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2,799 Works

Modeling of chemical reaction systems with detailed balance using gradient structures

Jan Maas & Alexander Mielke
We consider various modeling levels for spatially homogeneous chemical reaction systems, namely the chemical master equation, the chemical Langevin dynamics, and the reaction-rate equation. Throughout we restrict our study to the case where the microscopic system satisfies the detailed-balance condition. The latter allows us to enrich the systems with a gradient structure, i.e. the evolution is given by a gradient-flow equation. We present the arising links between the associated gradient structures that are driven by...

Extrapolated elliptic regularity and application to the van Roosbroeck system of semiconductor equations

Hannes Meinlschmidt & Joachim Rehberg
In this paper we present a general <i>extrapolated elliptic regularity</i> result for second order differential operators in divergence form on fractional Sobolev-type spaces of negative order X<sup>s-1,q</sup><sub>D</sub>(Ω) for s > 0 small, including mixed boundary conditions and with a fully nonsmooth geometry of Ω and the Dirichlet boundary part D. We expect the result to find applications in the analysis of nonlinear parabolic equations, in particular for quasilinear problems or when treating coupled systems of...

Near-optimal tensor methods for minimizing gradient norm

Pavel Dvurechensky, Alexander Gasnikov, Petr Ostroukhov, A. Cesar Uribe &
Motivated by convex problems with linear constraints and, in particular, by entropy-regularized optimal transport, we consider the problem of finding approximate stationary points, i.e. points with the norm of the objective gradient less than small error, of convex functions with Lipschitz p-th order derivatives. Lower complexity bounds for this problem were recently proposed in [Grapiglia and Nesterov, arXiv:1907.07053]. However, the methods presented in the same paper do not have optimal complexity bounds. We propose two...

Turbulent coherent structures in a long cavity semiconductor laser near the lasing threshold

Uday Gowda, Amy Roche, Alexander Pimenov, Andrei G. Vladimirov, Svetlana Slepneva, Evgeny A. Viktorov &
We report on the formation of novel turbulent coherent structures in a long cavity semiconductor laser near the lasing threshold. Experimentally, the laser emits a series of power dropouts within a roundtrip and the number of dropouts per series depends on a set of parameters including the bias current. At fixed parameters, the drops remain dynamically stable, repeating over many roundtrips. By reconstructing the laser electric field in the case where the laser emits one...

A numerical simulation of the Jominy end-quench test

Dietmar Hömberg
We present a numerical algorithm for simulating the Jominy end-quench test and deriving continuous cooling diagrams. The underlying mathematical model for the austenite-pearlite phase transition is based on Scheil's Additivity Rule and the Johnson-Mehl equation. For the formation of martensite we compare the Koistinen-Marburger formula with a rate law, which takes into account the irreversibility of this process. We carry out numerical simulations for the plain carbon steels C 1080 and C 100 W 1....

Stefan problems and the Penrose-Fife phase field model

Pierluigi Colli & Jürgen Sprekels
This paper is concerned with singular Stefan problems in which the heat flux is proportional to the gradient of the inverse absolute temperature. Both the standard interphase equilibrium conditions and phase relaxations are considered. These problems turn out to be the natural limiting cases of a thermodynamically consistent model for diffusive phase transitions proposed by Penrose and Fife. By supplying the systems of equations with suitable initial and boundary conditions, a rigorous asymptotic analysis is...

Stochastic systems of particles with weights and approximation of the Boltzmann equation. The Markov process in the spatially homogeneous case.

Wolfgang Wagner
A class of stochastic systems of particles with variable weights is studied. The corresponding empirical measures are shown to converge to the solution of the spatially homogeneous Boltzmann equation. In a certain sense, this class of stochastic processes generalizes the "Kac master process" ([4]).

Spectral density estimation via nonlinear wavelet methods for stationary non-Gaussian time series

Michael H. Neumann
In the present paper we consider nonlinear wavelet estimators of the spectral density ƒ of a zero mean stochastic process, which is stationary in the wide sense. It is known in the case of Gaussian regression that these estimators outperform traditional linear methods if the degree of smoothness of the regression function varies considerably over the interval of interest. Such methods are based on a nonlinear treatment of estimators of coefficients that arise from a...

