arXiv subjects

Christian Meyer

Publications and source records attributed to Christian Meyer.

35 records · Page 2Linked to original sources

arXiv2020

Optimal control of an abstract evolution variational inequality with application to homogenized plasticity

The paper is concerned with an optimal control problem governed by a state equation in form of a generalized abstract operator differential equation involving a maximal monotone operator. The state equation is uniquely solvable, but the associated solution operator is in general not Gâteaux-differentiable. In order to derive optimality conditions, we therefore regularize the state equation and its solution operator, respectively, by means of a (smoothed) Yosida approximation. We show convergence of global minimizers for regularization parameter tending to zero and derive necessary and sufficient optimality conditions for the regularized problems. The paper ends with an application of the abstract theory to optimal control of homogenized quasi-static elastoplasticity.

math.OC↗

arXiv2020

Optimal Control of Perfect Plasticity Part I: Stress Tracking

The paper is concerned with an optimal control problem governed by the rate-independent system of quasi-static perfect elasto-plasticity. The objective is to optimize the stress field by controlling the displacement at prescribed parts of the boundary. The control thus enters the system in the Dirichlet boundary conditions. Therefore, the safe load condition is automatically fulfilled so that the system admits a solution, whose stress field is unique. This gives rise to a well defined control-to-state operator, which is continuous but not Gateaux-differentiable. The control-to-state map is therefore regularized, first by means of the Yosida regularization and then by a second smoothing in order to obtain a smooth problem. The approximation of global minimizers of the original non-smooth optimal control problem is shown and optimality conditions for the regularized problem are established. A numerical example illustrates the feasibility of the smoothing approach.

math.OC↗

arXiv2020

Optimal Control of Perfect Plasticity Part II: Displacement Tracking

The paper is concerned with an optimal control problem governed by the rate-independent system of quasi-static perfect elasto-plasticity. The objective is optimize the displacement field in the domain occupied by the body by means of prescribed Dirichlet boundary data, which serve as control variables. The arising optimization problem is nonsmooth for several reasons, in particular, since the control-to-state mapping is not single-valued. We therefore apply a Yosida regularization to obtain a single-valued control-to-state operator. Beside the existence of optimal solutions, their approximation by means of this regularization approach is the main subject of this work. It turns out that a so-called reverse approximation guaranteeing the existence of a suitable recovery sequence can only be shown under an additional smoothness assumption on at least one optimal solution.

math.OC↗

arXiv2019

Quadratically regularized optimal transport

We investigate the problem of optimal transport in the so-called Kantorovich form, i.e. given two Radon measures on two compact sets, we seek an optimal transport plan which is another Radon measure on the product of the sets that has these two measures as marginals and minimizes a certain cost function. We consider quadratic regularization of the problem, which forces the optimal transport plan to be a square integrable function rather than a Radon measure. We derive the dual problem and show strong duality and existence of primal and dual solutions to the regularized problem. Then we derive two algorithms to solve the dual problem of the regularized problem: A Gauss-Seidel method and a semismooth quasi-Newton method and investigate both methods numerically. Our experiments show that the methods perform well even for small regularization parameters. Quadratic regularization is of interest since the resulting optimal transport plans are sparse, i.e. they have a small support (which is not the case for the often used entropic regularization where the optimal transport plan always has full measure).

math.OC↗

arXiv2019

A-priori error analysis of local incremental minimization schemes for rate-independent evolutions

This paper is concerned with a priori error estimates for the local incremental minimization scheme, which is an implicit time discretization method for the approximation of rate-independent systems with non-convex energies. We first show by means of a counterexample that one cannot expect global convergence of the scheme without any further assumptions on the energy. For the class of uniformly convex energies, we derive error estimates of optimal order, provided that the Lipschitz constant of the load is sufficiently small. Afterwards, we extend this result to the case of an energy, which is only locally uniformly convex in a neighborhood of a given solution trajectory. For the latter case, the local incremental minimization scheme turns out to be superior compared to its global counterpart, as a numerical example demonstrates.

math.NA↗

arXiv2018

Long-term monitoring of the internal energy distribution of isolated cluster systems

