# Post-gelation behavior of a spatial coagulation model

*Authors*

- Wagner, Wolfgang

*2010 Mathematics Subject Classification*

- 60K40

*Keywords*

- Spatial coagulation model, post-gelation behavior, stochastic particle systems

*Abstract*

A coagulation model on a finite spatial grid is considered. Particles of discrete masses jump randomly between sites and, while located at the same site, stick together according to some coagulation kernel. The asymptotic behavior (for increasing particle numbers) of this model is studied in the situation, when the coagulation kernel grows sufficiently fast so that the phenomenon of gelation is observed. Weak accumulation points of an appropriate sequence of measure-valued processes are characterized in terms of solutions of a nonlinear equation. A natural description of the behavior of the gel is obtained by using the one-point compactification of the size space. Two aspects of the limiting equation are of special interest. First, the formal extension of Smoluchowski's coagulation equation to the spatially inhomogeneous case has to be modified for a certain class of coagulation kernels. Second, due to spatial inhomogeneity, an equation for the time evolution of the gel mass density has to be added. The jump rates are assumed to vanish with increasing particle masses so that the gel is immobile. Two different gel growth mechanisms (active and passive gel) are found depending on the type of the coagulation kernel.

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# Attractors for the semiflow associated with a class of doubly nonlinear parabolic equations

*Authors*

- Schimperna, Giulio
- Segatti, Antonio

*2010 Mathematics Subject Classification*

- 35K55, 35B40, 35B41

*Keywords*

- Doubly nonlinear equation, singular potential, semiflow, global attractor, energy method, $omega$-limit

*Abstract*

A doubly nonlinear parabolic equation of the form $alpha(u_t)-Delta u +W'(u)ni f $, complemented with initial and either Dirichlet or Neumann homogeneous boundary conditions, is addressed. The two nonlinearities are given by the maximal monotone function $alpha$ and by the derivative $W'$ of a smooth but possibly nonconvex potential $W$; $f$ is a given known source. After defining a proper notion of solution and recalling a related existence result, we show that from any initial datum emanates at least one solution which gains further regularity for $t>0$. Such regularizing solutions contitute a semiflow $S$ for which unqueness is satisfied for strictly positive times and we can study long time behaviour properties,. In particular, we can prove existence of both global and exponential attractors and investigate the structure of $omega$-limits of single trajectories.

*Appeared in*

- Asymptot. Anal., 56 (2008) pp. 61 - 86.

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# Numerical approaches to rate-independent processes and applications in inelasticity

*Authors*

- Mielke, Alexander

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

*2010 Mathematics Subject Classification*

- 35K55 35K85 49J40 49S05 65J15 74C05 74F15 74H15 74H20 74N10 74R05 74S05

*Keywords*

- Rate-independent evolution, energetic solution, plasticity, damage, debonding, magnetostriction, martensitic transformation

*Abstract*

A general abstract approximation scheme for rate-independent processes in the energetic formulation is proposed and its convergence is proved under various rather mild data qualifications. The abstract theory is illustrated on several examples: plasticity with isotropic hardening, damage, debonding, magnetostriction, and two models of martensitic transformation in shape-memory alloys.

*Appeared in*

- ESAIM Math. Model. Numer. Anal., 43 (2009) pp. 399--429.

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# An iteration procedure for solving integral equations related to the American put options

*Authors*

- Belomestny, Denis
- Gapeev, Pavel

*2010 Mathematics Subject Classification*

- 65D15 91B28 60G40 65D30 60J60 60J65

*Keywords*

- American put option, Black-Scholes model, optimal stopping, Picard iterations, upper and lower bounds

*Abstract*

A new algorithm for pricing American put option in the Black-Scholes model is presented. It is based on a time discretization of the corresponding integral equation. The proposed iterative procedure for solving the discretized integral equation converges in a finite number of steps and delivers in each step a lower or an upper bound for the price of discretized option on the whole time interval. The method developed can be easily implemented and carried over to the case of more general optimal stopping problems.

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# Chaotic bound state of localized structures in the complex Ginzburg--Landau equation

*Authors*

- Turaev, Dmitry
- Vladimirov, Andrei
- Zelik, Sergey

*2010 Mathematics Subject Classification*

- 78A60 35Q60 35B32

*2008 Physics and Astronomy Classification Scheme*

- 42.65.Sf 05.45.-a 42.65.Tg

*Keywords*

- dissipative soliton, pulse interaction, homoclinic bifurcation

*Abstract*

A new type of stable dynamic bound state of dissipative localized structures is found. It is characterized by chaotic oscillations of distance between the localized structures, their phase difference, and the center of mass velocity.

*Appeared in*

- Phys. Rev. E, 75 (2007) pp. 045601(R).

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# Asymptotic behavior of a Neumann parabolic problem with hysteresis

*Authors*

- Eleuteri, Michela
- Krejčí, Pavel

*2010 Mathematics Subject Classification*

- 35K55 47J40 35B40

*Keywords*

- Parabolic equation, hysteresis, asymptotic behavior of solutions, Preisach model

*Abstract*

A parabolic equation in two or three space variables with a Preisach hysteresis operator and with homogeneous Neumann boundary conditions is shown to admit a unique global regular solution. A detailed investigation of the Preisach memory dynamics shows that the system converges to an equilibrium in the state space of all admissible Preisach memory configurations as time tends to infinity.

*Appeared in*

- ZAMM Z. Angew. Math. Mech., 87 (2007) pp. 261--277.

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# On the propagation of vector ultra-short pulses

*Authors*

- Pietrzyk, Monika
- Kanattšikow, Igor
- Bandelow, Uwe

ORCID: 0000-0003-3677-2347

*2010 Mathematics Subject Classification*

- 37K10 78A60 35Q60 35Q51

*2008 Physics and Astronomy Classification Scheme*

- 02.30.lk 42.65.-k 42.81.Gs

*Keywords*

- ultra-short optical pulses, short pulse equation, integrable systems, Manakov equation, birefringence

*Abstract*

A two component vector generalization of the Schäfer-Wayne short pulse equation which describes propagation of ultra-short pulses in optical fibers with Kerr nonlinearity beyond the slowly varying envelope approximation and takes into account the effects of anisotropy and polarization is presented. As a special case, the integrable two-component short pulse equations are constructed which represent the counterpart of the Manakov system in the case of ultra-short pulses.

*Appeared in*

- J. Nonlinear Math. Phys., 15 (2008) pp. 162-170.

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# Global and exponential attractors for 3-D wave equations with displacement dependent damping

*Authors*

- Pata, Vittorino
- Zelik, Sergey

*2010 Mathematics Subject Classification*

- 35B41 35L05 74K15

*Keywords*

- Wave equation, displacement depending damping coefficient, weak and strong attractors

*Abstract*

A weakly damped wave equation in the three-dimensional (3-D) space with a damping coefficient depending on the displacement is studied. This equation is shown to generate a dissipative semigroup in the energy phase space, which possesses finite-dimensional global and exponential attractors in a slightly weaker topology.

*Appeared in*

- Math. Methods Appl. Sci., 29 (2006) pp. 1291-1306.

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# Analytical and numerical methods for finite-strain elastoplasticity

*Authors*

- Gürses, Ercan
- Mainik, Andreas
- Miehe, Christian
- Mielke, Alexander

ORCID: 0000-0002-4583-3888

*2010 Mathematics Subject Classification*

- 74C15 74B20 49J40 49S05

*Keywords*

- Multiplicative plasticity, energetic formulation, time-incremental minimization, microstructure, energy relaxation

*Abstract*

An important class of finite-strain elastoplasticity is based on the multiplicative decomposition of the strain tensor $F=F_el F_pl$ and hence leads to complex geometric nonlinearities. This survey describes recent advances on the analytical treatment of time-incremental minimization problems with or without regularizing terms involving strain gradients. For a regularization controlling all of $nabla F_pl$ we provide an existence theory for the time-continuous rate-independent evolution problem, which is based on a recently developed energetic formulation for rate-independent systems in abstract topological spaces. In systems without gradient regularization one encounters the formation of microstructures, which can be described by sequential laminates or more general gradient Young measures. We provide a mathematical framework for the evolution of such microstructure and discuss algorithms for solving the associated space-time discretizations. We outline in a finite-step-sized incremental setting of standard dissipative materials details of relaxation-induced microstructure development for strain softening von Mises plasticity and single-slip crystal plasticity. The numerical implementations are based on simplified assumptions concerning the complexity of the microstructures.

*Appeared in*

- Multifield Problems in Solid and Fluid Mechanics, R. Helmig, A. Mielke, B. Wohlmuth, eds., vol. 28 of Lecture Notes in Applied and Computational Mechanics, Springer, Heidelberg, 2006, pp. 443--481

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# Accurate localization of brain activity in presurgical fMRI by structure adaptive smoothing

*Authors*

- Tabelow, Karsten

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

ORCID: 0000-0001-7471-2658 - Uluğ, Aziz M.
- Dyke, Jonathan P.
- Watts, Richard
- Heier, Linda A.
- Voss, Henning U.

*2010 Mathematics Subject Classification*

- 62P10 92C55 62G05 62G10

*Keywords*

- functional MRI, spatially adaptive smoothing, signal detection, presurgical planning

*Abstract*

An important problem of the analysis of fMRI experiments is to achieve some noise reduction of the data without blurring the shape of the activation areas. As a novel solution to this problem, the Propagation-Separation approach (PS), a structure adaptive smoothing method, has been proposed recently. PS adapts to different shapes of activation areas by generating a spatial structure corresponding to similarities and differences between time series in adjacent locations. In this paper we demonstrate how this method results in more accurate localization of brain activity. First, it is shown in numerical simulations that PS is superior over Gaussian smoothing with respect to the accurate description of the shape of activation clusters and and results in less false detections. Second, in a study of 37 presurgical planning cases we found that PS and Gaussian smoothing often yield different results, and we present examples showing aspects of the superiority of PS as applied to presurgical planning.

*Appeared in*

- IEEE Trans. Med. Imaging, 27 (2008) pp. 531--537.

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# Oscillatory instability in systems with delay

*Authors*

- Wolfrum, Matthias
- Yanchuk, Serhiy

*2010 Mathematics Subject Classification*

- 34K05 34K18 34K20 34K26

*2008 Physics and Astronomy Classification Scheme*

- 02.20.Ks

*Keywords*

- large delay, instability, Eckhaus, Ginzburg-Landau amplitude equation

*Abstract*

Any biological or physical system, which incorporates delayed feedback or delayed coupling, can be modeled by a dynamical system with delayed argument. We describe a standard oscillatory destabilization mechanism, which occurs in such systems.

*Appeared in*

- Phys. Rev. Lett., 96 (2006) pp. 220201/1--220201/4.

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# Compression limit by third-order dispersion in the normal dispersion regime

*Authors*

- Demircan, Ayhan
- Kroh, Marcel
- Bandelow, Uwe

ORCID: 0000-0003-3677-2347 - Hüttl, Bernd
- Weber, Hans-Georg

*2010 Mathematics Subject Classification*

- 35Q55 35Q60

*2008 Physics and Astronomy Classification Scheme*

- 42.65.Re 42.81.Dp

*Keywords*

- Nonlinear Schrödinger Equation, Optical Fiber

*Abstract*

Broad-band continua at gigahertz rates generated in high-nonlinear dispersion flattened fibers in the normal dispersion regime near the zero-dispersion wavelength can be used for a subsequent efficient pulse compression, leading to stable high-repetition-rate trains of femtosecond pulses. We show experimentally and theoretically that third-order dispersion defines a critical power, where beyond further compression is inhibited. This fundamental limit is caused by a pulse-breakup.

*Appeared in*

- IEEE Phot. Tech. Letter, 18 (2006) pp. 2353-2355.

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# Critical Galton--Watson processes: The maximum of total progenies within a large window

*Authors*

- Fleischmann, Klaus
- Vatutin, Vladimir A.
- Wachtel, Vitali

*2010 Mathematics Subject Classification*

- 60J80 60F17

*Keywords*

- Branching of index one plus alpha, limit theorem, conditional invariance principle, tail asymptotics, moving window, maximal total progeny, lower deviation probabilities

*Abstract*

Consider a critical Galton-Watson process Z=Z_n: n=0,1,... of index 1+alpha, alpha in (0,1]. Let S_k(j) denote the sum of the Z_n with n in the window [k,...,k+j), and M_m(j) the maximum of the S_k with k moving in [0,m-j]. We describe the asymptotic behavior of the expectation EM_m(j) if the window width j=j_m satisfies that j/m converges in [0,1] as m tends to infinity. This will be achieved via establishing the asymptotic behavior of the tail probabilities of M_infinity(j).

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# Adaptive smoothing of digital images: The R package adimpro

*Authors*

- Polzehl, Jörg

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

ORCID: 0000-0003-1274-9951

*2010 Mathematics Subject Classification*

- 62G05

*Keywords*

- adaptive weights, local structure, propagation, separation, image processing, denoising

*Abstract*

Digital imaging has become omnipresent in the past years with a bulk of applications ranging from medical imaging to photography. When pushing the limits of resolution and sensitivity noise has ever been a major issue. However, commonly used non-adaptive filters can do noise reduction at the cost of a reduced effective spatial resolution only. Here we present a new package adimpro for R, which implements the Propagation-Separation approach by Polzehl and Spokoiny (2006) for smoothing digital images. This method naturally adapts to different structures of different size in the image and thus avoids oversmoothing edges and fine structures. We extend the method for imaging data with spatial correlation. Furthermore we show how the estimation of the dependence between variance and mean value can be included. We illustrate the use of the package through some examples.

*Appeared in*

- J. Statist. Software, 19 (2007) pp. 1--17.

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# Scenario reduction in stochastic programming with respect to discrepancy distances

*Authors*

- Henrion, René
- Küchler, Christian
- Römisch, Werner

*2010 Mathematics Subject Classification*

- 90C15

*Keywords*

- Stochastic programming, chance constraints, two-stage, mixed-integer, scenario reduction, discrepancy, Kolmogorov metric

*Abstract*

Discrete approximations to chance constraints and mixed-integer two-stage stochastic programs require moderately sized scenario sets. The relevant distances of (multivariate) probability distributions for deriving quantitative stability results for such stochastic programs are B-discrepancies, where the class B of Borel sets depends on their structural properties. Hence, the optimal scenario reduction problem for such models is stated with respect to B-discrepancies. In this paper, upper and lower bounds, and some explicit solutions for optimal scenario reduction problems are derived. In addition, we develop heuristic algorithms for determining nearly optimally reduced probability measures, discuss the case of the cell discrepancy (or Kolmogorov metric) in some detail and provide some numerical experience.

