrical Engineering, Springer Verlag, 2006, pp. 203--214

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# Micro-macro transitions in the atomic chain via Whitham's modulation equation

*Authors*

- Dreyer, Wolfgang
- Herrmann, Michael
- Mielke, Alexander

ORCID: 0000-0002-4583-3888

*2010 Mathematics Subject Classification*

- 37K60 70F45 70K70 74A25 82C21

*Keywords*

- atomic chain, traveling waves, thermodynamic limit, modulation theory

*DOI*

*Abstract*

The subject matter of this paper is the thermodynamic description of the nonlinear atomic chain with temperature. For this reason we consider special approximate solutions of Newton's equations, in which the atoms perform microscopic oscillations in form of modulated traveling waves. We start with an existence result for periodic traveling wave with arbitrary large amplitudes, and study several examples including the harmonic chain, the hard sphere model, and the small-amplitude approximation. Then we discuss the thermodynamic properties of traveling waves, and derive the corresponding Gibbs equation. Afterwards we focus on the macroscopic evolution of modulated traveling waves. For this purpose we apply Whitham's modulation theory to the atomic chain, and derive the modulation equation, which turns out to be a system of four macroscopic conservation laws. The last part is devoted to the justification problem: We state a conjecture for the general case, and prove this conjecture for the harmonic chain and the hard sphere model.

*Appeared in*

- Nonlinearity, 19 (2006) pp. 471--500.

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# Monochromatic surface waves at the interface between poroelastic and fluid halfspaces

*Authors*

- Albers, Bettina

ORCID: 0000-0003-4460-9152

*2010 Mathematics Subject Classification*

- 74J15 76S05 74S99

*Keywords*

- Surface waves, flows in porous media, numerical analysis of dispersion relation

*DOI*

*Abstract*

The topic of a previous work was the study of monochromatic surface waves at the boundary between a porous medium and a vacuum. This article is an extension of this research to the propagation of surface waves on the interface between a porous halfspace and a fluid halfspace. Results for phase and group velocities and attenuations are shown in dependence on both the frequency and the surface permeability. In contrast to classical papers on surface waves where only the limits of the frequency (zero and infinity) and the limits of the surface permeability (fully sealed and fully open boundary) were studied, we investigate the problem in the full range of both parameters. For the analysis we use the ''simple mixture model'' which is a simplification of the classical Biot model for poroelastic media. The construction of a solution is shown and the dispersion relation solved numerically. There exist three surface waves for this boundary: a leaky Rayleigh wave and both a true and a leaky Stoneley wave. The true Stoneley wave exists only in a limited range of the surface permeability.

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# Lower deviation probabilities for supercritical Galton--Watson processes

*Authors*

- Fleischmann, Klaus
- Wachtel, Vitali

*2010 Mathematics Subject Classification*

- 60J80 60F10

*Keywords*

- Supercritical Galton-Watson process, local limit theorem, large deviation, Cramer transform, concentration function, Schroeder equation, Boettcher equation

*DOI*

*Abstract*

There is a well-known sequence of constants c_n describing the growth of supercritical Galton-Watson processes Z_n. With "lower derivation probabilities" we refer to P(Z_n = k_n) with k_n = o(c_n) as n increases. We give a detailed picture of the asymptotic behavior of such lower deviation probabilities. This complements and corrects results known from the literature concerning special cases. Knowledge on lower deviation probabilities is needed to describe large deviations of the ratio Z_n+1/Z_n. The latter are important in statistical inference to estimate the offspring mean. For our proofs, we adapt the well-known Cramer method for proving large deviations of sums of independent variables to our needs.

*Appeared in*

- Ann. Inst. H. Poincare Probab. Statist., 43 (2007) pp. 233-255.

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# Convergence of coercive approximations for strictly monotone quasistatic models in the inelastic deformation theory

*Authors*

- Chelminski, Krzysztof
- Gwiazda, Piotr

*2010 Mathematics Subject Classification*

- 35Q72

*Keywords*

- inelastic deformation theory, coercive approximation, Young measures, Orlicz spaces

*DOI*

*Abstract*

This article studies coercive approximation procedures in the infinitesimal inelastic deformation theory. For quasistatic, strictly monotone, viscoplastic models using the Young measures approach a convergence theorem in general Orlicz spaces is proved.

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# On convergence in elliptic shape optimization

*Authors*

- Eppler, Karsten
- Harbrecht, Helmut
- Schneider, Reinhold

*2010 Mathematics Subject Classification*

- 49Q10 49K20 49M15 65K10

*Keywords*

- shape optimization, shape calculus, existence and convergence of approximate solutions, optimality conditions

*DOI*

*Abstract*

This paper is aimed at analyzing the existence and convergence of approximate solutions in shape optimization. Two questions arise when one applies a Ritz-Galerkin discretization to solve the necessary condition: does there exists an approximate solution and how good does it approximate the solution of the original infinite dimensional problem? We motivate a general setting by some illustrative examples, taking into account the so-called two norm discrepancy. Provided that the infinite dimensional shape problem admits a stable second order optimizer, we are able to prove the existence of approximate solutions and compute the rate of convergence. Finally, we verify the predicted rate of convergence by numerical results.

