## Fraunhofer (ITWM)

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- numerical upscaling (7)
- Integer programming (4)
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- Darcy’s law (3)
- Heston model (3)
- Lagrangian mechanics (3)
- effective heat conductivity (3)
- facility location (3)
- non-Newtonian flow in porous media (3)
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In the literature, there are at least two equivalent two-factor Gaussian models for the instantaneous short rate. These are the original two-factor Hull White model (see [3]) and the G2++ one by Brigo and Mercurio (see [1]). Both these models first specify a time homogeneous two-factor short rate dynamics and then by adding a deterministic shift function '(·) fit exactly the initial term structure of interest rates. However, the obtained results are rather clumsy and not intuitive which means that a special care has to be taken for their correct numerical implementation.

In this paper we study the possibilities of sharing profit in combinatorial procurement auctions and exchanges. Bundles of heterogeneous items are offered by the sellers, and the buyers can then place bundle bids on sets of these items. That way, both sellers and buyers can express synergies between items and avoid the well-known risk of exposure (see, e.g., [3]). The reassignment of items to participants is known as the Winner Determination Problem (WDP). We propose solving the WDP by using a Set Covering formulation, because profits are potentially higher than with the usual Set Partitioning formulation, and subsidies are unnecessary. The achieved benefit is then to be distributed amongst the participants of the auction, a process which is known as profit sharing. The literature on profit sharing provides various desirable criteria. We focus on three main properties we would like to guarantee: Budget balance, meaning that no more money is distributed than profit was generated, individual rationality, which guarantees to each player that participation does not lead to a loss, and the core property, which provides every subcoalition with enough money to keep them from separating. We characterize all profit sharing schemes that satisfy these three conditions by a monetary flow network and state necessary conditions on the solution of the WDP for the existence of such a profit sharing. Finally, we establish a connection to the famous VCG payment scheme [2, 8, 19], and the Shapley Value [17].

In this paper we investigate the use of the sharp function known from functional analysis in image processing. The sharp function gives a measure of the variations of a function and can be used as an edge detector. We extend the classical notion of the sharp function for measuring anisotropic behaviour and give a fast anisotropic edge detection variant inspired by the sharp function. We show that these edge detection results are useful to steer isotropic and anisotropic nonlinear diffusion filters for image enhancement.

Determination of interaction between MCT1 and CAII via a mathematical and physiological approach
(2008)

The enzyme carbonic anhydrase isoform II (CAII), catalysing the hydration and dehydration of CO2, enhances transport activity of the monocarboxylate transporter isoform I (MCT1, SLC16A1) expressed in Xenopus oocytes by a mechanism that does not require CAII catalytic activity (Becker et al. (2005) J. Biol. Chem., 280). In the present study, we have investigated the mechanism of the CAII induced increase in transport activity by using electrophysiological techniques and a mathematical model of the MCT1 transport cycle. The model consists of six states arranged in cyclic fashion and features an ordered, mirror-symmetric, binding mechanism were binding and unbinding of the proton to the transport protein is considered to be the rate limiting step under physiological conditions. An explicit rate expression for the substrate °ux is derived using model reduction techniques. By treating the pools of intra- and extracellular MCT1 substrates as dynamic states, the time dependent kinetics are obtained by integration using the derived expression for the substrate °ux. The simulations were compared with experimental data obtained from MCT1-expressing oocytes injected with di®erent amounts of CAII. The model suggests that CAII increases the e®ective rate constants of the proton reactions, possibly by working as a proton antenna.

The level-set method has been recently introduced in the field of shape optimization, enabling a smooth representation of the boundaries on a fixed mesh and therefore leading to fast numerical algorithms. However, most of these algorithms use a Hamilton-Jacobi equation to connect the evolution of the level-set function with the deformation of the contours, and consequently they cannot create any new holes in the domain (at least in 2D). In this work, we propose an evolution equation for the level-set function based on a generalization of the concept of topological gradient. This results in a new algorithm allowing for all kinds of topology changes.