*Appeared in*

- Comput. Optim. Appl., 43 (2009) pp. 67--93.

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# In search of non-Gaussian components of a high-dimensional distribution

*Authors*

- Blanchard, Gilles
- Kawanabe, Motoaki
- Sugiyama, Masashi
- Spokoiny, Vladimir

ORCID: 0000-0002-2040-3427 - Müller, Klaus-Robert

*2010 Mathematics Subject Classification*

- 00A73 62H30 62H99 68Q32

*Keywords*

- Non-Gaussian components, dimension reduction

*Abstract*

Finding non-Gaussian components of high-dimensional data is an important preprocessing step for efficient information processing. This article proposes a new em linear method to identify the ``non-Gaussian subspace'' within a very general semi-parametric framework. Our proposed method, called NGCA (Non-Gaussian Component Analysis), is essentially based on the fact that we can construct a linear operator which, to any arbitrary nonlinear (smooth) function, associates a vector which belongs to the low dimensional non-Gaussian target subspace up to an estimation error. By applying this operator to a family of different nonlinear functions, one obtains a family of different vectors lying in a vicinity of the target space. As a final step, the target space itself is estimated by applying PCA to this family of vectors. We show that this procedure is consistent in the sense that the estimaton error tends to zero at a parametric rate, uniformly over the family, Numerical examples demonstrate the usefulness of our method.

*Appeared in*

- J. Mach. Learn. Res., 7 (2006) pp. 247--282.

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# Determination of stiffness and higher gradient coefficients by means of the embedded atom method: An approach for binary alloys

*Authors*

- Böhme, Thomas
- Dreyer, Wolfgang
- Müller, Wolfgang H.

*2010 Mathematics Subject Classification*

- 74A15 74N05 74N25 82B20 82B26

*Keywords*

- atomic potentials, crystals, elastic constants, higher gradient coefficients, phase diagram, diffusion, phase transformation, phase field theories

*Abstract*

For a quantitative theoretical description of phase separation and coarsening reliable data of stiffness constants and the so called Higher Gradient Coefficients (HGCs) are required. For that reason pair potentials of the Lennard-Jones type were used in [1] to provide a theoretical tool for their quantitative determination. Following up on this work these quantities are now calculated by means of the Embedded-Atom Method (EAM), a recently developed approach to describe interatomic potentials in metals. This is done, first, to achieve a better agreement between predicted and experimentally observed stiffness data as well as to avoid artifacts, such as the Cauchy paradox, and, second, to increase the trustworthiness of the HGCs for which experimental data are rarely available. After an introduction to the fundamentals of EAM it is outlined how it can be used for calculating stiffness constants and HGCs. In particular, Johnson's modification of EAM for nearest neighbor interactions [3] is applied to present explicit numerical results for a case study alloy, Ag-Cu, which has a ``simple" face-centered-cubic crystal structure and where it is comparatively easy to obtain all the required analysis data from the literature and to experimentally compare the predictions of mechanical data.

*Appeared in*

- Contin. Mech. Thermodyn., 18 (2007) pp. 411--441.

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# Scattering matrices and Weyl functions

*Authors*

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

*2010 Mathematics Subject Classification*

- 47B50

*Keywords*

- scattering system, scattering matrix, boundary triplet, (Titchmarsh-) Weyl function, spectral shift function, Krein-Birman formula, Sturm-Liouville operator, Dirac operator, Schroedinger operator

*Abstract*

For a scattering system consisting of two selfadjoint extensions of a symmetric operator A with finite deficiency indices, the scattering matrix and the spectral shift function are calculated in terms of the Weyl function associated with the boundary triplet for A* and a simple proof of the Krein-Birman formula is given. The results are applied to singular Sturm-Liouville operators with scalar- and matrix-valued potentials, to Dirac operators and to Schroedinger operators with point interactions.

*Appeared in*

- Proc. London Math. Soc. (3), 97 (2008) pp. 568--598.

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# Interpolation in variable Hilbert scales with application to inverse problems

*Authors*

- Mathé, Peter

ORCID: 0000-0002-1208-1421 - Tautenhahn, Ulrich

*2010 Mathematics Subject Classification*

- 65J20 46B70 65R30

*Keywords*

- Regularization, interpolation spaces, Hilbert scales

*Abstract*

For solving linear ill-posed problems with noisy data regularization methods are required. In the present paper regularized approximations in Hilbert scales are obtained by a general regularization scheme. The analysis of such schemes is based on new results for interpolation in Hilbert scales. Error bounds are obtained under general smoothness conditions.

*Appeared in*

- Inverse Problems, 22 (2006) pp. 2271--2297.

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# Energy release rate for cracks in finite-strain elasticity

*Authors*

- Knees, Dorothee
- Mielke, Alexander

ORCID: 0000-0002-4583-3888

*2010 Mathematics Subject Classification*

- 74B20 74R10

*Keywords*

- Griffith fracture criterion, energy release rate, finite-strain elasticity

*Abstract*

Griffith's fracture criterion describes in a quasistatic setting whether or not a pre-existing crack in an elastic body is stationary for given external forces. In terms of the energy release rate (ERR), which is the derivative of the deformation energy of the body with respect to a virtual crack extension, this criterion reads: If the ERR is less than a specific constant, then the crack is stationary, otherwise it will grow. In this paper, we consider geometrically nonlinear elastic models with polyconvex energy densities and prove that the ERR is well defined. Moreover, without making any assumption on the smoothness of minimizers, we derive rigorously the well-known Griffith formula and the $J$-integral, from which the ERR can be calculated. The proofs are based on a weak convergence result for Eshelby tensors.

*Appeared in*

- Math. Methods Appl. Sci., 31 (2008) pp. 501--528.

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# Regression methods in pricing American and Bermudan options using consumption processes

*Authors*

- Belomestny, Denis
- Milstein, Grigori N.
- Spokoiny, Vladimir

ORCID: 0000-0002-2040-3427

*2010 Mathematics Subject Classification*

- 60H30 65C05 91B28

*Keywords*

- American and Bermudan options, low and upper bounds, Monte Carlo method, consumption process, regression methods, optimal stopping times

*Abstract*

Here we develop methods for efficient pricing multidimensional discrete-time American and Bermudan options by using regression based algorithms together with a new approach towards constructing upper bounds for the price of the option. Applying sample space with payoffs at the optimal stopping times, we propose sequential estimates for continuation values, values of the consumption process, and stopping times on the sample paths. The approach admits constructing both low and upper bounds for the price by Monte Carlo simulations. The methods are illustrated by pricing Bermudan swaptions and snowballs in the Libor market model.

*Appeared in*

- Quant. Finance, 9 (2009) pp. 315--327.

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# On the large scale behavior of super-Brownian motion in three dimensions with a single point source

*Authors*

- Fleischmann, Klaus
- Mueller, Carl
- Vogt, Pascal

*2010 Mathematics Subject Classification*

- 60J80 60K35

*Keywords*

- Super-Brownian motion with singular mass creation, expected mass, Schrödinger equation with one-point-potential

*Abstract*

In a recent work, Fleischmann and Mueller (2004) showed the existence of a super-Brownian motion in R^d, d=2,3, with extra birth at the origin. Their construction made use of an analytical approach based on the fundamental solution of the heat equation with a one point potential worked out by Albeverio et al. (1995). The present note addresses two properties of this measure-valued process in the three-dimensional case, namely the scaling of the process and the large scale behavior of its mean.

*Appeared in*

- Commun. Stoch. Anal., 1 (2007) pp. 19-28.

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# Electronic states in semiconductor nanostructures and upscaling to semi-classical models

*Authors*

- Koprucki, Thomas

ORCID: 0000-0001-6235-9412 - Kaiser, Hans-Christoph
- Fuhrmann, Jürgen

ORCID: 0000-0003-4432-2434

*2010 Mathematics Subject Classification*

- 82D37 34L40

*Keywords*

- Semiconductor nanostructures, kp method, electronic states, band structure, semiclassical models, upscaling, quantum wells, semiconductor lasers

*Abstract*

In semiconductor devices one basically distinguishes three spatial scales: The atomistic scale of the bulk semiconductor materials (sub-Angstroem), the scale of the interaction zone at the interface between two semiconductor materials together with the scale of the resulting size quantization (nanometer) and the scale of the device itself (micrometer). The paper focuses on the two scale transitions inherent in the hierarchy of scales in the device. We start with the description of the band structure of the bulk material by kp Hamiltonians on the atomistic scale. We describe how the envelope function approximation allows to construct kp Schroedinger operators describing the electronic states at the nanoscale which are closely related to the kp Hamiltonians. Special emphasis is placed on the possible existence of spurious modes in the kp Schroedinger model on the nanoscale which are inherited from anomalous band bending on the atomistic scale. We review results of the mathematical analysis of these multi-band kp Schroedinger operators. Besides of the confirmation of the main facts about the band structure usually taken for granted, key results are conditions on the coefficients of the kp Schroedinger operator for the nanostructure, which exclude spurious modes and an estimate of the size of the band gap. Using these results, we give an overview of properties of the electronic band structure of strained quantum wells. Further, the assumption of flat-band conditions across the nanostructure allows for upscaling of quantum calculations to state equations for semi-classical models. We demonstrate this approach for parameters such as the quantum corrected band-edges, the effective density of states, the optical response, and the optical peak gain. Further, we apply the kp Schroedinger theory to low gap quantum wells, a case where a proper rescaling of the optical matrix element is necessary to avoid spurious modes. Finally, we discuss the application of the kp Schroedinger models to biased quantum wells, the operation mode of electro-optic modulators.

*Appeared in*

- Analysis, Modeling and Simulation of Multiscale Problems, A. Mielke, ed., Springer, Heidelberg, 2006, pp. 365--394

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# A shape calculus analysis for tracking type formulations in electrical impedance tomography

*Authors*

- Eppler, Karsten

*2010 Mathematics Subject Classification*

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

*Keywords*

- electrical impedance tomography, shape calculus, boundary integral equations, ill-posed problems, two norm discrepancy

*Abstract*

In the paper [17], the authors investigated the identification of an obstacle or void of perfectly conducting material in a two-dimensional domain by measurements of voltage and currents at the boundary. In particular, the reformulation of the given nonlinear identification problem was considered as a shape optimization problem using the Kohn and Vogelius criterion. The compactness of the complete shape Hessian at the optimal inclusion was proven, verifying strictly the ill-posedness of the identification problem. The aim of the paper is to present a similar analysis for the related least square tracking formulations. It turns out that the two-norm-discrepancy is of the same principal nature as for the Kohn and Vogelius objective. As a byproduct, the necessary first order optimality condition are shown to be satisfied if and only if the data are perfectly matching. Finally, we comment on possible consequences of the two-norm-discrepancy for the regularization issue.

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# Numerical analysis of Lavrentiev-regularized state constrained elliptic control problems

*Authors*

- Hinze, Michael
- Meyer, Christian

*2010 Mathematics Subject Classification*

- 49K20 49N10 49M20

*Keywords*

- Optimal control of elliptic equations, quadratic programming, pointwise state constraints, mixed constraints, Lavrentiev regularization

*Abstract*

In the present work, we apply semi-discretization proposed by the first author in [13] to Lavrentiev-regularized state constrained elliptic control problems. We extend the results of [17] and prove weak convergence of the adjoint states and multipliers of the regularized problems to their counterparts of the original problem. Further, we prove error estimates for finite element discretizations of the regularized problem and investigate the overall error imposed by the finite element discretization of the regularized problem compared to the continuous solution of the original problem. Finally we present numerical results which confirm our analytical findings.

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# Global attractors for semigroups of closed operators

*Authors*

- Pata, Vittorino
- Zelik, Sergey

*2010 Mathematics Subject Classification*

- 34D45 47H20 47J35

*Keywords*

- Semigroups of operators, abstract Cauchy problems, closed operators, global attractors, connected attractors

*Abstract*

In this note, we establish a general result on the existence of global attractors for semigroups $S(t)$ of operators acting on a Banach space $X$, where the strong continuity $S(t)in C(X,X)$ is replaced by the much weaker requirement that $S(t)$ be a closed map.

*Appeared in*

- Math. Methods Appl. Sci., 29 (2006) pp. 1291-1306.

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# New results on the stability of quasi-static paths of a single particle system with Coulomb friction and persistent contact

*Authors*

- Schmid, Florian
- Martins, João A. C.
- Rebrova, Natalia

*2010 Mathematics Subject Classification*

- 74H55 74M10 74B05 34D20

*Keywords*

- Coulomb friction, quasistatic, persistent contact, stability

*Abstract*

In this paper we announce some new mathematical results on the stability of quasi-static paths of a single particle linearly elastic system with Coulomb friction and persistent normal contact with a flat obstacle.A quasi-static path is said to be stable at some value of the load parameter if, for some finite interval of the load parameter thereafter, the dynamic solutions behave continuously with respect to the size of the initial perturbations (as in Lyapunov stability) and to the smallness of the rate of application of the external forces, $varepsilon$ (as in singular perturbation problems). In this paper we prove sufficient conditions for stability of quasi-static paths of a single particle linearly elastic system with Coulomb friction and persistent normal contact with a flat obstacle. The present system has the additional difficulty of its non-smoothness: the friction law is a multivalued operator and the dynamic evolutions of this system may have discontinuous accelerations.

*Appeared in*

- Topics on Mathematics for Smart Systems, B. Miara, G. Stavroulakis, V. Valente, eds., World Scientific Publishing Co. Pte. Ltd., 2007, pp. 208--217

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# Forward and reverse representations for Markov chains

*Authors*

- Milstein, Grigori N.
- Schoenmakers, John G. M.

ORCID: 0000-0002-4389-8266 - Spokoiny, Vladimir

ORCID: 0000-0002-2040-3427

*2010 Mathematics Subject Classification*

- 60J05 60H10 62G07 65C05

*Keywords*

- estimation of risk, transition density estimation, forward and reverse Markov chains, Monte Carlo simulation

*Abstract*

In this paper we carry over the concept of reverse probabilistic representations developed in Milstein, Schoenmakers, Spokoiny (2004) for diffusion processes, to discrete time Markov chains. We outline the construction of reverse chains in several situations and apply this to processes which are connected with jump-diffusion models and finite state Markov chains. By combining forward and reverse representations we then construct transition density estimators for chains which have root-N accuracy in any dimension and consider some applications.