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# Generation of the Maxwellian inflow distribution

*Authors*

- Garcia, Alejandro L.
- Wagner, Wolfgang

*2010 Mathematics Subject Classification*

- 65C05 76P05 82C80

*Keywords*

- Maxwellian inflow distribution, boundary conditions, rarefied gas dynamics, Direct Simulation Monte Carlo

*DOI*

*Abstract*

This paper presents several efficient, exact methods for generating the Maxwellian inflow distribution, the velocity distribution of gas molecules crossing a plane. The new methods are demonstrated to be computationally faster and more accurate than the schemes commonly used for open boundary conditions in particle simulations.

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# Maximal regularity for nonsmooth parabolic problems in Sobolev-Morrey spaces

*Authors*

- Griepentrog, Jens André

*2010 Mathematics Subject Classification*

- 35D10 35R05 35K20

*Keywords*

- Second order parabolic boundary value problems, instationary drift-diffusion problems, nonsmooth coefficients, unbounded lower order coefficients, mixed boundary conditions, LIPSCHITZ domains, regular sets, HARNACK-type inequality, global HOELDER continuity, maximal regularity, SOBOLEV-MORREY spaces, smooth dependence of the solutions

*DOI*

*Abstract*

This text is devoted to maximal regularity results for second order parabolic systems on LIPSCHITZ domains of space dimension greater or equal than three with diagonal principal part, nonsmooth coefficients, and nonhomogeneous mixed boundary conditions. We show that the corresponding class of initial boundary value problems generates isomorphisms between two scales of SOBOLEV-MORREY spaces for solutions and right hand sides introduced in the first part of our presentation. The solutions depend smoothly on the data of the problem. Moreover, they are HOELDER continuous in time and space up to the boundary for a certain range of MORREY exponents. Due to the complete continuity of embedding and trace maps these results remain true for a broad class of unbounded lower order coefficients.

*Appeared in*

- Adv. Differential Equations, 12 (2007) pp. 1031--1078.

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# Surface waves on permeable and impermeable boundaries of poroelastic media

*Authors*

- Albers, Bettina

ORCID: 0000-0003-4460-9152

*2010 Mathematics Subject Classification*

- 74J15 76S05 74S99

*Keywords*

- Surface waves, flows in porous media, numerical analysis of dispersion relation

*DOI*

*Abstract*

This work is devoted to the numerical analysis of surface waves in two-component saturated poroelastic media. We use the ßimple mixture model" which is a simplification of the classical Biot's model for poroelastic media. For the interface porous medium/vacuum there exist two surface waves in the whole range of freuencies - a leaky Rayleigh wave and a true Stoneley wave. For the interface porous medium/fluid one more surface wave appears - a leaky Stoneley wave. For this boundary velocities and attenuations of the waves are shown in dependence on the surface permeability. The true Stoneley wave exists only in a limited range of this parameter.

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# Probability of error deviations for the dependent sampling Monte Carlo methods: Exponential bounds in the uniform norm

*Authors*

- Kurbanmuradov, Orazgeldy
- Sabelfeld, Karl

*2010 Mathematics Subject Classification*

- 65C05 60F10 65C50

*2008 Physics and Astronomy Classification Scheme*

- 02.70.Lq

*Keywords*

- Moderately large deviations, dependent sampling Monte Carlo, exponential bound for probability of error deviation, optimal asymptotics

*DOI*

*Abstract*

Under Bernstein's condition a non-asymptotic exponential estimation for the probability of deviations of a sum of independent random fields in uniform norm is proposed. Application of this result to the problem of the error estimation for the dependent sampling Monte Carlo method is presented. It is shown that in the domain of moderately large deviations the suggested estimations have optimal asymptotics.

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# Influence of anisotropic thermal conductivity in the apparatus insulation for sublimation growth of SiC: Numerical investigation of heat transfer

*Authors*

- Geiser, Jürgen
- Klein, Olaf

ORCID: 0000-0002-4142-3603 - Philip, Peter

*2010 Mathematics Subject Classification*

- 80A20 80M25 74S10 76R50 74E10 35J60 35J65 65Z05

*2008 Physics and Astronomy Classification Scheme*

- 02.60.Cb 81.10.Bk 44.05.+e 47.27.Te

*Keywords*

- Numerical simulation, SiC single crystal, Physical vapor transport, Heat transfer, Anisotropic diffusion, Anisotropic thermal conductivity, Nonlinear elliptic PDE's

*DOI*

*Abstract*

Using a mathematical heat transfer model including anisotropic heat conduction, radiation, and RF heating, we use our software WIAS-HiTNIHS to perform numerical computations of the temperature field in axisymmetric growth apparatus during sublimation growth of silicon carbide (SiC) bulk single crystals by physical vapor transport (PVT) (modified Lely method). As it is not unusual for the thermal insulation of PVT growth apparatus to possess an anisotropic thermal conductivity, we numerically study the influence that this anisotropic thermal conductivity has on the temperature field in the growth chamber. Moreover, we also study the influence of the thickness of the insulation. Our results show that, depending on the insulation's orientation, even a moderate anisotropy in the insulation can result in temperature variations of more than 100 K at the growing crystal's surface, which should be taken into account when designing PVT growth apparatus.

*Appeared in*

- Crystal Growth Design, 6 (2006) pp. 2021--2028.