Im vorliegenden Bericht werden die Erfahrungen und Ergebnisse aus dem Projekt OptCast zusammengestellt. Das Ziel dieses Projekts bestand (a) in der Anpassung der Methodik der automatischen Strukturoptimierung für Gussteile und (b) in der Entwicklung und Bereitstellung von gießereispezifischen Optimierungstools für Gießereien und Ingenieurbüros. Gießtechnische Restriktionen lassen sich nicht vollständig auf geometrische Restriktionen reduzieren, da die lokalen Eigenschaften nicht nur von der geometrischen Form des Gussteils, sondern auch vom verwendeten Material abhängen. Sie sind jedoch über eine Gießsimulation (Erstarrungssimulation und Eigenspannungsanalyse) adäquat erfassbar. Wegen dieser Erkenntnis wurde ein neuartiges Topologieoptimierungsverfahren unter Verwendung der Level-Set-Technik entwickelt, bei dem keine variable Dichte des Materials eingeführt wird. In jeder Iteration wird ein scharfer Rand des Bauteils berechnet. Somit ist die Gießsimulation in den iterativen Optimierungsprozess integrierbar.

This report discusses two approaches for a posteriori error indication in the linear elasticity solver DDFEM: An indicator based on the Richardson extrapolation and Zienkiewicz-Zhu-type indicator. The solver handles 3D linear elasticity steady-state problems. It uses own input language to describe the mesh and the boundary conditions. Finite element discretization over tetrahedral meshes with first or second order shape functions (hierarchical basis) has been used to resolve the model. The parallelization of the numerical method is based on the domain decomposition approach. DDFEM is highly portable over a set of parallel computer architectures supporting the MPI-standard.

The rotational spinning of viscous jets is of interest in many industrial applications, including pellet manufacturing [4, 14, 19, 20] and drawing, tapering and spinning of glass and polymer fibers [8, 12, 13], see also [15, 21] and references within. In [12] an asymptotic model for the dynamics of curved viscous inertial fiber jets emerging from a rotating orifice under surface tension and gravity was deduced from the three-dimensional free boundary value problem given by the incompressible Navier-Stokes equations for a Newtonian fluid. In the terminology of [1], it is a string model consisting of balance equations for mass and linear momentum. Accounting for inner viscous transport, surface tension and placing no restrictions on either the motion or the shape of the jet’s center-line, it generalizes the previously developed string models for straight [3, 5, 6] and curved center-lines [4, 13, 19]. Moreover, the numerical results investigating the effects of viscosity, surface tension, gravity and rotation on the jet behavior coincide well with the experiments of Wong et.al. [20].

The optimal design of rotational production processes for glass wool manufacturing poses severe computational challenges to mathematicians, natural scientists and engineers. In this paper we focus exclusively on the spinning regime where thousands of viscous thermal glass jets are formed by fast air streams. Homogeneity and slenderness of the spun fibers are the quality features of the final fabric. Their prediction requires the computation of the fuidber-interactions which involves the solving of a complex three-dimensional multiphase problem with appropriate interface conditions. But this is practically impossible due to the needed high resolution and adaptive grid refinement. Therefore, we propose an asymptotic coupling concept. Treating the glass jets as viscous thermal Cosserat rods, we tackle the multiscale problem by help of momentum (drag) and heat exchange models that are derived on basis of slender-body theory and homogenization. A weak iterative coupling algorithm that is based on the combination of commercial software and self-implemented code for ow and rod solvers, respectively, makes then the simulation of the industrial process possible. For the boundary value problem of the rod we particularly suggest an adapted collocation-continuation method. Consequently, this work establishes a promising basis for future optimization strategies.

This work deals with the modeling and simulation of slender viscous jets exposed to gravity and rotation, as they occur in rotational spinning processes. In terms of slender-body theory we show the asymptotic reduction of a viscous Cosserat rod to a string system for vanishing slenderness parameter. We propose two string models, i.e. inertial and viscous-inertial string models, that differ in the closure conditions and hence yield a boundary value problem and an interface problem, respectively. We investigate the existence regimes of the string models in the four-parametric space of Froude, Rossby, Reynolds numbers and jet length. The convergence regimes where the respective string solution is the asymptotic limit to the rod turn out to be disjoint and to cover nearly the whole parameter space. We explore the transition hyperplane and derive analytically low and high Reynolds number limits. Numerical studies of the stationary jet behavior for different parameter ranges complete the work.

This paper analyzes and solves a patient transportation problem arising in several large hospitals. The aim is to provide an efficient and timely transport service to patients between several locations on a hospital campus. Transportation requests arrive in a dynamic fashion and the solution methodology must therefore be capable of quickly inserting new requests in the current vehicle routes. Contrary to standard dial-a-ride problems, the problem under study contains several complicating constraints which are specific to a hospital context. The paper provides a detailed description of the problem and proposes a two-phase heuristic procedure capable of handling its many features. In the first phase a simple insertion scheme is used to generate a feasible solution, which is improved in the second phase with a tabu search algorithm. The heuristic procedure was extensively tested on real data provided by a German hospital. Results show that the algorithm is capable of handling the dynamic aspect of the problem and of providing high quality solutions. In particular, it succeeded in reducing waiting times for patients while using fewer vehicles.

