98 Works

The thermodynamics of the Curie-Weiss model with random couplings.

Anton Bovier & Veronique Gayrard
We study the Curie-Weiss version of an Ising spin system with random, positively biased, couplings. In particular the case where the couplings ∈ij take the values one with probability p and zero with probability 1 - p which describes the Ising model on a random graph is considered. We prove that if p is allowed to decrease with the system size N in such a way that Np(N) ↑ ∞ as N ↑ ∞, then...

A nonlocal concave-convex problem with nonlocal mixed boundary data

Boumediene Abdellaoui, Abdelrazek Dieb & Enrico Valdinoci
The aim of this paper is to study a nonlocal equation with mixed Neumann and Dirichlet external conditions. We prove existence, nonexistence and multiplicity of positive energy solutions and analyze the interaction between the concave-convex nonlinearity and the Dirichlet-Neumann data.

Ocean rogue waves and their phase space dynamics in the limit of a linear interference model

Simon Birkholz, Carsten Brée, Ivan Veselić, Ayhan Demircan & Günter Steinmeyer
We reanalyse the probability for formation of extreme waves using the simple model of linear interference of a finite number of elementary waves with fixed amplitude and random phase fluctuations. Under these model assumptions no rogue waves appear when less than 10 elementary waves interfere with each other. Above this threshold rogue wave formation becomes increasingly likely, with appearance frequencies that may even exceed long-term observations by an order of magnitude. For estimation of the...

On maximal parabolic regularity for non-autonomous parabolic operators

Karoline Disser, A. F. M. Ter Elst & Joachim Rehberg
We consider linear inhomogeneous non-autonomous parabolic problems associated to sesquilinear forms, with discontinuous dependence of time. We show that for these problems, the property of maximal parabolic regularity can be extrapolated to time integrability exponents r ≠ 2. This allows us to prove maximal parabolic Lr-regularity for discontinuous non-autonomous second-order divergence form operators in very general geometric settings and to prove existence results for related quasilinear equations.

A note on the Green's function for the transient random walk without killing on the half lattice, orthant and strip

Alberto Chiarini & Alessandra Cipriani
In this note we derive an exact formula for the Green's function of the random walk on different subspaces of the discrete lattice (orthants, including the half space, and the strip) without killing on the boundary in terms of the Green's function of the simple random walk on $\Z^d$, $d\ge 3$.

Calibration methods for gas turbine performance models

Jürgen Borchardt, Peter Mathé & Galina Printsypar
The WIAS software package BOP is used to simulate gas turbine models. In order to make accurate predictions the underlying models need to be calibrated. This study compares different strategies of model calibration. These are the deterministic optimization tools as non-linear least squares (MSO) and the sparsity promoting variant LASSO, but also the probabilistic (Bayesian) calibration. The latter allows for the quantification of the inherent uncertainty, and it gives rise to a surrogate uncertainty measure...

The modeling of reactive solute transport with sorption to mobile and immobile sorbents.

Peter Knabner, I. Kögel-Knabner & K.U. Totsche
This paper presents a mathematical model to describe the transport of reactive solutes with sorption to mobile and immobile sorbents. The mobile sorbent is considered to be reactive, too. The sorption processes mentioned are equilibrium and nonequilibrium processes. A transformation of the model in terms of total concentrations of solute and mobile sorbents is presented which simplifies the mathematical formulation. Effective isotherms, which describe the sorption to the immobile sorbent in the presence of a...

Discretisation and error analysis for a mathematical model of milling processes

Dietmar Hömberg, Oliver Rott & Kevin Sturm
We investigate a mathematical model for milling where the cutting tool dynamics is considered together with an elastic workpiece model. Both are coupled by the cutting forces consisting of two dynamic components representing vibrations of the tool and of the workpiece, respectively, at the present and previous tooth periods. We develop a numerical solution algorithm and derive error estimates both for the semi-discrete and the fully discrete numerical scheme. Numerical computations in the last section...

Efficient all-optical control of solitons

Sabrina Pickartz, Uwe Bandelow & Shalva Amiranashvili
We consider the phenomenon of an optical soliton controlled (eg. amplified) by a much weaker second pulse which is efficiently scattered at the soliton. An important problem in this context is to quantify the small range of parameters at which the interaction takes place. This has been achieved by using adiabatic ODEs for the soliton characteristics, which is much faster than an empirical scan of the full propagation equations for all parameters in question.


Doan Duy Hai, Patricio Farrell, Jürgen Fuhrmann, Thomas Koprucki, Markus Kantner & Nella Rotundo
ddfermi is an open-source software prototype which simulates drift diffusion processes in semiconductor materials. Its strength lies in the ability to simulate multidimensional devices under general statistics (Fermi-Dirac, Gauss-Fermi, Blakemore and Boltzmann) using state-of-the-art flux approximations. It implements a finite volume discretization of the basic semiconductor equations (the van Roosbroeck system), employing thermodynamically consistent extensions of the well-known Scharfetter-Gummel scheme. The code is based on pdelib and interfaced via Python or Lua.