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# Transient numerical study of temperature gradients during sublimation growth of SiC: Dependence on apparatus design

*Authors*

- Geiser, Jürgen
- Klein, Olaf

ORCID: 0000-0002-4142-3603 - Philip, Peter

*2010 Mathematics Subject Classification*

- 80A20 80M25 76R50 65Z05 35K55

*2008 Physics and Astronomy Classification Scheme*

- 02.60.Cb 81.10.Bk 44.05.+e 47.27.Te

*Keywords*

- Numerical simulation, SiC single crystal, Physical vapor transport, Heat transfer, Temperature gradients, Nonlinear parabolic PDE's

*DOI*

*Abstract*

Using a transient mathematical heat transfer model including heat conduction, radiation, and radio frequency (RF) induction heating, we numerically investigate the time evolution of temperature gradients in axisymmetric growth apparatus during sublimation growth of silicon carbide (SiC) bulk single crystals by physical vapor transport (PVT) (modified Lely method). Temperature gradients on the growing crystal's surface can cause defects. Here, the evolution of these gradients is studied numerically during the heating process, varying the apparatus design, namely the amount of the source powder charge as well as the size of the upper blind hole used for cooling of the seed. Our results show that a smaller upper blind hole can reduce the temperature gradients on the surface of the seed crystal without reducing the surface temperature itself.

*Appeared in*

- J. Crystal Growth, 297 (2006) pp. 20-32.

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# Extensions of multiscale Gaussian random field simulation algorithms

*Authors*

- Kramer, Peter
- Kurbanmuradov, Orazgeldy
- Sabelfeld, Karl

*2010 Mathematics Subject Classification*

- 65C05 65C20 65T60

*2008 Physics and Astronomy Classification Scheme*

- 02.50.Ng, 02.60.Cb

*Keywords*

- Randomization method, Fourier-wavelet representation, multiscale random fields

*DOI*

*Abstract*

We analyze and compare the efficiency and accuracy of two simulation methods for homogeneous random fields with multiscale resolution. We consider in particular the Fourier-wavelet method and three variants of the Randomization method: (A) without any stratified sampling of wavenumber space, (B) with stratified sampling of wavenumbers with equal energy subdivision, (C) stratified sampling with a logarithmically uniform subdivision. We focus on fractal Gaussian random fields with Kolmogorov-type spectra. As noted in previous work [3,6], variants (A) and (B) of the Randomization method are only able to generate a self-similar structure function over three to four decades with reasonable computational effort. By contrast, variant (C), suggested by [34,22], along with the Fourier-wavelet method developed by [6], is able to reproduce accurate self-similar scaling of the structure function over a number of decades increasing linearly with computational effort (for our examples we will show that nine decades can be reproduced). We provide some conceptual and numerical comparison of the various cost contributions to each random field simulation method (overhead, cost per realization, cost per evaluation). When evaluating ensemble averaged quantities like the correlation and structure functions, as well as some multi-point statistical characteristics, the Randomization method can provide good accuracy with considerably less cost than the Fourier-wavelet method. The Fourier-wavelet method, however, has better ergodic properties, and hence becomes more efficient for the computation of spatial (rather than ensemble) averages which may be important in simulating the solutions to partial differential equations with random field coefficients.

*Appeared in*

- J. Comput. Phys., 226 (2007) pp. 897--924.

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# A dissipative discretization scheme for a nonlocal phase segregation model

*Authors*

- Gajewski, Herbert
- Gärtner, Klaus

*2010 Mathematics Subject Classification*

- 65M12 65P30 35K65

*Keywords*

- Cahn-Hilliard equation, initial boundary value problem, Lyapunov function, stable and unstable steady states, classical thermodynamics, nonlocal phase segregation model

*DOI*

*Abstract*

We are interested in finite volume discretization schemes and numerical solutions for a nonlocal phase segregation model, suitable for large times and interacting forces. Our main result is a scheme with definite discrete dissipation rate proportional to the square of the driving force for the evolution, i. e., the discrete antigradient of the chemical potential v. Steady states are characterized by constant v and satisfy a nonlocal stationary equation. A numerical bifurcation analysis of that stationary equation explains the observed global behavior of numerically computed trajectories of the evolution equation. For strong interaction forces the model shows steady states distinguished by small deformations of the 'mushy region' or 'interface states'. One essential open question in the discrete case is the global boundedness of v.

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# Threshold to liquefaction in granular materials as a formation of strong wave discontinuity in poroelastic media

*Authors*

- Wilmanski, Krzysztof

*2010 Mathematics Subject Classification*

- 74J30 76S05 74N25

*Keywords*

- Shock waves in porous materials, flow instability in granular materials (fluidization)

*DOI*

*Abstract*

We consider a one-dimensional problem of propagation of acoustic waves in a nonlinear poroelastic saurated material. Stress-strain relations in the skeleton are described by Signorini-type constitutitve equations. Material parameters depend on the current porosity. The governing set of equations describes changes of extension of the skeleton, and of the mass density of the fluid, partial velocities of the skeleton and of the fluid and a porosity. We rely on a second order approximation. Relations of the critical time to an initial porosity and to an initial amplitude are discussed. The connection to the threshold of liquefaction in granular materials is indicated.

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# Bragg localized structures in a passive cavity with transverse refractive index modulation

*Authors*

- Vladimirov, Andrei
- Skryabin, Dmitry
- Kozyreff, Gregory
- Mandel, Paul
- Tlidi, Mustapha

*2010 Mathematics Subject Classification*

- 35Q60 37L10

*Keywords*

- localized structures, bistability, photonic band gap

*DOI*

*Abstract*

We consider a passive nonlinear optical cavity containing a photonic crystal inside it. The cavity is driven by a superposition of the two coherent beams forming a periodically modulated pump. Using a coupled mode reduction and direct numerical modeling of the full system we demonstrate existence of resting and moving transversely localized structures of light in this system.

