Density and current of a dissipative Schrödinger operator
Authors
- Kaiser, Hans-Christoph
- Neidhardt, Hagen
- Rehberg, Joachim
2010 Mathematics Subject Classification
- 47A20 34B24 47A55 47B44
Keywords
- open Schrödinger-Poisson systems, carrier and current density, dissipative Schrödinger operator, self-adjoint dilation, generalized eigenfunctions, characteristic function, density matrix
DOI
Abstract
A net current flow through an open 1-dimensional Schrödinger-Poisson system is modeled by replacing self-adjoint boundary conditions by dissipative ones. To give a rigorous definition of carrier and current densities the well-known dilation theory of dissipative operators is used where the self-adjoint dilation is regarded as the Hamiltonian of a larger closed system which contains the open one. The carrier density turns out to be performed by the generalized eigenstates of the dilation while the current density is related to the characteristic function of the dissipative operator. A rigorous setup of a dissipative Schrödinger-Poisson system is outlined.
Appeared in
- Journal of Mathematical Physics, 2002, 43, 11, p 5325-5350
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WIAS Preprint No. 2003, (2002)An approximation method for Navier-Stokes equations based on probabilistic approach
Authors
- Belopolskaya, Yana
- Milstein, Grigori N.
2010 Mathematics Subject Classification
- 76D05 60H30 65M99
Keywords
- Numerical analysis of Navier-Stokes equations, probabilistic representations for equations of mathematical physics, weak approximation of solutions of stochastic differential equations
DOI
Abstract
A new layer method solving the space-periodic problem for the Navier-Stokes equations is constructed by using probabilistic representations of their solutions. The method exploits the ideas of weak sense numerical integration of stochastic differential equations. Despite its probabilistic nature this method is nevertheless deterministic. A convergence theorem is proved.
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WIAS Preprint No. 2003, (2002)Homoclinic bifurcations and dimension of attractors for damped nonlinear hyperbolic equations
Authors
- Turaev, Dmitry
- Zelik, Sergey
2010 Mathematics Subject Classification
- 35B41 37G20 35B45 37D45
Keywords
- Damped hyperbolic equations, Global attractors, Homoclinic bifurcations, Lyapunov dimension, Fractal dimension
DOI
Abstract
A new method of obtaining lower bounds for the attractor's dimension is suggested which involves analysis of homoclinic bifurcations. The method is applied for obtaining sharp estimates of the attractor's dimension for a class of abstract damped wave equations which are beyond the reach of the classical methods.
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WIAS Preprint No. 2003, (2002)A Lagrangian stochastic model for the transport in statistically homogeneous porous media
Authors
- Kurbanmuradov, Orazgeldy
- Sabelfeld, Karl K.
- Smidts, Olivier F.
- Vereecken, Henry
2010 Mathematics Subject Classification
- 65C05 76S05
2008 Physics and Astronomy Classification Scheme
- 47.55.Mh
Keywords
- Porous media, Lognormal hydraulic conductivity, Stochastic and turbulent flows, stochastic Eulerian and Lagrangian models
DOI
Abstract
A new type of stochastic simulation models is developed for solving transport problems in saturated porous media which is based on a generalized Langevin stochastic differential equation. A detailed derivation of the model is presented in the case when the hydraulic conductivity is assumed to be a random field with a lognormal distribution, being statistically isotropic in space. To construct a model consistent with this statistical information, we use the well-mixed condition which relates the structure of the Langevin equation and the probability density function of the Eulerian velocity field. Numerical simulations of various statistical characteristics like the mean displacement, the displacement covariance tensor and the Lagrangian correlation function are presented. These results are compared against the conventional random displacement method.
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WIAS Preprint No. 2003, (2002)Super-Brownian motion with extra birth at one point
Authors
- Fleischmann, Klaus
- Mueller, Carl
2010 Mathematics Subject Classification
- 60J80 60K35
Keywords
- Super-Brownian motion, measure-valued processes, heat equation, singular potential, one-point potential
DOI
Abstract
A super-Brownian motion in two and three dimensions is constructed where "particles" give birth at a higher rate, if they approach the origin. Via a log-Laplace approach, the construction is based on Albeverio et al. (1995) who calculated the fundamental solutions of the heat equation with one-point potential in dimensions less than four.
Appeared in
- SIAM J. Math. Anal., 2004, vol. 36, no. 3, pp. 740-772
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WIAS Preprint No. 2003, (2002)On safe crack shapes in elastic bodies
Authors
- Hömberg, Dietmar
ORCID: 0000-0001-9460-5729 - Khludnev, Alexander M.
2010 Mathematics Subject Classification
- 74G65 74P99 74R99
2008 Physics and Astronomy Classification Scheme
- 46.50.+a
Keywords
- Griffith criterion, nonlinear crack, shape derivative
DOI
Abstract
According to the Griffith criterion, a crack propagation occurs provided that the derivative of the energy functional with respect to the crack length reaches some critical value. We consider a generalization of this criterion to the case of nonlinear cracks satisfying a non-penetration condition and investigate the dependence of the shape derivative of the energy functional on the crack shape. In the paper, we find the crack shape which gives the maximal deviation of the energy functional derivative from a given critical value and, in particular, prove that this optimality problem admits a solution.
Appeared in
- European J. Mech. A/Solids, 21 (2002), pp. 991-998
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WIAS Preprint No. 2003, (2002)Delayed loss of stability and excitation of oscillations in nonautonomous differential equations with retarded argument
Authors
- Lani-Wayda, Bernhard
- Schneider, Klaus R.
2010 Mathematics Subject Classification
- 34K12 34K06
Keywords
- Nonautonomous delay equations, slowly changing parameters, delayed loss of stability, excitation of oscillations
DOI
Abstract
Assume that zero is a stable equilibrium of an ODE ẋ = ƒ(𝑥, λ) for parameter values λ < λ0, and becomes unstable for λ > λ0. If we suppose that λ(t) varies slowly with t, then, under some conditions, the trajectories of the nonautonomous ODE ẋ = ƒ(𝑥, λ (t)) stay close to zero even long after λ(t) has crossed the value λ0. This phenomenon is called νdelayed loss of stabilityν and is well-known for ODEs. In this paper, we describe an analogous phenomenon for delay equations of the form ẋ(t) = ƒ(t, 𝑥 (t-1)). Further, we point out a difference between delay equations and ODEs: The inhomogeneity 𝒽 in the linear equation ẋ (t) = c𝑥 (t-1) + 𝒽(t) inevitably leads to an excitation of the most unstable modes of oscillation of the homogeneous equation, even if all segments 𝒽t are contained in a space of more rapidly decaying solutions for the homogeneous equation.
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WIAS Preprint No. 2003, (2002)On the velocity of the Biot slow wave in a porous medium: Uniform asymptotic expansion
Authors
- Edelman, Inna
2010 Mathematics Subject Classification
- 35C20 35L45 74F10 74J10 76M45
Keywords
- Porous media, bulk waves, asymptotics, bifurcation
DOI
Abstract
Asymptotic behavior of the Biot slow wave is investigated. Formulae for short- and long-wave approximations of phase velocity of the P2 wave are presented. These asymptotic expansions are compared with exact solution, constructed numerically. It is shown that both expansions fit very well the real velocity of the P2 mode. Procedure for matching of short- and long-wave asymptotic expansions is suggested.
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WIAS Preprint No. 2003, (2002)Identical synchronization of time-continuous chaotic oscillators
Authors
- Yanchuk, Serhiy
- Maistrenko, Yuri
- Mosekilde, Erik
2010 Mathematics Subject Classification
- 34C15 34D05 34D20
Keywords
- chaotic synchronization, coupled oscillators, Roessler oscillators
DOI
Abstract
Considering a system of two coupled identical chaotic oscillators, the paper first establishes the conditions of transverse stability for the fully synchronized chaotic state. Periodic orbit threshold theory is applied to determine the bifurcations through which low-periodic orbits embedded in the fully synchronized state lose their transverse stability, and the appearance of globally and locally riddled basins of attraction is discussed in terms of the sub-, respectively supercritical nature of the riddling bifurcations. We show how the introduction of a small parameter mismatch between the interacting chaotic oscillators causes a shift of the synchronization manifold. The presence of a coupling asymmetry is found to lead to further modifications of the destabilization process. Finally, the paper considers the problem of partial synchronization in a system of four coupled Rössler oscillators.
Appeared in
- Chaos, 13 (2003), pp. 388-400
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WIAS Preprint No. 2003, (2002)Existence of the Stoneley surface wave at vacuum/porous medium interface: Low-frequency range
Authors
- Edelman, Inna
2010 Mathematics Subject Classification
- 35C20 74J15 35L50 74F10 76M45 35B32
Keywords
- porous medium, interface, surface waves, asymptotics, bifurcation
DOI
Abstract
Existence and asymptotic behavior of the Stoneley surface wave at vacuum/porous medium interface are investigated in the low frequency range. It is shown that the Stoneley wave possesses a bifurcation in the vicinity of critical wave number 𝑘cr. It is proven also that within the 𝑘-domain of existence, the Stoneley wave cannot appear for certain values of elastic moduli of the solid phase. Asymptotic formulae for the phase velocity of the Stoneley wave are presented.
Appeared in
- Wave Motion 39 (2004), No. 2, pp. 111--127
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WIAS Preprint No. 2003, (2002)Asymptotic analysis of surface waves at vacuum/porous medium interface: Low-frequency range
Authors
- Edelman, Inna
2010 Mathematics Subject Classification
- 35C20 35L50 74J15 74J10 35B32
Keywords
- porous media, bulk waves, surface waves, asymptotics, bifurcation
DOI
Abstract
Existence and propagation of the surface waves at a free interface of a saturated porous medium are investigated in the low-frequency range. Similar to the high-frequency range, two types of surface waves are proven to be possible: the generalized Rayleigh wave, which exists always and propagates almost without attenuation and the Stoneley wave, which exists for a limited range of wave numbers and is strongly attenuated. Bifurcation behavior of both the Stoneley wave and the Biot slow bulk wave depending on wave number is revealed.
Appeared in
- Wave Motion 39 (2004), pp. 111--127
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WIAS Preprint No. 2003, (2002)Flow and thermal convection in full-zone liquid bridges of wide-ranging aspect ratio
Authors
- Davis, Dominic
- Smith, Frank
2010 Mathematics Subject Classification
- 35Q30 35B40 65M60 65N06 75D45
Keywords
- liquid bridges, crystal growth, floating-zone, axisymmetric, Navier-Stokes equations, thermocapillarity, buoyancy, asymptotic, finite-element
DOI
Abstract
Flow and thermal effects concerned with liquid crystal growth are studied nonlinearly for low Prandtl numbers, in an axisymmetric steady configuration with endwalls present. Full solutions are obtained by finite element simulation to examine the influences of aspect ratio, Marangoni number and Rayleigh number. In particular a `lemonhead' phenomenon is found in which the velocity profiles acquire a very localised bell-shaped form as jet like flow starts to emerge, when the relative size of the Marangoni number increases above a finite critical value which is identified. Wide-domain and narrow-domain analyses are then presented, showing favourable agreement with the full solutions and explaining the impact of the end walls on the motion, especially through slender flow modelling in the narrow-domain context. Sensitive scalings for the relative effects of the Marangoni and Rayleigh numbers emerge, and the lemonhead phenomenon is also accounted for as a substantial change in the convective flow structure.
Appeared in
- Theoret. Comput. Fluid Dynamics, No.17(2), 2003, pp. 113-146
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WIAS Preprint No. 2003, (2002)Trimmed trees and embedded particle systems
Authors
- Fleischmann, Klaus
- Swart, Jan
2010 Mathematics Subject Classification
- 60J80 60G57 60J60 60K35
Keywords
- (Historical) superprocess, binary branching, Poissonization, embedded particle system, trimmed tree, compensated h-transform, finite ancestry property
DOI
Abstract
In a supercritical branching particle system, the trimmed tree consists of those particles which have descendants at all times. We develop this concept in the superprocess setting. For a class of continuous superprocesses, we identify the trimmed tree, which turns out to be a binary splitting particle system with a new underlying motion that is a compensated h-transform of the old one. We show how trimmed trees may be estimated from above by embedded binary branching particle systems.