*Appeared in*

- Stochastic Process. Appl., 117 (2007) pp. 1052--1075.

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# Fractional-splitting and domain-decomposition methods for parabolic problems and applications

*Authors*

- Daoud, Daoud
- 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, domain-decomposition, parabolic partial differential equations, convergence analysis, error analysis

*Abstract*

In this paper we consider the first order fractional splitting method to solve decomposed complex equations with multi-physical processes for applications in porous media and phase-transitions. The first order fractional splitting method is also considered as basic solution for the overlapping Schwarz-Waveform-Relaxation method for an overlapped subdomains. The accuracy and the efficiency of the methods are investigated through the solution of different model problems of scalar, coupling and decoupling systems of convection reaction diffusion equation.

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# Maximal convergence theorems for functions of squared modulus holomorphic type in $R^2$ and various applications

*Authors*

- Kraus, Christiane

*2010 Mathematics Subject Classification*

- 41A17 41A10 41A60 41A63 41A25 30E10 30C35

*Keywords*

- Polynomial approximation in 2-space, maximal convergence, Bernstein-Walsh's type theorems, real-analytic functions

*Abstract*

In this paper we extend the theory of maximal convergence introduced by Walsh to functions of squared modulus holomorphic type. We introduce in accordance to the well-known complex maximal convergence number for holomorphic functions a real maximal convergence number for functions of squared modulus holomorphic type and prove several maximal convergence theorems. We achieve that the real maximal convergence number for F is always greater or equal than the complex maximal convergence number for g and equality occurs if L is a closed disk in R^2. Among other various applications of the resulting approximation estimates we show that for functions F of squared holomorphic type which have no zeros in a closed disk B_r the relation $$ limsup_n to infty sqrt[n] E_n( B_r,F) = limsup_n to infty sqrt[n]E_n( partial B_r,F) $$ is valid, where E_n is the polynomial approximation error.

*Appeared in*

- J. Approx. Theory, 147 (2007) pp. 47-66.

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# Quasilinear parabolic systems with mixed boundary conditions

*Authors*

- Hieber, Matthias
- Rehberg, Joachim

*2010 Mathematics Subject Classification*

- 35A05 35B65 35K15 35K20

*Keywords*

- Partial differential equations, parabolic operators, maximal regularity, nonsmooth domains, discontinuous coefficients, mixed boundary conditions

*Abstract*

In this paper we investigate quasilinear systems of reaction-diffusion equations with mixed Dirichlet-Neumann bondary conditions on non smooth domains. Using techniques from maximal regularity and heat-kernel estimates we prove existence of a unique solution to systems of this type.

*Appeared in*

- SIAM J. Math. Anal., 40 (2008) pp. 292--305.

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# Time discretization and Markovian iteration for coupled FBSDEs

*Authors*

- Bender, Christian
- Zhang, Jianfeng

*2010 Mathematics Subject Classification*

- 65C30 60H10 60H30 65C05

*Keywords*

- forward backward SDE, numerics, time discretization, Monte Carlo simulation

*Abstract*

In this paper we lay the foundation for a numerical algorithm to simulate high-dimensional coupled FBSDEs under weak coupling or monotonicity conditions. In particular we prove convergence of a time discretization and a Markovian iteration. The iteration differs from standard Picard iterations for FBSDEs in that the dimension of the underlying Markovian process does not increase with the number of iterations. This feature seems to be indispensable for an efficient iterative scheme from a numerical point of view. We finally suggest a fully explicit numerical algorithm and present some numerical examples with up to 10-dimensional state space.

*Appeared in*

- Ann. Appl. Probab., 18 (2008) pp. 143--177.

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# A jump-diffusion Libor model and its robust calibration

*Authors*

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

ORCID: 0000-0002-4389-8266

*2010 Mathematics Subject Classification*

- 60G51 62G20 60H05 60H10 90A09 91B28

*Keywords*

- jump-diffusion Libor models, calibration, stability, correlation structure

*Abstract*

In this paper we propose a jump-diffusion Libor model with jumps in a high-dimensional space and test a stable non-parametric calibration algorithm which takes into account a given local covariance structure. The algorithm returns smooth and simply structured Lévy densities, and penalizes the deviation from the Libor market model. In practice, the procedure is FFT based, thus fast, easy to implement, and yields good results, particularly in view of the ill-posedness of the underlying inverse problem.

*Appeared in*

- Quant. Finance, 11 (2011) pp. 529--546 .

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# Large deviations for sums defined on a Galton--Watson process

*Authors*

- Fleischmann, Klaus
- Wachtel, Vitali

*2010 Mathematics Subject Classification*

- 60J80 60F10

*Keywords*

- Large deviation probabilities, supercritical Galton-Watson processes, lower deviation probabilities, Schröder case, Böttcher case, Lotka-Nagaev estimator

*Abstract*

In this paper we study the large deviation behavior of sums of i.i.d. random variables X_i defined on a supercritical Galton-Watson process Z. We assume the finiteness of the moments EX_1^2 and EZ_1log Z_1. The underlying interplay of the partial sums of the X_i and the lower deviation probabilities of Z is clarified. Here we heavily use lower deviation probability results on Z we recently published in [FW06].

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# Two-particle models for the estimation of the mean and standard deviation of concentrations in coastal waters

*Authors*

- Spivakovskaya, Daria
- Heemink, Arnold W.
- Schoenmakers, John G. M.

ORCID: 0000-0002-4389-8266

*2010 Mathematics Subject Classification*

- 62G07 60H10 65C05

*Keywords*

- two-particle model, Lagrangian models, Advection-diffusion equation

*Abstract*

In this paper we study the mean and standard deviation of concentrations using random walk models. Two-particle models that takes into account the space correlation of the turbulence are introduced and some properties of the distribution of the particle concentration are studied. In order to reduce the CPU time of the calculation a new estimator based on reverse time diffusion is applied. This estimator has been introduced recently by Milstein, Schoenmakers, and Spokoiny (2004). Some numerical aspects of the implementation are discussed for relative simple test problems and finally a realistic application to predict the spreading of the pollutant in the Dutch coastal zone is described.

*Appeared in*

- Stochastic Environmental Research and Risk Assessment, Vol. 21 Number 3, (2007) 235-251

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# The von Mises model for one-dimensional elastoplastic beams and Prandtl--Ishlinskii hysteresis operators

*Authors*

- Krejčí, Pavel
- Sprekels, Jürgen

*2010 Mathematics Subject Classification*

- 74C05 35Q72 74N30 34C55 47J40

*Keywords*

- elastoplasticity, beam equation, hysteresis operators, Prandtl-Ishlinskii model, von Mises model

*Abstract*

In this paper, the one-dimensional equation for the transversal vibrations of an elastoplastic beam is derived from a general three-dimensional system. The plastic behavior is modeled using the classical three-dimensional von Mises plasticity model. It turns out that this single-yield model without hardening leads after a dimensional reduction to a multi-yield one-dimensional hysteresis model with kinematic hardening, given by a hysteresis operator of Prandtl-Ishlinskii type whose density function can be determined explicitly. This result indicates that the use of Prandtl-Ishlinskii hysteresis operators in the modeling of elastoplasticity is not just a questionable phenomenological approach, but in fact quite natural. In addition to the derivation of the model, it is shown that the resulting partial differential equation with hysteresis can be transformed into an equivalent system for which the existence and uniqueness of a strong solution is proved. The proof employs techniques from the mathematical theory of hysteresis operators.

*Appeared in*

- Math. Methods Appl. Sci., 30 (2007) pp. 2371--2393.

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# Optimal boundary control of a phase field system modeling nonisothermal phase transitions

*Authors*

- Lefter, Catalin
- Sprekels, Jürgen

*2010 Mathematics Subject Classification*

- 49K20 35K55 80A22 49J20

*Keywords*

- Phase-field system, phase transition, optimal control, necessary conditions

*Abstract*

In this paper, we study an optimal control problem for a singular system of partial differential equations that models a nonisothermal phase transition with a nonconserved order parameter. The control acts through a third boundary condition for the absolute temperature and plays the role of the outside temperature. It is shown that the corresponding control-to-state mapping is well defined, and the existence of an optimal control and the first-order optimality conditions for a quadratic cost functional of Bolza type are established.

*Appeared in*

- Adv. Math. Sci. Appl., 17 (2007) pp. 181-194.

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# A remark on the weakly damped wave equation

*Authors*

- Pata, Vittorino
- Zelik, Sergey

*2010 Mathematics Subject Classification*

- 35B33 35B41 35L05 35Q40

*Keywords*

- Weakly damped wave equation, critical nonlinearity, global attractor

*Abstract*

In this short note we present a direct method to establish the optimal regularity of the attractor for the semilinear damped wave equation with a nonlinearity of critical growth.

*Appeared in*

- Commun. Pure Appl. Anal., 5 (2006) pp. 609--614.

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# Slip-controlled thin film dynamics

*Authors*

- Fetzer, Renate
- Rauscher, Markus
- Münch, Andreas
- Wagner, Barbara
- Jacobs, Karin

*2010 Mathematics Subject Classification*

- 76D08 76E17 74A55

*Keywords*

- Polymer melts, slip boundary effects, interfacial and free surface flows

*Abstract*

In this study, we present a novel method to assess the slip length and the viscosity of thin films of highly viscous Newtonian liquids. We quantitatively analyse dewetting fronts of low molecular weight polystyrene melts on Octadecyl- (OTS) and Dodecyltrichlorosilane (DTS) polymer brushes. Using a thin film (lubrication) model derived in the limit of large slip lengths, we can extract slip length and viscosity. We study polymer films with thicknesses between 50 nm and 230 nm and various temperatures above the glass transition. We find slip lengths from 100 nm up to 1 $mu$m on OTS and between 300 nm and 10 $mu$m on DTS covered silicon wafers. The slip length decreases with temperature. The obtained values for the viscosity are consistent with independent measurements.

*Appeared in*

- Europhys. Lett., 75 (2006) pp. 638-644.

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# Sensitivity analysis for indirect measurement in scatterometry and the reconstruction of periodic grating structures

*Authors*

- Gross, Hermann
- Rathsfeld, Andreas

*2010 Mathematics Subject Classification*

- 78A46 65N30 65K05

*2008 Physics and Astronomy Classification Scheme*

- 42.25.Fx

*Keywords*

- Diffraction gratings, inverse problems, sensitivity analysis

*Abstract*

In this work, we discuss some aspects of numerical algorithms for the determination of periodic surface structures (gratings) from light diffraction patterns. With decreasing structure details of lithography masks, increasing demands on suitable metrology techniques arise. Methods like scatterometry as a non-imaging indirect optical method are applied to simple periodic line structures in order to evaluate the quality of the manufacturing process. Using scatterometry, geometrical parameters of periodic structures including period (pitch), side-wall angles, heights, top and bottom widths of trapezoid shaped bridges can be determined. The mathematical model for the scattering is based on the time-harmonic Maxwell's equations and reduces in case of grating structures to the Helmholtz equation. For the numerical simulation, e.g. finite element methods can be applied to solve the corresponding boundary value problems. More challenging is the inverse problem, where the grating geometry is to be reconstructed from the measured diffraction patterns. Restricting the class of gratings and the set of measurements, the inverse problem can be reformulated as a non-linear operator equation in Euclidean spaces. The operator maps the parameters describing the grating to special efficiencies of plane wave modes diffracted by the grating. We employ a Newton type iterative method to solve this operator equation. The reconstruction properties and the convergence of the numerical algorithm, however, is controlled by the local conditioning of the non-linear mapping, i.e. by the condition numbers of its Jacobian matrix. To improve the convergence of the iteration and the accuracy of the reconstruction, we determine optimal sets of efficiencies for the measurements by optimizing the condition numbers of the corresponding Jacobians. Numerical examples for a chrome-glass mask and for an inspecting light of wave length 632.8 nm confirm that an optimization of the measurement data results in better solutions.

*Appeared in*

- Waves Random Complex Media, 18 (2008) pp. 129--149.

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# Mathematical modelling of indirect measurements in periodic diffractive optics and scatterometry

*Authors*

- Gross, Hermann
- Model, Regine
- Bär, Markus
- Wurm, Matthias
- Bodermann, Bernd
- Rathsfeld, Andreas

*2010 Mathematics Subject Classification*

- 78A46 65N30 65K10

*Keywords*

- indirect measurements, mathematical modelling, inverse methods, diffractive optics

*Abstract*

In this work, we illustrate the benefits and problems of mathematical modelling and effective numerical algorithms to determine the diffraction of light by periodic grating structures. Such models are required for reconstruction of the grating structure from the light diffraction patterns. With decreasing structure dimensions on lithography masks, increasing demands on suitable metrology techniques arise. Methods like scatterometry as a non-imaging indirect optical method offer access to the geometrical parameters of periodic structures including pitch, side-wall angles, line heights, top and bottom widths. The mathematical model for scatterometry is based on the Helmholtz equation derived as a time-harmonic solution of Maxwell's equations. It determines the incident and scattered electric and magnetic fields, which fully specify the light propagation in a periodic two-dimensional grating structure. For numerical simulations of the diffraction patterns, a standard finite element method (FEM) or a generalized finite element method (GFEM) is used for solving the elliptic Helmholtz equation. In a first step, we performed systematic forward calculations for different varying structure parameters to evaluate the applicability and sensitivity of different scatterometric measurement methods. Furthermore our programs include several iterative optimization methods for reconstructing the geometric parameters of the grating structure by the minimization of a functional. First reconstruction results for different test data sets are presented.

*Appeared in*

- Measurement, 39 (2006) pp. 782--794.

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# Experimental investigations on the suppression of Q-switching in monolithic 40 GHz mode-locked semiconductor lasers

*Authors*

- Hüttl, Bernd
- Kaiser, Ronald
- Kindel, Christian
- Fidorra, Sybille
- Rehbein, Wolfgang
- Stolpe, Heiko
- Sahin, Gabriel
- Bandelow, Uwe

ORCID: 0000-0003-3677-2347 - Radziunas, Mindaugas
- Vladimirov, Andrei
- Heidrich, Helmut

*2010 Mathematics Subject Classification*

- 78A60

*2008 Physics and Astronomy Classification Scheme*

- 42.55.Px 42.60.Fc 42.60.Mi 42.60.Gd

*Keywords*

- monolithic semiconductor lasers, mode-locking, Q-switching

*Abstract*

Inherent Q-switching as a source of intra-cavity pulse energy modulations, i.e. unwanted amplitude noise, is still a challenging task in order to fabricate monolithic mode-locked semiconductor lasers in view of different commercial applications. In this paper, the results of experimental investigations on the influence of the quantum well number on the occurrence and suppression of Q-switching in 40 GHz mode-locked multiple quantum well buried heterostructure lasers are presented. Improved mode-locked lasers emit short optical pulses (<=1.6 ps) with very low amplitude noise (1-2%) and timing jitter (50-100 fs).