Pendulum with positive and negative dry friction. Continuum of homoclinic orbits

Gennadii A. Leonov
A two-order differential equation of pendulum with dry friction is considered. The existence of a continuum of homoclinic orbits with various homotopic properties on the cylinder is proven.

Immediate exchange of stabilities in singularly perturbed systems

Nikolai N. Nefedow & Klaus R. Schneider
We study the initial value problem for singularly perturbed systems of ordinary differential equations whose associated systems have two transversally intersecting families of equilibria (transcritical bifurcation) which exchange their stabilities. By means of the method of upper and lower solutions we derive a sufficient condition for the solution of the initial value problem to exhibit an immediate exchange of stabilities. Concerning its asymptotic behavior with respect to ε we prove that an immediate exchange of...

Finite time extinction of super-Brownian motions with catalysts

Donald A. Dawson, Klaus Fleischmann & Carl Mueller
Consider a catalytic super-Brownian motion X = X<sup>Γ</sup> with finite variance branching. Here "catalytic" means that branching of the reactant X is only possible in the presence of some catalyst. Our intrinsic example of a catalyst is a stable random measure Γ on R of index 0 < γ < 1. Consequently, here the catalyst is located in a countable dense subset of R. Starting with a finite reactant mass X<sub>0</sub> supported by a compact...

On the existence of transition layers of spike type in reaction-diffusion-convection equations

Klaus R. Schneider & Adelaida B. Vasil´Eva
We investigate steady state solutions to a class of systems of reaction-diffusion-convection equations with small diffusion and small convection, and which depend on one space variable. Our main concern is to prove the existence of a solution with an interior layer of spike type for higher order systems without taking into consideration the influence of boundary conditions. To this end we combine two methods of the theory of singular perturbations: the method of integral manifolds...

The retrieval phase of the Hopfield model: A rigorous analysis of the overlap distribution

Anton Bovier & Véronique Gayrard
Standard large deviation estimates or the use of the Hubbard-Stratonovich transformation reduce the analysis of the distribution of the overlap parameters essentially to that of an explicitly known random function ΦN,β on ℝM. In this article we present a rather careful study of the structure of the minima of this random function related to the retrieval of the stored patterns. We denote by m* (β) the modulus of the spontaneous magnetization in the Curie-Weiss model...

Solving joint chance constrained problems using regularization and Benders' decomposition

Lukáš Adam, Martin Branda, Holger Heitsch & René Henrion
In this paper we investigate stochastic programms with joint chance constraints. We consider discrete scenario set and reformulate the problem by adding auxiliary variables. Since the resulting problem has a difficult feasible set, we regularize it. To decrease the dependence on the scenario number, we propose a numerical method by iteratively solving a master problem while adding Benders cuts. We find the solution of the slave problem (generating the Benders cuts) in a closed form...

Numerical methods for accurate description of ultrashort pulses in optical fibers

Shalva Amiranashvili, Mindaugas Radziunas, Uwe Bandelow & Raimondas Čiegis
We consider a one-dimensional first-order nonlinear wave equation (the so-called forward Maxwell equation, FME) that applies to a few-cycle optical pulse propagating along a preferred direction in a nonlinear medium, e.g., ultrashort pulses in nonlinear fibers. The model is a good approximation to the standard second-order wave equation under assumption of weak nonlinearity. We compare FME to the commonly accepted generalized nonlinear Schrödinger equation, which quantifies the envelope of a quickly oscillating wave field based...

How 1-dimensional hyperbolic attractors determine their basins

Hans Günter Bothe
Two attractors Λi (i=1,2) of diffeomorphisms ƒi : Mi → Mi will be called intrinsically equivalent if there is a homeomorphism h: Λ1 → Λ2 satisfying ƒ2h = hƒ1. If we can find a homeomorphism g: WsΛ1 → WsΛ2 of the basins WsΛi of Λi such that ƒ2g = gƒ1, then we say that Λ1, Λ2 are basin equivalent. Let Λ1, Λ2 be transversely tame 1-dimensional hyperbolic attractors which are intrinsically equivalent. Then, if WsΛ1,...

