1,258 Works

Optimal choice of observation window for Poisson observations

Yury Kutoyants & Vladimir Spokoiny
We consider the possibility of optimal choice of observation window in the problem of parameter estimation by the observations of an inhomogeneous Poisson process. A minimax lower bound is proposed for the risk of estimation under an arbitrary choice of observation window. Then the adaptive procedure is proposed which is asymptotically efficient in the sense of this bound.

A stochastic log-Laplace equation

Jie Xiong
We study a nonlinear stochastic partial differential equation whose solution is the conditional log-Laplace functional of a superprocess in a random environment. We establish its existence and uniqueness by smoothing out the nonlinear term and making use of the particle system representation developed by Kurtz and Xiong (1999). We also derive the Wong-Zakai type approximation for this equation. As an application, we give a direct proof of the moment formulas of Skoulakis and Adler (2001).

Hysteresis and phase transition in many-particle storage systems

Wolfgang Dreyer, Clemens Guhlke & Michael Herrmann
We study the behavior of systems consisting of ensembles of interconnected storage particles. Our examples concern the storage of lithium in many-particle electrodes of rechargeable lithium-ion batteries and the storage of air in a system of interconnected rubber balloons. We are particularly interested in those storage systems whose constituents exhibit non-monotone material behavior leading to transitions between two coexisting phases and to hysteresis. In the current study we consider the case that the time to...

Efficient coupling of inhomogeneous current spreading and dynamic electro-optical models for broad-area edge-emitting semiconductor devices

Mindaugas Radziunas, Anissa Zeghuzi, Jürgen Fuhrmann, Thomas Koprucki, Hans-Jürgen Wünsche, Hans Wenzel & Uwe Bandelow
We extend a 2 (space) + 1 (time)-dimensional traveling wave model for broad-area edge-emitting semiconductor lasers by a model for inhomogeneous current spreading from the contact to the active zone of the laser. To speedup the performance of the device simulations, we suggest and discuss several approximations of the inhomogeneous current density in the active zone.

Heterogeneous dynamic process flowsheet simulation of chemical plants

Friedrich Grund, Klaus Ehrhardt, Jürgen Borchardt & Dietmar Horn
For large-scale dynamic simulation problems in chemical process engineering, a heterogeneous simulation concept is described which allows to distribute the solution of the models of coupled dynamic subprocesses to a computer network. The main principle of such a technique is to solve the submodels of an overall model independently of each other on subsequent time intervals. This is done by estimating the vector of input variables of the submodels, calculating the corresponding time behaviour of...

Rate-independent evolution of sets

Riccarda Rossi, Ulisse Stefanelli & Marita Thomas
The goal of this work is to analyze a model for the rate-independent evolution of sets with finite perimeter. The evolution of the admissible sets is driven by that of a given time-dependent set, which has to include the admissible sets and hence is to be understood as an external loading. The process is driven by the competition between perimeter minimization and minimization of volume changes. In the mathematical modeling of this process, we distinguish...

Linearized plasticity is the evolutionary Γ-limit of finite plasticity

Alexander Mielke & Ulisse Stefanelli
We provide a rigorous justification of the classical linearization approach in plasticity. By taking the small-deformations limit, we prove via Γ-convergence for rate-independent processes that energetic solutions of the quasi-static finite-strain elastoplasticity system converge to the unique strong solution of linearized elastoplasticity

On convergence rates of suprema in the presence of non-negligible trends

Valentin Konakov
We investigate the convergence rates for the maximal deviation distribution of kernel estimates from the true density. The convergence rates for related Gaussian fields are also investigated. We consider the optimal choice of the smoothing parameter in the sense of Konakov and Piterbarg (1994) and in doing so we take into account a non-negligible trend. It is shown that the convergence rates depend on the asymptotic behaviour of the Laplace type integrals over a small...