*Appeared in*

- Appl Phys. Lett. 88, art. no. 221104, 2006.

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# The existence of triangulations of non-convex polyhedra without new vertices

*Authors*

- Si, Hang

*2010 Mathematics Subject Classification*

- 52B55 65D18

*Keywords*

- non-convex polyhedron, regular subdivision, triangulation, Steiner points

*Abstract*

It is well known that a simple three-dimensional non-convex polyhedron may not be triangulated without using new vertices (so-called it Steiner points). In this paper, we prove a condition that guarantees the existence of a triangulation of a non-convex polyhedron (of any dimension) without Steiner points. We briefly discuss algorithms for efficiently triangulating three-dimensional polyhedra.

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# Regularity of elastic fields in composites

*Authors*

- Knees, Dorothee
- Sändig, Anna-Margarete

*2010 Mathematics Subject Classification*

- 35B65 74B05 35L55 35J60 49N60 74B20 74R10

*Keywords*

- regularity, elasticity, composite, stress singularity, Ramberg-Osgod model

*Abstract*

It is well known that high stress concentrations can occur in elastic composites in particular due to the interaction of geometrical singularities like corners, edges and cracks and structural singularities like jumping material parameters. In the project C5 "Stress concentrations in heterogeneous materials" of the SFB 404 "Multifield Problems in Solid and Fluid Mechanics" it was mathematically analyzed where and which kind of stress singularities in coupled linear and nonlinear elastic structures occur. In the linear case asymptotic expansions near the geometrical and structural peculiarities are derived, formulae for generalized stress intensity factors included. In the nonlinear case such expansions are unknown in general and regularity results are proved for elastic materials with power-law constitutive equations with the help of the difference quotient technique combined with a quasi-monotone covering condition for the subdomains and the energy densities. Furthermore, some applications of the regularity results to shape and structure optimization and the Griffith fracture criterion in linear and nonlinear elastic structures are discussed. Numerical examples illustrate the results.

*Appeared in*

- Multifield Problems in Solid and Fluid Mechanics, R. Helmig, A. Mielke, B. Wohlmuth, eds., vol. 28 of Lecture Notes in Applied and Computational Mechanics, Springer, Heidelberg, 2006, pp. 321--350

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# Exact controllability on a curve for a semilinear parabolic equation

*Authors*

- Hömberg, Dietmar
- Yamamoto, Masahiro

*2010 Mathematics Subject Classification*

- 93B05 80A22 35K55

*Keywords*

- Exact controllability, parabolic equation, laser material treatment

*Abstract*

Motivated by the growing number of industrially important laser material treatments we investigate the controllability on a curve for a semilinear parabolic equation. We prove the local exact controllability and a global stability result in the twodimensional setting. As an application we consider the control of laser surface hardening. We show that our theory applies to this situation and present numerical simulations for a PID control of laser hardening. Moreover, the result of an industrial case study is presented confirming that the numerically derived temperature in the hot-spot of the laser can indeed be used as set-point for the machine-based process control.

*Appeared in*

- Inverse Problems, 22 (2006) pp. 1855--1867.

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# Scattering theory for open quantum systems

*Authors*

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

*2010 Mathematics Subject Classification*

- 47A40 47B25 47A55 47B44 47E05

*Keywords*

- Scattering theory, open quantum system, maximal dissipative operator, pseudo-Hamiltonian, quasi-Hamiltonian, Lax-Phillips scattering, scattering matrix, characteristic function, boundary triplet, Weyl function, Sturm-Liouville operator

*Abstract*

Quantum systems which interact with their environment are often modeled by maximal dissipative operators or so-called Pseudo-Hamiltonians. In this paper the scattering theory for such open systems is considered. First it is assumed that a single maximal dissipative operator $A_D$ in a Hilbert space $sH$ is used to describe an open quantum system. In this case the minimal self-adjoint dilation $widetilde K$ of $A_D$ can be regarded as the Hamiltonian of a closed system which contains the open system $[A_D,sH]$, but since $widetilde K$ is necessarily not semibounded from below, this model is difficult to interpret from a physical point of view. In the second part of the paper an open quantum system is modeled with a family $[A(mu)]$ of maximal dissipative operators depending on energy $mu$, and it is shown that the open system can be embedded into a closed system where the Hamiltonian is semibounded. Surprisingly it turns out that the corresponding scattering matrix can be completely recovered from scattering matrices of single Pseudo-Hamiltonians as in the first part of the paper. The general results are applied to a class of Sturm-Liouville operators arising in dissipative and quantum transmitting Schrödinger-Poisson systems.

*Appeared in*

- Math. Phys. Anal. Geom., 10 (2007) pp. 313--358.

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# A mathematical framework for standard generalized materials in the rate-independent case

*Authors*

- Mielke, Alexander

ORCID: 0000-0002-4583-3888

*2010 Mathematics Subject Classification*

- 35K85 49S05 74C15 74N15

*Keywords*

- Rate independence, energetic formulation, Gamma-convergence, relaxation, shape-memory material, magnetostriction, piezoelectricity

*Abstract*

Standard generalized materials are described by an elastic energy density and a dissipation potential. The latter gives rise to the evolution equation (flow law) for the internal variables. The energetic formulation provides a very weak, derivative-free form of this flow law. It is based on a global stability condition and an energy balance. Using time-incremental minimization problems, which allow for the usage of the rich theory in the direct method of the calculus of variations, it is possible to establish general, abstract existence results as well as convergence for numerical approximations. Applications to shape-memory materials and to magnetostrictive or piezoelectric materials are surveyed.

*Appeared in*

- Multifield Problems in Solid and Fluid Mechanics, R. Helmig, A. Mielke, B. Wohlmuth, eds., vol. 28 of Lecture Notes in Applied and Computational Mechanics, Springer, Heidelberg, 2006, pp. 351--379

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# Random walk on fixed spheres for Laplace and Lamé equations

*Authors*

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

*2010 Mathematics Subject Classification*

- 65C05 65C20 74B05

*2008 Physics and Astronomy Classification Scheme*

- 02.70.Lq

*Keywords*

- Poisson integral formula, random walk on fixed spheres, Lamé equation, successive over relaxation method, divergent Neumann series, discrete random walks

*Abstract*

The Random Walk on Fixed Spheres (RWFS) introduced in our previous paper is presented in details for Laplace and Lamé equations governing static elasticity problems. The approach is based on the Poisson type integral formulae written for each disc of a domain consisting of a family of overlapping discs. The original differential boundary value problem is equivalently reformulated in the form of a system of integral equations defined on the intersection surfaces (arches, in 2D, and caps, if generalized to 3D spheres). To solve the obtained system of integral equations, a Random Walk procedure is constructed where the random walks are living on the intersecting surfaces. Since the spheres are fixed, it is convenient to construct also discrete random walk methods for solving the system of linear equations approximating the system of integral equations. We develop here two classes of special Monte Carlo iterative methods for solving these systems of linear algebraic equations which are constructed as a kind of randomized versions of the Chebyshev iteration method and Successive Over Relaxation (SOR) method. It is found that in this class of randomized SOR methods, the Gauss-Seidel method has a minimal variance. In our prevoius paper we have concluded that in the case of classical potential theory, the Random Walk on Fixed Spheres considerably improves the convergence rate of the standard Random Walk on Spheres method. More interesting, we succeeded there to extend the algorithm to the system of Lamé equations which cannot be solved by the conventional Random Walk on Spheres method. We present here a series of numerical experiments for 2D domains consisting of 5, 10, and 17 discs, and analyze the dependence of the variance on the number of discs and elastic constants. Further generalizations to Neumann and Dirichlet-Neumann boundary conditions are also possible.

*Appeared in*

- Monte Carlo Methods Appl., 12 (2006) pp. 55--93.

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# Discrete logistic branching populations and the canonical diffusion of adaptive dynamics

*Authors*

- Champagnat, Nicolas
- Lambert, Amaury

*2010 Mathematics Subject Classification*

- 60J80 60J25 60J70 60J75 60J85 60K35 92D10 92D15 92D25 92D40

*Keywords*

- Logistic branching process, population dynamics, density-dependence, competition, fixation probability, genetic drift, weak selection, adaptive dynamics, invasion fitness, timescale separation, trait substitution sequence, diffusion approximation, harmonic equations, convergence of measure-valued processes

*Abstract*

The biological theory of adaptive dynamics proposes a description of the long-time evolution of an asexual population, based on the assumptions of large population, rare mutations and small mutation steps, that lead to a deterministic ODE, called `canonical equation of adaptive dynamics'. However, in order to include the effect of genetic drift in this description, we have to apply a limit of weak selection to a finite stochastically fluctuating discrete population subject to competition in the logistic branching fashion. We start with the study of the particular case of two competing subpopulations (resident and mutant) and seek explicit first-order formulae for the probability of fixation of the mutant, also interpreted as the mutant's fitness, in the vicinity of neutrality. In particular, the first-order term is a linear combination of products of functions of the initial mutant frequency times functions of the initial total population size, called invasibility coefficients (fertility, defence, aggressiveness, isolation, survival). Then we apply a limit of rare mutations to a population subject to mutation, birth and competition where the number of coexisting types may fluctuate, while keeping the population size finite. This leads to a jump process, the so-called `trait substitution sequence', where evolution proceeds by successive invasions and fixations of mutant types. Finally, we apply a limit of weak selection (small mutation steps) to this jump process, that leads to a diffusion process of evolution, called `canonical diffusion of adaptive dynamics', in which genetic drift is combined with directional selection driven by the fitness gradient.

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# From synchronization to Lyapunov exponents and back

*Authors*

- Politi, Antonio
- Ginelli, Francesco
- Yanchuk, Serhiy
- Maistrenko, Yuri

*2008 Physics and Astronomy Classification Scheme*

- 05.45.Xt, 05.45.Ac

*Keywords*

- Lyapunov exponents, synchronization

*Abstract*

The goal of this paper is twofold. In the first part we discuss a general approach to determine Lyapunov exponents from ensemble- rather than time-averages. The approach passes through the identification of locally stable and unstable manifolds (the Lyapunov vectors), thereby revealing an analogy with generalized synchronization. The method is then applied to a periodically forced chaotic oscillator to show that the modulus of the Lyapunov exponent associated to the phase dynamics increases quadratically with the coupling strength and it is therefore different from zero already below the onset of phase-synchronization. The analytical calculations are carried out for a model, the generalized special flow, that we construct as a simplified version of the periodically forced Rössler oscillator.

*Appeared in*

- Phys. D, 224 (2006) pp. 90-101.

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# On a fast integral equation method for diffraction gratings

*Authors*

- Rathsfeld, Andreas
- Schmidt, Gunther
- Kleemann, Bernd H.

*2010 Mathematics Subject Classification*

- 78M15 78A45 65R99

*Keywords*

- Diffraction gratings, integral equation method, preconditioning, fundamental solution

*Abstract*

The integral equation method for the simulation of the diffraction by optical gratings is an efficient numerical tool if profile gratings determined by simple cross-section curves are considered. This method in its recent version is capable to tackle profile curves with corners, gratings with thin coated layers, and diffraction scenarios with unfavorably large ratios period over wavelength. We discuss special implementational issues including the efficient evaluation of the quasi-periodic Green kernels, the quadrature algorithm, and the iterative solution of the arising systems of linear equations. Finally, as application we present the simulation of coated echelle gratings which demonstrates the efficency of our approach.

*Appeared in*

- Commun. Comput. Phys., 1 (2006) pp. 984-1009.

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# Invasion and adaptive evolution for individual-based spatially structured populations

*Authors*

- Champagnat, Nicolas
- Méléard, Sylvie

*2010 Mathematics Subject Classification*

- 60J85 60K35 92D15 92D25 35K60

*Keywords*

- spatially structured population, adaptive evolution, stochastic individual-based process, birth-and-death point process, reflected diffusion, mutation and selection, nonlinear reaction-diffusion equation, nonlocal and local interactions, clustering and polymorphism, invasion and evolution

*Abstract*

The interplay between space and evolution is an important issue in population dynamics, that is in particular crucial in the emergence of polymorphism and spatial patterns. Recently, biological studies suggest that invasion and evolution are closely related. Here we model the interplay between space and evolution starting with an individual-based approach and show the important role of parameter scalings on clustering and invasion. We consider a stochastic discrete model with birth, death, competition, mutation and spatial diffusion, where all the parameters may depend both on the position and on the trait of individuals. The spatial motion is driven by a reflected diffusion in a bounded domain. The interaction is modelled as a trait competition between individuals within a given spatial interaction range. First, we give an algorithmic construction of the process. Next, we obtain large population approximations, as weak solutions of nonlinear reaction-diffusion equations with Neumann's boundary conditions. As the spatial interaction range is fixed, the nonlinearity is nonlocal. Then, we make the interaction range decrease to zero and prove the convergence to spatially localized nonlinear reaction-diffusion equations, with Neumann's boundary conditions. Finally, simulations based on the microscopic individual-based model are given, illustrating the strong effects of the spatial interaction range on the emergence of spatial and phenotypic diversity (clustering and polymorphism) and on the interplay between invasion and evolution. The simulations focus on the qualitative differences between local and nonlocal

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# A comparison of analytical cutting force models

*Authors*

- Rott, Oliver
- Hömberg, Dietmar
- Mense, Carsten

*2010 Mathematics Subject Classification*

- 70E50 70J25

*Keywords*

- Milling, regenerative chatter, time domain simulation, stability, machine dynamics

*Abstract*

The modeling of dynamic processes in milling and the determination of stable cutting conditions have become increasingly important for the optimization of manufacturing processes. Analytic approaches and time domain simulations based on simplified dynamic systems are used to identify chatter-free machining conditions. Stresses applied to the system are generally estimated by cutting force models. The goal of this paper is to compare the influence of the cutting force models on the stability limits. Numerical simulations of a simplified, generic milling machine model are therefore performed, while varying the cutting force approach. In order to distinguish stable from unstable cutting conditions a numerical stability criterion is used. The resulting stability charts are then investigated and analyzed to show the effect of the different cutting force models.