*Appeared in*

- Optics Express, 14 (2006) pp. 1--6.

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# Entrainment of modulation frequency: A case study

*Authors*

- Recke, Lutz
- Schneider, Klaus R.

*2010 Mathematics Subject Classification*

- 34E10 34C14 34C27 34C29 34C30 34C25

*Keywords*

- frequency entrainment, modulated wave, equivariance, approximate synchronization

*DOI*

*Abstract*

We consider a system of autonomous ODE's which is $S^1$-equivariant and has a family of asymptotically stable modulated wave solutions with wave frequency $alpha_0$ and modulation frequency $beta_0$. This system will be perturbed, where the applied nonautonomous force also represents a modulated wave, but with wave frequency $alpha$ and modulations frequency $beta$. The strength of this perturbation is not necessarily small. Our goal is to look for conditions such that the perturbed system exhibits an approximate entrainment of the modulation frequency $beta$ on any given finite time interval, where the approximation error can be controlled by the wave frequency.

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# Progressively refining penalized gradient projection method for semilinear parabolic optimal control problems

*Authors*

- Chryssoverghi, Ion
- Geiser, Juergen
- Al-Hawasy, Jamil

*2010 Mathematics Subject Classification*

- 93C83 74S05 35K20 47A05

*2008 Physics and Astronomy Classification Scheme*

- 02.60.Cb 44.05.+e

*Keywords*

- Optimal Control, Semilinear Parabolic Systems, Discretisation, Finite Element Method, Theta-Scheme, Discrete Penalized Gradient Projection Methods

*DOI*

*Abstract*

We consider an optimal control problem defined by semilinear parabolic partial differential equations, with control and state constraints, where the state constraints and cost functional involve also the state gradient. The problem is discretized by using a finite element method in space and an implicit -scheme in time for state approximation, while the controls are approximated by blockwise constant ones. We propose a discrete penalized gradient projection method, which is applied to the continuous problem and progressively refines the discretization during the iterations, thus reducing computing time and memory. We prove that strong accumulation points in of sequences generated by this method are admissible and weakly extremal for the continuous problem. Finally, numerical examples are given.

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# Hydrodynamic limit fluctuations of super-Brownian motion with a stable catalyst

*Authors*

- Fleischmann, Klaus
- Mörters, Peter
- Wachtel, Vitali

*2010 Mathematics Subject Classification*

- 60G57 60J80 60K35

*Keywords*

- Catalyst, reactant, superprocess, critical scaling, refined law of large numbers, catalytic branching, stable medium, random environment, supercritical dimension, generalised stable Ornstein-Uhlenbeck process, index jump, Anderson model with stable random potential, infinite overall density

*DOI*

*Abstract*

We consider the behaviour of a continuous super-Brownian motion catalysed by a random medium with infinite overall density under the hydrodynamic scaling of mass, time, and space. We show that, in supercritical dimensions, the scaled process converges to a macroscopic heat flow, and the appropriately rescaled random fluctuations around this macroscopic flow are asymptotically bounded, in the sense of log-Laplace transforms, by generalised stable Ornstein-Uhlenbeck processes. The most interesting new effect we observe is the occurrence of an index-jump from a 'Gaussian' situation to stable fluctuations of index 1+gamma, where gamma is an index associated to the medium.

*Appeared in*

- Electron. J. Probab., 11 (2006) pp. 723-767.

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# Dispersive evolution of pulses in oscillator chains with general interaction potentials

*Authors*

- Giannoulis, Johannes
- Mielke, Alexander

ORCID: 0000-0002-4583-3888

*2010 Mathematics Subject Classification*

- 34E13 34C20 37K60 70F45 70K70

*Keywords*

- Nonlinear oscillator chain, multiscale ansatz, nonlinear Schroedinger equation, justification of modulation equations, normal form transformation, nonresonance conditions

*DOI*

*Abstract*

We consider the dispersive evolution of a single pulse in a nonlinear oscillator chain embedded in a background field. We assume that each atom of the chain interacts pairwise with an arbitrary but finite number of neighbours. The pulse is modeled as a macroscopic modulation of the exact spatiotemporally periodic solutions of the linearized model. The scaling of amplitude, space and time is chosen in such a way that we can describe how the envelope changes in time due to dispersive effects. By this multiscale ansatz we find that the macroscopic evolution of the amplitude is given by the nonlinear Schroedinger equation. The main part of the work is focused on the justification of the formally derived equation: We show that solutions which have initially the form of the assumed ansatz preserve this form over time-intervals with a positive macroscopic length. The proof is based on a normal form transformation constructed in Fourier space, and the results depend on the validity of suitable nonresonance conditions.

*Appeared in*

- Discrete Contin. Dyn. Syst. Ser. B, 6 (2006) pp. 493--523.