Appeared in
- Ann. Probab., 32 (2004), pp. 2179--222
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WIAS Preprint No. 2003, (2002)A mathematical model for induction hardening including mechanical effects
Authors
- Hömberg, Dietmar
ORCID: 0000-0001-9460-5729
2010 Mathematics Subject Classification
- 74F05 74F10 74D10 77N25
2008 Physics and Astronomy Classification Scheme
- 64.70.Kg 81.40.Gh
Keywords
- joule heating, thermoviscoelasticity, phase transitions
DOI
Abstract
In most structural components in mechanical engineering, there are surface parts, which are particularly stressed. The aim of surface hardening is to increase the hardness of the corresponding boundary layers by rapid heating and subsequent quenching. This heat treatment leads to a change in the microstructure, which produces the desired hardening effect. The mathematical model accounts for electromagnetic effects that lead to the heating of the workpiece as well as thermomechanical effects that cause the hardening of the workpiece. The new contribution of this paper is that we put a special emphasis on the thermomechanical effects caused by the phase transitions. We formulate a consistent model which takes care of effects like transformation strain and transformation plasticity induced by the phase transitions and allows for physical parameters depending on the respective phase volume fractions. The coupling between the electromagnetic and the thermomechanical part of the model is given through the temperature-dependent electric conductivity on the one hand and through the Joule heating term on the other hand, which appears in the energy balance and leads to the rise in temperature. Owing to the quadratic Joule heat term and a quadratic mechanical dissipation term in the energy balance, we obtain a parabolic equation with L1 data. We prove existence of a weak solution to the complete system using a truncation argument.
Appeared in
- Nonlinear Anal. Real World Appl., 5 (2004), pp. 55-90
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WIAS Preprint No. 2003, (2002)A generalized non-square Cholesky decomposition algorithm with applications to finance
Authors
- Reiß, Oliver
2010 Mathematics Subject Classification
- 15-04 65F30 91-08
Keywords
- Modified Cholesky decomposition, LDLT decomposition, Semi--positive matrices, Covariance matrix, Correlation matrix, Value at risk
DOI
Abstract
In several applications there is the need to compute a Cholesky decomposition of a given symmetric matrix. The usual Cholesky decomposition algorithm will fail if the given matrix is semi-positive, although such a decomposition exists. To overcome this problem there exists a LDLT decomposition for semi-positive matrices. In the case that the given symmetric matrix is not semi-positive, no Cholesky decomposition exists. In such a situation one aims to approximate this matrix by a (semi-)positive one and computes the Cholesky decomposition of the approximation. From the context of numerical optimization there exist algorithms by Gill, Murray and Wright and a refinement by Eskow and Schnabel. Both methods basicly return a Cholesky decomposition of a positive approximation of an indefinite input matrix. In this paper we extend the LDLT algorithm such that it coincides for a semi-definite input with the LDLT decomposition and for indefinite input it gives the decomposition of a semi-positive approximation. In contrast to the algorithms mentioned before, for indefinite input matrices our algorithm gives a decomposition, which has a lower rank. This gives the important opportunity to introduce a dimension reduction, if possible, and we will show that this algorithm can save computation time in several applications in finance, especially for risk management.
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WIAS Preprint No. 2003, (2002)On asymptotic minimaxity of kernel based tests
Authors
- Ermakov, Mikhail S.
2010 Mathematics Subject Classification
- 62G10 62G20
Keywords
- Nonparametric hypothesis testing, kernel-based-tests, goodness-of-fit, efficiency, kernel estimator
DOI
Abstract
In the problem of signal detection in Gaussian white noise we show asymptotic minimaxity of kernel-based tests. The test statistics equal L2-norms of kernel estimates. The sets of alternatives are essentially nonparametric and are defined as the sets of all signals such that L2-norms of signal smoothed by the kernels exceed some constants ρε > 0. The constant ρε depends on the power ε of noise and ρε → 0 as ε → 0. Similar statements are proved also if an additional information on a signal smoothness is given. By theorems on asymptotic equivalence of statistical experiments these results are extended on the problems of testing nonparametric hypothesis on density and regression. The exact asymptotically minimax lower bounds of type II error probabilities are pointed out for all these settings. Similar results are also obtained in the problems of testing parametric hypothesis versus nonparametric sets of alternatives.
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WIAS Preprint No. 2003, (2002)Transient temperature phenomena during sublimation growth of silicon carbide single crystals
Authors
- Klein, Olaf
ORCID: 0000-0002-4142-3603 - Philip, Peter
2010 Mathematics Subject Classification
- 80A20 65C20 65Z05
2008 Physics and Astronomy Classification Scheme
- 81.10.Bk 44.05.+e 02.60.Cb
Keywords
- Temperature evolution, heating stage, sublimation growth, physical vapor transport, modified Lely method, SiC single crystal, SiC powder source, temperature difference, transient modeling, numerical simulation
DOI
Abstract
In this article, we use numerical simulation to investigate transient temperature phenomena during sublimation growth of SiC single crytals via physical vapor transport (also called the modified Lely method). We consider the evolution of temperatures at the SiC source and at the SiC seed crystal, which are highly relevant to the quality of the grown crystals, but inaccessible to direct measurements. The simulations are based on a transient mathematical model for the heat transport, including heat conduction, radiation, and radio frequency (RF) induction heating. Varying the position of the induction coil as well as the heating power, it is shown that the measurable temperature difference between the bottom and the top of the growth apparatus can usually not be used as a simple indicator for the respective temperature difference between SiC source and seed. Moreover, it is shown that there can be a time lack of 1.5 hours between the heating of the temperature measuring points and the heating of the interior of the SiC source.
Appeared in
- Journal of Crystal Growwth, 249 (2003), pp. 514-522
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WIAS Preprint No. 2003, (2002)Confidence estimation of the covariance function of stationary and locally stationary processes
Authors
- Giurcanu, Mihai
- Spokoiny, Vladimir
ORCID: 0000-0002-2040-3427
2010 Mathematics Subject Classification
- 62M10 62G15
Keywords
- covariance function estimation, confidence intervals, local stationarity
DOI
Abstract
In this note we consider the problem of confidence estimation of the covariance function of a stationary or locally stationary zero mean Gaussian process. The constructed confidence intervals are based on the usual empirical covariance estimate and a special estimate of its variance. The results about coverage probability are stated in nonasymptotic way and apply for small and moderate sample size under mild conditions on the model. The presented numerical results are in agreement with the theoretical issues and demonstrate applicability of the method.
Appeared in
- Statist. Decisions, 22 (2004) pp. 283--300.
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WIAS Preprint No. 2003, (2002)Duality formula for the bridges of a Brownian diffusion. Application to gradient drifts
Authors
- Rœlly, Sylvie
- Thieullen, Michèle
2010 Mathematics Subject Classification
- 60G15 60G60 60H10 60J60
Keywords
- reciprocal processes, stochastic bridge, mixture of bridges, integration by parts formula, Malliavin calculus, entropy, time reversal, reversible process
DOI
Abstract
In this paper we consider families of time Markov fields (or reciprocal classes) which have the same bridges as a Brownian diffusion. We characterize each class as the set of solutions of an integration by parts formula on the space of continuous paths C([0; 1]; ℝd ). Our techniques provide a characterization of gradient diffusions by a duality formula and, in case of reversibility, a generalization of a result of Kolmogorov
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WIAS Preprint No. 2003, (2002)A direct simulation Monte Carlo method for the Uehling-Uhlenbeck-Boltzmann equation
Authors
- Garcia, Alejandro L.
- Wagner, Wolfgang
2010 Mathematics Subject Classification
- 65C05 76P05 82C40
Keywords
- Uehling-Uhlenbeck-Boltzmann equation, Fermi-Dirac case, Bose-Einstein case, DSMC algorithm, numerical experiments
DOI
Abstract
In this paper we describe a DSMC algorithm for the Uehling-Uhlenbeck-Boltzmann equation in terms of Markov processes. This provides a unifying framework for both the classical Boltzmann case as well as the Fermi-Dirac and Bose-Einstein cases. We establish the foundation of the algorithm by demonstrating its link to the kinetic equation. By numerical experiments we study its sensitivity to the number of simulation particles and to the discretization of the velocity space, when approximating the steady state distribution.
Appeared in
- Physical Review E,68, 056703 (2003), 11 pages
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WIAS Preprint No. 2003, (2002)Global existence result for pair diffusion models
Authors
- Glitzky, Annegret
ORCID: 0000-0003-1995-5491 - Hünlich, Rolf
2010 Mathematics Subject Classification
- 35K45 35K57 35R05 35D05 35B45 80A30
Keywords
- Reaction-diffusion systems for charged particles, pair diffusion models, global existence, a priori estimates, fixed point theorems
DOI
Abstract
In this paper we prove a global existence result for pair diffusion models in two dimensions. Such models describe the transport of charged particles in semiconductor heterostructures. The underlying model equations are continuity equations for mobile and immobile species coupled with a nonlinear Poisson equation. The continuity equations for the mobile species are nonlinear parabolic PDEs involving drift, diffusion and reaction terms, the corresponding equations for the immobile species are ODEs containing reaction terms only. Forced by applications to semiconductor technology these equations have to be considered with non-smooth data and kinetic coefficients additionally depending on the state variables. Our proof is based on regularizations, on a priori estimates which are obtained by energy estimates and Moser iteration as well as on existence results for the regularized problems. These are obtained by applying the Banach Fixed Point Theorem for the equations of the immobile species, and the Schauder Fixed Point Theorem for the equations of the mobile species.
Appeared in
- SIAM J. MATH. ANAL., Vol. 36, No. 4, pp. 1200-1225
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WIAS Preprint No. 2003, (2002)Rigorous results on some simple spin glass models
Authors
- Bovier, Anton
- Kurkova, Irina
2010 Mathematics Subject Classification
- 82B44 60G70 60K35
Keywords
- Gaussian processes, generalized random energy model, continuous hierarchies, spin glasses, Poisson cascades, probability cascades, Ghirlanda-Guerra identities
DOI
Abstract
In this paper we review some recent rigorous results that provide an essentially complete solution of a class of spin glass models introduced by Derrida in the 1980ies. These models are based on Gaussian random processes on {-1,1}N whose covariance is a function of a ultrametric distance on that set. We prove the convergence of the free energy as well as the Gibbs measures in an appropriate sense. These results confirm fully the predictions of the replica method including in situations where continuous replica symmetry breaking takes place.
Appeared in
- Markov Proc. Rel. Fields, 9 (2003), pp.209-242
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WIAS Preprint No. 2003, (2002)Uniaxial, extensional flows in liquid bridges
Authors
- Bänsch, Eberhard
ORCID: 0000-0003-2743-1612 - Berg, Christian P.
- Ohlhoff, Antje
2010 Mathematics Subject Classification
- 76D 76E
Keywords
- Uniaxial extensional flow, bridge stretching, microgravity, capillary forces, Navier-Stokes equations, free surface flow, finite elements, flow simulation
DOI
Abstract
In this paper, we discuss the possibility to generate homogeneous flows with a nearly constant strain rate. This is achieved by stretching an almost cylindrical liquid bridge under microgravity. One key issue is the adaptation of the disk diameters in order to have always ideal boundary conditions. We first study the stretching of two different fluids both, by numerical and experimental means. The numerical results are compared with these experimental data resulting in a very good agreement. The numerical method is then used to study the behavior of liquid bridges for quite a large range of the flow parameters Capillary number Ca and Weber number We and detect those regimes with most suitable flow conditions.
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WIAS Preprint No. 2003, (2002)On the necessity of some constraint qualification conditions in convex programming
Authors
- Tiba, Dan
- Zălinescu, Constantin
2010 Mathematics Subject Classification
- 49K27 90C25
Keywords
- convex function, constraint qualification, Lagrange multiplier, metric regularity, normal cone
DOI
Abstract
In this paper, we realize a study of various constraint qualification conditions for the existence of Lagrange multipliers for convex minimization problems in general normed vector spaces, it is based on a new formula for the normal cone to the constraint set, on local metric regularity and a metric regularity property on bounded subsets. As a by-product, we obtain a characterization of the metric regularity of a finite family of closed convex sets.
Appeared in
- J. Convex Anal., 11 (2004) pp. 95--110.