Within this paper we review image distortion measures. A distortion measure is a criterion that assigns a "quality number" to an image. We distinguish between mathematical distortion measures and those distortion measures in-cooperating a priori knowledge about the imaging devices ( e.g. satellite images), image processing algorithms or the human physiology. We will consider representative examples of different kinds of distortion measures and are going to discuss them.

The objective of this paper is to bridge the gap between location theory and practice. To meet this objective focus is given to the development of software capable of addressing the different needs of a wide group of users. There is a very active community on location theory encompassing many research fields such as operations research, computer science, mathematics, engineering, geography, economics and marketing. As a result, people working on facility location problems have a very diverse background and also different needs regarding the software to solve these problems. For those interested in non-commercial applications (e. g. students and researchers), the library of location algorithms (LoLA can be of considerable assistance. LoLA contains a collection of efficient algorithms for solving planar, network and discrete facility location problems. In this paper, a detailed description of the functionality of LoLA is presented. In the fields of geography and marketing, for instance, solving facility location problems requires using large amounts of demographic data. Hence, members of these groups (e. g. urban planners and sales managers) often work with geographical information too s. To address the specific needs of these users, LoLA was inked to a geographical information system (GIS) and the details of the combined functionality are described in the paper. Finally, there is a wide group of practitioners who need to solve large problems and require special purpose software with a good data interface. Many of such users can be found, for example, in the area of supply chain management (SCM). Logistics activities involved in strategic SCM include, among others, facility location planning. In this paper, the development of a commercial location software tool is also described. The too is embedded in the Advanced Planner and Optimizer SCM software developed by SAP AG, Walldorf, Germany. The paper ends with some conclusions and an outlook to future activities.

We propose a constraint-based approach for the two-dimensional rectangular packing problem with orthogonal orientations. This problem is to arrange a set of rectangles that can be rotated by 90 degrees into a rectangle of minimal size such that no two rectangles overlap. It arises in the placement of electronic devices during the layout of 2.5D System-in-Package integrated electronic systems. Moffitt et al. [8] solve the packing without orientations with a branch and bound approach and use constraint propagation. We generalize their propagation techniques to allow orientations. Our approach is compared to a mixed-integer program and we provide results that outperform it.

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.

The Discrete Ordered Median Problem (DOMP) generalizes classical discrete location problems, such as the N-median, N-center and Uncapacitated Facility Location problems. It was introduced by Nickel [16], who formulated it as both a nonlinear and a linear integer program. We propose an alternative integer linear programming formulation for the DOMP, discuss relationships between both integer linear programming formulations, and show how properties of optimal solutions can be used to strengthen these formulations. Moreover, we present a specific branch and bound procedure to solve the DOMP more efficiently. We test the integer linear programming formulations and this branch and bound method computationally on randomly generated test problems.

In this paper, a stochastic model [5] for the turbulent fiber laydown in the industrial production of nonwoven materials is extended by including a moving conveyor belt. In the hydrodynamic limit corresponding to large noise values, the transient and stationary joint probability distributions are determined using the method of multiple scales and the Chapman-Enskog method. Moreover, exponential convergence towards the stationary solution is proven for the reduced problem. For special choices of the industrial parameters, the stochastic limit process is an Ornstein{Uhlenbeck. It is a good approximation of the fiber motion even for moderate noise values. Moreover, as shown by Monte{Carlo simulations, the limiting process can be used to assess the quality of nonwoven materials in the industrial application by determining distributions of functionals of the process.

Radiation therapy planning is always a tight rope walk between dangerous insufficient dose in the target volume and life threatening overdosing of organs at risk. Finding ideal balances between these inherently contradictory goals challenges dosimetrists and physicians in their daily practice. Today’s planning systems are typically based on a single evaluation function that measures the quality of a radiation treatment plan. Unfortunately, such a one dimensional approach cannot satisfactorily map the different backgrounds of physicians and the patient dependent necessities. So, too often a time consuming iteration process between evaluation of dose distribution and redefinition of the evaluation function is needed. In this paper we propose a generic multi-criteria approach based on Pareto’s solution concept. For each entity of interest - target volume or organ at risk a structure dependent evaluation function is defined measuring deviations from ideal doses that are calculated from statistical functions. A reasonable bunch of clinically meaningful Pareto optimal solutions are stored in a data base, which can be interactively searched by physicians. The system guarantees dynamical planning as well as the discussion of tradeoffs between different entities. Mathematically, we model the upcoming inverse problem as a multi-criteria linear programming problem. Because of the large scale nature of the problem it is not possible to solve the problem in a 3D-setting without adaptive reduction by appropriate approximation schemes. Our approach is twofold: First, the discretization of the continuous problem is based on an adaptive hierarchical clustering process which is used for a local refinement of constraints during the optimization procedure. Second, the set of Pareto optimal solutions is approximated by an adaptive grid of representatives that are found by a hybrid process of calculating extreme compromises and interpolation methods.