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# Macroscopic behavior of microscopic oscillations in harmonic lattices via Wigner--Husimi transforms

*Authors*

- Mielke, Alexander

ORCID: 0000-0002-4583-3888

*2010 Mathematics Subject Classification*

- 37K60 70F45 74Q10 74A25

*Keywords*

- Harmonic lattice, discrete microscopic systems, macroscopic continuum limit, Gamma-limit, Wigner measure, Husimi transform

*DOI*

*Abstract*

We consider the dynamics of infinite harmonic lattices in the limit of the lattice distance epsilon tending to 0. We allow for general polyatomic crystals but assume exact periodicity such that the system can be solved in principle by Fourier transform and linear algebra. Our aim is to derive macroscopic continuum limit equations for epsilon --> 0. For the weak limit of displacements and velocities we find the equation of linear elastodynamics, where the elasticity tensor is obtained as a Gamma-limit. The weak limit of the local energy density can be described by generalizations of the Wigner-Husimi measure which satisfies a transport equation on the product of physical space and Fourier space. The concepts are illustrated via several examples and via a comparison to Whitham's modulation equation.

*Appeared in*

- Arch. Ration. Mech. Anal., 181 (2006) pp. 401--448.

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# Uniqueness in determining polygonal periodic structures

*Authors*

- Elschner, Johannes
- Yamamoto, Masahiro

*2010 Mathematics Subject Classification*

- 78A46 35R30

*Keywords*

- Diffraction grating, periodic Helmholtz equation, inverse Dirichlet and Neumann problems, polygonal grating profile

*DOI*

*Abstract*

We consider the inverse problem of recovering a two-dimensional perfectly reflecting diffraction grating from scattered waves measured above the structure. We establish the uniqueness within the class of general polygonal grating profiles by a minimal number of incoming plane waves, without excluding Rayleigh frequencies and further geometric constraints on the profile. This extends and improves the uniqueness results of [10].

*Appeared in*

- Z. Anal. Anwendungen 26 (2007) pp. 165-177

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# Existence and uniqueness results for general rate-independent hysteresis problems

*Authors*

- Mielke, Alexander

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

*2010 Mathematics Subject Classification*

- 35K85 35K90 49J53 49J40 49S05

*Keywords*

- Rate independent systems, doubly nonlinear equation, subdifferential inclusion, incremental problems, uniform convexity

*DOI*

*Abstract*

We consider the special case of doubly nonlinear differential inclusions which are rate independent. The new feature is that the dissipation potential depends not only on the rate but also on the state itself. The energy potential is assumed to be uniformly convex. This corresponds to evolutionary quasivariational inequalities where the constraint set depends on the state itself. A priori estimates are obtained using a special convexity condition for the sum of the energy potential and the directional derivative of the dissipation potential. Using this, an existence result is derived under the additional assumption that the dissipation potential satisfies certain weak continuity properties. Our uniqueness result generalizes previous of [MT04,BKS04] relies on differentiability conditions and a one-sided Lipschitz estimate, also called structure condition in [MT04].

*Appeared in*

- Math. Models Methods Appl. Sci., 17 (2007) pp. 81--123.

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# Uniqueness in determining polygonal sound-hard obstacles with a single incoming wave

*Authors*

- Elschner, Johannes
- Yamamoto, Masahiro

*2010 Mathematics Subject Classification*

- 35R30 35B60

*Keywords*

- inverse scattering problem, uniqueness, sound-hard, polygonal obstacle

*DOI*

*Abstract*

We consider the two dimensional inverse scattering problem of determining a sound-hard obstacle by the far field pattern. We establish the uniqueness within the class of polygonal domains by a single incoming plane wave.

*Appeared in*

- Inverse Problems, 22 (2006) pp. 355--364.

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# A bi-Lipschitz, volume preserving map from the unit ball onto a cube

*Authors*

- Griepentrog, Jens André
- Höppner, Wolfgang
- Kaiser, Hans-Christoph
- Rehberg, Joachim

*2010 Mathematics Subject Classification*

- 53A55 26B10 57R50

*Keywords*

- Lipschitz-homeomorphism, invariant sets, measure preserving maps

*DOI*

*Abstract*

We construct two bi-Lipschitz, volume preserving maps from Euclidean space onto itself which map the unit ball onto a cylinder and onto a cube, respectively. Moreover, we characterize invariant sets of these mappings.

*Appeared in*

- Note Mat., 28 (2008) pp. 185--201.

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# A thin-film equation for viscoelastic liquids of Jeffreys type

*Authors*

- Rauscher, Markus
- Münch, Andreas
- Wagner, Barbara
- Blossey, Ralf

*2010 Mathematics Subject Classification*

- 76D08 76E17 74D05

*2008 Physics and Astronomy Classification Scheme*

- 83.60.Bc 47.50.+d 68.15.+e

*Keywords*

- linear viscoelasticity, non-Newtonian fluid flows, lubrication approximation, interfacial instability

*DOI*

*Abstract*

We derive a novel thin film equation for linear viscoelastic media describable by generalized Maxwell or Jeffreys models. As a first application of this equation we discuss the shape of a liquid rim near a dewetting front. Although the dynamics of the liquid is equivalent to that of a phenomenological model recently proposed by Herminghaus et al. [19], the liquid rim profile in our model always shows oscillatory behaviour, contrary to that obtained in the former. Our finding supports recent conclusions, based on calculations for Newtonian liquids, that the monotonely decaying rim profiles are a consequence of large slip effects in thin polymer films.

*Appeared in*

- Eur. Phys. J. E. Soft Matter 17 (2005), pp. 373-379.