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WIAS Preprint No. 2003, (2002)Analogues of non-Gibbsianness in joint measures of disordered mean-field models
Authors
- Külske, Christof
2010 Mathematics Subject Classification
- 82B44 82B26 82B20
2008 Physics and Astronomy Classification Scheme
- 05.50.+q 61.43.-j 02.50.Cw
Keywords
- Disordered systems, non-Gibbsian measures, mean field models, Morita-approach, random field model, decimation transformation, diluted ferromagnet
DOI
Abstract
It is known that the joint measures on the product of spin-space and disorder space are very often non-Gibbsian measures, for lattice systems with quenched disorder, at low temperature. Are there reflections of this non-Gibbsianness in the corresponding mean-field models? We study the continuity properties of the conditional expectations in finite volume of the following mean field models: a) joint measures of random field Ising, b) joint measures of dilute Ising, c) decimation of ferromagnetic Ising. For a) we find 1) discontinuous dependence on the conditioning for almost any realization and 2) dependence of the conditional expectation on the phase. In contrast to that we see continuous behavior for b) and c), for almost any realization. This is in complete analogy to the behavior of the corresponding lattice models in high dimensions. It shows that non-Gibbsian behavior which seems a genuine lattice phenomenon can be partially understood already on the level of mean-field models.
Appeared in
- J. Statist. Phys., 112 (2003), pp. 1079-1108. Electron. J.
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WIAS Preprint No. 2003, (2002)Polynomial approximations of symplectic dynamics and richness of chaos in non-hyperbolic area-preserving maps
Authors
- Turaev, Dmitry
2010 Mathematics Subject Classification
- 37J10 37C15 37E30 37G25
Keywords
- homoclinic tangency, Hénon map, standard map, Cremona group, Newhouse phenomenon
DOI
Abstract
It is shown that every symplectic map of $R^2n$ can be approximated, in the $C^infty$-topology, on any compact set, by some iteration of some map of the form $(x,y)mapsto (y+eta, -x +Phi(y))$ where $xin R^n$, $yin R^n$, and $Phi$ is a polynomial $R^nrightarrow R^n$ and $etain R^n$ is a constant vector. For the case of area-preserving maps (i.e. $n=1$), it is shown how this result can be applied to prove that $C^r$-universal maps (a map is universal if its iterations approximate dynamics of all $C^r$-smooth area-preserving maps altogether) are dense (in the $C^r$-topology) in the Newhouse regions.
Appeared in
- Nonlinearity, 16 (2003), pp. 1-13
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WIAS Preprint No. 2003, (2002)Self-regularization of projection methods with a posteriori discretization level choice for severely ill-posed problems
Authors
- Bruckner, Gottfried
- Pereverzev, Sergei V.
2010 Mathematics Subject Classification
- 65R30 65J20
Keywords
- severely ill-posed problems, regularization by discretization, projection methods, integral equation of the first kind
DOI
Abstract
It is well known that projection schemes for certain linear ill-posed problems A𝓍 = y can be regularized by a proper choice of the discretization level only, where no additional regularization is needed. The previous study of this self-regularization phenomenon was restricted to the case of so-called moderately ill-posed problems, i.e., when the singular values σ𝑘(A), 𝑘 = 1,2,..., of the operator A tend to zero with polynomial rate. The main accomplishment of the present paper is a new strategy for a discretization level choice that provides optimal order accuracy also for severely ill-posed problems, i.e., when σ𝑘(A) tend to zero exponentially. The proposed strategy does not require a priori information regarding the solution smoothness and the exact rate of σ𝑘(A).
Appeared in
- Inverse Problems 19, (2003), pp. 147-156
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WIAS Preprint No. 2003, (2002)Numerical methods for Langevin type equations based on symplectic integrators
Authors
- Milstein, Grigori N.
- Tretyakov, Michael V.
2010 Mathematics Subject Classification
- 65C30 65P10 82C31
Keywords
- Langevin equations, stochastic Hamiltonian systems, symplectic and quasi-symplectic numerical methods, mean-square and weak schemes
DOI
Abstract
Langevin type equations are an important and fairly large class of systems close to Hamiltonian ones. The constructed mean-square and weak quasi-symplectic methods for such systems degenerate to symplectic methods when a system degenerates to a stochastic Hamiltonian one. In addition, quasi-symplectic methods' law of phase volume contractivity is close to the exact law. The methods derived are based on symplectic schemes for stochastic Hamiltonian systems. Mean-square symplectic methods were obtained in citehadd,hmul while symplectic methods in the weak sense are constructed in this paper. Special attention is paid to Hamiltonian systems with separable Hamiltonians, with additive noise, and with colored noise. Some numerical tests of both symplectic and quasi-symplectic methods are presented. They demonstrate superiority of the proposed methods in comparison with standard ones.
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WIAS Preprint No. 2003, (2002)Excitability and self-pulsations near homoclinic bifurcations in semiconductor laser systems
Authors
- Krauskopf, Bernd
- Schneider, Klaus R.
- Sieber, Jan
- Wieczorek, Sebastian
- Wolfrum, Matthias
ORCID: 0000-0002-4278-2675
2010 Mathematics Subject Classification
- 34C37 78A60 34C23 34C25
2008 Physics and Astronomy Classification Scheme
- 42.50.Ne 42.55.Px 05.45.+b
Keywords
- excitability, self-pulsations, semiconductor laser, homoclinic bifurcations
DOI
Abstract
Many laser systems show self-pulsations with a large amplitude that are born suddenly in a homoclinic bifurcation. Just before the onset of these self-pusations the laser is excitable where the excitability threshold is formed by the stable manifold of a saddle point. We show that there exists a special configuration, a codimension-two bifurcation called non-central saddle-ode homoclinic orbit, that acts as a organizing center of excitability in lasers. It is the key to understanding excitability in laser systems as diverse as lasers with saturable absorbers, lasers with optical injection and lasers with optical feedback.
Appeared in
- Optics Communications, 215 (2003), pp 367-379
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WIAS Preprint No. 2003, (2002)Dissipative Schrödinger-type operator as a model for generation and recombination
Authors
- Baro, Michael
- Neidhardt, Hagen
2010 Mathematics Subject Classification
- 47B44 47E05 47A20 47A55
Keywords
- open quantum system, dissipative Schroedinger operator, delta perturbation, dilation, characteristic function, generalized eigenfunctions, carrier and current densities, density matrix
DOI
Abstract
Non-selfadjoint operators play an important role in the modeling of open quantum systems. We consider a one-dimensional Schroedinger-type operator with dissipative boundary conditions and dissipative delta potentials. An explicit description of the characteristic function, the minimal dilation and the generalized eigenfunctions of the dilation is given. The quantities of carrier and current densities are rigorously defined. Furthermore we will show that the current is not constant and that the variation of the current depend essentially on the chosen density matrix and imaginary parts of the delta potentials. This correspondence can be used to model a recombination-generation rate in the open quantum system.
Appeared in
- J. Math. Phys. 44 (2003), no. 6, pp. 2373-2401
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WIAS Preprint No. 2003, (2002)Dynamics of multi-section DFB semiconductor laser: Traveling wave and mode approximation models
Authors
- Radziunas, Mindaugas
ORCID: 0000-0003-0306-1266 - Wünsche, Hans-Jürgen
2010 Mathematics Subject Classification
- 78A60 35P10 37N20 37G35
Keywords
- traveling wave, mode approximation, modeling, bifurcation, multi-section DFB laser
DOI
Abstract
Nonlinear dynamical effects of a multi-section DFB semiconductor laser such as self-pulsations or hysteresis can be described by the traveling wave model. The present paper demonstrates that such a model can be effectively approximated by a low dimensional system of ordinary differential equations where only few dynamically varying longitudinal modes of optical field are taken into account. A bifurcation analysis of the reduced models allow us to identify the mechanisms of switching on and switching off of the self-pulsations by tuning model parameters. An explanation of hysteresis is given as well.
Appeared in
- SPIE proceedings Series, vol. 4646, pp. 27-37, (2002)
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WIAS Preprint No. 2003, (2002)On the bifurcation of the Biot slow wave in a porous medium
Authors
- Edelman, Inna
2010 Mathematics Subject Classification
- 35C20 35L50 74J15
Keywords
- asymptotic expansions, waves in porous media
DOI
Abstract
Propagation of the slow Biot wave is investigated within the low-frequency range. For the first time it is proven theoretically that longitudinal wave of the second kind is not propagatory if its wave number is lower than some critical value. This critical wave number is a bifurcation point, above which longitudinal wave of the second kind becomes to be propagatory. Asymptotical formulae for phase velocity and attenuation of P2 wave are derived.
Appeared in
- Dokl. Akad. Nauk, Ross. Akad. Nauk, 388 (2003), pp. 812--816 (in Russian)
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WIAS Preprint No. 2003, (2002)Surface diffusion of graphs: Variational formulation, error analysis and simulation
Authors
- Bänsch, Eberhard
ORCID: 0000-0003-2743-1612 - Morin, Pedro
- Nochetto, Ricardo H.
2010 Mathematics Subject Classification
- 35K55 65M12 65M15 65M60 65Z05
Keywords
- Surface diffusion, fourth-order parabolic problem, finite elements, a priori error estimates, Schur complement, smoothing effect
DOI
Abstract
Surface diffusion is a (4th order highly nonlinear) geometric driven motion of a surface with normal velocity proportional to the surface Laplacian of mean curvature. We present a novel variational formulation for graphs and derive a priori error estimates for a time-continuous finite element discretization. We also introduce a semi-implicit time discretization and a Schur complement approach to solve the resulting fully discrete, linear systems. After computational verification of the orders of convergence for polynomial degrees 1 and 2, we show several simulations in 1d and 2d with and without forcing which explore the smoothing effect of surface diffusion as well as the onset of singularities in finite time, such as infinite slopes and cracks.
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WIAS Preprint No. 2003, (2014)Hausdorff metric BV discontinuity of sweeping processes
Authors
- Klein, Olaf
ORCID: 0000-0002-4142-3603 - Recupero, Vincenzo
2010 Mathematics Subject Classification
- 34A60 34C55 34G25, 47H30, 74C05
Keywords
- Sweeping process,, discontinuity,, bounded variation,, Hausdorff metric
DOI
Abstract
Sweeping processes are a class of evolution differential inclusions arising in elastoplasticity and were introduced by J.J. Moreau in the early seventies. The solution operator of the sweeping processes represents a relevant example of emphrate independent operator containing as a particular case the so called emphplay operator which is widely used in hysteresis. The continuity properties of these operators were studied in several works. In this note we address the continuity with respect to the strict metric in the space of functions of bounded variation with values in the metric space of closed convex subsets of a Hilbert space. We provide a counterexample showing that the solution operator of the sweeping process is not continuous when its domain is endowed with the strict topology of $BV$ and its codomain is endowed with the $L^1$-topology. This is at variance with the case of the play operator which instead is continuous in this sense.
Appeared in
- 727 of Journal of Physics: Conference Series, 2016, pp. 012006/1--012006/12.
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WIAS Preprint No. 2003, (2002)Asymptotic behaviour for a phase-field model with hysteresis in one-dimensional thermo-visco-plasticity
Authors
- Klein, Olaf
ORCID: 0000-0002-4142-3603
2010 Mathematics Subject Classification
- 74N30 35B40 47J40 34C55 35K60 74K05
Keywords
- Phase-field systems, phase transitions, hysteresis operators, thermo-visco-plasticity, asymptotic behaviour
DOI
Abstract
The asymptotic behaviour for t → ∞ of the solutions to a one-dimensional model for thermo-visco-plastic behaviour is investigated in this paper. The model consists of a coupled system of nonlinear partial differential equations, representing the equation of motion, the balance of the internal energy, and a phase evolution equation, determining the evolution of a phase variable. The phase evolution equation can be used to deal with relaxation processes. Rate-independent hysteresis effects in the strain-stress law and also in the phase evolution equation are described by using the mathematical theory of hysteresis operators.
Appeared in
- Appl. Math. 49 (2004) pp. 309--341
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WIAS Preprint No. 2003, (2002)Towards rigorous micro-macro transitions: The microscopic oscillator motion
Authors
- Dreyer, Wolfgang
- Herrmann, Michael
2010 Mathematics Subject Classification
- 70F45 82C22 35L65
Keywords
- Conservation laws, Many particle systems, Micro-macro transitions, Multiscale problems, Young measures
DOI
Abstract
The atomic chain whose dynamics evolve according to Newton's equations of motion serves as a simple microscopic many-particle system for an analysis of the macroscopic or thermodynamic limit. If the interaction potential has a sufficient strong repulsive part, it is possible to create a special microscopic motion, that we call oscillator motion, which is simpler as the classical thermal motion. However, also the oscillator motion leads to temperature, a Gibbs equation and an entropy. In the current paper we derive the thermodynamics for the oscillator motion without the subtle replacement of the many-particle sytem by a single equation of motion as it is done in [4]. Furthermore we introduce a different mathematical setting for micro-macro transitions as in [4], which is better suited for a rigorous treatment.