A method is presented to monitor the internal energy distribution of cluster anions via delayed electron detachment by pulsed photoexcitation and demonstrated on Co$_4{}^-$ in an electrostatic ion beam trap. In cryogenic operation, we calibrate the detachment delay to internal energy. By laser frequency scans, at room temperature, we reconstruct the time-dependent internal energy distribution of the clusters. The mean energies of ensembles from a cold and a hot ion source both approach thermal equilibrium. Our data yield a radiative emission law and the absorptivity of the cluster for thermal radiation.

physics.atm-clus↗

arXiv2018

RUSSE: The First Workshop on Russian Semantic Similarity

The paper gives an overview of the Russian Semantic Similarity Evaluation (RUSSE) shared task held in conjunction with the Dialogue 2015 conference. There exist a lot of comparative studies on semantic similarity, yet no analysis of such measures was ever performed for the Russian language. Exploring this problem for the Russian language is even more interesting, because this language has features, such as rich morphology and free word order, which make it significantly different from English, German, and other well-studied languages. We attempt to bridge this gap by proposing a shared task on the semantic similarity of Russian nouns. Our key contribution is an evaluation methodology based on four novel benchmark datasets for the Russian language. Our analysis of the 105 submissions from 19 teams reveals that successful approaches for English, such as distributional and skip-gram models, are directly applicable to Russian as well. On the one hand, the best results in the contest were obtained by sophisticated supervised models that combine evidence from different sources. On the other hand, completely unsupervised approaches, such as a skip-gram model estimated on a large-scale corpus, were able score among the top 5 systems.

cs.CL↗

arXiv2018

A non-smooth trust-region method for locally Lipschitz functions with application to optimization problems constrained by variational inequalities

We propose a nonsmooth trust-region method for solving optimization problems with locally Lipschitz continuous functions, with application to problems constrained by variational inequalities of the second kind. Under suitable assumptions on the model functions, convergence of the general algorithm to a C-stationary point is verified. For variational inequality constrained problems, we are able to properly characterize the Bouligand subdifferential of the reduced cost function and, based on that, we propose a computable trust-region model which fulfills the convergence hypotheses of the general algorithm. The article concludes with the experimental study of the main properties of the proposed method based on two different numerical instances.

math.OC↗

arXiv2017

Optimal control of a non-smooth semilinear elliptic equation

This paper is concerned with an optimal control problem governed by a non-smooth semilinear elliptic equation. We show that the control-to-state mapping is directionally differentiable and precisely characterize its Bouligand subdifferential. By means of a suitable regularization, first-order optimality conditions including an adjoint equation are derived and afterwards interpreted in light of the previously obtained characterization. In addition, the directional derivative of the control-to-state mapping is used to establish strong stationarity conditions. While the latter conditions are shown to be stronger, we demonstrate by numerical examples that the former conditions are amenable to numerical solution using a semi-smooth Newton method.

math.OC↗

arXiv2016

The Cryogenic Storage Ring CSR

An electrostatic cryogenic storage ring, CSR, for beams of anions and cations with up to 300 keV kinetic energy per unit charge has been designed, constructed and put into operation. With a circumference of 35 m, the ion-beam vacuum chambers and all beam optics are in a cryostat and cooled by a closed-cycle liquid helium system. At temperatures as low as (5.5 $\pm$ 1) K inside the ring, storage time constants of several minutes up to almost an hour were observed for atomic and molecular, anion and cation beams at an energy of 60 keV. The ion-beam intensity, energy-dependent closed-orbit shifts (dispersion) and the focusing properties of the machine were studied by a system of capacitive pickups. The Schottky-noise spectrum of the stored ions revealed a broadening of the momentum distribution on a time scale of 1000 s. Photodetachment of stored anions was used in the beam lifetime measurements. The detachment rate by anion collisions with residual-gas molecules was found to be extremely low. A residual-gas density below 140 cm$^{-3}$ is derived, equivalent to a room-temperature pressure below 10$^{-14}$ mbar. Fast atomic, molecular and cluster ion beams stored for long periods of time in a cryogenic environment will allow experiments on collision- and radiation-induced fragmentation processes of ions in known internal quantum states with merged and crossed photon and particle beams.