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# A fast stratified sampling simulation of coagulation processes

*Authors*

- Sabelfeld, Karl
- Levykin, Alexander I.
- Privalova, Tatiana

*2010 Mathematics Subject Classification*

- 65C05 65C35 65Z05

*Keywords*

- Smoluchowski coagulation equation, stratified sampling, 2D diffusion controlled coagulation

*DOI*

*Abstract*

We develop a new version of the direct simulation Monte Carlo method [3] for coagulation processes governed by homogeneous Smoluchowsky equations. The method is based on a subdivision of the set of particle pairs into classes, and on an efficient algorithm for sampling from a discrete distribution, the so-called Walker's alias method [4]. The efficiency of the new method is drastically increased compared to the conventional methods, especially when the coagulation kernel is strongly varying. The method is applied to solving a problem of islands formation on a surface due to a diffusion controlled coagulation.

*Appeared in*

- Monte Carlo Methods Appl., 13 (2007) pp. 71--88.

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# Numerical simulation of heat transfer in materials with anisotropic thermal conductivity: A finite volume scheme to handle complex geometries

*Authors*

- Geiser, Juergen
- Klein, Olaf

ORCID: 0000-0002-4142-3603 - Philip, Peter

*2010 Mathematics Subject Classification*

- 80A20 80M25 74S10 76R50 35J60 35J65 65M99 65Z05

*2008 Physics and Astronomy Classification Scheme*

- 02.60.Cb 44.05.+e 47.27.Te

*Keywords*

- Numerical simulation, Heat transfer. Anisotropic diffusion, Anisotropic thermal conductivity, Finite volume method, Delaunay triangulation, Nonlinear elliptic PDE's

*DOI*

*Abstract*

We devise a finite volume scheme for nonlinear heat transfer in materials with anisotropic thermal conductivity. We focus on the difficulties arising from the discretization of complex domains which are typical in the simulation of industrially relevant processes. For polyhedral domains in two dimensions, we consider Cartesian as well as cylindrical coordinates. Our finite volume scheme is based on unstructured constrained Delaunay triangulations of the domain. For simplicity, we assume that the thermal conductivity tensor has vanishing off-diagonal entries and that the anisotropy is independent of the temperature. We present numerical simulations, verifying our finite volume scheme in cases where a closed-form solution is available. Further results demonstrate the effectiveness of the method in computing the heat transfer in a complex growth apparatus used in crystal growth.

*Appeared in*

- Adv. Math. Sci. Appl., 18 (2008) pp. 43--67.

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# An energetic material model for time-dependent ferroelectric behavior: Existence and uniqueness

*Authors*

- Mielke, Alexander

ORCID: 0000-0002-4583-3888 - Timofte, Aida M.

*2010 Mathematics Subject Classification*

- 82D45 74G25 74C05

*Keywords*

- Ferroelectric material model, existence of solutions, energetic formulation.

*DOI*

*Abstract*

We discuss rate-independent engineering models for multi-dimensional behavior of ferroelectric materials. These models capture the non-linear and hysteretic behavior of such materials. We show that these models can be formulated in an energetic framework which is based on the elastic and the electric displacements as reversible variables and interior, irreversible variables like the remanent polarization. We provide quite general conditions on the constitutive laws which guarantee the existence of a solution. Under more restrictive assumptions we are also able to establish uniqueness results.

*Appeared in*

- Math. Methods Appl. Sci., 29 (2006) pp. 1393--1410.

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# Optimal control of the sweeping process

*Authors*

- Colombo, Giovanni
- Henrion, René
- Hoang, Nguyen D.
- Mordukhovich, Boris S.

*2010 Mathematics Subject Classification*

- 49J52 49J53 45K24 45M25 90C30

*Keywords*

- Sweeping process, optimal control, dissipative differential inclusions, variational analysis, generalized differentiation

*DOI*

*Abstract*

We formulate and study an optimal control problem for the sweeping (Moreau) process, where control functions enter the moving sweeping set. To the best of our knowledge, this is the first study in the literature devoted to optimal control of the sweeping process. We first establish an existence theorem of optimal solutions and then derive necessary optimality conditions for this optimal control problem of a new type, where the dynamics is governed by discontinuous differential inclusions with variable right-hand sides. Our approach to necessary optimality conditions is based on the method of discrete approximations and advanced tools of variational analysis and generalized differentiation. The final results obtained are given in terms of the initial data of the controlled sweeping process and are illustrated by nontrivial examples.

*Appeared in*

- Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 19 (2012) pp. 117--159.

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# Forward simulation of backward SDEs

*Authors*

- Bender, Christian
- Denk, Robert

*2010 Mathematics Subject Classification*

- 65C05 65C30 91B28

*Keywords*

- BSDE, Numerics, Monte-Carlo simulation, Picard iteration, Finance

*DOI*

*Abstract*

We introduce a forward scheme to simulate backward SDEs and analyze the error of the scheme. Finally, we demonstrate the strength of the new algorithm by solving some financial problems numerically.

*Appeared in*

- Stochastic Process. Appl., 117 (2007) pp. 1793--1812.

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# Zeno product formula revisited

*Authors*

- Exner, Pavel
- Ichinose, Takashi
- Neidhardt, Hagen
- Zagrebnov, Valentin

*2010 Mathematics Subject Classification*

- 47A55 47B25 47D03 81Q10

*Keywords*

- Zeno dynamics, product formulæ, resolvents, convergence, generalized observables

*DOI*

*Abstract*

We introduce a new product formula which combines an orthogonal projection with a complex function of a non-negative operator. Under certain assumptions on the complex function the strong convergence of the product formula is shown. Under more restrictive assumptions even operator-norm convergence is verified. The mentioned formula can be used to describe Zeno dynamics in the situation when the usual non-decay measurement is replaced by a particular generalized observables in the sense of Davies.