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WIAS Preprint No. 2003, (2002)Perfectly matched layers in transmission lines
Authors
- Hebermehl, Georg
- Hübner, Friedrich-Karl
- Schlundt, Rainer
ORCID: 0000-0002-4424-4301 - Tischler, Thorsten
- Zscheile, Horst
- Heinrich, Wolfgang
2010 Mathematics Subject Classification
- 35Q60 65F15 65N22
Keywords
- Microwave device, Optoelectronic device, Simulation, Maxwell's equation, PML boundary condition, Eigenvalue problem
DOI
Abstract
The field distribution at the ports of the transmission line structure is computed by applying Maxwell's equations to the structure and solving an eigenvalue problem. The high dimensional sparse system matrix is complex in the presence of losses and Perfectly Matched Layer. A method is presented which preserves sparseness and delivers only the small number of interesting modes out with the smallest attenuation. The modes are found solving a sequence of eigenvalue problems of modified matrices with the aid of the invert mode of the Arnoldi iteration using shifts. A new strategy is described which allows the application of the method, first developed for microwave structures, to optoelectronic devices.
Appeared in
- Numerical Mathematics and Advanced Applications, ENUMATH 2001, Springer Italia, Eds: Brezzi, F., Buffa, A., Corsaro, S., Murli, A., pp. 281-290
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WIAS Preprint No. 2003, (2002)Eigen mode computation of microwave and laser structures including PML
Authors
- Hebermehl, Georg
- Hübner, Friedrich-Karl
- Schlundt, Rainer
ORCID: 0000-0002-4424-4301 - Tischler, Thorsten
- Zscheile, Horst
- Heinrich, Wolfgang
2010 Mathematics Subject Classification
- 35Q60 65F15 65N22
Keywords
- Microwave device, Optoelectronic device, Simulation, Maxwell's equations, PML boundary condition, Eigenvalue problem
DOI
Abstract
The field distribution at the ports of the transmission line structure is computed by applying Maxwell's equations to the structure. Assuming longitudinal homogeneity an eigenvalue problem can be derived, whose solutions correspond to the propagation constants of the modes. The nonsymmetric sparse system matrix is complex in the presence of losses and Perfectly Matched Layer. The propagation constants are found solving a sequence of eigenvalue problems of modified matrices with the aid of the invert mode of the Arnoldi method. Using coarse and fine grids, and a new parallel sparse linear solver, the method, first developed for microwave structures, can be applied also to high dimensional problems of optoelectronics.
Appeared in
- Scientific Computing in Electrical Engineering, Eds. W. H. A. Schilders, E. J. W. ter Maten, St. H. M. J. Houben, Mathematics in Industry, Springer Verlag, Vol. 4, pp. 196--205, 2004
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WIAS Preprint No. 2003, (2002)On thermodynamics of nonlinear poroelastic materials
Authors
- Wilmanski, Krzysztof
2010 Mathematics Subject Classification
- 74A15 74E30 74L05
Keywords
- Continuum thermodynamics, porous media, mixtures
DOI
Abstract
The paper contains a brief presentation of a macroscopical thermodynamic model of poroelastic materials with many fluid components. A particular emphasis is placed on a Lagrangian formulation of the model and, consequently, on a consistent formulation of field equations on the reference configuration of the skeleton (solid phase of the mixture). It is demonstrated that the model possesses an identical structure as that in the pioneering work of C. A. Truesdell on the continuum mixture of fluids. An issue of porosity as an additional microstructural variable is particularly exposed.
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WIAS Preprint No. 2003, (2002)Note on weak discontinuity waves in linear poroelastic materials. Part I: Acoustic waves in saturated porous media
Authors
- Wilmański, Krzysztof
2010 Mathematics Subject Classification
- 35C20 35L50 74J10 74F10
Keywords
- waves in porous media, monochromatic waves
DOI
Abstract
The paper contains the analysis of the propagation of acoustic waves in two-component poroelastic media. It is shown that the existence of P2-mode as a wave in the range of low frequencies depends on the way in which the wave is excited. This property as well as properties of other bulk modes are discussed on practical examples of soil mechanics.
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WIAS Preprint No. 2003, (2002)Outwards pointing hysteresis operators and asymptotic behaviour of evolution equations
Authors
- Klein, Olaf
ORCID: 0000-0002-4142-3603 - Krejčí, Pavel
ORCID: 0000-0002-7579-6002
2010 Mathematics Subject Classification
- 34C55 35B40 74N30 47J40 35K60 74K05
Keywords
- hysteresis operators, Prandtl-Ishlinskii operator, asymptotic behaviour, visco-elasto-plasticity
DOI
Abstract
The paper deals with the long-time behaviour of evolution systems described by ODEs and PDEs with hysteresis operators. The analysis is based on two concepts. The first one is the outward pointing property of the involved hysteresis operators which implies uniform a priori bounds for solutions, the second one is related to the hysteresis modelling itself and consists in introducing a class of thermodynamically consistent generalized Prandtl-Ishlinskii operators as a model for a nonlinear elastoplastic material law. A stability result for solutions in one-dimensional visco-elasto-plasticity is derived as an illustration of the theory.
Appeared in
- Nonlinear Anal. Real World Applications Volume 4, no. 5, pp 755-785. DOI 10.1016/S1468-1218(03)00013-0 .
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WIAS Preprint No. 2003, (2002)Dissipative Schrödinger-Poisson systems
Authors
- Baro, Michael
- Kaiser, Hans-Christoph
- Neidhardt, Hagen
- Rehberg, Joachim
2010 Mathematics Subject Classification
- 47B44 47E05 35J05
Keywords
- dissipative Schrödinger-type operators, dissipative Schrödinger-Poisson systems, carrier and current densities, density matrices, a priori estimates
DOI
Abstract
The paper is devoted to the dissipative Schrödinger-Poisson system. We prove that the system always admits a solution and that all solutions of a given Schrödinger-Poisson system are included in a uniform ball whose radius depends only on the data of the system.
Appeared in
- J. Math. Phys. 45 (2004), no. 1, pp. 21-43
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WIAS Preprint No. 2003, (2002)Relaxation analysis and linear stability vs. adsorption in porous materials
Authors
- Albers, Bettina
ORCID: 0000-0003-4460-9152
2010 Mathematics Subject Classification
- 76E20 76S05 74J05
Keywords
- stability of geophysical flows, flows in porous media, linear waves
DOI
Abstract
The paper presents a linear stability analysis of a 1D stationary flow through a poroelastic medium. This base flow is perturbed in four ways: by longitudinal (1D) disturbances without and with mass exchange and by transversal (2D) disturbances without and with mass exchange. The eigenvalue problem for the first step field equations is solved using a finite-difference-scheme. For both disturbances without mass exchange results are confirmed by an analytical solution. We present the stability and relaxation properties in dependence on the two most important model parameters, namely the bulk and surface permeability coefficients.
Appeared in
- Continuum Mech. Thermodyn,. 15 (2003) 1, pp. 73-95
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WIAS Preprint No. 2003, (2002)Local likelihood modeling by adaptive weights smoothing
Authors
- Polzehl, Jörg
ORCID: 0000-0001-7471-2658 - Spokoiny, Vladimir
ORCID: 0000-0002-2040-3427
2010 Mathematics Subject Classification
- 62G08
Keywords
- adaptive weights, local likelihood, exponential family, density estimation, volatility, tail index, classification
DOI
Abstract
The paper presents a unified approach to local likelihood estimation for a broad class of nonparametric models, including e.g. the regression, density, Poisson and binary response model. The method extends the adaptive weights smoothing (AWS) procedure introduced in Polzehl and Spokoiny (2000) in context of image denoising. Performance of the proposed procedure is illustrated by a number of numerical examples and applications to estimation of the tail index parameter, classification, density and volatility estimation.
Appeared in
- Probab. Theory Related Fields, 135 (2006) pp. 335--362.
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WIAS Preprint No. 2003, (2002)Numerical algorithms to calculate periodic solutions of the Sivashinsky equation
Authors
- Karlin, Vladimir
- Maz'ya, Vladimir
- Schmidt, Gunther
2010 Mathematics Subject Classification
- 65M70 76E17 65G50
Keywords
- saturated asymptotic approximations, Sivashinsky equation, flame fronts, stability, round-off errors
DOI
Abstract
The primary aim of this work is the accurate calculation of periodic solutions to the Sivashinsky equation, which models dynamics of the long wave flame instability. A highly accurate computational algorithm has been developed in both one and two spatial dimensions and its crucial implementation details has been presented. The algorithm is based on the concept of "approximate approximations" which also can be referred to as saturated asymptotic approximations. The given computations support the idea of the instability of steady solutions to the Sivashinsky equation in large domains through huge linear amplification of nonmodal perturbations. Unlike the presentation of the algorithm is given for a particular equation, the evaluations have been carried out in a very general manner and the algorithm can be straightforwardly applied to a wide variety of nonlinear integro-differential equations.
Appeared in
- J. Comput. Phys. 188 (2003), pp. 209--231; under the new title: High accuracy periodic solutions to the Sivashinsky equation
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WIAS Preprint No. 2003, (2002)Numerical techniques in the simulation of microwave and laser structures including PML
Authors
- Hebermehl, Georg
- Hübner, Friedrich-Karl
- Schlundt, Rainer
ORCID: 0000-0002-4424-4301 - Tischler, Thorsten
- Zscheile, Horst
- Heinrich, Wolfgang
2010 Mathematics Subject Classification
- 35Q60 65F10 65F15 65N22
Keywords
- Microwave device, Optoelectronic device, Maxwell's equations, PML boundary condition, Eigenvalue problem, Systems of linear algebraic equations
DOI
Abstract
The properties of circuit structures can be described in terms of their scattering matrix. For the simulation of these structures, we use a Finite Difference Frequency Domain (FDFD) method in order to solve the three dimensional boundary value problem, governed by Maxwells equations. For the computation of the discrete grid equations, advanced preconditioning techniques are applied to reduce the dimension and the number of iterations solving the large-scale systems of linear algebraic equations by means of a block Krylov subspace method. The computational domain is truncated by electric or magnetic walls, open structures are treated using the Perfectly Matched Layer (PML) absorbing boundary condition. Calculating the excitation at the structures ports, one obtains an eigenvalue problem and thus large-scale systems of linear algebraic equations. The interesting modes of smallest attenuation are found solving a sequence of eigenvalue problems of modified matrices. Non-physical PML modes are detected by checking the eigenfunctions. Due to the high wavenumbers that have to be treated in optoelectronic device simulations, the number of modified eigenvalue problems as well as the dimension of the problem grows substantially in comparison to microwave structures. To reduce the execution times a coarse and a fine grid and parallelization techniques are used.
Appeared in
- Proceedings of the 10th International IGTE Symposium "Numerical Field Calculation in Electrical Engineering", CD-ROM, September 16-18, 2002, Technical University Graz, Austria
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WIAS Preprint No. 2003, (2002)Simulation of microwave and semiconductor laser structures including absorbing boundary conditions
Authors
- Hebermehl, Georg
- Hübner, Friedrich-Karl
- Schlundt, Rainer
ORCID: 0000-0002-4424-4301 - Tischler, Thorsten
- Zscheile, Horst
- Heinrich, Wolfgang
2010 Mathematics Subject Classification
- 35Q60 65N22 65F15 65F10
Keywords
- Microwave device, Semiconductor Laser, Simulation, Maxwell's equations, PML boundary condition, Eigenvalue problem, Linear algebraic equations
DOI
Abstract
The transmission properties of microwave and optical structures can be described in terms of their scattering matrix using a three-dimensional boundary value problem for Maxwell's equations. The computational domain is truncated by electric or magnetic walls, open structures are treated using the Perfectly Matched Layer (PML) Absorbing Boundary Condition. The boundary value problem is solved by a finite-volume scheme. This results in a two-step procedure: an eigenvalue problem for general complex matrices and the solution of a large-scale system of linear equations with indefinite symmetric complex matrices. The modes of smallest attenuation are located in a longsome region bounded by two parabolas, and are found solving a sequence of eigenvalue problems of modified matrices. To reduce the execution times a coarse and a fine grid, and two levels of parallelization can be used. For the computation of the discrete grid equations, advanced preconditioning techniques are applied to reduce the dimension and the number of iterations solving the large-scale systems of linear algebraic equations. These matrix problems need to be solved repeatedly for different right-hand sides, but with the same coefficient matrix. The used block quasi-minimal residual algorithm is a block Krylov subspace iterative method that incorporates deflation to delete linearly and almost linearly dependent vectors in the block Krylov sequences. Special attention is paid to the PML which causes significantly increased number of iterations within Krylov subspace methods.
