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We consider a highly-qualified individual with respect to her choice between two distinct career paths. She can choose between a mid-level management position in a large company and an executive position within a smaller listed company with the possibility to directly affect the company’s share price. She invests in the financial market includ- ing the share of the smaller listed company. The utility maximizing strategy from consumption, investment, and work effort is derived in closed form for logarithmic utility. The power utility case is discussed as well. Conditions for the individual to pursue her career with the smaller listed company are obtained. The participation constraint is formulated in terms of the salary differential between the two posi- tions. The smaller listed company can offer less salary. The salary shortfall is offset by the possibility to benefit from her work effort by acquiring own-company shares. This gives insight into aspects of optimal contract design. Our framework is applicable to the pharma- ceutical and financial industry, and the IT sector.

In this expository article, we give an introduction into the basics of bootstrap tests in general. We discuss the residual-based and the wild bootstrap for regression models suitable for applications in signal and image analysis. As an illustration of the general idea, we consider a particular test for detecting differences between two noisy signals or images which also works for noise with variable variance. The test statistic is essentially the integrated squared difference between the signals after denoising them by local smoothing. Determining its quantile, which marks the boundary between accepting and rejecting the hypothesis of equal signals, is hardly possible by standard asymptotic methods whereas the bootstrap works well. Applied to the rows and columns of images, the resulting algorithm not only allows for the detection of defects but also for the characterization of their location and shape in surface inspection problems.

We present a two-scale finite element method for solving Brinkman’s and Darcy’s equations. These systems of equations model fluid flows in highly porous and porous media, respectively. The method uses a recently proposed discontinuous Galerkin FEM for Stokes’ equations byWang and Ye and the concept of subgrid approximation developed by Arbogast for Darcy’s equations. In order to reduce the “resonance error” and to ensure convergence to the global fine solution the algorithm is put in the framework of alternating Schwarz iterations using subdomains around the coarse-grid boundaries. The discussed algorithms are implemented using the Deal.II finite element library and are tested on a number of model problems.

Lithium-ion batteries are broadly used nowadays in all kinds of portable electronics, such as laptops, cell phones, tablets, e-book readers, digital cameras, etc. They are preferred to other types of rechargeable batteries due to their superior characteristics, such as light weight and high energy density, no memory effect, and a big number of charge/discharge cycles. The high demand and applicability of Li-ion batteries naturally give rise to the unceasing necessity of developing better batteries in terms of performance and lifetime. The aim of the mathematical modelling of Li-ion batteries is to help engineers test different battery configurations and electrode materials faster and cheaper. Lithium-ion batteries are multiscale systems. A typical Li-ion battery consists of multiple connected electrochemical battery cells. Each cell has two electrodes - anode and cathode, as well as a separator between them that prevents a short circuit.
Both electrodes have porous structure composed of two phases - solid and electrolyte. We call macroscale the lengthscale of the whole electrode and microscale - the lengthscale at which we can distinguish the complex porous structure of the electrodes. We start from a Li-ion battery model derived on the microscale. The model is based on nonlinear diffusion type of equations for the transport of Lithium ions and charges in the electrolyte and in the active material. Electrochemical reactions on the solid-electrolyte interface couple the two phases. The interface kinetics is modelled by the highly nonlinear Butler-Volmer interface conditions. Direct numerical simulations with standard methods, such as the Finite Element Method or Finite Volume Method, lead to ill-conditioned problems with a huge number of degrees of freedom which are difficult to solve. Therefore, the aim of this work is to derive upscaled models on the lengthscale of the whole electrode so that we do not have to resolve all the small-scale features of the porous microstructure thus reducing the computational time and cost. We do this by applying two different upscaling techniques - the Asymptotic Homogenization Method and the Multiscale Finite Element Method (MsFEM). We consider the electrolyte and the solid as two self-complementary perforated domains and we exploit this idea with both upscaling methods. The first method is restricted only to periodic media and periodically oscillating solutions while the second method can be applied to randomly oscillating solutions and is based on the Finite Element Method framework. We apply the Asymptotic Homogenization Method to derive a coupled macro-micro upscaled model under the assumption of periodic electrode microstructure. A crucial step in the homogenization procedure is the upscaling of the Butler-Volmer interface conditions. We rigorously determine the asymptotic order of the interface exchange current densities and we perform a comprehensive numerical study in order to validate the derived homogenized Li-ion battery model. In order to upscale the microscale battery problem in the case of random electrode microstructure we apply the MsFEM, extended to problems in perforated domains with Neumann boundary conditions on the holes. We conduct a detailed numerical investigation of the proposed algorithm and we show numerical convergence of the method that we design. We also apply the developed technique to a simplified two-dimensional Li-ion battery problem and we show numerical convergence of the solution obtained with the MsFEM to the reference microscale one.