*Appeared in*

- Integral Equations Operator Theory, 57 (2007) pp. 67--81.

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# Linear stability of a ridge

*Authors*

- King, John R.
- Münch, Andreas
- Wagner, Barbara

*2010 Mathematics Subject Classification*

- 76A20 76D27 76M45 34B15 65M06

*Keywords*

- Lubrication theory, numerics, asymptotic approximation, sharp interface

*DOI*

*Abstract*

We investigate the stability of the three-phase contact-line of a thin liquid ridge on a hydrophobic substrate for flow driven by surface tension and van der Waals forces. We study the role of slippage in the emerging instability at the three-phase contact-line by comparing the lubrication models for no-slip and slip-dominated conditions at the liquid/substrate interface. For both cases we derive a sharp-interface model via matched asymptotic expansions and derive the eigenvalues from a linear stability analysis of the respective reduced models. We compare our asymptotic results with the eigenvalues obtained numerically for the full lubrication models.

*Appeared in*

- Nonlinearity, 19 (2006) pp. 2813-2831.

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# Nonlocal temperature-dependent phase-field models for non-isothermal phase transitions

*Authors*

- Krejčí, Pavel
- Rocca, Elisabetta
- Sprekels, Jürgen

*2010 Mathematics Subject Classification*

- 80A22 35K50 45K05 35B50

*Keywords*

- Phase transition, nonlocal model, integrodifferential heat equation

*DOI*

*Abstract*

We propose a model for non-isothermal phase transitions with non-conserved order parameter driven by a spatially nonlocal free energy with respect to both the temperature and the order parameter. The resulting system of equations is shown to be thermodynamically consistent and to admit a strong solution.

*Appeared in*

- J. London Math. Soc. (2), 76 (2007) pp. 197-210.

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# Iterating snowballs and related path dependent callables in a multi-factor Libor model

*Authors*

- Bender, Christian
- Kolodko, Anastasia
- Schoenmakers, John G. M.

ORCID: 0000-0002-4389-8266

*2010 Mathematics Subject Classification*

- 91B28 62L15 65C05

*Keywords*

- optimal stopping, path dependent derivative, Libor market model

*DOI*

*Abstract*

We propose a valuation method for callable structures in a multi-factor Libor model which are path-dependent in the sense that, after calling, one receives a sequence of cash-flows in the future, instead of a well specified cash-flow at the calling date. The method is based on a Monte Carlo procedure for standard Bermudans recently developed in citetKSc, and is applied to the cancelable snowball interest rate swap. The proposed procedure is quite generic, straightforward to implement, and can be easily adapted to other related path-dependent products.

*Appeared in*

- RISK, September 2006 pp. 126--130, under the new title: Iterating cancellable snowballs and related exotics.

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# Existence and asymptotic analysis of a phase field model for supercooling

*Authors*

- Klein, Olaf

ORCID: 0000-0002-4142-3603 - Luterotti, Fabio
- Rossi, Riccarda

*2010 Mathematics Subject Classification*

- 80A22 36K60 35R35

*Keywords*

- Phase field system, supercooling, Young measures

*DOI*

*Abstract*

We prove an existence result for an initial-boundary value problem which models a perturbation of a phase transition phenomenon with supercooling effects. When the perturbation parameter goes to 0, an asymptotic analysis is performed. It leads to an existence result for a slight modification of the original problem in the framework of Young measures.

*Appeared in*

- Quart. Appl. Math., 64 (2006) pp. 291-319.

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# Functional central limit theorem for the occupation time of the origin for branching random walks $dge 3$

*Authors*

- Birkner, Matthias
- Zähle, Iljana

*2010 Mathematics Subject Classification*

- 60K35

*Keywords*

- Branching random walk, occupation time, functional central limit theorem

*DOI*

*Abstract*

We show that the centred occupation time process of the origin of a system of critical binary branching random walks in dimension $d ge 3$, started off either from a Poisson field or in equilibrium, when suitably normalised, converges to a Brownian motion in $d ge 4$. In $d=3$, the limit process is fractional Brownian motion with Hurst parameter $3/4$ when starting in equilibrium, and a related Gaussian process when starting from a Poisson field.

*Appeared in*

- Ann. Probab., 35 (2007) pp. 2063-2090.

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# On the approximation of periodic traveling waves for the nonlinear atomic chain

*Authors*

- Dreyer, Wolfgang
- Herrmann, Michael

*2010 Mathematics Subject Classification*

- 34K07 34K28 37J45

*Keywords*

- atomic chain, periodic traveling waves

*DOI*

*Abstract*

We study a scheme from citeFV99, which allows to approximate periodic traveling waves in the nonlinear atomic chain with nearest neighbour interactions. We prove a compactness result for this scheme, and derive some generalizations. Moreover, we discuss the thermodynamic properties of traveling waves.