Appeared in
- Challenges in Scientific Computing - CISC 2002, Ed. E. Baensch, Lecture Notes in Computational Science and Engineering, Springer Verlag, Vol. 35, pp. 131--159, 2003
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WIAS Preprint No. 2003, (2002)Longtime behavior of the traveling-wave model for semiconductor lasers
Authors
- Sieber, Jan
2010 Mathematics Subject Classification
- 78A60 37L10 35P10
Keywords
- laser dynamics, invariant manifold theory, hyperbolic systems of partial differential equations
DOI
Abstract
The traveling-wave model is a popular tool for investigating longitudinal dynamical effects in semiconductor lasers, e.g., sensitivity to delayed optical feedback. This model consists of a hyperbolic linear system of partial differential equations (PDEs) with one spatial dimension which is nonlinearly coupled with a slow subsystem of ordinary differential equations (ODEs). Firstly, we prove the basic statements about the existence of solutions of the initial-boundary-value problem and their smooth dependence on initial values and parameters. Hence, the model constitutes a smooth infinite-dimensional dynamical system. Then, we exploit this fact and the particular slow-fast structure of the system to construct a low-dimensional attracting invariant manifold for certain parameter constellations. The flow on this invariant manifold is described by a system of ODEs which is accessible to classical bifurcation theory and numerical tools like, e.g., AUTO.
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WIAS Preprint No. 2003, (2002)Simulation of static and dynamic properties of edge-emitting multi quantum well lasers
Authors
- Bandelow, Uwe
ORCID: 0000-0003-3677-2347 - Hünlich, Rolf
- Koprucki, Thomas
ORCID: 0000-0001-6235-9412
2010 Mathematics Subject Classification
- 78A60 68U20
2008 Physics and Astronomy Classification Scheme
- 42.55.Px 73.20.Dx 85.60.Bt 78.66.Fd
Keywords
- semiconductor lasers, quantum wells, device simulation, multi section lasers
DOI
Abstract
This paper demonstrates simulation tools for edge-emitting multi quantum well (MQW) lasers. Properties of the strained MQW active region are simulated by eight-band kp calculations. Then, a 2D simulation along the transverse cross section of the device is performed based on a drift-diffusion model, which is self-consistently coupled to heat transport and equations for the optical field. Furthermore, a method is described, which allows for an efficient quasi 3D simulation of dynamic properties of multi-section edge-emitting lasers.
Appeared in
- IEEE journal of selected topics in quantum electronics vol. 9, 2003, pp. 798 - 806
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WIAS Preprint No. 2003, (2002)A simple but rigorous micro-macro transition
Authors
- Dreyer, Wolfgang
- Herrmann, Michael
2010 Mathematics Subject Classification
- 35D05
Keywords
- Micro-macro transitions, Multiscale problems, Young measures
DOI
Abstract
This paper is devoted to a case study of micro-macro transitions. The main objective is the mathematically rigorous description of the macroscopic behavior of highly oscillating microscopic variables. In particular, we show that the theory of Young measures provides an elegant approach to this problem. A nontrivial application of the results is given in WIAS-Preprint 724.
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WIAS Preprint No. 2003, (2002)Convergence of the stochastic weighted particle method for the Boltzmann equation
Authors
- Matheis, Ingo
- Wagner, Wolfgang
2010 Mathematics Subject Classification
- 65C05 76P05 82C80
Keywords
- Boltzmann equation, stochastic weighted particle method, convergence, variance reduction, numerical experiment
DOI
Abstract
This paper studies convergence of the stochastic weighted particle method for the Boltzmann equation. First the method is extended by introducing new stochastic reduction procedures, in order to control the number of simulation particles. Then, under rather general conditions, convergence to the solution of the Boltzmann equation is proved. Finally, numerical experiments are performed illustrating both convergence and considerable variance reduction, for the specific problem of calculating tails of the velocity distribution.
Appeared in
- SIAM J. Sci. Comput., 24 (2003), pp. 1589-1609
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WIAS Preprint No. 2003, (2002)On the diffraction by biperiodic anisotropic structures
Authors
- Schmidt, Gunther
2010 Mathematics Subject Classification
- 35Q60 78A45 35J50 35J20
Keywords
- Maxwell equations, diffraction, strongly elliptic variational formulation, existence and uniqueness of solutions
DOI
Abstract
This paper studies the scattering of electromagnetic waves by a nonmagnetic biperiodic structure. The structure consists of anisotropic optical materials and separates two regions with constant dielectric coefficients. The time harmonic Maxwell equations are transformed to an equivalent strongly elliptic variational problem for the magnetic field in a bounded biperiodic cell with nonlocal boundary conditions. This guarantees the existence of quasiperiodic magnetic fields in 𝐻1 and electric fields in 𝐻(curl) solving Maxwell's equations. The uniqueness is proved for all frequencies excluding possibly a discrete set. The analytic dependence of these solutions on frequency and incident angles is studied.
Appeared in
- Applicable Analysis, 82 (2003), pp. 75-92
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WIAS Preprint No. 2003, (2002)Classical solutions of quasilinear parabolic systems on two-dimensional domains
Authors
- Kaiser, Hans-Christoph
- Neidhardt, Hagen
- Rehberg, Joachim
2010 Mathematics Subject Classification
- 35K40 35K45 35K57
Keywords
- Partial differential equations, quasilinear parabolic systems, nonsmooth domains, mixed boundary conditions, discontinuous coefficients, local classical solutions, reaction-diffusion systems
DOI
Abstract
Using a classical theorem of Sobolevskii on equations of parabolic type in a Banach space and recently obtained results on elliptic operators with discontinuous coefficients including mixed boundary conditions we prove that quasilinear parabolic systems in diagonal form admit a local, classical solution in the space of 𝑝-integrable functions, for some 𝑝 > 1, over a bounded two dimensional space domain. As applications we have in mind systems of reaction diffusion equations, e.g. van Roosbroeck's system. The treatment of such equations in a space of integrable functions enables us to define the normal component of the flow across any part of the Dirichlet boundary by Gauss' theorem.
Appeared in
- NoDEA Nonlinear Differential Equations Appl., 13 (2006) pp. 287-310.
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WIAS Preprint No. 2003, (2002)Instabilities of lasers with moderately delayed optical feedback
Authors
- Wolfrum, Matthias
ORCID: 0000-0002-4278-2675 - Turaev, Dmitry
Keywords
- delay differential equations, bifurcation analysis
DOI
Abstract
We Perform a bifurcation analysis of the Lang-Kobayashi system for a laser with delayed optical feedback in the situation of moderate delay times. Using scaling methods, we are able to calculate the primary bifurcations, leading to instability of the stationary lasing state. We classify different types of pulsations and identify a codimension two bifurcation of fold-Hopf interaction type as the organizing centre for the appearance of more complicated dynamics.
Appeared in
- Optics Communications, 213, (2002) pp. 127-138
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WIAS Preprint No. 2003, (2002)On multichannel signal detection
Authors
- Ingster, Yuri I.
- Lepskii, Oleg V.
2010 Mathematics Subject Classification
- 62G10 62G20
Keywords
- Multichannel signal detection, minimax hypothesis testing, adaptive hypothesis testing, distinguishability conditions
DOI
Abstract
We consider 𝑛-channel signal detection system. Each 𝑖th channel could contain (or not contain) a signal. We suppose a signal is a function ƒ𝑖(t), t ∈ (0,1) observing in the white Gaussian noise of level ε > 0. Let 𝑘 be a number of channels which contain the signals. This number could be known or unknown. The functions ƒ𝑖 could be known or unknown as well. If shapes of functions ƒ𝑖 are unknown, then we consider nonparametric case. We suppose that functions ƒ𝑖 belongs to the Sobolev ball 𝚂σ where the smoothness parameter σ > 0 could be known or unknown as well. The cases, when 𝑘 or σ are unknown, lead to the "adaptive" problems. We are interested in the following problems:
(1) How large the signals ƒ𝑖 should be in order to detect these signals with vanishing errors, as the number of channels 𝑛 tends to infinity?
(2) What are the structures of test procedures which provide the detection of signals with the vanishing errors, if it is possible?
We show that there are two main types of results in the problems which, roughly, correspond to the cases either 𝑘 is "large" (this means 𝑘 >> 𝑛½ in the problem) or 𝑘 is "small" (this means 𝑘 << 𝑛½).
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WIAS Preprint No. 2003, (2002)Micro-macro transitions by interpolation, smoothing, averaging and scaling of particle trajectories
Authors
- Dreyer, Wolfgang
- Guckel, Ralf
2010 Mathematics Subject Classification
- 35D05
Keywords
- Micro-macro transitions, Multiscale problems, Young measures
DOI
Abstract
We consider a Newtonian system of many diatomicmolecules each of which consisting of two atoms of equal mass whichare separated by a fixed distance. The barycenters are allowed to movealong some fixed straight line. Moreover each molecule has an additionalrotational degree of freedom. The atoms of neighbouring moleculesinteract to each other by a generic pair potential. By means of thisexample we propose a new method for deriving macroscopic models from microscopic ones. The method is based on the definition of macroscopicobservables and the derivation of corresponding balance laws by interpolation smoothing/averaging and subsequent scaling of particle trajectories.
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WIAS Preprint No. 2003, (2002)Delayed exchange of stabilities in a class of singularly perturbed parabolic problems
Authors
- Nefedov, Nikolai N.
- Schneider, Klaus R.
2010 Mathematics Subject Classification
- 35B25 35K20
Keywords
- Singularly perturbed parabolic problem, delayed exchange of stabilities, upper and lower solutions
DOI
Abstract
We consider a class of singularly perturbed parabolic problems in case of exchange of stabilities, that is, the corresponding degenerate equation has two intersecting roots. By means of the technique of asymptotic lower and upper solutions we prove that the considered initial-boundary value problem has a unique solution exhibiting the phenomenon of delayed exchange of stabilities. Thus, the problem under consideration has a canard solution.
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WIAS Preprint No. 2003, (2002)On polynomial collocation for second kind integral equations with fixed singularities of Mellin type
Authors
- Mastroianni, Giuseppe
- Frammartino, Carmelina
- Rathsfeld, Andreas
ORCID: 0000-0002-2029-5761
2010 Mathematics Subject Classification
- 65R20 45L10 65N38
Keywords
- integral equation of the second kind, Mellin kernel, polynomial collocation, convergence rate, quadrature, recursion
DOI
Abstract
We consider a polynomial collocation for the numerical solution of a second kind integral equation with an integral kernel of Mellin convolution type. Using a stability result by Junghanns and one of the authors, we prove that the error of the approximate solution is less than a logarithmic factor times the best approximation and, using the asymptotics of the solution, we derive the rates of convergence. Finally, we describe an algorithm to compute the stiffness matrix based on simple Gauss quadratures and an alternative algorithm based on a recursion in the spirit of Monegato and Palamara Orsi. All together an almost best approximation to the solution of the integral equation can be computed with O(n^2[log n]^2) resp. O(n^2) operations, where n is the dimension of the polynomial trial space.
Appeared in
- Numer. Math. 94, 2003, pp. 333-365
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WIAS Preprint No. 2003, (2002)Coalescence in a random background
Authors
- Barton, Nick H.
- Etheridge, Alison M.
- Sturm, Anja K.
2010 Mathematics Subject Classification
- 60J80 60J85 60J70 60K35
Keywords
- coalescent, selection, recombination, identity by descent, random environment
DOI
Abstract
We consider a single genetic locus which carries two alleles, labelled 𝑃 and Ｑ. This locus experiences selection and mutation. It is linked to a second neutral locus with recombination rate 𝑟. If 𝑟 = 0, this reduces to the study of a single selected locus. Assuming a Moran model for the population dynamics, we pass to a diffusion approximation and, assuming that the allele frequencies at the selected locus have reached stationarity, establish the joint generating function for the genealogy of a sample from the population and the frequency of the 𝑃 allele. In essence this is the joint generating function for a coalescent and the random background in which it evolves. We use this to characterise, for the diffusion approximation, the probability of identity in state at the neutral locus of a sample of two individuals (whose type at the selected locus is known) as solutions to a system of ordinary differential equations. The only subtlety is to find the boundary conditions for this system. Finally, numerical examples are presented that illustrate the accuracy and predictions of the diffusion approximation. In particular, a comparison is made between this approach and one in which the frequencies at the selected locus are estimated by their mean and a classical structured coalescent model is used.