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# Multiple disorder problems for Wiener and compound Poisson processes with exponential jumps

*Authors*

- Gapeev, Pavel

*2010 Mathematics Subject Classification*

- 60G40 62M20 34K10 62C10 60J60 60J75

*Keywords*

- Multiple disorder problem, Wiener process, compound Poisson process, optimal switching, coupled optimal stopping problem, (integro-differential) coupled free-boundary problem, smooth and continuous fit, Itô-Tanaka-Meyer formula

*Abstract*

The multiple disorder problem consists of finding a sequence of stopping times which are as close as possible to the (unknown) times of 'disorder' when the distribution of an observed process changes its probability characteristics. We present a formulation and solution of the multiple disorder problem for a Wiener and a compound Poisson process with exponential jumps. The method of proof is based on reducing the initial optimal switching problems to the corresponding coupled optimal stopping problems and solving the equivalent coupled free-boundary problems by means of the smooth- and continuous-fit conditions.

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# Time splitting error in DSMC schemes for the inelastic Boltzmann equation

*Authors*

- Rjasanow, Sergej
- Wagner, Wolfgang

*2010 Mathematics Subject Classification*

- 82C40 82C80 65R20

*Keywords*

- Granular matter, Boltzmann equation, Stochastic numerics

*Abstract*

The paper is concerned with the numerical treatment of the uniformly heated inelastic Boltzmann equation by the direct simulation Monte Carlo (DSMC) method. This technique is presently the most widely used numerical method in kinetic theory. We consider three modifications of the DSMC method and study them with respect to their efficiency and convergence properties. Convergence is investigated both with respect to the number of particles and to the time step. The main issue of interest is the time step discretization error due to various splitting strategies. A scheme based on the Strang-splitting strategy is shown to be of second order with respect to time step, while there is only first order for the commonly used Euler-splitting scheme. On the other hand, a no-splitting scheme based on appropriate Markov jump processes does not produce any time step error. It is established in numerical examples that the no-splitting scheme is about two orders of magnitude more efficient than the Euler-splitting scheme. The Strang-splitting scheme reaches almost the same level of efficiency compared to the no-splitting scheme, since the deterministic time step error vanishes sufficiently fast.

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# Structural properties of linear probabilistic constraints

*Authors*

- Henrion, René

*2010 Mathematics Subject Classification*

- 90C15

*Keywords*

- probabilistic constraints, stochastic programming, chance constraints, stochastic optimization

*Abstract*

The paper provides a structural analysis of the feasible set defined by linear probabilistic constraints. Emphasis is laid on single (individual) probabilistic constraints. A classical convexity result by Van de Panne/Popp and Kataoka is extended to a broader class of distributions and to more general functions of the decision vector. The range of probability levels for which convexity can be expected is exactly identified. Apart from convexity, also nontriviality and compactness of thefeasible set are precisely characterized at the same time. The relation between feasible sets with negative and nonnegative right-hand side is revealed. Finally, an existence result is formulated for the more difficult case of joint probabilistic constraints.

*Appeared in*

- Optimization, 56 (2007) pp. 425--440.

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# On calculating the normal cone to a finite union of convex polyhedra

*Authors*

- Henrion, René
- Outrata, Jiří

*2010 Mathematics Subject Classification*

- 49J52 90C31

*Keywords*

- limitng normal cone, convex polyhedra, union of polyhedral cones

*Abstract*

The paper provides formulae for calculating the limiting normal cone introduced by Mordukhovich to a finite union of convex polyhedra. In the first part, special cases of independent interest are considered (almost disjoint cones, half spaces, orthants). The second part focusses on unions of general polyhedra. Due to the local nature of the normal cone, one may restrict considerations without loss of generality to finite unions of polyhedral cones. First, an explicit formula for the normal cone is provided in the situation of two cones. An algorithmic approach is presented along with a refined, more efficient formula. Afterwards, a general formula for the union of N cones is derived. Finally, an application to the stability analysis of a special type of probabilistic constraints is provided.

*Appeared in*

- Optimization, 57 (2008) pp. 57--78.

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# Continuum descriptions for the dynamics in discrete lattices: Derivation and justification

*Authors*

- Giannoulis, Johannes
- Herrmann, Michael
- Mielke, Alexander

ORCID: 0000-0002-4583-3888

*2010 Mathematics Subject Classification*

- 34C20 34E13 37K60 37K05 70F45 70K70

*Keywords*

- Discrete lattice systems, modulation equations, Hamiltonian structure, multiscale ansatz, dispersive wave propagation

*Abstract*

The passage from microscopic systems to macroscopic ones is studied by starting from spatially discrete lattice systems and deriving several continuum limits. The lattice system is an infinite-dimensional Hamiltonian system displaying a variety of different dynamical behavior. Depending on the initial conditions one sees quite different behavior like macroscopic elastic deformations associated with acoustic waves or like propagation of optical pulses. We show how on a formal level different macroscopic systems can be derived such as the Korteweg-de Vries equation, the nonlinear Schroedinger equation, Whitham's modulation equation, the three-wave interaction model, or the energy transport equation using the Wigner measure. We also address the question how the microscopic Hamiltonian and the Lagrangian structures transfer to similar structures on the macroscopic level. Finally we discuss rigorous analytical convergence results of the microscopic system to the macroscopic one by either weak-convergence methods or by quantitative error bounds.

*Appeared in*

- Analysis, Modeling and Simulation of Multiscale Problems, A. Mielke, ed., Springer, Heidelberg, 2006, pp. 435--466

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# A solution of Braess' approximation problem on powers of the distance function

*Authors*

- Kraus, Christiane

*2010 Mathematics Subject Classification*

- 41A25 41A10 41A60 41A63 41A17

*Keywords*

- Polynomial approximation in 2-space, maximal convergence, Bernstein-Walsh's type theorems

*Abstract*

The polynomial approximation behaviour of the class of functions $$ F_s: R^2(x_0, y_0 ) -> R, F_s(x,y) = ( (x-x_0)^2 + (y-y_0)^2 )^(-s), s in (0, infty),$$ is studied in [Bra01]. There it is claimed that the obtained results can be embedded in a more general setting. This conjecture will be confirmed and complemented by a different approach than in [Bra01]. The key is to connect the approximation rate of F_s with its holomorphic continuability for which the classical Bernstein approximation theorem is linked with the convexity of best approximants. Approximation results of this kind also play a vital role in the numerical treatment of elliptic differential equations [Sau].

*Appeared in*

- Constructive Approximation, 27, p. 323-327, 2008 under the new title: Multivariate Polynomial Approximation of Powers of the Euclidian Distance Function

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# A new fictitious domain method in shape optimization

*Authors*

- Eppler, Karsten
- Harbrecht, Helmut
- Mommer, Mario

*2010 Mathematics Subject Classification*

- 49Q10 49M15 65N30 65K10 49K20

*Keywords*

- Ficticious domain method, shape optimization, least square solution, Quasi--Newton method

*Abstract*

The present paper is concerned with investigating the capability of the smoothness preserving fictitious domain method from [22] to shape optimization problems. We consider the problem of maximizing the Dirichlet energy functional in the class of all simply connected domains with fixed volume, where the state equation involves an elliptic second order differential operator with non-constant coefficients. Numerical experiments in two dimensions validate that we arrive at a fast and robust algorithm for the solution of the considered class of problems. The proposed method keeps applicable for three dimensional shape optimization problems.

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# Analysis of profile functions for general linear regularization methods

*Authors*

- Mathé, Peter

ORCID: 0000-0002-1208-1421 - Hofmann, Bernd

*2010 Mathematics Subject Classification*

- 47A52 65J20 65F22 65R30

*Keywords*

- linear ill-posed problems, regularization, distance function, convergence rates, index function, source condition, qualification, range inclusion

*Abstract*

The stable approximate solution of ill-posed linear operator equations in Hilbert spaces requires regularization. Tight bounds for the noise-free part of the regularization error are constitutive for bounding the overall error. Norm bounds of the noise-free part which decrease to zero along with the regularization parameter are called profile functions and are subject of our analysis. The interplay between properties of the regularization and certain smoothness properties of solution sets, which we shall describe in terms of source-wise representations is crucial for the decay of associated profile functions. On the one hand, we show that a given decay rate is possible only if the underlying true solution has appropriate smoothness. On the other hand, if smoothness fits the regularization, then decay rates are easily obtained. If smoothness does not fit, then we will measure this in terms of some distance function. Tight bounds for these allow us to obtain profile functions. Finally we study the most realistic case when smoothness is measured with respect to some operator which is related to the one governing the original equation only through a link condition. In many parts the analysis is done on geometric basis, extending classical concepts of linear regularization theory in Hilbert spaces. We emphasize intrinsic features of linear ill-posed problems which are frequently hidden in the classical analysis of such problems.

*Appeared in*

- SIAM J. Numer. Anal., 45 (2007) pp. 1122--1141.

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# A rate-independent model for the isothermal quasi-static evolution of shape-memory materials

*Authors*

- Auricchio, Ferdinando
- Mielke, Alexander

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

*2010 Mathematics Subject Classification*

- 74C05 49J40

*Keywords*

- Shape-memory materials, rate-independent evolution, time and space discretization, Gamma convergence, asymptotics

*Abstract*

This note addresses a three-dimensional model for isothermal stress-induced transformation in shape-memory polycrystalline materials. We treat the problem within the framework of the energetic formulation of rate-independent processes and investigate existence and continuous dependence issues at both the constitutive relation and quasi-static evolution level. Moreover, we focus on time and space approximation as well as on regularization and parameter asymptotics.

*Appeared in*

- Math. Methods Appl. Sci., 18 (2008) pp. 125--164.

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# Adaptive tetrahedral mesh generation by constrained Delaunay refinement

*Authors*

- Si, Hang

*2010 Mathematics Subject Classification*

- 65M50 65N50

*Keywords*

- tetrahedral mesh generation, adaptive mesh generation, constrained Delaunay, mesh refinement, mesh quality

*Abstract*

This paper discusses the problem of refining a constrained Delaunay tetrahedralization (CDT) for adaptive numerical simulation. A simple and efficient algorithm which makes use of the classical Delaunay refinement scheme is proposed. It generates an isotropic tetrahedral mesh corresponding to a sizing function which can be either user-specified or automatically derived from the input CDT. The quality of the produced meshes is guaranteed, i.e., most output tetrahedra have their circumradius-to-shortest-edge ratios bounded except those in the neighborhood of small input angles. Good mesh conformity can be obtained for smoothly changing sizing information. The algorithm has been implemented. Various examples are provided to illustrate its theoretical aspects as well as practical performance.

*Appeared in*

- Internat. J. Numer. Methods Engrg., 75 (2008) pp. 856--880.

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# Smooth attractors for strongly damped wave equations

*Authors*

- Pata, Vittorino
- Zelik, Sergey

*2010 Mathematics Subject Classification*

- 35B33 35B40 35L05 35M10

*Keywords*

- Strongly damped wave equation, critical and supercritical growths, compact global attractors, regularity

*Abstract*

This paper is concerned with the semilinear strongly damped wave equation $$ptt u-Delta pt u-Delta u+varphi(u)=f.$$ The existence of compact global attractors of optimal regularity is proved for nonlinearities $varphi$ of critical and supercritical growth.

*Appeared in*

- Nonlinearity, 19 (2006) pp. 1495--1506.

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# Two-scale homogenization for evolutionary variational inequalities via the energetic formulation

*Authors*

- Mielke, Alexander

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

*2010 Mathematics Subject Classification*

- 35J85 49J40 74C05 74Q05

*Keywords*

- Weak two-scale convergence, strong two-scale convergence, two-scale Gamma limit, rate-independent evolution, energetic formulation, elastoplasticity with hardening

*Abstract*

This paper is devoted to the two-scale homogenization for a class of rate-independent systems described by the energetic formulation or equivalently by an evolutionary variational inequality. In particular, we treat the classical model of linearized elastoplasticity with hardening. The associated nonlinear partial differential inclusion has periodically oscillating coefficients, and the aim is to find a limit problem for the case that the period tends to 0. Our approach is based on the notion of energetic solutions which is phrased in terms of a stability condition and an energy balance of an energy-storage functional and a dissipation functional. Using the recently developed method of weak and strong two-scale convergence via periodic unfolding, we show that these two functionals have a suitable two-scale limit, but now involving the macroscopic variable in the physical domain as well as the microscopic variable in the periodicity cell. Moreover, relying on an abstract theory of Gamma convergence for the energetic formulation using so-called joint recovery sequences it is possible to show that the solutions of the problem with periodicity converge to the energetic solution associated with the limit functionals.

*Appeared in*

- SIAM J. Math. Anal., 39 (2007) pp. 642--668.

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# Spatially adaptive estimation via fitted local likelihood techniques

*Authors*

- Katkovnik, Vladimir
- Spokoiny, Vladimir

ORCID: 0000-0002-2040-3427

*2010 Mathematics Subject Classification*

- 62G05 62G20

*Keywords*

- local model selection, fitted likelihood, adaptive estimation

*Abstract*

This paper offers a new technique for spatially adaptive estimation. The local likelihood is exploited for nonparametric modelling of observations and estimated signals. The approach is based on the assumption of a local homogeneity of the signal: for every point there exists a neighborhood in which the signal can be well approximated by a constant. The fitted local likelihood statistics is used for selection of an adaptive size of this neighborhood. The algorithm is developed for quite a general class of observations subject to the exponential distribution. The estimated signal can be uni- and multivariable. We demonstrate a good performance of the new algorithm for Poissonian image denoising and compare of the new method versus the intersection of confidence interval $(ICI) $ technique that also exploits a selection of an adaptive neighborhood for estimation.

*Appeared in*

- IEEE Trans. Signal Process., 56 (2008) pp. 873--886.

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# A class of minimum principles for characterizing the trajectories and the relaxation of dissipative systems

*Authors*

- Mielke, Alexander

ORCID: 0000-0002-4583-3888 - Ortiz, Michael

*2010 Mathematics Subject Classification*

- 35K90 49J40 49S05

*Keywords*

- Gamma convergence, relaxation, energetic formulation, rate-independent problems, microstructure

*Abstract*

This work is concerned with the reformulation of evolutionary problems in a weak form enabling consideration of solutions that may exhibit evolving microstructures. This reformulation is accomplished by expressing the evolutionary problem in variational form, i.e., by identifying a functional whose minimizers represent entire trajectories of the system. The particular class of functionals under consideration is derived by first defining a sequence of time-discretized minimum problems and subsequently formally passing to the limit of continuous time. The resulting functionals may be regarded as elliptic regularizations of the original evolutionary problem. We find that the $Gamma$-limits of interest are highly degenerate and provide limited information regarding the limiting trajectories of the system. Instead we seek to characterize the minimizing trajectories directly. The special class of problems characterized by a rate-independent dissipation functional is amenable to a particularly illuminating analysis. For these systems it is possible to derive a priori bounds that are independent of the regularizing parameter, whence it is possible to extract convergent subsequences and find the limiting trajectories. Under general assumptions on the functionals, we show that all such limits satisfy the energetic formulation (S) & (E) for rate-independent systems. Moreover, we show that the accumulation points of the regularized solutions solve the associated limiting energetic formulation.