physics.atom-ph↗

arXiv2016

Optimal Control of the Thermistor Problem in Three Spatial Dimensions

This paper is concerned with the state-constrained optimal control of the three-dimensional thermistor problem, a fully quasilinear coupled system of a parabolic and elliptic PDE with mixed boundary conditions. This system models the heating of a conducting material by means of direct current. Local existence, uniqueness and continuity for the state system are derived by employing maximal parabolic regularity in the fundamental theorem of Prüss. Global solutions are addressed, which includes analysis of the linearized state system via maximal parabolic regularity, and existence of optimal controls is shown if the temperature gradient is under control. The adjoint system involving measures is investigated using a duality argument. These results allow to derive first-order necessary conditions for the optimal control problem in form of a qualified optimality system. The theoretical findings are illustrated by numerical results.

math.OC↗

arXiv2014

Strong stationarity conditions for a class of optimization problems governed by variational inequalities of the 2nd kind

We investigate optimality conditions for optimization problems constrained by a class of variational inequalities of the second kind. Based on a nonsmooth primal-dual reformulation of the governing inequality, the differentiability of the solution map is studied. Directional differentiability is proved both for finite-dimensional and function space problems, under suitable assumptions on the active set. A characterization of B- and strong stationary optimal solutions is obtained thereafter. Finally, based on the obtained first-order information, a trust-region algorithm is proposed for the solution of the optimization problems.

math.OC↗

arXiv2010

Recursive Numerical Evaluation of the Cumulative Bivariate Normal Distribution

We propose an algorithm for evaluation of the cumulative bivariate normal distribution, building upon Marsaglia's ideas for evaluation of the cumulative univariate normal distribution. The algorithm is mathematically transparent, delivers competitive performance and can easily be extended to arbitrary precision.

math.NA↗

arXiv2009

The Bivariate Normal Copula

We collect well known and less known facts about the bivariate normal distribution and translate them into copula language. In addition, we prove a very general formula for the bivariate normal copula, we compute Gini's gamma, and we provide improved bounds and approximations on the diagonal.

math.PR↗

arXiv2007

Imaging correlated wave functions of few-electron quantum dots: Theory and scanning tunneling spectroscopy experiments

We show both theoretically and experimentally that scanning tunneling spectroscopy (STS) images of semiconductor quantum dots may display clear signatures of electron-electron correlation. We apply many-body tunneling theory to a realistic model which fully takes into account correlation effects and dot anisotropy. Comparing measured STS images of freestanding InAs quantum dots with those calculated by the full configuration interaction method, we explain the wave function sequence in terms of images of one- and two-electron states. The STS map corresponding to double charging is significantly distorted by electron correlation with respect to the non-interacting case.

cond-mat.mes-hall↗

arXiv2007

Correlation Effects in Wave Function Mapping of Molecular Beam Epitaxy Grown Quantum Dots

We investigate correlation effects in the regime of a few electrons in uncapped InAs quantum dots by tunneling spectroscopy and wave function (WF) mapping at high tunneling currents where electron-electron interactions become relevant. Four clearly resolved states are found, whose approximate symmetries are roughly s and p, in order of increasing energy. Because the major axes of the p-like states coincide, the WF sequence is inconsistent with the imaging of independent-electron orbitals. The results are explained in terms of many-body tunneling theory, by comparing measured maps with those calculated by taking correlation effects into account.

cond-mat.mes-hall↗

arXiv2005

The mirror quintic as a quintic

The general quintic hypersurface in ${\mathbb P}^4$ is the most famous example of a Calabi--Yau threefold for which mirror symmetry has been investigated in detail. There is a description of the mirror as a hypersurface in a certain weighted projective space. In this note we present a model for the mirror which is again (the resolution of) a quintic hypersurface in ${\mathbb P}^4$. We also deal with the special members in the respective families. They lead to rigid Calabi--Yau threefolds with interesting arithmetical properties. In the last section we try to find a similarly nice model for the mirror of the complete intersection of two cubics in ${\mathbb P}^5$. We also formulate a general question about mirror models.

math.AG↗