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WIAS Preprint No. 2003, (2002)A filtered no arbitrage model for term structures from noisy data
Authors
- Gombani, Andrea
- Jaschke, Stefan R.
- Runggaldier, Wolfgang R.
2010 Mathematics Subject Classification
- 93E11 60G35
Keywords
- term structure of interest rates, linear estimation, Kalman filter
DOI
Abstract
We consider an affine term structure model of interest rates, where the factors satisfy a linear diffusion equation. We assume that the information available to an agent comes from observing the yields of a finite number of traded bonds and that this information is not sufficient to reconstruct exactly the factors. We derive a method to obtain arbitrage-free prices of illiquid or non traded bonds that are compatible with the available incomplete information. The method is based on an application of the Kalman filter for linear Gaussian systems.
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WIAS Preprint No. 2003, (2002)Global uniqueness in determining rectangular periodic structures by scattering data with a single wave number
Authors
- Elschner, Johannes
- Schmidt, Gunther
- Yamamoto, Masahiro
2010 Mathematics Subject Classification
- 78A46 35J05 35R30
Keywords
- periodic structure, inverse scattering, transverse electric polarization, transverse magnetic polarization, analyticity, unique continuation
DOI
Abstract
We consider an inverse scattering problem of determining a periodic structure by near-field observations of the total field. We prove the global uniqueness results in both cases of the transverse electric polarization and the transverse magnetic polarization within the class of rectangular periodic structures by a single choice of any wave number. The proof is based on the analyticity of solutions to the Helmholtz equation.
Appeared in
- J. Inv. Ill-Posed Problems 11 (2003), pp. 235-244
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WIAS Preprint No. 2003, (2002)Regularity properties of potentials for joint measures of random spin systems
Authors
- Külske, Christof
2010 Mathematics Subject Classification
- 82B44 82B26 82B20
2008 Physics and Astronomy Classification Scheme
- 05.50.+q 61.43.-j 02.50.Cw
Keywords
- Disordered systems, Gibbs measures, non-Gibbsian measures, joint measures, random field model
DOI
Abstract
We consider general quenched disordered lattice spin models on compact local spin spaces with possibly dependent disorder. We discuss their corresponding joint measures on the product space of disorder variables and spin variables in the infinite volume. These measures often possess pathologies in a low temperature region reminiscent of renormalization group pathologies in the sense that they are not Gibbs measures on the product space. Often the joint measures are not even almost Gibbs, but it is known that there is always a potential for their conditional expectations that may however only be summable on a full measures set, and not everywhere. In this note we complement the picture from the non-pathological side. We show regularity properties for the potential in the region of interactions where the joint potential is absolutely summable everywhere. We prove unicity and Lipschitz-continuity, much in analogy to the two fundamental regularity theorems proved by van Enter, Fernandez, Sokal for renormalization group transformations.
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WIAS Preprint No. 2003, (2002)Existence and approximation of slow integral manifolds in some degenerate cases
Authors
- Schneider, Klaus R.
- Sobolev, Vladimir A.
2010 Mathematics Subject Classification
- 34C45 34D15 34E15
Keywords
- singular perturbations, integral manifolds, degenerate cases
DOI
Abstract
We consider singularly perturbed differential systems whose degenerate equations have an isolated but not simple solution. In that case, the standard theory to establish a slow integral manifold near this solution does not work. Applying scaling transformations and using the technique of gauge functions we reduce the original singularly perturbed problem to a regularized one such that the existence of slow integral manifolds can be established by means of the standard theory of singular perturbations. We illustrate our method by several examples.
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WIAS Preprint No. 2003, (2002)Geometric singular perturbation theory for stochastic differential equations
Authors
- Berglund, Nils
- Gentz, Barbara
2010 Mathematics Subject Classification
- 37H20 34E15 60H10
Keywords
- Singular perturbations, slow-fast systems, invariant manifolds, dynamic bifurcations, stochastic differential equations, first-exit times, concentration of measure
DOI
Abstract
We consider slow-fast systems of differential equations, in which both the slow and fast variables are perturbed by additive noise. When the deterministic system admits a uniformly asymptotically stable slow manifold, we show that the sample paths of the stochastic system are concentrated in a neighbourhood of the slow manifold, which we construct explicitly. Depending on the dynamics of the reduced system, the results cover time spans which can be exponentially long in the noise intensity squared (that is, up to Kramers' time). We give exponentially small upper and lower bounds on the probability of exceptional paths. If the slow manifold contains bifurcation points, we show similar concentration properties for the fast variables corresponding to non-bifurcating modes. We also give conditions under which the system can be approximated by a lower-dimensional one, in which the fast variables contain only bifurcating modes.
Appeared in
- J. Differential Equations 191, No. 1, 1-54 (2003), DOI 10.1016/S0022-0396(03)00020-2 .
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WIAS Preprint No. 2003, (2002)Grating profile reconstruction based on finite elements and optimization techniques
Authors
- Elschner, Johannes
- Hsiao, George C.
- Rathsfeld, Andreas
ORCID: 0000-0002-2029-5761
2010 Mathematics Subject Classification
- 35R30 35J05 35J05 78A46 78M50
Keywords
- Diffraction grating, profile reconstruction, optimization method, conjugate gradient algorithm
DOI
Abstract
We consider the inverse diffraction problem to recover a two-dimensional periodic structure from scattered waves measured above and beneath the structure. The task is reformulated in form of an optimization problem including special regularization terms. The solvability and the dependence on the parameter of regularization is analyzed. Numerical results for synthetic data demonstrate the practicability of the inversion algorithm.
Appeared in
- SIAM J. Appl. Math. 64 (2003), pp.525-545
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WIAS Preprint No. 2003, (2002)A two-step algorithm for the reconstruction of perfectly reflecting periodic profiles
Authors
- Bruckner, Gottfried
- Elschner, Johannes
2010 Mathematics Subject Classification
- 35R30 35J05 78A46 78M50
Keywords
- diffraction grating, profile reconstruction, Tikhonov regularization, optimization method, Gauss-Newton method
DOI
Abstract
We consider the inverse problem of recovering a 2D periodic structure from scattered waves measured above the structure. First, following [5], the inverse problem is reformulated as an optimization problem which consists of two parts: a linear severely ill-posed problem and a nonlinear well--posed one. Then, contrary to [5], here the two problems are solved separately to diminish the computational effort by exploiting their special properties. Numerical results for exact and noisy data demonstrate the practicability of the inversion algorithm.
Appeared in
- Inverse Problems 19 (2003), pp. 315-329
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WIAS Preprint No. 2003, (2002)On the shape-from-moments problem and recovering edges from noisy Radon data
Authors
- Goldenshluger, Alexander
- Spokoiny, Vladimir
ORCID: 0000-0002-2040-3427
2010 Mathematics Subject Classification
- 62C20 62G20 94A12
Keywords
- minimax estimation, optimal rates of convergence, shape, moments, support function, Radon transform, tomography
DOI
Abstract
We consider the problem of reconstructing a planar convex set from noisy observations of its moments. An estimation method based on pointwise recovering of the support function of the set is developed. We study intrinsic accuracy limitations in the shape-from-moments estimation problem by establishing a lower bound on the rate of convergence of the mean squared error. It is shown that the proposed estimator is near-optimal in the sense of the order. An application to tomographic reconstruction is discussed, and it is indicated how the proposed estimation method can be used for recovering edges from noisy Radon data.
Appeared in
- Probab. Theory Related Fields, vol 128 (2004), no 1, pp. 123-140
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WIAS Preprint No. 2003, (2002)Second kind similarity solutions of the modified porous medium equation
Authors
- Wagner, Barbara
ORCID: 0000-0001-8306-3645
2010 Mathematics Subject Classification
- 76M60 76S05 76M45
Keywords
- Lie group methods, similarity solutions, asymptotics, porous medium flow
DOI
Abstract
We consider the problem of a spreading ground water mound of liquid in a porous medium, situated on an impermeable horizontal solid layer. The mathematical formulation for this problem is given by the modified porous medium equation. We derive a global condition in form of an energy integral, describing the loss of liquid in the porous medium. This yields the necessary condition that determines the similarity exponents for the similarity solution of second kind, describing the long time behavior of the mound. We further apply our method to the problem when instead of an energy integral another conservation law, such as the first moment integral, obeyed by a family of antisymmetric solutions, is violated. Here, we consider as an application the problem of the impact of a flood infiltrating a porous medium. In all cases we will also solve and compare our analytical results with our numerical solution, and, if available, with examples existing in the literature.
Appeared in
- J. Engrg. Math., 53 (2005) pp. 201-220.
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WIAS Preprint No. 2003, (2002)Positive feedback control of Rayleigh-Bénard convection
Authors
- Wagner, Barbara
ORCID: 0000-0001-8306-3645 - Bertozzi, Andrea
- Howle, Laurens
2010 Mathematics Subject Classification
- 37L65 93A30 76E15
Keywords
- Hydrodynamic stability, Galerkin method, Mathematical modeling
DOI
Abstract
We consider the problem of active feedback control of rbc via shadowgraphic measurement. Our theoretical studies show, that when the feedback control is positive, i.e. is tuned to advance the onset of convection, there is a critical threshold beyond which the system becomes linearly ill-posed so that short-scale disturbances are greatly amplified. Experimental observation suggests that finite size effects become important and we develop a theory to explain these contributions. As an efficient modelling tool for studying the dynamics of such a controlled pattern forming system, we use a Galerkin approximation to derive a dimension reduced model.
Appeared in
- Discrete Contin. Dyn. Syst. Ser. B, 3 (2003), pp. 619--642.
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WIAS Preprint No. 2003, (2002)An inverse problem in periodic diffractive optics: Reconstruction of Lipschitz grating profiles
Authors
- Elschner, Johannes
- Yamamoto, Masahiro
2010 Mathematics Subject Classification
- 35R30 35J05 78A46 78M50
Keywords
- Diffraction grating, profile reconstruction, optimization method, convergence analysis
DOI
Abstract
We consider the problem of recovering a two-dimensional periodic structure from scattered waves measured above the structure. Following an approach by Kirsch and Kress, this inverse problem is reformulated as a nonlinear optimization problem. We develop a theoretical basis for the reconstruction method in the case of an arbitrary Lipschitz grating profile. The convergence analysis is based on new perturbation and stability results for the forward problem.
Appeared in
- Applicable Analysis 81 (2002), pp. 1307-1328
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WIAS Preprint No. 2003, (2002)On convergence of population processes in random environments to the stochastic heat equation with colored noise
Authors
- Sturm, Anja K.
2010 Mathematics Subject Classification
- 60H15 60K35 60K37 60J80 60F05
Keywords
- Heat equation, colored noise, stochastic partial differential equation, superprocess, weak convergence, particle representation, random environment, existence theorem
DOI
Abstract
We consider the stochastic heat equation with a multiplicative colored noise term on ℝ2 d ≥ 1 in dimensions greater or equal to one. First, we prove convergence of a branching particle system in a random environment, to the stochastic heat equation with a linear noise term. For this stochastic partial differential equation with more general non-Lipschitz noise coefficients we show convergence of associated lattice systems, which are infinite dimensional stochastic differential equations with correlated noise terms, provided that uniqueness of the limit is known. In the course of the proof, we establish existence and uniqueness of solutions to the lattice systems, as well as a new existence result for solutions to the stochastic heat equation. The latter are shown to be jointly continuous in time and space under some mild additional assumptions.