*Appeared in*

- ESAIM Control Optim. Calc. Var., 14 (2008) pp. 494--516.

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# Γ-limits and relaxations for rate-independent evolutionary problems

*Authors*

- Mielke, Alexander

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

*2010 Mathematics Subject Classification*

- 49J40 49S05 35K90

*Keywords*

- Rate-independent problems, energetic formulation, Gamma convergence, relaxation, time-incremental minimization, joint recovery sequence

*DOI*

*Abstract*

This work uses the energetic formulation of rate-independent systems that is based on the stored-energy functionals ε and the dissipation distance D. For sequences (ε _{k})_{k ∈ ℕ} and (D _{k})_{k ∈ ℕ} we address the question under which conditions the limits q_{∞} of solutions q_{k}: [0,T] → Q satisfy a suitable limit problem with limit functionals ε_{∞} and D_{∞}, which are the corresponding Γ-limits. We derive a sufficient condition, called emphconditional upper semi-continuity of the stable sets, which is essential to guarantee that q_{∞} solves the limit problem. In particular, this condition holds if certain emphjoint recovery sequences exist. Moreover, we show that time-incremental minimization problems can be used to approximate the solutions. A first example involves the numerical approximation of functionals using finite-element spaces. A second example shows that the stop and the play operator convergece if the yield sets converge in the sense of Mosco. The third example deals with a problem developing microstructure in the limit k → ∞, which in the limit can be described by an effective macroscopic model.

*Appeared in*

- Calc. Var. Partial Differ. Equ., 31 (2008) pp. 387--416.

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# Slip vs. viscoelasticity in dewetting thin films

*Authors*

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

*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, thin liquid films

*Abstract*

Ultrathin polymer films on non-wettable substrates display dynamic features which have been attributed to either viscoelastic or slip effects. Here we show that in the weak and strong slip regime effects of viscoelastic relaxation are either absent or essentially indistinguishable from slip effects. Strong-slip modifies the fastest unstable mode in a rupturing thin film, which questions the standard approach to reconstruct the effective interface potential from dewetting experiments.

*Appeared in*

- Eur. Phys. J. E -- Soft Matter, 20 (2006) pp. 267-271.

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# Attractors and their regularity for 2-D wave equations with nonlinear damping

*Authors*

- Pata, Vittorino
- Zelik, Sergey

*2010 Mathematics Subject Classification*

- 35B41 35L05 74K15

*Keywords*

- Wave equation, nonlinear damping, compact global attractor, exponential attractor

*Abstract*

We address the study of a weakly damped wave equation in space-dimension two, with a damping coefficient depending on the displacement. The equation is shown to generate a semigroup possessing a compact global attractor of optimal regularity, as well as an exponential attractor.

*Appeared in*

- Adv. Math. Sci. Appl. 17 (2007), no. 1, 225--237.

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# Attractors for semilinear equations of viscoelasticity with very low dissipation

*Authors*

- Gatti, Stefania
- Miranville, Alain
- Pata, Vittorino
- Zelik, Sergey

*2010 Mathematics Subject Classification*

- 35B40 35L70 37L45 45K05 74D99

*Keywords*

- Hyperbolic equation with memory, dynamical system, Lyapunov function, gradient system, global attractor

*Abstract*

We analyze a differential system arising in the theory of isothermal viscoelasticity. This system is equivalent to an integrodifferential equation of hyperbolic type with a cubic nonlinearity, where the dissipation mechanism is contained only in the convolution integral, accounting for the past history of the displacement. In particular, we consider here a convolution kernel which entails an extremely weak dissipation. In spite of that, we show that the related dynamical system possesses a global attractor of optimal regularity.

*Appeared in*

- Rocky Mountain J. Math., 38 (2008) pp. 1117-1138

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# Error estimates for the finite-element approximation of an elliptic control problem with pointwise state and control constraints

*Authors*

- Meyer, Christian

*2010 Mathematics Subject Classification*

- 49K20 49M25 65N30

*Keywords*

- Linear-quadratic optimal control problems, elliptic equations, state constraints, numerical approximation

*Abstract*

We consider a linear-quadratic elliptic optimal control problem with pointwise state constraints. The problem is fully discretized using linear ansatz functions for state and control. Based on a Slater-type argument, we investigate the approximation behavior for mesh size tending to zero. The obtained convergence order for the $L^2$-error of the control and for $H^1$-error of the state amounts $1-ve$ in the two-dimensional case and $1/2-ve$ in three dimension. In a second step, a state-constrained problem with additional control constraints is considered. Here, the control is discretized by constant ansatz functions. It is shown that the convergence theory can be adapted to this case yielding the same order of convergence. The theoretical findings are confirmed by numerical examples.

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# Well posedness results for a class of partial differential equations with hysteresis arising in electromagnetism

*Authors*

- Eleuteri, Michela

*2010 Mathematics Subject Classification*

- 35Q60 47J40

*Keywords*

- partial differential equations, hysteresis, electromagnetic processes

*Abstract*

We consider an evolutionary P.D.E. motivated by models for electromagnetic processes in ferromagnetic materials. Magnetic hysteresis is represented by means of a hysteresis operator. Under suitable assumptions, an existence and uniqueness theorem is obtained, together with the Lipschitz continuous dependence on the data and some further regularity results. The discussion of the behaviour of the solution in dependence on physical parameters of the problem is also outlined.

*Appeared in*

- Nonlinear Anal. Real World Appl., 8 (2007) pp. 1494-1511.

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# Invariant manifolds for random dynamical systems with slow and fast variables

*Authors*

- Schmalfuss, Björn
- Schneider, Klaus R.

*2010 Mathematics Subject Classification*

- 37H10 37L25 70K70

*Keywords*

- Random dynamical systems, fast-slow system, slow manifold, inertial manifold

*Abstract*

We consider random dynamical systems with slow and fast variables driven by two independent metric dynamical systems modelling stochastic noise. We establish the existence of a random inertial manifold eliminating the fast variables. If the scaling parameter tends to zero, the inertial manifold tends to another manifold which is called the slow manifold. We achieve our results by means of a fixed point technique based on a random graph transform. To apply this technique we need an asymptotic gap condition.

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# Existence results for a contact problem with varying friction coefficient and nonlinear forces

*Authors*

- Schmid, Florian
- Mielke, Alexander

ORCID: 0000-0002-4583-3888

*2010 Mathematics Subject Classification*

- 74M10 74M15 74H20 74B20

*Keywords*

- Coulomb friction, varying friction coefficient, unilateral contact, quasistatic, energetic formulation, nonlinear elasticity, existence, finite-dimensional

*Abstract*

We consider the rate-independent problem of a particle moving in a three - dimensional half space subject to a time-dependent nonlinear restoring force having a convex potential and to Coulomb friction along the flat boundary of the half space, where the friction coefficient may vary along the boundary. Our existence result allows for solutions that may switch arbitrarily often between unconstrained motion in the interior and contact where the solutions may switch between sticking and frictional sliding. However, our existence result is local and guarantees continuous solutions only as long as the convexity of the potential is strong enough to compensate the variation of the friction coefficient times the contact pressure. By simple examples we show that our sufficient conditions are also necessary. Our method is based on the energetic formulation of rate-independent systems as developed by Mielke and co-workers. We generalize the time-incremental minimization procedure of Mielke and Rossi for the present situation of a non-associative flow rule.

*Appeared in*

- ZAMM, Volume 87, Issue 8-9, 2007, pp. 616-631

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# A thin film model for corotational Jeffreys fluids under strong slip

*Authors*

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

*2010 Mathematics Subject Classification*

- 76D08 76E17 74A55

*2008 Physics and Astronomy Classification Scheme*

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

*Keywords*

- Viscoelasticity, non-newtonian fluid flows, interfacial and free surface flows

*Abstract*

We derive a thin film model for viscoelastic liquids under strong slip which obey the stress tensor dynamics of corotational Jeffreys fluids.

*Appeared in*

- Eur. Phys. J. E -- Soft Matter, 20 (2006) pp. 365-368.

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# Modeling of drift-diffusion systems

*Authors*

- Stephan, Holger

*2010 Mathematics Subject Classification*

- 35K55 80A20 82D37 82C31 35B50 35G25

*Keywords*

- drift-diffusion systems, energy models, free energy, Lyapunov functions, positive solutions, inverse Hessian, cross diffusion

*Abstract*

We derive drift-diffusion systems describing transport processes starting from free energy and equilibrium solutions by a unique method. We include several statistics, heterostructures and cross diffusion. The resulting systems of nonlinear partial differential equations conserve mass and positivity, and have a Lyapunov function (free energy). Using the inverse Hessian as mobility, non-degenerate diffusivity matrices turn out to be diagonal, or - in the case of cross diffusion - even constant.

*Appeared in*

- Zeitschrift fuer Angewandte Mathematik und Physik (ZAMP) 60 (2009) pp.33-53

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# Potentials of Gaussians and approximate wavelets

*Authors*

- Maz'ya, Vladimir
- Schmidt, Gunther

*2010 Mathematics Subject Classification*

- 41A55 41A63 65D30

*Keywords*

- Multidimensional cubature, wavelets, harmonic, diffraction, elastic, hydrodynamic potentials

*Abstract*

We derive new formulas for harmonic, diffraction, elastic, and hydrodynamic potentials acting on anisotropic Gaussians and approximate wavelets. These formulas can be used to construct accurate cubature formulas for these potentials.

*Appeared in*

- Math. Nachr., 280 (2007) pp. 1176--1189.

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# Stationary solutions to an energy model for semiconductor devices where the equations are defined on different domains

*Authors*

- Glitzky, Annegret
- Hünlich, Rolf

*2010 Mathematics Subject Classification*

- 35J55 35A07 35R05 80A20

*Keywords*

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

*Abstract*

We discuss a stationary energy model from semiconductor modelling. We accept the more realistic assumption that the continuity equations for electrons and holes have to be considered only in a subdomain $Omega_0$ of the domain of definition $Omega$ of the energy balance equation and of the Poisson equation. Here $Omega_0$ corresponds to the region of semiconducting material, $OmegasetminusOmega_0$ represents passive layers. Metals serving as contacts are modelled by Dirichlet boundary conditions. We prove a local existence and uniqueness result for the two-dimensional stationary energy model. For this purpose we derive a $W^1,p$-regularity result for solutions of systems of elliptic equations with different regions of definition and use the Implicit Function Theorem.

*Appeared in*

- Math. Nachr., 281 (2008) pp. 1676--1693.

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# On modeling, analytical study and homogenization for smart materials

*Authors*

- Timofte, Aida

*2010 Mathematics Subject Classification*

- 74H20 74H25 35J85 74C05 74Q05

*Keywords*

- Ferroelectrics, shape memory alloys, energetic formulation, evolutionary variational inequalities, homogenization, existence, uniqueness

*Abstract*

We discuss existence, uniqueness, regularity, and homogenization results for some nonlinear time-dependent material models. One of the methods for proving existence and uniqueness is the so-called energetic formulation, based on a global stability condition and on an energy balance. As for the two-scale homogenization we use the recently developed method of periodic unfolding and periodic folding. We also take advantage of the abstract Gamma convergence theory for rate-independent evolutionary problems.

*Appeared in*

- Topics on mathematics for smart systems, 237--252, World Sci, Publ., Hackensack, NJ, 2007

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# Improving the modulation bandwidth in semiconductor lasers by passive feedback

*Authors*

- Radziunas, Mindaugas
- Glitzky, Annegret
- Bandelow, Uwe

ORCID: 0000-0003-3677-2347 - Wolfrum, Matthias
- Troppenz, Ute
- Kreissl, Jochen
- Rehbein, Wolfgang

*2010 Mathematics Subject Classification*

- 78A60 35B35 37L65 35B60 35-04

*2008 Physics and Astronomy Classification Scheme*

- 42.60.Fc 42.30.Lr 42.55.Px 73.40.-c

*Keywords*

- semiconductor laser, passive feedback, modulation bandwidth, direct modulation, damping, small signal response, large signal response, eye diagram

*Abstract*

We explore the concept of passive-feedback lasers for direct signal modulation at 40 Gbit/s. Based on numerical simulation and bifurcation analysis, we explain the main mechanisms in these devices which are crucial for modulation at high speed. The predicted effects are demonstrated experimentally by means of correspondingly designed devices. In particular a significant improvement of the modulation bandwidth at low injection currents can be demonstrated.

*Appeared in*

- IEEE JSTQE 13(1), pp 136-142, 2007.

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# Limit for pulse compression by pulse splitting

*Authors*

- Demircan, Ayhan
- Bandelow, Uwe

ORCID: 0000-0003-3677-2347

*2010 Mathematics Subject Classification*

- 35Q55 35Q60 78A60

*Keywords*

- Nonlinear Schrödinger equation, optical fiber, pulse compression

*Abstract*

We have detected a fundamental pulse-compression limit for high-nonlinear fibers in the normal dispersion regime near the zero-dispersion wavelength. The desired generation of a broadband continuum by self-phase modulation is limited by already small amounts of third-order dispersion, which results in pulse splitting above a critical pulse power. We investigate the critical fiber length in dependence on pulse- and fiber parameters.

*Appeared in*

- Optical and Quantum Electronics 38 (12 - 14): 1167-1172, Sep 2006

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# External cavity modes in Lang--Kobayashi and traveling wave models

*Authors*

- Radziunas, Mindaugas
- Wünsche, Hans-Jürgen
- Krauskopf, Bernd
- Wolfrum, Matthias

*2010 Mathematics Subject Classification*

- 37M20 37N20 35B60 78A60

*2008 Physics and Astronomy Classification Scheme*

- 42.55.Px 42.65.Sf 42.60.Da 02.30.Jr 02.30.Ks 02.30.Mv

*Keywords*

- Lang--Kobayashi model, traveling wave model, feedback laser, cavity modes

*Abstract*

We investigate a semiconductor laser with delayed optical feedback due to an external cavity formed by a regular mirror. We discuss similarities and differences of the well-known Lang--Kobayashi delay differential equation model and the traveling wave partial differential equation model. For comparison we locate the continuous wave states in both models and analyze their stability.