This work presents a new framework for Gröbner basis computations with Boolean polynomials. Boolean polynomials can be modeled in a rather simple way, with both coefficients and degree per variable lying in {0, 1}. The ring of Boolean polynomials is, however, not a polynomial ring, but rather the quotient ring of the polynomial ring over the field with two elements modulo the field equations x2 = x for each variable x. Therefore, the usual polynomial data structures seem not to be appropriate for fast Gröbner basis computations. We introduce a specialized data structure for Boolean polynomials based on zero-suppressed binary decision diagrams (ZDDs), which is capable of handling these polynomials more efficiently with respect to memory consumption and also computational speed. Furthermore, we concentrate on high-level algorithmic aspects, taking into account the new data structures as well as structural properties of Boolean polynomials. For example, a new useless-pair criterion for Gröbner basis computations in Boolean rings is introduced. One of the motivations for our work is the growing importance of formal hardware and software verification based on Boolean expressions, which suffer – besides from the complexity of the problems – from the lack of an adequate treatment of arithmetic components. We are convinced that algebraic methods are more suited and we believe that our preliminary implementation shows that Gröbner bases on specific data structures can be capable to handle problems of industrial size.

In this work we extend the multiscale finite element method (MsFEM)
as formulated by Hou and Wu in [14] to the PDE system of linear elasticity.
The application, motivated from the multiscale analysis of highly heterogeneous
composite materials, is twofold. Resolving the heterogeneities on
the finest scale, we utilize the linear MsFEM basis for the construction of
robust coarse spaces in the context of two-level overlapping Domain Decomposition
preconditioners. We motivate and explain the construction
and present numerical results validating the approach. Under the assumption
that the material jumps are isolated, that is they occur only in the
interior of the coarse grid elements, our experiments show uniform convergence
rates independent of the contrast in the Young's modulus within the
heterogeneous material. Elsewise, if no restrictions on the position of the
high coefficient inclusions are imposed, robustness can not be guaranteed
any more. These results justify expectations to obtain coefficient-explicit
condition number bounds for the PDE system of linear elasticity similar to
existing ones for scalar elliptic PDEs as given in the work of Graham, Lechner
and Scheichl [12]. Furthermore, we numerically observe the properties
of the MsFEM coarse space for linear elasticity in an upscaling framework.
Therefore, we present experimental results showing the approximation errors
of the multiscale coarse space w.r.t. the fine-scale solution.

For the numerical simulation of a mechanical multibody system (MBS), dynamical loads are needed as input data, such as a road profile. With given input quantities, the equations of motion of the system can be integrated. Output quantities for further investigations are calculated from the integration results. In this paper, we consider the corresponding inverse problem: We assume, that a dynamical system and some reference output signals are given. The general task is to derive an input signal, such that the system simulation produces the desired reference output. We present the state-of-the-art method in industrial applications, the iterative learning control method (ILC) and give an application example from automotive industry. Then, we discuss three alternative methods based on optimal control theory for differential algebraic equations (DAEs) and give an overview of their general scheme.

Optimal control methods for the calculation of invariant excitation signals for multibody systems
(2010)

Input signals are needed for the numerical simulation of vehicle multibody systems. With these input data, the equations of motion can be integrated numerically and some output quantities can be calculated from the simulation results. In this work we consider the corresponding inverse problem: We assume that some reference output signals are available, typically gained by measurement and focus on the task to derive the input signals that produce the desired reference output in a suitable sense. If the input data is invariant, i.e., independent of the specific system, it can be transferred and used to excite other system variants. This problem can be formulated as optimal control problem. We discuss solution approaches from optimal control theory, their applicability to this special problem class and give some simulation results.