Appeared in
- Electron. J. Probab. 8 (2003), 39 pp. (electronic)
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WIAS Preprint No. 2003, (2002)Sound and surface waves in poroelastic media
Authors
- Albers, Bettina
ORCID: 0000-0003-4460-9152 - Wilmanski, Krzysztof
2010 Mathematics Subject Classification
- 74J10 74J15 74F10
Keywords
- Waves in porous media, monochromatic waves, surface waves
DOI
Abstract
We consider two problems of propagation of weak discontinuity waves in porous materials. In the first part we present basic properties of bulk waves in fully saturated materials. These materials are modelled by a two-component immiscible mixture. We present general propagation conditions for such a model which yield three modes of propagation: P1-, S-, and P2-waves. Then we discuss the dispersion relation and we show that results are strongly dependent on the way in which waves are excited. In the second part we present some properties of surface waves. We begin with the classical Rayleigh and Love problems and then we extend them on heterogeneous materials important in practical applications. Subsequently we proceed to surface waves in two-component porous materials on the contact surface with vacuum (impermeable boundary) and with a liquid (permeable boundary). We show the existence of different modes of surface waves in the high frequency limit as well as the degeneration of the problem in the low frequency limit.
Appeared in
- Dynamic Response of Granular and Porous Materials under Large and Catastrophic Deformations, K. Hutter, N. Kirchner (eds.), Lecture Notes in Applied and Computational Mechanics, Vol. 11, Springer, Berlin, pp. 285--314, 2003
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WIAS Preprint No. 2003, (2002)Metastability in reversible diffusion processes II. Precise asymptotics for small eigenvalues
Authors
- Bovier, Anton
- Gayrard, Véronique
- Klein, Markus
2010 Mathematics Subject Classification
- 82C44 60K35
Keywords
- Metastability, diffusion processes, spectral theory, potential theory, capacity, exit times
DOI
Abstract
We continue the analysis of the problem of metastability for reversible diffusion processes, initiated in [BEGK3], with a precise analysis of the low-lying spectrum of the generator. Here we consider only the generic situation where the depths of all local minima are different. We show that in general the exponentially small part of the spectrum is given, up to multiplicative errors tending to one, by the eigenvalues of the classical capacity matrix of the array of capacitors made of small balls centered at the positions of the local minima of F. We also get very precise uniform control on the corresponding eigenfunctions. Moreover, these eigenvalues can be identified with the same precision with the inverse mean metastable exit times from each minimum. In [BEGK3] it was proven that these mean times are given, again up to multiplicative errors that tend to one, by the classical Eyring-Kramers formula.
Appeared in
- J.Eur.Math.Soc. (JEMS) vol. 7, pp. 69--99, 2005
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WIAS Preprint No. 2003, (2002)Concentration inequalities for functions of Gibbs fields with application to diffraction and random Gibbs measures
Authors
- Külske, Christof
2010 Mathematics Subject Classification
- 82B44 78A45 60F10 82B20
2008 Physics and Astronomy Classification Scheme
- 05.50.+q 61.10.Dp
Keywords
- Gibbs measures, disordered systems, diffraction theory, random scatterers, random point sets, quasicrystals, large deviations
DOI
Abstract
We derive useful general concentration inequalities for functions of Gibbs fields in the uniqueness regime. We also consider expectations of random Gibbs measures that depend on an additional disorder field, and prove concentration w.r.t the disorder field. Both fields are assumed to be in the uniqueness regime, allowing in particular for non-independent disorder field. The modification of the bounds compared to the case of an independent field can be expressed in terms of constants that resemble the Dobrushin contraction coefficient, and are explicitly computable. On the basis of these inequalities, we obtain bounds on the deviation of a diffraction pattern created by random scatterers located on a general discrete point set in the Euclidean space, restricted to a finite volume. Here we also allow for thermal dislocations of the scatterers around their equilibrium positions. Extending recent results for independent scatterers, we give a universal upper bound on the probability of a deviation of the random scattering measures applied to an observable from its mean. The bound is exponential in the number of scatterers with an upper bound rate that involves only the minimal distance between points in the point set.
Appeared in
- Comm. Math. Phys., 239 (2003), pp. 29-51
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WIAS Preprint No. 2003, (2002)Unsteady thermal convection in the North-East-German basin
Authors
- Bayer, Ulf
- Clausnitzer, Volker
- Fuhrmann, Jürgen
ORCID: 0000-0003-4432-2434
2010 Mathematics Subject Classification
- 65N99 76R10
2008 Physics and Astronomy Classification Scheme
- 91.35.Dc 91.95.Gf
Keywords
- Thermal Convection, Finite Volumes, Nonlinear Time Series Analysis
DOI
Abstract
We describe a Voronoi box based finite volume method for the numerical simulation of thermal convection in sedimental basins. The method shows a temperature maximum principle and consistent velocity approximation. We present results of simulation runs in vertical slices of the North-East German basin. These indicate that the system is far from a stationary state, it shows quasi-periodic, and possibly chaotic behaviour. The chaos hypothesis is formulated based on the analysis of the Nusselt number time series obtaind from simulation runs.
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WIAS Preprint No. 2003, (2002)Space-time asymptotics of an infinite-dimensional diffusion having a long-range memory
Authors
- Rœlly, Sylvie
- Sortais, Michel
2010 Mathematics Subject Classification
- 60H10 60K35 82C22 82C44
Keywords
- Random Field Ising Model, Langevin Dynamics, Interacting Diffusion Processes, Space-Time Cluster Expansions
DOI
Abstract
We develop a cluster expansion in space-time for an infinite-dimensional system of interacting diffusions where the drift term of each diffusion depends on the whole past of the trajectory, these interacting diffusions arise when considering the Langevin dynamics of a ferromagnetic system submitted to a disordered external magnetic field.
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WIAS Preprint No. 2003, (2002)Soft billiards with corners
Authors
- Turaev, Dmitry
- Rom-Kedar, Vered
2010 Mathematics Subject Classification
- 58F15 82C05 34C37 58F05 58F13 58F14
Keywords
- singular Hamiltonians, scattering potentials, elliptic islands
DOI
Abstract
We develop a framework for dealing with smooth approximations to billiards with corners in the two-dimensional setting. Let a polygonal trajectory in a billiard start and end up at the same billiard's corner point. We prove that smooth Hamiltonian flows which limit to this billiard have a nearby periodic orbit if and only if the polygon angles at the corner are "acceptable". The criterion for a corner polygon to be acceptable depends on the smooth potential behavior at the corners, which is expressed in terms of a scattering function. We define such an asymptotic scattering function and prove the existence of it, explain how it can be calculated and predict some of its properties. In particular, we show that it is non-monotone for some potentials in some phase space regions. We prove that when the smooth system has a limiting periodic orbit it is hyperbolic provided the scattering function is not extremal there. We then prove that if the scattering function is extremal, the smooth system has elliptic periodic orbits limiting to the corner polygon, and, furthermore, that the return map near these periodic orbits is conjugate to a small perturbation of the Henon map and therefore has elliptic islands. We find from the scaling that the island size is typically algebraic in the smoothing parameter and exponentially small in the number of reflections of the polygon orbit.
Appeared in
- J. Statist. Phys., 112 (2003), pp. 765-813
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WIAS Preprint No. 2003, (2002)Metastability in reversible diffusion processes I. Sharp asymptotics for capacities and exit times
Authors
- Bovier, Anton
- Eckhoff, Michael
- Gayrard, Véronique
- Klein, Markus
2010 Mathematics Subject Classification
- 82C44 60K35
Keywords
- Metastability, diffusion processes, potential theory, capacity, exit times
DOI
Abstract
We develop a potential theoretic approach to the problem of metastability for reversible diffusion processes in potentials F given by a smooth function with finitely many local minima. In analogy to previous work in discrete Markov chains, we show that metastable exit times from the attractive domains of the minima of F can be related, up to multiplicative errors that tend to one when the noise strength tends to zero to the capacities of suitably constructed sets. We show that this capacities can be computed, again up to multiplicative errors that tend to one, in terms of local characteristics of F at the starting minimum and the relevant saddle points. As a result, we are able to give the first rigorous proof of the classical Eyring-Kramers formula in dimension larger than 1. The estimates on capacities make use of their variational representation and monotonicity properties of Dirichlet forms. The methods developed here are extensions of our earlier work on discrete Markov chains to continuous diffusion processes.
Appeared in
- J. Eur. Math. Soc. (JEMS), 6 (2004), pp. 399--424
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WIAS Preprint No. 2003, (2002)On modification of the Newton's law of gravity at very large distances
Authors
- Kirillov, Alexandr A.
- Turaev, Dmitry
Keywords
- dark matter problem
DOI
Abstract
We discuss a Modified Field Theory (MOFT) in which the number of fields can vary. It is shown that when the number of fields is conserved MOFT reduces to the standard field theory but interaction constants undergo an additional renormalization and acquire a dependence on spatial scales. In particular, the renormalization of the gravitational constant leads to the deviation of the law of gravity from the Newton's law in some range of scales rmin< r < rmax, in which the gravitational potential shows essentially logarithmic ln r (instead of 1/r) behavior. In this range, the renormalized value of the gravitational constant G increases and atscales r > rmax acquires a new constant value G' Grmaxrmin. From the dynamical standpoint this looks as if every point source is surrounded with a halo of dark matter. It is also shown that if the maximal scale rmax is absent, the homogeneity of the dark matter in the Universe is consistent with a fractal distribution of baryons in space, in which the luminous matter is located on thin two-dimensional surfaces separated by empty regions of ever growing size.
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WIAS Preprint No. 2003, (2002)Excitability of a DFB laser with short external cavity
Authors
- Radziunas, Mindaugas
ORCID: 0000-0003-0306-1266 - Wünsche, Hans-Jürgen
- Brox, Olaf
- Henneberger, Fritz
2010 Mathematics Subject Classification
- 78A60 78-05 35P10
Keywords
- DFB laser, excitability, modes, traveling wave equations
DOI
Abstract
We discuss some aspects of the excitability in a semiconductor laser with short external cavity. It is demonstrated both theoretically and experimentally how a two-section semiconductor laser consisting of a DFB section and an integrated passive phase tuning section performs an excitable response to optical injection. A mode analysis of the model equations allows to understand and explain the origin of the excitability.
Appeared in
- SPIE Proceedings Series, vol. 4646, pp. 420-428, (2002)
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WIAS Preprint No. 2003, (2002)Calibration of LIBOR models to caps and swaptions: A way around intrinsic instabilities via parsimonious structures and a collateral market criterion
Authors
- Schoenmakers, John G. M.
ORCID: 0000-0002-4389-8266
2010 Mathematics Subject Classification
- 60H05 60H10 90A09
Keywords
- parsimonious LIBOR models, calibration, stability, correlation structures
DOI
Abstract
We expose an intrinsic stability problem in joint calibration of a LIBOR market model to caps and swaptions by direct least squares calibration. This problem typically encounters if one tries to identify jointly the volatility norm behaviour and the correlation structure of the forward LIBORs. As a remedy we propose collateral incorporation of a 'Market Swaption Formula', a rule-of-thumb formula which practitioners tend to use, in the calibration routine. It is shown by experiments with practical data that with this new calibration procedure and suitably parametrized volatility structures LIBOR model calibration to caps and swaptions is stable. The involved calibration routine is based on standard swaption approximation or its refinements by Hull & White, Jäckel & Rebonato. We deal with the issue of differently settled caps and swaptions by accordingly adapting the swap rate formula and give a respective modification of Jäckel and Rebonato's refined swaption approximation formula.
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WIAS Preprint No. 2003, (2002)Space-time regularity of catalytic super-Brownian motion
Authors
- Zähle, Henryk
2010 Mathematics Subject Classification
- 60H15 60K35 60G57 60J80
Keywords
- catalytic super-Brownian motion, cumulant equation, martingale problem, collision local time, stochastic partial differential equation
DOI
Abstract
We focus on the question, for which catalysts does the catalytic super-Brownian motion in ℝ1 have a jointly continuous space-time Lebesgue density? As it turns out, nearly all non-atomic catalysts provide such a regular density which can be characterized as the unique solution to a stochastic partial differential equation driven by a degenerated space-time white noise.
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WIAS Preprint No. 2003, (2002)Wavelet approximation of correlated wavefunctions. II. Hyperbolic wavelets and adaptive approximation schemes
Authors
- Luo, Hongjun
- Kolb, Dietmar
- Flad, Heinz-Jürgen
- Hackbusch, Wolfgang
- Koprucki, Thomas
ORCID: 0000-0001-6235-9412
2010 Mathematics Subject Classification
- 65T60 81V70
Keywords
- wavelets, adaptive schemes, correlated wavefunctions, Jastrow factor, electron-electron cusp, groundstate
DOI
Abstract
We have studied various aspects concerning the use of hyperbolic wavelets and adaptive approximation schemes for wavelet expansions of correlated wavefunctions. In order to analyze the consequences of reduced regularity of the wavefunction at the electron-electron cusp, we first considered a realistic exactly solvable many-particle model in one dimension. Convergence rates of wavelet expansions, with respect to L2 and H1 norms and the energy, were established for this model. We compare the performance of hyperbolic wavelets and their extensions through adaptive refinement in the cusp region, to a fully adaptive treatment based on the energy contribution of individual wavelets. Although hyperbolic wavelets show an inferior convergence behavior, they can be easily refined in the cusp region yielding an optimal convergence rate for the energy. Preliminary results for the helium atom are presented, which demonstrate the transferability of our observations to more realistic systems. We propose a contraction scheme for wavelets in the cusp region, which reduces the number of degrees of freedom and yields a favorable cost to benefit ratio for the evaluation of matrix elements.
