182 Works

Estimation of the infinitesimal generator by square-root approximation

Luca Donati, Martin Heida, Marcus Weber & Bettina Keller
For the analysis of molecular processes, the estimation of time-scales, i.e., transition rates, is very important. Estimating the transition rates between molecular conformations is -- from a mathematical point of view -- an invariant subspace projection problem. A certain infinitesimal generator acting on function space is projected to a low-dimensional rate matrix. This projection can be performed in two steps. First, the infinitesimal generator is discretized, then the invariant subspace is approximated and used for...

Hybrid finite-volume/finite-element schemes for $p(x)$-Laplace thermistor models

Jürgen Fuhrmann, Annegret Glitzky & Matthias Liero
We introduce an empirical PDE model for the electrothermal description of organic semiconductor devices by means of current and heat flow. The current flow equation is of p(x)-Laplace type, where the piecewise constant exponent p(x) takes the non-Ohmic behavior of the organic layers into account. Moreover, the electrical conductivity contains an Arrhenius-type temperature law. We present a hybrid finite-volume/finite-element discretization scheme for the coupled system, discuss a favorite discretization of the p(x)-Laplacian at hetero interfaces,...

The weighted energy-dissipation principle and evolutionary Γ-convergence for doubly nonlinear problems

Matthias Liero & Stefano Melchionna
We consider a family of doubly nonlinear evolution equations that is given by families of convex dissipation potentials, nonconvex energy functionals, and external forces parametrized by a small parameter ε. For each of these problems, we introduce the so-called weighted energy-dissipation (WED) functional, whose minimizer correspond to solutions of an elliptic-in-time regularization of the target problems with regularization parameter δ. We investigate the relation between the Γ-convergence of the WED functionals and evolutionary Γ-convergence of...

The Hausdorff dimension of certain attractors.

Hans Guenther Bothe
For the solid torus V = S1 x D2 and a C1 embedding ƒ : V → V given by ƒ(t,x1,x2) = (φ(t),λ1(t)·x1+z1(t), λ2(t)·x2+z2(t)) with dφ⁄dt > 1, 0 < λi(t) < 1 the attractor Λ= ∩∞i=0 ƒi (V) is a solenoid, and for each disk D(t) = {t} x D2 (t∈S1) the intersection Λ(t) = Λ ∩ D(t) is a Cantor set. It is the aim of the paper to find conditions under which...

Global bifurcation analysis of a class of planar systems

Alexander Grin & Klaus R. Schneider
We consider planar autonomous systems $dx/dt =P(x,y,la), ; dy/dt =Q(x,y,la)$ depending on a scalar parameter $la$. We present conditions on the functions $P$ and $Q$ which imply that there is a parameter value $la_0$ such that for $la > la_0$ this system has a unique limit cycle which is hyperbolic and stable. Dulac-Cherkas functions, rotated vector fields and singularly perturbed systems play an important role in the proof.

3D electrothermal simulations of organic LEDs showing negative differential resistance

Matthias Liero, Jürgen Fuhrmann, Annegret Glitzky & Thomas Koprucki
Organic semiconductor devices show a pronounced interplay between temperature-activated conductivity and self-heating which in particular causes inhomogeneities in the brightness of large-area OLEDs at high power. We consider a 3D thermistor model based on partial differential equations for the electrothermal behavior of organic devices and introduce an extension to multiple layers with nonlinear conductivity laws, which also take the diode-like behavior in recombination zones into account. We present a numerical simulation study for a red...

On really locking-free mixed finite element methods for the transient incompressible Stokes equations

Naveed Ahmed, Alexander Linke & Christian Merdon
Inf-sup stable mixed methods for the steady incompressible Stokes equations that relax the divergence constraint are often claimed to deliver locking-free discretizations. However, this relaxation leads to a pressure-dependent contribution in the velocity error, which is proportional to the inverse of the viscosity, thus giving rise to a (different) locking phenomenon. However, a recently proposed modification of the right hand side alone leads to a discretization that is really locking-free, i.e., its velocity error converges...

On a Cahn-Hilliard system with convection and dynamic boundary conditions

Pierluigi Colli, Gianni Gilardi & Jürgen Sprekels
This paper deals with an initial and boundary value problem for a system coupling equation and boundary condition both of Cahn--Hilliard type; an additional convective term with a forced velocity field, which could act as a control on the system, is also present in the equation. Either regular or singular potentials are admitted in the bulk and on the boundary. Both the viscous and pure Cahn--Hilliard cases are investigated, and a number of results is...