Input loads are essential for the numerical simulation of vehicle multibody system
(MBS)- models. Such load data is called invariant, if it is independent of the specific system under consideration. A digital road profile, e.g., can be used to excite MBS models of different
vehicle variants. However, quantities efficiently obtained by measurement such as wheel forces
are typically not invariant in this sense. This leads to the general task to derive invariant loads
on the basis of measurable, but system-dependent quantities. We present an approach to derive
input data for full-vehicle simulation that can be used to simulate different variants of a vehicle
MBS model. An important ingredient of this input data is a virtual road profile computed by optimal control methods.

The performance of oil filters used in the automotive industry can be significantly improved, especially when computer simulation is an essential component of the design process. In this paper, we consider parallel numerical algorithms for solving mathematical models describing the process of filtration, filtering out solid particles from liquid oil. The Navier-Stokes-Brinkmann system of equations is used to describe the laminar flow of incompressible isothermal oil. The space discretization in the complicated filter geometry is based on the finite-volume method. Special care is taken for an accurate approximation of velocity and pressure on the interface between the fluid and the porous media. The time discretization used here is a proper modification of the fractional time step discretization (cf. Chorin scheme) of the Navier-Stokes equations, where the Brinkmann term is considered at both, prediction and correction substeps. A data decomposition method is used to develop a parallel algorithm, where the domain is distributed among processors by using a structured reference grid. The MPI library is used to implement the data communication part of the algorithm. A theoretical model is proposed for the estimation of the complexity of the given parallel algorithm and a scalability analysis is done on the basis of this model. Results of computational experiments are presented, and the accuracy and efficiency of the parallel algorithm is tested on real industrial geometries.

In this paper we consider numerical algorithms for solving a system of nonlinear PDEs arising in modeling of liquid polymer injection. We investigate the particular case when a porous preform is located within the mould, so that the liquid polymer flows through a porous medium during the filling stage. The nonlinearity of the governing system of PDEs is due to the non-Newtonian behavior of the polymer, as well as due to the moving free boundary. The latter is related to the penetration front and a Stefan type problem is formulated to account for it. A finite-volume method is used to approximate the given differential problem. Results of numerical experiments are presented. We also solve an inverse problem and present algorithms for the determination of the absolute preform permeability coefficient in the case when the velocity of the penetration front is known from measurements. In both cases (direct and inverse problems) we emphasize on the specifics related to the non-Newtonian behavior of the polymer. For completeness, we discuss also the Newtonian case. Results of some experimental measurements are presented and discussed.

The capacitated single-allocation hub location problem revisited: A note on a classical formulation
(2009)

Denote by G = (N;A) a complete graph where N is the set of nodes and A is the set of edges. Assume that a °ow wij should be sent from each node i to each node j (i; j 2 N). One possibility is to send these °ows directly between the corresponding pairs of nodes. However, in practice this is often neither e±cient nor costly attractive because it would imply that a link was built between each pair of nodes. An alternative is to select some nodes to become hubs and use them as consolidation and redistribution points that altogether process more e±ciently the flow in the network. Accordingly, hubs are nodes in the graph that receive tra±c (mail, phone calls, passengers, etc) from di®erent origins (nodes) and redirect this tra±c directly to the destination nodes (when a link exists) or else to other hubs. The concentration of tra±c in the hubs and its shipment to other hubs lead to a natural decrease in the overall cost due to economies of scale.

In this paper, an extension to the classical capacitated single-allocation hub location problem is studied in which the size of the hubs is part of the decision making process. For each potential hub a set of capacities is assumed to be available among which one can be chosen. Several formulations are proposed for the problem, which are compared in terms of the bound provided by the linear programming relaxation. Di®erent sets of inequalities are proposed to enhance the models. Several preprocessing tests are also presented with the goal of reducing the size of the models for each particular instance. The results of the computational experiments performed using the proposed models are reported.

Das Smart Grid, „intelligentes Stromnetz“, ist eines der Themen, welche von der Politik und natürlich auch der Stromwirtschaft immer wieder in den Vordergrund gestellt werden. Das Potential der erneuerbaren Energien reicht aus, um Deutschland und Europa zuverlässig mit Strom zu versorgen. Der Umbau der Stromnetze ist dabei von zentraler Bedeutung und bedarf einer Anstrengung der gesamten Gesellschaft. Leider kommt dabei der Stromkunde zu kurz — die Bedürfnisse von Stromkunden werden weitgehend ignoriert und der Datenschutz wird oft ausser acht gelassen. Aber auch kleinere Stadtwerke haben mit dieser Entwicklung Probleme: Aufgrund politischer Vorgaben müssen sie zum Beispiel Smart Meter einführen, obwohl ihnen dadurch Kosten entstehen, die sie nicht direkt auf den Kunden umlegen können. Die Bereitschaft der Kunden, für ein Smart Grid mehr Geld zu bezahlen, ist wohl kaum vorhanden. Gleichzeitig ist es aber notwendig, die bestehenden Stromnetze zu flexibilisieren und auf einen weiter steigenden Anteil von erneuerbaren Energiequellen vorzubereiten