Appeared in
- Journal of Chemical Physics 117 (2002), pp. 3625-3638
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WIAS Preprint No. 2003, (2002)Optimization of ordinary differential systems with hysteresis
Authors
- Sprekels, Jürgen
ORCID: 0009-0000-0618-8604 - Tiba, Dan
2010 Mathematics Subject Classification
- 34A34 34B15 49J15
Keywords
- Peano-type existence theorem, continuous hysteresis operators, Clarke's generalized gradient
DOI
Abstract
We investigate general control problems governed by ordinary differential systems involving hysteresis operators. Our main hypotheses are of continuity type, and we discuss existence results, discretization methods, and approximation approaches.
Appeared in
- ``Analysis and Optimization of Differential Systems'' (V. Barbu, I. Lasiecka, D. Tiba, C. Varsan, eds.), Kluver Acad. Publishers, 2003, Boston/Dordrecht/London, pp. 387-398
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WIAS Preprint No. 2003, (2002)Stochastic interacting particle systems and nonlinear kinetic equations
Authors
- Eibeck, Andreas
- Wagner, Wolfgang
2010 Mathematics Subject Classification
- 60K40 65C35
Keywords
- Stochastic particle systems, regularity of jump processes, kinetic equations, existence of solutions, coagulation, fragmentation, source and efflux, dissipative collisions
DOI
Abstract
We present the stochastic approach to nonlinear kinetic equations (without gradient terms) in a unifying general framework, which covers many interactions important in applications, like coagulation, fragmentation, inelastic collisions, as well as source and efflux terms. We provide conditions for the existence of corresponding stochastic particle systems in the sense of regularity (non-explosion) of a jump process with unbounded intensity. Using an appropriate space of measure-valued functions, we prove relative compactness of the sequence of processes and characterize the weak limits in terms of solutions to the nonlinear equation. As a particular application, we derive existence theorems for Smoluchowski's coagulation equation with fragmentation, efflux and source terms, and for the Boltzmann equation with dissipative collisions.
Appeared in
- Ann. Appl. Probab., 13 (2003), pp. 845-889
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WIAS Preprint No. 2003, (2002)On the uniqueness of solutions for nonlinear elliptic-parabolic equations
Authors
- Gajewski, Herbert
- Skrypnik, Igor V.
2010 Mathematics Subject Classification
- 35B45 35K15 35K20 35K65
Keywords
- Nonlinear parabolic equations, bounded solutions, uniqueness, nonstandard assumptions, degenerate type
DOI
Abstract
We prove existence, boundedness and uniqueness of solutions to Cauchy-Dirichlet problems for elliptic-parabolic systems, where the specific coupling is such that the solution of the elliptic equation forms a drift term in the parabolic equation. Such systems arise as mathematical models of various applied problems, for instance, drift-diffusion processes in semiconductors and segregation processes in alloys. Our basic assumptions are thermodynamically motivated.
Appeared in
- J. Evol. Equ., 3 (2003), pp. 247-281
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WIAS Preprint No. 2003, (2002)An existence result for infinite-dimensional Brownian diffusions with non-regular and non-Markovian drift
Authors
- Dai Pra, Paolo
- Rœlly, Sylvie
2010 Mathematics Subject Classification
- 60G15 60G60 60H10 60J60
Keywords
- infinite-dimensional Brownian diffusion, space-time Gibbs field, cluster expansion
DOI
Abstract
We prove in this paper an existence result for infinite-dimensional stationary interactive Brownian diffusions. The interaction is supposed to be small in the norm · ∞ but otherwise is very general, being possibly non-regular and non-Markovian. Our method consists in using the characterization of such diffusions as space-time Gibbs fields so that we construct them by space-time cluster expansions in the small coupling parameter.
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WIAS Preprint No. 2003, (2002)Infinitely many elliptic periodic orbits in four-dimensional symplectic maps with a homoclinic tangency
Authors
- Gonchenko, Sergey V.
- Shilnikov, Leonid P.
- Turaev, Dmitry
2010 Mathematics Subject Classification
- 37G25 37J45 37J20
Keywords
- Hamiltonian dynamics, Henon map, renormalization
DOI
Abstract
We show that systems having infinitely many coexisting generic 2-elliptic periodic orbits are dense among the four-dimensional symplectic maps with an orbit of homoclinic tangency to a saddle-focus.
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WIAS Preprint No. 2003, (2002)Sensitivity analysis of an eddy current problem related to induction heating
Authors
- Hömberg, Dietmar
ORCID: 0000-0001-9460-5729 - Sokołowski, Jan
2010 Mathematics Subject Classification
- 49K35 78A55 74N25
Keywords
- sensitivity analysis, Maxwell equations, material derivatives
DOI
Abstract
We study a mathematical model for the inductive heating of steel.It consists of a vector potential formulation of Maxwells equations coupled with a heat equation and an evolution equation for the volume fraction of high temperature phase in steel. An important task for practical applications of induction heating it to find the optimal coupling distance between inductor and workpiece. To this end, we employ the speed method to investigate the sensitivity of solutions to the state equations with respect to perturbations of the inductor coil. We show the existence of strong material derivatives for the state variables and apply the structure theorem to characterize the Eulerian derivative of the cost functional.
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WIAS Preprint No. 2003, (2002)Describing a class of global attractors via symbol sequences
Authors
- Härterich, Jörg
- Wolfrum, Matthias
ORCID: 0000-0002-4278-2675
2010 Mathematics Subject Classification
- 35B40 34E15 37C29
Keywords
- singular perturbation, global attractor, transition layer, heteroclinic orbit
DOI
Abstract
We study a singularly perturbed scalar reaction-diffusion equation on a bounded interval with a spatially inhomogeneous bistable nonlinearity. For certain nonlinearities, which are piecewise constant in space on 𝑘 subintervals, it is possible to characterize all stationary solutions for small ε by means of sequences of 𝑘 symbols, indicating the behavior of the solution in each subinterval. Determining also Morse-indices and zero numbers of the equilibria in terms of the symbol sequences, we are able to give a criterion for heteroclinic connections and a description of the associated global attractor for all 𝑘.
Appeared in
- Discrete Contin. Dyn. Syst., 12, (2005) pp. 531-554
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WIAS Preprint No. 2003, (2002)On dynamical properties of diffeomorphisms with homoclinic tangencies
Authors
- Gonchenko, Sergey V.
- Shilnikov, Leonid P.
- Turaev, Dmitry
2010 Mathematics Subject Classification
- 37G25 37D45 37C15 37G30 34C20 34C27
Keywords
- Newhouse regions, Henon map, renormalization, strange attractor, chaos, moduli, stable periodic orbits
DOI
Abstract
We study bifurcations of a homoclinic tangency to a saddle fixed point without non-leading multipliers. We give criteria for the birth of an infinite set of stable periodic orbits, an infinite set of coexisting saddle periodic orbits with different instability indices, non-hyperbolic periodic orbits with more than one multiplier on the unit circle, and an infinite set of stable closed invariant curves (invariant tori). The results are based on the rescaling of the first-return map near the orbit of homoclinic tangency, which is shown to bring the map close to one of four standard quadratic maps, and on the analysis of the bifurcations in these maps.
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WIAS Preprint No. 2003, (2002)Extinction versus explosion in a supercritical super-Wright-Fisher diffusion
Authors
- Fleischmann, Klaus
- Swart, Jan
2010 Mathematics Subject Classification
- 60J80 60G57 35K55 35K15 60J57 60J60
Keywords
- Binary splitting, binary branching, weighted superprocess, semilinear Cauchy problem, semilinear parabolic PDE, finite ancestry property, trimmed tree, h-transform, extinction, explosion
DOI
Abstract
We study mild solutions to a semilinear Cauchy problem related to a supercritical superprocess taking values in the finite measures on the unit interval, whose underlying motion is the Wright-Fisher diffusion. We establish a dichotomy in the long-time behavior of this superprocess. When a parameter related to the growth of the process is smaller than or equal to a certain critical value (in our case one), the mass in the interior of the unit interval dies out after a finite random time, while for larger values of the growth parameter, the mass in the interior explodes with positive probability as time tends to infinity. In the case of explosion, the mass in the interior grows exponentially and is approximately uniformly distributed over the unit interval. We apply these results to show that the semilinear Cauchy problem has precisely four fixed points when the growth parameter is small and five fixed points when the growth parameter is large, and determine their domains of attraction.
Appeared in
- Stoc. Proc. Appl., vol.106, number 1, pp. 141-165, under new title: Extinction versus Exponential Growth in a Supercritical Super-Wright- Fischer Diffusion
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WIAS Preprint No. 2003, (2002)Competing species superprocesses with infinite variance
Authors
- Fleischmann, Klaus
- Mytnik, Leonid
2010 Mathematics Subject Classification
- 60K35 60G57 60J80
Keywords
- Superprocess with killing, competing superprocesses, interactive superprocesses, superprocess with immigration, measure-valued branching, interactive branching, state-dependent branching, collision measure, collision local time, martingale problem
DOI
Abstract
We study pairs of interacting measure-valued branching processes (superprocesses) with (1 + β)-branching mechanism. The interaction is realized via some killing procedure. The collision local time for such processes is constructed as a limit of approximating collision local times. For certain dimensions this convergence holds uniformly over all pairs of such interacting superprocesses. We use this uniformity to prove existence of a solution to a competing species martingale problem under a natural dimension restriction. The fact that the branching mechanism does not have finite variance requires the development of new methods for handling the collision local time which we believe are of some independent interest.
Appeared in
- Electr. J. Probab. , vol. 8, no. 8, 59 pp
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WIAS Preprint No. 2003, (2002)Hölder and Lipschitz stability of solution sets in programs with probabilistic constraints
Authors
- Henrion, René
ORCID: 0000-0001-5572-7213 - Römisch, Werner
2010 Mathematics Subject Classification
- 90C15 90C31
Keywords
- probabilistic constraints, chance constraints, Lipschitz stability, stochastic optimization
DOI
Abstract
We study perturbations of a stochastic program with a probabilistic constraint and 𝑟-concave original probability distribution. First we improve our earlier results substantially and provide conditions implying Hölder continuity properties of the solution sets w.r.t. the Kolmogorov distance of probability distributions. Secondly, we derive an upper Lipschitz continuity property for solution sets under more restrictive conditions on the original program and on the perturbed probability measures. The latter analysis applies to linear-quadratic models and is based on work by Bonnans and Shapiro. The stability results are illustrated by numerical tests showing the different asymptotic behaviour of parametric and nonparametric estimates in a program with a normal probabilistic constraint.
Appeared in
- Mathematical Programming 100 (2004), 589-611
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WIAS Preprint No. 2003, (2002)On a nonlocal model of image segmentation
Authors
- Gajewski, Herbert
- Gärtner, Klaus
2010 Mathematics Subject Classification
- 35K45 35K65 35B40 80A22 74N25
Keywords
- Cahn-Hilliard equation, initial boundary value problem, Perrona-Malik model, a priori estimates, Lyapunov function, equilibria, asymptotic behaviour, classical thermodynamics, nonlocal phase separation model, image reconstruction and separation
DOI
Abstract
We understand an image as binary grey 'alloy' of a black and a white component and use a nonlocal phase separation model to describe image segmentation. The model consists in a degenerate nonlinear parabolic equation with a nonlocal drift term additionally to the familiar Perona-Malik model. We formulate conditions for the model parameters to guarantee global existence of a unique solution that tends exponentially in time to a unique steady state. This steady state is solution of a nonlocal nonlinear elliptic boundary value problem and allows a variational characterization. Numerical examples demonstrate the properties of the model.
Appeared in
- ZAMP Z. Angew. Math. Phys., 56 (2005) pp. 572--591.
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