Eigenvalue fluctuations for lattice Anderson Hamiltonians: Unbounded potentials

Marek Biskup, Ryoki Fukushima & Wolfgang König
We consider random Schrödinger operators with Dirichlet boundary conditions outside lattice approximations of a smooth Euclidean domain and study the behavior of its lowest-lying eigenvalues in the limit when the lattice spacing tends to zero. Under a suitable moment assumption on the random potential and regularity of the spatial dependence of its mean, we prove that the eigenvalues of the random operator converge to those of a deterministic Schrödinger operator. Assuming also regularity of the...

Optimal stopping via deeply boosted backward regression

Denis Belomestny, John G.M. Schoenmakers, Vladimir Spokoiny & Yuri Tavyrikov
In this note we propose a new approach towards solving numerically optimal stopping problems via boosted regression based Monte Carlo algorithms. The main idea of the method is to boost standard linear regression algorithms in each backward induction step by adding new basis functions based on previously estimated continuation values. The proposed methodology is illustrated by several numerical examples from finance.

The asymptotic behavior of semi-invariants for linear stochastic systems

Grigori N. Milstein
The asymptotic behavior of semi-invariants of the random variable ln |X(t, 푥)|, where X(t,푥) is a solution of a linear system of stochastic differential equations, is connected with the moment Lyapunov exponent g(푝). Namely, it is obtained that the 푛-th semi-invariant is asymptotically proportional to the time 헍 with the coefficient of proportionallity g(n)(0). The proof is based on the concept of analytic characteristic functions. It is also shown that the asymptotic behavior of the...

Orthogonality of fluxes in general nonlinear reaction networks

D. R. Michiel Renger & Johannes Zimmer
We consider the chemical reaction networks and study currents in these systems. Reviewing recent decomposition of rate functionals from large deviation theory for Markov processes, we adapt these results for reaction networks. In particular, we state a suitable generalisation of orthogonality of forces in these systems, and derive an inequality that bounds the free energy loss and Fisher information by the rate functional.

Analytic-numerical investigation of delayed exchange of stabilities in singularly perturbed parabolic problems

Nikolai N. Nefedov, Mindaugas Radziunas & Klaus R. Schneider
We consider a class of singularly perturbed parabolic problem in case of exchange of stabilities, that is, the corresponding degenerate equation has two intersecting roots. We present an analytic result about the phenomenon of delayed exchange of stabilities and compare it with numerical investigations of some examples.

Monte Carlo difference schemes for the wave equation

Sergej M. Ermakov & Wolfgang Wagner
The paper is concerned with Monte Carlo algorithms for iteration processes. A recurrent procedure is introduced, where information on various iteration levels is stored. Stability in the sense of boundedness of the correlation matrix of the component estimators is studied. The theory is applied to difference schemes for the wave equation. The results are illustrated by numerical examples.

A proof of a Shilnikov theorem for C^1-smooth dynamical systems

Mikhail Shashkov & Dmitry Turaev
Dynamical systems with a homoclinic loop to a saddle equilibrium state are considered. Andronov and Leontovich have shown (see [1939], [1959]) that a generic bifurcation of a two-dimensional C1-smooth dynamical system with a homoclinic loop leads to appearance of a unique periodic orbit. This result holds true in the multi-dimensional setting if some additional conditions are satisfied, which was proved by Shilnikov [1962, 1963, 1968] for the case of dynamical systems of sufficiently high smoothness....

Weak solutions and weak-strong uniqueness for a thermodynamically consistent phase-field model

Robert Lasarzik, Elisabetta Rocca & Giulio Schimperna
In this paper we prove the existence of weak solutions for a thermodynamically consistent phase-field model introduced in [26] in two and three dimensions of space. We use a notion of solution inspired by [18], where the pointwise internal energy balance is replaced by the total energy inequality complemented with a weak form of the entropy inequality. Moreover, we prove existence of local-in-time strong solutions and, finally, we show weak-strong uniqueness of solutions, meaning that...