*Appeared in*

- SPIE Proceedings Series, vol. 6184, art. no. 61840X, 2006.

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# Resolvent estimates in $W^-1,p$ related to strongly coupled linear parabolic systems with coupled nonsmooth capacities

*Authors*

- Glitzky, Annegret
- Hünlich, Rolf

*2010 Mathematics Subject Classification*

- 35K45 35R05 35B65

*Keywords*

- Linear parabolic systems, coupled capacities, strong coupling, nonsmooth data, resolvent estimates, regularity, existence, uniqueness

*Abstract*

We investigate linear parabolic systems with coupled nonsmooth capacities and mixed boundary conditions. We prove generalized resolvent estimates in $W^-1,p$ spaces. The method is an appropriate modification of a technique introduced by Agmon to obtain $L^p$ estimates for resolvents of elliptic differential operators in the case of smooth boundary conditions. Moreover, we establish an existence and uniqueness result.

*Appeared in*

- Math. Methods Appl. Sci., 30 (2007) pp. 2215--2232.

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# Moderate deviations for random walk in random scenery

*Authors*

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

*2010 Mathematics Subject Classification*

- 60F10 60K37

*Keywords*

- Moderate deviation principles, self-intersection local times, concentration inequalities, large deviations, moderate deviation regimes, maximum of local times, precise asymptotics, annealed probabilities, Cramér's condition

*Abstract*

We investigate random walks in independent, identically distributed random sceneries under the assumption that the scenery variables satisfy Cramér's condition. We prove moderate deviation principles in dimensions $dge 2$, covering all those regimes where rate and speed do not depend on the actual distribution of the scenery. In the case $dge 4$ we even obtain precise asymptotics for the annealed probability of a moderate deviation, extending a classical central limit theorem of Kesten and Spitzer. In $dge 3$, an important ingredient in the proofs are new concentration inequalities for self-intersection local times of random walks, which are of independent interest, whilst in $d=2$ we use a recent moderate deviation result for self-intersection local times, which is due to Bass, Chen and Rosen.

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# Analysis of the interplay between soliton fission and modulation instability in supercontinuum generation

*Authors*

- Demircan, Ayhan
- Bandelow, Uwe

ORCID: 0000-0003-3677-2347

*2010 Mathematics Subject Classification*

- 35Q55 35Q60 78A60

*2008 Physics and Astronomy Classification Scheme*

- 48.81.Dp; 42.65.Re

*Keywords*

- Nonlinear Schrödinger Equation, optical Fiber, Four-Wave-Mixing

*Abstract*

We investigate the generation mechanisms for ultrawide spectra in nonlinear optical fibers. Soliton fission and modulation instability represent fundamental mechanisms for the generation process. The primary origin of the spectral broadening changes with the pump-pulse duration. Soliton fission dominates for low input power and short pulses. Its efficiency for supercontinuum generation and especially the extend to the blue side can be increased by proper design of the dispersion profile. The modulation instability has a strong impact for high input powers and greatly enhances the generation process, but leads to a degradation of the coherence properties. Also for short pulses with durations of 60 fs the modulation instability is present and can hardly be suppressed. The interplay between these two effects leads to various characteristics of the resulting spectra, which are modified by to the relative impact of the modulation instability.

*Appeared in*

- Appl. Phys. B, 86 (2007) pp. 31-39.

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# True upper bounds for Bermudan products via non-nested Monte Carlo

*Authors*

- Belomestny, Denis
- Bender, Christian
- Schoenmakers, John G. M.

ORCID: 0000-0002-4389-8266

*2010 Mathematics Subject Classification*

- 60H30 65C05 91B28

*Keywords*

- Bermudan options, Monte Carlo method, primal-dual method, martingale representation theorem, regression

*Abstract*

We present a generic non-nested Monte Carlo procedure for computing true upper bounds for Bermudan products, given an approximation of the Snell envelope. The pleonastic ``true'' stresses that, by construction, the estimator is biased above the Snell envelope. The key idea is a regression estimator for the Doob martingale part of the approximative Snell envelope, which preserves the martingale property. The so constructed martingale may be employed for computing dual upper bounds without nested simulation. In general, this martingale can also be used as a control variate for simulation of conditional expectations. In this context, we develop a variance reduced version of the nested primal-dual estimator (Anderson & Broadie (2004)) and nested consumption based (Belomestny & Milstein (2006)) methods . Numerical experiments indicate the efficiency of the non-nested Monte Carlo algorithm and the variance reduced nested one.

*Appeared in*

- Math. Finance, 19 (2009) pp. 53--71.

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# On maximal inequalities for some jump processes

*Authors*

- Gapeev, Pavel

*2010 Mathematics Subject Classification*

- 60G40 34K10 60E15 60J60 60J75

*Keywords*

- Jump process, stochastic differential equation, maximum process, optimal stopping problem, compound Poisson process, Ito's formula, integro-differential free-boundary problem, normal reflection, continuous and smooth fit, maximality principle, maximal inequalities

*Abstract*

We present a solution to the considered in [5] and [22] optimal stopping problem for some jump processes. The method of proof is based on reducing the initial problem to an integro-differential free-boundary problem where the normal reflection and smooth fit may break down and the latter then be replaced by the continuous fit. The derived result is applied for determining the best constants in maximal inequalities for a compound Poisson process with linear drift and exponential jumps.

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# Control of unstable steady states by strongly delayed feedback

*Authors*

- Yanchuk, Serhiy
- Wolfrum, Matthias
- Hövel, Philipp
- Schöll, Eckehard

*2008 Physics and Astronomy Classification Scheme*

- 05.45.-a 05.45.Gg 02.30.Ks

*Keywords*

- delayed feedback, large delay, control

*Abstract*

We present an asymptotic analysis of time-delayed feedback control of steady states for large delay time. By scaling arguments, and a detailed comparison with exact solutions, we establish the parameter ranges for successful stabilization of an unstable fixed point of focus type. Insight into the control mechanism is gained by analysing the eigenvalue spectrum, which consists of a pseudo-continuous spectrum and up to two strongly unstable eigenvalues. Although the standard control scheme generally fails for large delay, we find that if the uncontrolled system is sufficiently close to its instability threshold, control does work even for relatively large delay times.

*Appeared in*

- Phys. Rev. E, 74 (2006) pp. 026201/1--026201/7.

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# Wave trains, solitons and modulation theory in FPU chains

*Authors*

- Dreyer, Wolfgang
- Herrmann, Michael
- Rademacher, Jens D. M.

*2010 Mathematics Subject Classification*

- 34E13 37K60 70F45 70K70 82C21

*Keywords*

- FPU chain, traveling waves, multiscale ansatz, modulation theory, dispersive shocks

*Abstract*

We present an overview of recent results concerning wave trains, solitons and their modulation in FPU chains. We take a thermodynamic perspective and use hyperbolic scaling of particle index and time in order to pass to a macroscopic continuum limit. While strong convergence yields the well-known p-system of mass and momentum conservation, we generally obtain a weak form of it in terms of Young measures. The modulation approach accounts for microscopic oscillations, which we interpret as temperature, causing convergence only in a weak, average sense. We present the arising Whitham modulation equations in a thermodynamic form, as well as analytic and numerical tools for the resolution of the modulated wave trains. As a prototype for the occurrence of temperature from oscillation-free initial data, we discuss various Riemann problems, and the arising dispersive shock fans, which replace Lax-shocks. We predict scaling and jump conditions assuming a generic soliton at the shock front.

*Appeared in*

- Analysis, Modeling and Simulation of Multiscale Problems, A. Mielke, ed., Springer, Heidelberg, 2006, pp. 467--500

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# Discounted optimal stopping for maxima of some jump-diffusion processes

*Authors*

- Gapeev, Pavel

*2010 Mathematics Subject Classification*

- 60G40 34K10 91B70 60J60 60J75 91B28

*Keywords*

- Discounted optimal stopping problem, Brownian motion, compound Poisson process, maximum process, integro-differential free-boundary problem, continuous and smooth fit, normal reflection, a change-of-variable formula with local time on surfaces, perpetual lookback American options

*Abstract*

We present solutions to some discounted optimal stopping problems for the maximum process in a model driven by a Brownian motion and a compound Poisson process with exponential jumps. The method of proof is based on reducing the initial problems to integro-differential free-boundary problems where the normal reflection and smooth fit may break down and the latter then be replaced by the continuous fit. The results can be interpreted as pricing perpetual American lookback options with fixed and floating strikes in a jump-diffusion model.

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# A model for mode-locking in quantum dot lasers

*Authors*

- Viktorov, Evgeny
- Mandel, Paul
- Vladimirov, Andrei
- Bandelow, Uwe

ORCID: 0000-0003-3677-2347

*2010 Mathematics Subject Classification*

- 34K60 78A60

*2008 Physics and Astronomy Classification Scheme*

- 42.55.Px 42.60.Fc 42.65.Sf

*Keywords*

- mode-locking, quantum dot lasers, delay differential equations

*Abstract*

We propose a model for passive mode-locking in quantum dot laser and report on specific dynamical properties of the regime which is characterized by a fast gain recovery. No Q-switching instability has been found accompanying the mode-locking. Bistability can occur between the mode-locking regime and zero intensity steady state.

*Appeared in*

- Applied Physics Letters, 88: 201102, 2006

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# A model for temperature-induced phase transformations in finite-strain elasticity

*Authors*

- Mielke, Alexander

ORCID: 0000-0002-4583-3888

*2010 Mathematics Subject Classification*

- 49J40 49S05 74F05 74M05 74N30

*Keywords*

- shape-memory alloys, rate-independent energetic formulation, temperature-induced phase transformation, polyconvexity, time-dependent Dirichlet conditions

*Abstract*

We propose a model for phase transformations that are driven by changes in the temperature. We consider the temperature as a prescribed prescribed quantity like an applied load. The model is based on the energetic formulation for rate-independent systems and thus allows for finite-strain elasticity. Time-dependent Dirichlet boundary conditions can be treated by decomposing the deformation as a composition of a given deformation satisfying the time-dependent boundary conditions and a part coinciding with the identity on the Dirichlet boundary.

*Appeared in*

- IMA J. Appl. Math., 72 (2007) pp. 644--658.

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# Stabilized finite element schemes for incompressible flow using Scott--Vogelius elements

*Authors*

- Burman, Erik
- Linke, Alexander

ORCID: 0000-0002-0165-2698

*2010 Mathematics Subject Classification*

- 76M10 65M12 65M15

*Keywords*

- mixed finite elements, stabilized methods, solenoidal finite elements, edge stabilization, interior penalty, local projection, Oseen's equation, LBB-condition, a-priori estimate

*Abstract*

We propose a stabilized finite element method based on the Scott-Vogelius element in combination with either a local projection stabilization or an edge oriented stabilization based on a penalization of the gradient jumps over element edges. We prove a discrete inf-sup condition leading to optimal a priori error estimates. The theoretical considerations are illustrated by some numerical examples.

*Appeared in*

- Appl. Numer. Math., 58 (2008) pp. 1704--1719.

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# Optimal regularity for elliptic transmission problems including $C^1$ interfaces

*Authors*

- Elschner, Johannes
- Rehberg, Joachim
- Schmidt, Gunther

*2010 Mathematics Subject Classification*

- 35B45 35B65 35D10 35J15 35Q40

*Keywords*

- Partial differential equations, elliptic operators, nonsmooth domains, transmission problems, discontinuous coefficients

*Abstract*

We prove an optimal regularity result for elliptic operators $-nabla cdot mu nabla:W^1,q_0 rightarrow W^-1,q$ for a $q>3$ in the case when the coefficient function $mu$ has a jump across a $C^1$ interface and is continuous elsewhere. A counterexample shows that the $C^1$ condition cannot be relaxed in general. Finally, we draw some conclusions for corresponding parabolic operators.

*Appeared in*

- Interfaces Free Bound., 9 (2007) pp. 233--252.

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# Principle of linearized stability and smooth center manifold theorem for semilinear hyperbolic systems

*Authors*

- Lichtner, Mark

*2010 Mathematics Subject Classification*

- 35L40 35B30 37C05 34D09 37C75 37L05 37L10 47D03 47D06 37D10 35L05 35L40 35L50 35L60

*Keywords*

- Semilinear hyperbolic systems, spectral mapping theorem, semigroups, exponential dichotomy, center manifolds, smooth dependence on data, stability

*Abstract*

We prove principle of linearized stability and smooth center manifold theorem for a general class of semilinear hyperbolic systems $mathrm(SH)$ in one space dimension, which are of the following form: For $0 < x < l$ and $t > 0$ $$ mathrm(SH) left beginarrayl partial over partial t beginpmatrix u(t,x) v(t,x) endpmatrix + K(x) partial over partial x beginpmatrix u(t,x) v(t,x) endpmatrix + H(x, u(t,x), v(t,x)) = 0, d over dt left [ v(t,l) - D u(t,l) right ] = F(u(t,cdot),v(t,cdot)), u(t,0) = E , v(t,0), u(0,x) = u_0(x), ; v(0,x) = v_0(x), endarray right . $$ where $u(t,x) in R^n_1$, $v(t,x) in R^n_2$, $K(x) = mathrmdiag , left( k_i(x) right )_1 le i le n$ is a diagonal matrix of functions $k_i in C^1left( [0,l], R right)$, $k_i(x) > 0$ for $i = 1, dots, n_1$ and $k_i(x) < 0$ for $i = n_1+1, dots, n=n_1+n_2$, and $D$,$E$ are matrices. First we prove that weak solutions to $mathrm(SH)$ form a smooth semiflow in a Banach space $X$ of continuous functions under natural conditions on the nonlinearities $H$ and $F$. Then we show a spectral gap mapping theorem for linearizations of $mathrm(SH)$ in the complexification of $X$, which implies that growth and spectral bound coincide. Consequently we obtain principle of linearized stability for $mathrm(SH)$. Moreover, the spectral gap mapping theorem characterizes exponential dichotomy in terms of a spectral gap of the infinitesimal generator for linearized hyperbolic systems. This resolves a key problem in applying invariant manifold theory to prove smooth center manifold theorem for $mathrm(SH)$.