Spatially adaptive regression estimation: Propagation-separation approach

Jörg Polzehl & Vladimir Spokoiny
Polzehl and Spokoiny (2000) introduced the adaptive weights smoothing (AWS) procedure in the context of image denoising. The procedure has some remarkable properties like preservation of edges and contrast, and (in some sense) optimal reduction of noise. The procedure is fully adaptive and dimension free. Simulations with artificial images show that AWS is superior to classical smoothing techniques especially when the underlying image function is discontinuous and can be well approximated by a piecewise constant...

Uniform second order convergence of a complete flux scheme on unstructured 1D grids for a singularly perturbed advection-diffusion equation and some multidimensional extensions

Patricio Farrell & Alexander Linke
The accurate and efficient discretization of singularly perturbed advection-diffusion equations on arbitrary 2D and 3D domains remains an open problem. An interesting approach to tackle this problem is the complete flux scheme (CFS) proposed by G. D. Thiart and further investigated by J. ten Thije Boonkkamp. For the CFS, uniform second order convergence has been proven on structured grids. We extend a version of the CFS to unstructured grids for a steady singularly perturbed advection-diffusion...

The space of bounded variation with infinite-dimensional codomain

Martin Heida, Robert I.A.Robert Patterson & D. R. Michiel Renger
We study functions of bounded variation with values in a Banach or in a metric space. We provide several equivalent notions of variations and provide the notion of a time derivative in this abstract setting. We study four distinct topologies on the space of bounded variations and provide some insight into the structure of these topologies. In particular, we study the meaning of convergence, duality and regularity for these topologies and provide some useful compactness...

Higher-order discontinuous Galerkin time stepping and local projection stabilization techniques for the transient Stokes problem

Naveed Ahmed, Simon Becher & Gunar Matthies
We introduce and analyze discontinuous Galerkin time discretizations coupled with continuous finite element methods based on equal-order interpolation in space for velocity and pressure in transient Stokes problems. Spatial stability of the pressure is ensured by adding a stabilization term based on local projection. We present error estimates for the semi-discrete problem after discretization in space only and for the fully discrete problem. The fully discrete pressure shows an instability in the limit of small...

Localization of the principal Dirichlet eigenvector in the heavy-tailed random conductance model

Franziska Flegel
We study the asymptotic behavior of the principal eigenvector and eigenvalue of the random conductance Laplacian in a large domain of Zd (d ≥ 2) with zero Dirichlet condition. We assume that the conductances w are positive i.i.d. random variables, which fulfill certain regularity assumptions near zero. If γ = sup q ≥ 0; E [w^-q]¼ the top of the Dirichlet spectrum homogenizes. Our proofs are based on a spatial extreme value analysis of the...

Spectral properties of one-dimensional Schrödinger operators with potentials generated by substitutions.

Anton Bovier & Jean-Michel Ghez
We investigate one-dimensional discrete Schrödinger operators whose potentials are invariant under a substitution rule. The spectral properties of these operators can be obtained from the analysis of a dynamical system, called the trace map. We give a careful derivation of these maps in the general case and exhibit some specific properties. Under an additional, easily verifiable hypothesis concerning the structure of the trace map we present an analysis of their dynamical properties that allows us...

Existence and uniqueness results for equations modelling transport of dopants in semiconductors.

Annegret Glitzky, Konrad Gröger & Rolf Hünlich
This paper is devoted to the analytical investigation of some non-linear reaction-diffusion system modelling the transport of dopants in semiconductors. Estimates by the energy functional and L∞-estimates obtained by a modified De Giorgi method imply global existence and uniqueness as well as results concerning the asymptotic behaviour.

Piecewise polynomial collocation for the double layer potential equation over polyhedral boundaries. Part I: The wedge, Part II: The cube.

Andreas Rathsfeld
In this paper we consider a piecewise polynomial method for the solution of the double layer potential equation corresponding to Lapalce's equation in a three-dimensional wedge. We prove the stability for our method in case of special triangulations over the boundaty.

Chance constraints in PDE constrained optimization

Mohammad Hassan Farshbaf Shaker, René Henrion & Dietmar Hömberg
Chance constraints represent a popular tool for finding decisions that enforce a robust satisfaction of random inequality systems in terms of probability. They are widely used in optimization problems subject to uncertain parameters as they arise in many engineering applications. Most structural results of chance constraints (e.g., closedness, convexity, Lipschitz continuity, differentiability etc.) have been formulated in a finite-dimensional setting. The aim of this paper is to generalize some of these well-known semi-continuity and convexity...

Zur direkten Lösung linearer Gleichungssysteme auf Shared und Distributed Memory Systemen.

Georg Hebermehl
Der Gaußsche Algorithtmus zur Lösung linearer Gleichungssysteme Ax = b wird für Shared Memory Systeme als Rank-r LU Update Verfahren und bei blockzyklischer Aufteilung von A für Distributed Memory Systeme als Message Passing Implementation vorgestellt.

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