A stochastic algorithm without time discretization error for the Wigner equation

Orazio Muscato & Wolfgang Wagner
Stochastic particle methods for the numerical treatment of the Wigner equation are considered. The approximation properties of these methods depend on several numerical parameters. Such parameters are the number of particles, a time step (if transport and other processes are treated separately) and the grid size (used for the discretization of the position and the wave-vector). A stochastic algorithm without time discretization error is introduced. Its derivation is based on the theory of piecewise deterministic...

A semismooth Newton method with analytical path-following for the $H^1$-projection onto the Gibbs simplex

Lukáš Adam, Michael Hintermüller & Thomas M. Surowiec
An efficient, function-space-based second-order method for the $H^1$-projection onto the Gibbs-simplex is presented. The method makes use of the theory of semismooth Newton methods in function spaces as well as Moreau-Yosida regularization and techniques from parametric optimization. A path-following technique is considered for the regularization parameter updates. A rigorous first and second-order sensitivity analysis of the value function for the regularized problem is provided to justify the update scheme. The viability of the algorithm is...

Analysis of improved Nernst--Planck--Poisson models of compressible isothermal electrolytes. Part II: Approximation and a priori estimates

Wolfgang Dreyer, Pierre-Étienne Druet, Paul Gajewski & Clemens Guhlke
We consider an improved Nernst--Planck--Poisson model first proposed by Dreyer et al. in 2013 for compressible isothermal electrolytes in non equilibrium. The model takes into account the elastic deformation of the medium that induces an inherent coupling of mass and momentum transport. The model consists of convection--diffusion--reaction equations for the constituents of the mixture, of the Navier-Stokes equation for the barycentric velocity, and of the Poisson equation for the electrical potential. Due to the principle...

A degenerating Cahn--Hilliard system coupled with complete damage processes

Christian Heinemann & Christiane Kraus
In this work, we analytically investigate a degenerating PDE system for phase separation and complete damage processes considered on a nonsmooth time-dependent domainwith mixed boundary conditions. The evolution of the system is described by a degenerating Cahn-Hilliard equation for the concentration, a doubly nonlinear differential inclusion for the damage variable and a degenerating quasi-static balance equation for the displacement field. All these equations are highly nonlinearly coupled. Because of the doubly degenerating character of the...

On bond price dynamics.

Eckhard Platen & R. Rebolledo
This article proposes a new approach to bond price dynamics. By means of exponential formulae and a notion of forward derivatives we construct a general theoretical framework, which allows to include most of known bond price properties. In particular, we perform a new analysis of no arbitrage conditions together with their consequences on the corresponding return premium. An expression for the general bond price is obtained which also turns out to be computationally convenient. Finally,...

Geometric properties of cones with applications on the Hellinger--Kantorovich space, and a new distance on the space of probability measures

Vaios Laschos & Alexander Mielke
By studying general geometric properties of cone spaces, we prove the existence of a distance on the space of Probability measures that turns the Hellinger--Kantorovich space into a cone space over the space of probabilities measures. Here we exploit a natural two-parameter scaling property of the Hellinger-Kantorovich distance. For the new space, we obtain a full characterization of the geodesics. We also provide new geometric properties for the original space, including a two-parameter rescaling and...

Anisotropic solid-liquid interface kinetics in silicon: An atomistically informed phase-field model

Sibylle Bergmann, Daniel A. Barragan-Yani, Elke Flegel, Karsten Albe & Barbara Wagner
We present an atomistically informed parametrization of a phase-field model for describing the anisotropic mobility of liquid-solid interfaces in silicon. The model is derived from a consistent set of atomistic data and thus allows to directly link molecular dynamics and phase field simulations. Expressions for the free energy density, the interfacial energy and the temperature and orientation dependent interface mobility are systematically fitted to data from molecular dynamics simulations based on the Stillinger-Weber interatomic potential....