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# Spectral mapping theorem for linear hyperbolic systems

*Authors*

- Lichtner, Mark

*2010 Mathematics Subject Classification*

- 47D03 47D06 34D09 35P20 37L10 37D10

*Keywords*

- Linear hyperbolic systems, estimates for spectrum and resolvent, spectral mapping theorem, $C_0$ semigroups, exponential dichotomy, invariant manifolds

*Abstract*

We prove spectral mapping theorem for linear hyperbolic systems of PDEs. The system is of the following form: For $0 < x < l$ and $t > 0$ $$ rm(H) quad left beginarrayl displaystyle partial over partial t beginpmatrix u(t,x) v(t,x) endpmatrix + K(x) partial over partial x beginpmatrix u(t,x) v(t,x) endpmatrix + C(x) beginpmatrix u(t,x) v(t,x) endpmatrix = 0, displaystyle d over dt left [ v(t,l) - D u(t,l) right ] = F u(t,cdot) + G v(t,cdot) , displaystyle u(t,0) = E v(t,0), endarray right . $$ where $u(t,x) in C^n_1$, $v(t,x) in C^n_2$, $K(x) = mathrmdiag , left( k_i(x) right )_1 le i le n$ is a diagonal matrix of functions $k_i in C^1left( [0,l], R right)$, $k_i(x) > 0$ for $i = 1, dots, n_1$ and $k_i(x) < 0$ for $i = n_1+1, dots, n=n_1+n_2$, and $D$,$E$ are matrices. We show high frequency estimates of spectra and resolvents in terms of reduced (block)diagonal systems. Let $A$ denote the infinitesimal generator for $mathrm(H)$ which generates $C_0$ semigroup $e^At$ on $L^2 times C^n_2$. Our main result is the following spectral mapping theorem $$sigma(e^At) setminus 0 = overlinee^sigma(A)t setminus 0 .$$

*Appeared in*

- Proc. Amer. Math. Soc., 136 (2008) pp. 2091-2101.

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# A nonlocal phase-field model with nonconstant specific heat

*Authors*

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

*2010 Mathematics Subject Classification*

- 80A22 35K50 35B50 35B65 45J05 45K05

*Keywords*

- phase transitions, nonlocal models, integrodifferential equations

*Abstract*

We prove the existence, uniqueness, thermodynamic consistency, global boundedness from both above and below, and continuous data dependence for a strong solution to an integrodifferential model for nonisothermal phase transitions under nonhomogeneous mixed boundary conditions. The specific heat is allowed to depend on the order parameter, and the convex component of the free energy may or may not be singular.

*Appeared in*

- Interfaces Free Bound., 9 (2007) pp. 285--306.

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# Feedback stabilization of magnetohydrodynamic equations

*Authors*

- Lefter, Cătălin

*2010 Mathematics Subject Classification*

- 93D15 35Q35 76W05 35Q30 35Q60 93B07

*Keywords*

- Magnetohydrodynamic equations, feedback stabilization, Carleman estimates

*Abstract*

We prove the local exponential stabilizability for the MHD system, with internally distributed feedback controllers. These controllers take values in a finite dimensional space which is the unstable manifold of the elliptic part of the linearized operator. The stabilization of the linear system is derived using a unique continuation property for systems of parabolic and elliptic equations, as well as the equivalence between controllability and feedback stabilizability in the case of finite dimensional systems. The feedback that stabilizes the linearized system is also stabilizing the nonlinear system in the domain of a fractional power of the elliptic operator.

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# Classical solutions of drift-diffusion equations for semiconductor devices: The 2D case

*Authors*

- Kaiser, Hans-Christoph
- Neidhardt, Hagen
- Rehberg, Joachim

*2010 Mathematics Subject Classification*

- 35K45 35K50 35K55 35K57 78A35

*Keywords*

- Initial boundary value problem, reaction-diffusion processes, quasi-linear prarbolic systems

*Abstract*

We regard drift-diffusion equations for semiconductor devices in Lebesgue spaces. To that end we reformulate the (generalized) van Roosbroeck system as an evolution equation for the potentials to the driving forces of the currents of electrons and holes. This evolution equation falls into a class of quasi-linear parabolic systems which allow unique, local in time solution in certain Lebesgue spaces. In particular, it turns out that the divergence of the electron and hole current is an integrable function. Hence, Gauss' theorem applies, and gives the foundation for space discretization of the equations by means of finite volume schemes. Moreover, the strong differentiability of the electron and hole density in time is constitutive for the implicit time discretization scheme. Finite volume discretization of space, and implicit time discretization are accepted custom in engineering and scientific computing. ---This investigation puts special emphasis on non-smooth spatial domains, mixed boundary conditions, and heterogeneous material compositions, as required in electronic device simulation.

*Appeared in*

- Nonlinear Anal., 71 (2009) pp. 1584--1605.

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# No-arbitrage pricing beyond semimartingales

*Authors*

- Bender, Christian
- Sottinen, Tommi
- Valkeila, Esko

*2010 Mathematics Subject Classification*

- 91B28 91B70 60G15 60H05

*Keywords*

- arbitrage, pricing, quadratic variation, robust hedging, stylized facts

*Abstract*

We show how no-arbitrage pricing can be extended to some non-semimartingale models by restricting the class of admissible strategies. However, this restricted class is big enough to cover hedges for relevant options. Moreover, we show that the hedging prices depend essentially only on a path property of the stock price process, viz. on the quadratic variation. As a consequence we can incorporate many stylized facts to a pricing model without changing the option prices.

*Appeared in*

- Finance Stoch., 12 pp. 441--468.

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# Energy estimates for electro-reaction-diffusion systems with partly fast kinetics

*Authors*

- Glitzky, Annegret

*2010 Mathematics Subject Classification*

- 35B40 35K45 35K57 78A35 35R05

*Keywords*

- Reaction--diffusion systems, drift--diffusion processes, motion of charged particles, energy estimates, asymptotic behaviour, reduction of model equations

*Abstract*

We start from a basic model for the transport of charged species in heterostructures containing the mechanisms diffusion, drift and reactions in the domain and at its boundary. Considering limit cases of partly fast kinetics we derive reduced models. This reduction can be interpreted as some kind of projection scheme for the weak formulation of the basic electro--reaction--diffusion system. We verify assertions concerning invariants and steady states and prove the monotone and exponential decay of the free energy along solutions to the reduced problem and to its fully implicit discrete-time version by means of the results of the basic problem. Moreover we make a comparison of prolongated quantities with the solutions to the basic model.

*Appeared in*

- Discrete Contin. Dyn. Syst., 25 (2009) pp. 159--174.

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# Survival and complete convergence for a spatial branching system with local regulation

*Authors*

- Birkner, Matthias
- Depperschmidt, Andrej

*2010 Mathematics Subject Classification*

- 60K35 92D40

*Keywords*

- Regulated population, survival, coexistence, complete convergence

*Abstract*

We study a discrete time spatial branching system on $Z^d$ with logistic-type local regulation at each deme depending on a weighted average of the population in neighbouring demes. We show that the system survives for all time with positive probability if the competition term is small enough. For a restricted set of parameter values, we also obtain uniqueness of the non-trivial equilibrium and complete convergence, as well as long-term coexistence in a related two-type model.

*Appeared in*

- Ann. Probab., 17 (2007) pp. 1777-1807.

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# Hydrodynamic limit for the $A+Bto emptyset$ model

*Authors*

- Bovier, Anton
- Černý, Jiri

*2010 Mathematics Subject Classification*

- 82D30 82C41 60F17

*Keywords*

- hydrodynamic limit, large deviations, order book dynamics

*Abstract*

We study a two-species interacting particle model on a subset of $Z$ with open boundaries. The two species are injected with time dependent rate on the left, resp. right boundary. Particles of different species annihilate when they try to occupy the same site. This model has been proposed as a simple model for the dynamics of an ``order book'' on a stock market. We consider the hydrodynamic scaling limit for the empirical process and prove a large deviation principle that implies convergence to the solution of a non-linear parabolic equation.

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# Nonadditive disorder problems for some diffusion processes

*Authors*

- Gapeev, Pavel

*2010 Mathematics Subject Classification*

- 60G40 62M20 34K10 62C10 62L15 60J60

*Keywords*

- Quickest 'disorder'(change-point) detection, diffusion process, optimal stopping, sufficient statistic, free-boundary problem, smooth-fit and normal-entrance conditions, Itô's formula

*Abstract*

We study nonadditive Bayesian problems of detecting a change in drift of an observed diffusion process where the cost function of the detection delay has the same structure as in [27] and construct a finite-dimensional Markovian sufficient statistic for that case. We show that when the cost function is linear the optimal stopping time is found as the first time when the a posteriori probability process hits a stochastic boundary depending on the observation process. It is shown that under some nontrivial relationships on the coefficients of the observed diffusion the problem admits a closed form solution. The method of proof is based on embedding the initial problem into a two-dimensional optimal stopping problem and solving the equivalent free-boundary problem by means of the smooth-fit conditions.

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# Multi-pulse evolution and space-time chaos in dissipative systems

*Authors*

- Zelik, Sergey
- Mielke, Alexander

ORCID: 0000-0002-4583-3888

*2010 Mathematics Subject Classification*

- 35Q30 37L30

*Keywords*

- dissipative systems, unbounded domains, multi-pulses, normal hyperbolicity, center-manifold reduction, space-time chaos, Bernoulli shifts

*Abstract*

We study semilinear parabolic systems on the full space $R^n$ that admit a family of exponentially decaying pulse-like steady states obtained via translations. The multi-pulse solutions under consideration look like the sum of infinitely many such pulses which are well separated. We prove a global center-manifold reduction theorem for the temporal evolution of such multi-pulse solutions and show that the dynamics of these solutions can be described by an infinite systems of ODEs for the positions of the pulses. As an application of the developed theory, we verify the existence of Sinai-Bunimovich space-time chaos in 1D space-time periodically forced Swift--Hohenberg equation.

*Appeared in*

- Mem. Amer. Math. Soc., 198 (2009) pp. 1--97.

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# Sequential testing problems for some diffusion processes

*Authors*

- Gapeev, Pavel

*2010 Mathematics Subject Classification*

- 60G40 62M20 34K10 62C10 62L15 60J60

*Keywords*

- Sequential testing, diffusion process, optimal stopping, free-boundary problem, smooth-fit condition, Itô's formula

*Abstract*

We study the Bayesian problem of sequential testing of two simple hypotheses about the local drift of an observed diffusion process. The optimal stopping time is found as the first time when the a posteriori probability process leaves the region defined by two stochastic boundaries depending on the observation process. It is shown that under some nontrivial relationships on the coefficients of the observed diffusion the problem admits a closed form solution. The method of proof is based on embedding the initial problem into a two-dimensional optimal stopping problem and solving the equivalent free-boundary problem by means of the smooth-fit conditions.

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# The sharp interface limit of the van der Waals--Cahn--Hilliard phase model for fixed and time dependent domains

*Authors*

- Dreyer, Wolfgang
- Kraus, Christiane

*2010 Mathematics Subject Classification*

- 82B26 49Q05 35R35

*Keywords*

- Van der Waals-Cahn-Hilliard theory of phase transitions, two-phase fluid, asymptotic expansion of the density, local energy estimates, mechanical equilibrium condition, phase equilibrium condition, Gibbs-Thompson relation, surface tension, curvature, perimeter, minimal area, entropy, thermodynamic consistency

*Abstract*

We study the equilibria of liquid--vapor phase transitions of a single substance at constant temperature and relate the sharp interface model of classical thermodynamics to a phase field model that determines the equilibria by the stationary van der Waals--Cahn--Hilliard theory.

For two reasons we reconsider this old problem. 1. Equilibria in a two phase system can be established either under fixed total volume of the system or under fixed external pressure. The latter case implies that the domain of the two--phase system varies. However, in the mathematical literature rigorous sharp interface limits of phase transitions are usually considered under fixed volume. This brings the necessity to extend the existing tools for rigorous sharp interface limits to changing domains since in nature most processes involving phase transitions run at constant pressure. 2. Thermodynamics provides for a single substance two jump conditions at the sharp interface, viz. the continuity of the specific Gibbs free energies of the adjacent phases and the discontinuity of the corresponding pressures, which is balanced by the mean curvature. The existing estimates for rigorous sharp interface limits show only the first condition. We identify the cause of this phenomenon and develop a strategy that yields both conditions up to the first order.

The necessary information on the equilibrium conditions are achieved by an asymptotic expansion of the density which is valid for an arbitrary double well potential. We establish this expansion by means of local energy estimates, uniform convergence results of the density and estimates on the Laplacian of the density.

*Appeared in*

- Proc. Roy. Soc. Edinburgh Sect. A, 140 A (2010) pp. 1161--1186.

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# Simple Monte Carlo and the Metropolis algorithm

*Authors*

- Mathé, Peter

ORCID: 0000-0002-1208-1421 - Novak, Erich

*2010 Mathematics Subject Classification*

- 65C05 65Y20 68Q17 82B80

*Keywords*

- Monte Carlo methods, Metropolis algorithm, log-concave density, rapidly mixing Markov chains, optimal algorithms, adaptivity, complexity

*Abstract*

We study the integration of functions with respect to an unknown density. Information is available as oracle calls to the integrand and to the non-normalized density function. We are interested in analyzing the integration error of optimal algorithms (or the complexity of the problem) with emphasis on the variability of the weight function. For a corresponding large class of problem instances we show that the complexity grows linearly in the variability, and the simple Monte Carlo method provides an almost optimal algorithm. Under additional geometric restrictions (mainly log-concavity) for the density functions, we establish that a suitable adaptive local Metropolis algorithm is almost optimal and outperforms any non-adaptive algorithm.

*Appeared in*

- J. Complexity, 23 (2007) pp. 673--696.

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# Intermediate-asymptotic structure of a dewetting rim with strong slip

*Authors*

- Evans, Peter L.
- King, John R.
- Münch, Andreas

*2010 Mathematics Subject Classification*

- 76D08 76A20 34E10

*Keywords*

- Lubrication theory, evolution, slip models

*Abstract*

When a thin viscous liquid film dewets, it typically forms a rim which spreads outwards, leaving behind a growing dry region. We consider the dewetting behaviour of a film, when there is strong slip at a liquid-substrate interface. The film can be modelled by two coupled partial differential equations (PDEs) describing the film thickness and velocity. Using asymptotic methods, we describe the structure of the rim as it evolves in time, and the rate of dewetting, in the limit of large slip lengths. An inner region emerges, closest to the dewetted region, where surface tension is important; in an outer region, three subregions develop. This asymptotic description is compared with numerical solutions of the full system of PDEs.

*Appeared in*

- Appl. Math. Res eXpress, (2006) pp. 25262/1--25262/25.

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