Waveform iteration and one-sided Lipschitz conditions

Ingo Bremer

Discrete qualocation methods for logarithmic-kernel integral equations on a piecewise smooth boundary

Johannes Elschner, Youngmok Jeon, Ian H. Sloan & Ernst P. Stephan
We consider a fully discrete qualocation method for Symm's integral equation. The method is that of Sloan and Burn [14], for which a complete analysis is available in the case of smooth curves. The convergence for smooth curves can be improved by a subtraction of singularity (Jeon and Kimn [10]). In this paper we extend these results for smooth boundaries to polygonal boundaries. The analysis uses a mesh grading transformation method for Symm's integral equation,...

Asymptotic minimaxity of chi-square tests

Michael S. Ermakov
We show that the sequence of chi-square tests is asymptotically minimax if a number of cells increases with increasing sample size. The proof utilizes Theorem about asymptotic normality of chi-square test statistics obtained under new compact assumptions.

Symmetry breaking in dynamical systems

Reiner Lauterbach
Symmetry breaking bifurcations and dynamical systems have obtained a lot of attention over the last years. This has several reasons: real world applications give rise to systems with symmetry, steady state solutions and periodic orbits may have interesting patterns, symmetry changes the notion of structural stability and introduces degeneracies into the systems as well as geometric simplifications. Therefore symmetric systems are attractive to those who study specific applications as well as to those who are...

Stability of pulses on optical fibers with phase-sensitive amplifiers

James C. Alexander, Manoussos G. Grillakis, Christopher K.R.T. Jones & Björn Sandstede
Pulse stability is crucial to the effective propagation of information in a soliton-based optical communication system. It is shown in this paper that pulses in optical fibers, for which attenuation is compensated by phase-sensitive amplifiers, are stable over a large range of parameter values. A fourth-order nonlinear diffusion model due to Kath and co-workers is used. The stability proof invokes a number of mathematical techniques, including the Evans function and Grillakis' functional analytic approach.

Semiparametric single index versus fixed link function modelling

Wolfgang Härdle, Stefan Sperlich & Vladimir Spokoiny
Discrete choice models are frequently used in statistical and econometric practice. Standard models such as logit models are based on exact knowledge of the form of the link and linear index function. Semiparametric models avoid possible misspecification but often introduce a computational burden. It is therefore interesting to decide between approaches. Here we propose a test of semiparametric versus parametric single index modelling. Our procedure allows that the (linear) index of the semiparametric alternative is...

Optimization of a multiphysics problem in semiconductor laser design

Lukáš Adam, Michael Hintermüller, Dirk Peschka & Thomas M. Surowiec
A multimaterial topology optimization framework is suggested for the simultaneous optimization of mechanical and optical properties to be used in the development of optoelectronic devices. Based on the physical aspects of the underlying device, a nonlinear multiphysics model for the elastic and optical properties is proposed. Rigorous proofs are provided for the sensitivity of the fundamental mode of the device with respect to the changes in the underlying topology. After proving existence and optimality results,...

Optimal change-point estimation in inverse problems

Michael H. Neumann
We develop a method of estimating change-points of a function in the case of indirect noisy observations. As two paradigmatic problems we consider deconvolution and errors-in-variables regression. We estimate the scalar products of our indirectly observed function with appropriate test functions, which are shifted over the interval of interest. An estimator of the change point is obtained by the extremal point of this quantity. We derive rates of convergence for this estimator. They depend on...

Highly accurate quadrature-based Scharfetter--Gummel schemes for charge transport in degenerate semiconductors

Matteo Patriarca, Patricio Farrell, Jürgen Fuhrmann & Thomas Koprucki
We introduce a family of two point flux expressions for charge carrier transport described by drift-diffusion problems in degenerate semiconductors with non-Boltzmann statistics which can be used in Voronoï finite volume discretizations. In the case of Boltzmann statistics, Scharfetter and Gummel derived such fluxes by solving a linear two point boundary value problem yielding a closed form expression for the flux. Instead, a generalization of this approach to the nonlinear case yields a flux value...

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  • Weierstrass Institute for Applied Analysis and Stochastics