Optimal design of the tweezer control for chimera states

Iryna Omelchenko, Oleh E. Omel'chenko, Anna Zakharova & Eckehard Schöll
Chimera states are complex spatio-temporal patterns, which consist of coexisting domains of spatially coherent and incoherent dynamics in systems of coupled oscillators. In small networks, chimera states usually exhibit short lifetimes and erratic drifting of the spatial position of the incoherent domain. A tweezer feedback control scheme can stabilize and fix the position of chimera states. We analyse the action of the tweezer control in small nonlocally coupled networks of Van der Pol and FitzHugh--Nagumo...

A Hamilton--Jacobi point of view on mean-field Gibbs-non-Gibbs transitions

Richard Kraaij, Frank Redig & Willem B. Van Zuijlen
We study the loss, recovery, and preservation of differentiability of time-dependent large deviation rate functions. This study is motivated by mean-field Gibbs-non-Gibbs transitions. The gradient of the rate-function evolves according to a Hamiltonian flow. This Hamiltonian flow is used to analyze the regularity of the time dependent rate function, both for Glauber dynamics for the Curie-Weiss model and Brownian dynamics in a potential. We hereby create a unifying framework for the treatment of mean-field Gibbs-non-Gibbs...

Large deviations for the capacity in dynamic spatial relay networks

Christian Hirsch & Benedikt Jahnel
We derive a large deviation principle for the space-time evolution of users in a relay network that are unable to connect due to capacity constraints. The users are distributed according to a Poisson point process with increasing intensity in a bounded domain, whereas the relays are positioned deterministically with given limiting density. The preceding work on capacity for relay networks by the authors describes the highly simplified setting where users can only enter but not...

A kinetic equation for the distribution of interaction clusters in rarefied gases

Robert I. A. Patterson, Sergio Simonella & Wolfgang Wagner
We consider a stochastic particle model governed by an arbitrary binary interaction kernel. A kinetic equation for the distribution of interaction clusters is established. Under some additional assumptions a recursive representation of the solution is found. For particular choices of the interaction kernel (including the Boltzmann case) several explicit formulas are obtained. These formulas are confirmed by numerical experiments. The experiments are also used to illustrate various conjectures and open problems.

From nonlinear to linear elasticity in a coupled rate-dependent/independent system for brittle delamination

Riccarda Rossi & Marita Thomas
We revisit the weak, energetic-type existence results obtained in [Rossi/Thomas-ESAIM-COCV-21(1):1-59,2015] for a system for rate-independent, brittle delamination between two visco-elastic, physically nonlinear bulk materials and explain how to rigorously extend such results to the case of visco-elastic, linearly elastic bulk materials. Our approximation result is essentially based on deducing the Mosco-convergence of the functionals involved in the energetic formulation of the system. We apply this approximation result in two different situations: Firstly, to pass from...

Comparison of thermodynamically consistent charge carrier flux discretizations for Fermi--Dirac and Gauss--Fermi statistics

Patricio Farrell, Matteo Patriarca, Jürgen Fuhrmann & Thomas Koprucki
We compare three thermodynamically consistent Scharfetter--Gummel schemes for different distribution functions for the carrier densities, including the Fermi--Dirac integral of order 1/2 and the Gauss--Fermi integral. The most accurate (but unfortunately also most costly) generalized Scharfetter--Gummel scheme requires the solution of an integral equation. We propose a new method to solve this integral equation numerically based on Gauss quadrature and Newton's method. We discuss the quality of this approximation and plot the resulting currents for...

Gibbs states of the Hopfield model with extensively many patterns

Anton Bovier, Véronique Gayrard & Pierre Picco
We consider the Hopfield model with M(N) = αN patterns, where N is the number of neurons. We show that if α is sufficiently small and the temperature sufficiently low, then there exist disjoint Gibbs states for each of the stored patterns, almost surely with respect to the distribution of the random patterns. This solves a problem left open in previous work [BGPl]. The key new ingredient is a self averaging result on the free...

Non-equilibrium thermodynamical principles for chemical reactions with mass-action kinetics

Alexander Mielke, Robert I. A. Patterson, Mark A. Peletier & D. R. Michiel Renger
We study stochastic interacting particle systems that model chemical reaction networks on the micro scale, converging to the macroscopic Reaction Rate Equation. One abstraction level higher, we study the ensemble of such particle systems, converging to the corresponding Liouville transport equation. For both systems, we calculate the corresponding large deviations and show that under the condition of detailed balance, the large deviations induce a non-linear relation between thermodynamic fluxes and free energy driving force.

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