In the ground vehicle industry it is often an important task to simulate full vehicle models based on the wheel forces and moments, which have been measured during driving over certain roads with a prototype vehicle. The models are described by a system of differential algebraic equations (DAE) or ordinary differential equations (ODE). The goal of the simulation is to derive section forces at certain components for a durability assessment. In contrast to handling simulations, which are performed including more or less complex tyre models, a driver model, and a digital road profile, the models we use here usually do not contain the tyres or a driver model. Instead, the measured wheel forces are used for excitation of the unconstrained model. This can be difficult due to noise in the input data, which leads to an undesired drift of the vehicle model in the simulation.

One approach to multi-criteria IMRT planning is to automatically calculate a data set of Pareto-optimal plans for a given planning problem in a first phase, and then interactively explore the solution space and decide for the clinically best treatment plan in a second phase. The challenge of computing the plan data set is to assure that all clinically meaningful plans are covered and that as many as possible clinically irrelevant plans are excluded to keep computation times within reasonable limits. In this work, we focus on the approximation of the clinically relevant part of the Pareto surface, the process that consititutes the first phase. It is possible that two plans on the Parteto surface have a very small, clinically insignificant difference in one criterion and a significant difference in one other criterion. For such cases, only the plan that is clinically clearly superior should be included into the data set. To achieve this during the Pareto surface approximation, we propose to introduce bounds that restrict the relative quality between plans, so called tradeoff bounds. We show how to integrate these trade-off bounds into the approximation scheme and study their effects.

Territory design may be viewed as the problem of grouping small geographic areas into larger geographic clusters called territories in such a way that the latter are acceptable according to relevant planning criteria. In this paper we review the existing literature for applications of territory design problems and solution approaches for solving these types of problems. After identifying features common to all applications we introduce a basic territory design model and present in detail two approaches for solving this model: a classical location–allocation approach combined with optimal split resolution techniques and a newly developed computational geometry based method. We present computational results indicating the efficiency and suitability of the latter method for solving large–scale practical problems in an interactive environment. Furthermore, we discuss extensions to the basic model and its integration into Geographic Information Systems.

For the numerical simulation of 3D radiative heat transfer in glasses and glass melts, practically applicable mathematical methods are needed to handle such problems optimal using workstation class computers. Since the exact solution would require super-computer capabilities we concentrate on approximate solutions with a high degree of accuracy. The following approaches are studied: 3D diffusion approximations and 3D ray-tracing methods.

We will present a rigorous derivation of the equations and interface conditions for ion, charge and heat transport in Li-ion insertion batteries. The derivation is based exclusively on universally accepted principles of nonequilibrium thermodynamics and the assumption of a one step intercalation reaction at the interface of electrolyte and active particles. Without loss of generality the transport in the active particle is assumed to be isotropic. The electrolyte is described as a fully dissociated salt in a neutral solvent. The presented theory is valid for transport on a spatial scale for which local charge neutrality holds i.e. beyond the scale of the diffuse double layer. Charge neutrality is explicitely used to determine the correct set of thermodynamically independent variables. The theory guarantees strictly positive entropy production. The various contributions to the Peltier coeficients for the interface between the active particles and the electrolyte as well as the contributions to the heat of mixing are obtained as a result of the theory.

In this paper we develop a network location model that combines the characteristics of ordered median and gradual cover models resulting in the Ordered Gradual Covering Location Problem (OGCLP). The Gradual Cover Location Problem (GCLP) was specifically designed to extend the basic cover objective to capture sensitivity with respect to absolute travel distance. Ordered Median Location problems are a generalization of most of the classical locations problems like p-median or p-center problems. They can be modeled by using so-called ordered median functions. These functions multiply a weight to the cost of fulfilling the demand of a customer which depends on the position of that cost relative to the costs of fulfilling the demand of the other customers. We derive Finite Dominating Sets (FDS) for the one facility case of the OGCLP. Moreover, we present efficient algorithms for determining the FDS and also discuss the conditional case where a certain number of facilities are already assumed to exist and one new facility is to be added. For the multi-facility case we are able to identify a finite set of potential facility locations a priori, which essentially converts the network location model into its discrete counterpart. For the multi-facility discrete OGCLP we discuss several Integer Programming formulations and give computational results.