On decomposition of embedded prismatoids in ℝ³ without additional points

Hang Si
This paper considers three-dimensional <i>prismatoids</i> which can be embedded in ℝ³ A subclass of this family are <i>twisted prisms</i>, which includes the family of non-triangulable Scönhardt polyhedra [12, 10]. We call a prismatoid <i>decomposable</i> if it can be cut into two smaller prismatoids (which have smaller volumes) without using additional points. Otherwise it is <i>indecomposable</i>. The indecomposable property implies the non-triangulable property of a prismatoid but not vice versa.
In this paper we prove...

Effective diffusion in thin structures via generalized gradient systems and EDP-convergence

Thomas Frenzel & Matthias Liero
The notion of Energy-Dissipation-Principle convergence (EDP-convergence) is used to derive effective evolution equations for gradient systems describing diffusion in a structure consisting of several thin layers in the limit of vanishing layer thickness. The thicknesses of the sublayers tend to zero with different rates and the diffusion coefficients scale suitably. The Fokker--Planck equation can be formulated as gradient-flow equation with respect to the logarithmic relative entropy of the system and a quadratic Wasserstein-type gradient structure....

Computational modelling and simulation of cancer growth and migration within a 3D heterogeneous tissue: The effects of fibre and vascular structure

Cicely K. Macnamara1, Alfonso Caiazzo, I Ramis-Conde & Mark A. J. Chaplain
The term cancer covers a multitude of bodily diseases, broadly categorised by having cells which do not behave normally. Since cancer cells can arise from any type of cell in the body, cancers can grow in or around any tissue or organ making the disease highly complex. Our research is focused on understanding the specific mechanisms that occur in the tumour microenvironment via mathematical and computational modeling. We present a 3D individual-based model which allows...

Towards thermodynamic modeling of nucleation and growth of droplets in crystals

Wolfgang Dreyer & Frank Duderstadt
Stress assisted diffusion in single crystal Gallium Arsenide (GaAs) leads to the formation and growth of unwanted liquid arsenic droplets in a solid matrix. This process happens during the heat treatment of single crystal GaAs, which is needed for its application in opto-electronic devices, and it is of crucial importance to pose and answer the question if the appearance of droplets can be avoided. To this end we start a thermodynamic simulation of this process....

Reconstruction of quasi-local numerical effective models from low-resolution measurements

Alfonso Caiazzo, Roland Maier & Daniel Peterseim
We consider the inverse problem of reconstructing an effective model for a prototypical diffusion process in strongly heterogeneous media based on low-resolution measurements. We rely on recent quasi-local numerical effective models that, in contrast to conventional homogenized models, are provably reliable beyond periodicity assumptions and scale separation. The goal of this work is to show that the identification of the matrix representation of these effective models is possible. Algorithmic aspects of the inversion procedure and...

Component identification and estimation in nonlinear high-dimensional regression models by structural adaption

Alexander Samarov, Vladimir Spokoiny & Celine Vial
This article proposes a new method of analysis of a partially linear model whose nonlinear component is completely unknown. The target of analysis is identification of the set of regressors which enter in a nonlinear way in the model function, and the complete estimation of the model including slope coefficients of the linear component and the link function of the nonlinear component. The procedure also allows for selecting the significant regression variables. As a by-product,...

Longtime behavior for a generalized Cahn--Hilliard system with fractional operators

Pierluigi Colli, Gianni Gilardi & Sprekels, ##!Error: Attribute Unknown!##
In this contribution, we deal with the longtime behavior of the solutions to the fractional variant of the Cahn--Hilliard system, with possibly singular potentials, which we recently investigated in the paper "Well-posedness and regularity for a generalized fractional CahnHilliard system". More precisely, we give a complete characterization of the Omega-limit of the phase parameter. The characterization depends on the first eigenvalue of one of the involved operators: if this eigenvalue is positive, then the chemical...

Dissipative Schrödinger-Poisson systems

Michael Baro, Hans-Christoph Kaiser, Hagen Neidhardt & Joachim Rehberg
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.

An inverse problem in periodic diffractive optics: Reconstruction of Lipschitz grating profiles

Johannes Elschner & Masahiro Yamamoto
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.

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