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# Harmonic mode-locking in monolithic semiconductor lasers: Theory, simulations and experiment

*Authors*

- Bandelow, Uwe

ORCID: 0000-0003-3677-2347 - Radziunas, Mindaugas
- Vladimirov, Andrei
- Hüttl, Bernd
- Kaiser, Ronald

*2010 Mathematics Subject Classification*

- 78A60 34C23 35-04

*2008 Physics and Astronomy Classification Scheme*

- 42.60.Fc 42.55.Px 42.60.Mi

*Keywords*

- mode-locking, harmonic mode-locking, traveling wave equations, delay-differential equations

*DOI*

*Abstract*

We study both theoretically and experimentally typical operation regimes of 40 GHz monolithic mode-locked lasers. The underlying Traveling Wave Equation model reveals quantitative agreement for characteristics of the fundamental mode-locking as pulse width and repetition frequency tuning, as well as qualitative agreement with the experiments for other dynamic regimes. Especially the appearance of stable harmonic mode-locking at 80 GHz has been predicted theoretically and confirmed by measurements. Furthermore, we derive and apply a simplified Delay-Differential Equation model which guides us to a qualitative analysis of bifurcations responsible for the appearance and the breakup of different mode-locking regimes. Higher harmonics of mode-locking are predicted by this model as well.

*Appeared in*

- Optical and Quantum Electronics 38, pp. 495-512, 2006.

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# Dynamic coalescence kinetics of facetted 2D islands

*Authors*

- Kaganer, Vladimir
- Ploog, Klaus
- Sabelfeld, Karl

*2010 Mathematics Subject Classification*

- 65C05

*2008 Physics and Astronomy Classification Scheme*

- 81.10.Aj, 05.10.Ln, 68.43.Jk, 81.15.-z

*Keywords*

- Coalescence of islands, atomic-scale Monte Carlo simulations, crystal surface, screening effect, Smoluchowski equation

*DOI*

*Abstract*

We study the coalescence of 2D islands on a crystal surface by atomic-scale kinetic Monte Carlo simulations on an ensemble of meandering islands. The Brownian motion of islands is due to the motion of atoms within the islands, with the escape of atoms from islands prohibited by the presence of a step edge barrier. We find that the diffusion of individual islands and their size distribution qualitatively change for large bond energies or low temperatures, when the islands develop straight edges (facets). The island diffusion coefficient becomes size-independent and the size distribution becomes monotonously decreasing. The results of the kinetic Monte Carlo simulations are supported by numerical solutions of the Smoluchowski equations. We derive the kernel of the Smoluchowski equations for the 2D case taking into account the screening effects and find that the screening essentially alters the island size distribution.

*Appeared in*

- Phys. Rev. B., 73 (2006) pp. 115425--1--8.

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# Semiconductor laser under resonant feedback from a Fabry--Perot: Stability of continuous wave operation

*Authors*

- Tronciu, Vasile Z.
- Wünsche, Hans-Jürgen
- Radziunas, Mindaugas
- Wolfrum, Matthias

*2008 Physics and Astronomy Classification Scheme*

- 42.65.Sf, 42.55.Px, 42.60.Da, 02.30.Lt

*Keywords*

- semiconductor laser, delayed feedback control, stability analysis

*DOI*

*Abstract*

We study the continuous-wave (CW) operation of a semiconductor laser subject to optical feedback from a Fabry-Perot resonator in a case where the emission is resonant to a reflection minimum of the resonator. This configuration is treated in the framework of Lang-Kobayashi equations. The nature of bifurcations and the stability of steady state solutions is analyzed in dependence on magnitude and phase of the feedback. In contrast to conventional optical feedback from a single mirror, the locus of external cavity modes is no more elliptic but represents a tilted eight with possible satellite bubbles. Below a critical feedback strength, which is analytically given, only one single mode exists representing the completely unchanged CW emission of the laser. In this weak-feedback regime, the feedback phase allows a noninvasive control of the CW emission and a tailoring of its small-signal response within wide limits. The obtained results are prototype for all-optical realizations of delayed feedback control.

*Appeared in*

- Phys. Rev. E 73, art. no. 046205, 2006.

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# Discretization of frequencies in delay coupled chaotic oscillators

*Authors*

- Yanchuk, Serhiy

*2010 Mathematics Subject Classification*

- 34K23 34K26 34K28 34C15

*2008 Physics and Astronomy Classification Scheme*

- 05.4.Xt

*Keywords*

- coupled oscillators, time delay, large delay, feedback

*DOI*

*Abstract*

We study the dynamics of two mutually coupled oscillators with a time delayed coupling. Due to the delay, the allowed frequencies of the oscillators are shown to be discretized. The phenomenon is observed in the case when the delay is much larger than the characteristic period of the solitary uncoupled oscillator.

*Appeared in*

- Phys. Rev. E 72 (2005) 036205

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# Pulse interaction via gain and loss dynamics in passive mode-locking

*Authors*

- Nizette, Michel
- Rachinskii, Dmitrii I.
- Vladimirov, Andrei
- Wolfrum, Matthias

*2010 Mathematics Subject Classification*

- 78A60 34C23

*2008 Physics and Astronomy Classification Scheme*

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

*Keywords*

- mode-locking, bifurcations, pulse interaction, Q-switching

*DOI*

*Abstract*

We study theoretically the effects of pulse interactions mediated by the gain and absorber dynamics in a passively mode-locked laser containing a slow saturable absorber, and operating in a regime with several pulses coexisting in the cavity.

*Appeared in*

- Phys. D, 218 (2006) pp. 95--104.

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