The scope of this paper is to enhance the model for the own-company stockholder (given in Desmettre, Gould and Szimayer (2010)), who can voluntarily performance-link his personal wealth to his management success by acquiring stocks in the own-company whose value he can directly influence via spending work effort. The executive is thereby characterized by a parameter of risk aversion and the two work effectiveness parameters inverse work productivity and disutility stress. We extend the model to a constant absolute risk aversion framework using an exponential utility/disutility set-up. A closed-form solution is given for the optimal work effort an executive will apply and we derive the optimal investment strategies of the executive. Furthermore, we determine an up-front fair cash compensation applying an indifference utility rationale. Our study shows to a large extent that the results previously obtained are robust under the choice of the utility/disutility set-up.

We develop a framework for analyzing an executive’s own-company stockholding and work effort preferences. The executive, characterized by risk aversion and work effectiveness parameters, invests his personal wealth without constraint in the financial market, including the stock of his own company whose value he can directly influence with work effort. The executive’s utility-maximizing personal investment and work effort strategy is derived in closed-form, and an indifference utility rationale is demonstrated to determine his required compensation. Our results have implications for the practical and theoretical assessment of executive quality and the benefits of performance contracting. Assuming knowledge of the company’s non-systematic risk, our executive’s unconstrained own-company investment identifies his work effectiveness (i.e. quality), and also reflects work effort that establishes a base-level that performance contracting should seek to exceed.

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.

This report describes the calibration and completion of the volatility cube in the SABR model. The description is based on a project done for Assenagon GmbH in Munich. However, we use fictitious market data which resembles realistic market data. The problem posed by our client is formulated in section 1. Here we also motivate why this is a relevant problem. The SABR model is briefly reviewed in section 2. Section 3 discusses the calibration and completion of the volatility cube. An example is presented in section 4. We conclude by suggesting possible future research in section 5.

In this work we use the Parsimonious Multi–Asset Heston model recently developed in [Dimitroff et al., 2009] at Fraunhofer ITWM, Department Financial Mathematics, Kaiserslautern (Germany) and apply it to Quanto options. We give a summary of the model and its calibration scheme. A suitable transformation of the Quanto option payoff is explained and used to price Quantos within the new framework. Simulated prices are given and compared to market prices and Black–Scholes prices. We find that the new approach underprices the chosen options, but gives better results than the Black–Scholes approach, which is prevailing in the literature on Quanto options.

We present two heuristic methods for solving the Discrete Ordered Median Problem (DOMP), for which no such approaches have been developed so far. The DOMP generalizes classical discrete facility location problems, such as the p-median, p-center and Uncapacitated Facility Location problems. The first procedure proposed in this paper is based on a genetic algorithm developed by Moreno Vega [MV96] for p-median and p-center problems. Additionally, a second heuristic approach based on the Variable Neighborhood Search metaheuristic (VNS) proposed by Hansen & Mladenovic [HM97] for the p-median problem is described. An extensive numerical study is presented to show the efficiency of both heuristics and compare them.

Reliable methods for the analysis of tolerance-affected analog circuits are of great importance in nowadays microelectronics. It is impossible to produce circuits with exactly those parameter specifications proposed in the design process. Such component tolerances will always lead to small variations of a circuit’s properties, which may result in unexpected behaviour. If lower and upper bounds to parameter variations can be read off the manufacturing process, interval arithmetic naturally enters the circuit analysis area. This paper focuses on the frequency-response analysis of linear analog circuits, typically consisting of current and voltage sources as well as resistors, capacitances, inductances, and several variants of controlled sources. These kind of circuits are still widely used in analog circuit design as equivalent circuit diagrams for representing in certain application tasks Interval methods have been applied to analog circuits before. But yet this was restricted to circuit equations only, with no interdependencies between the matrix elements. But there also exist formulations of analog circuit equations containing dependent terms. Hence, for an efficient application of interval methods, it is crucial to regard possible dependencies in circuit equations. Part and parcel of this strategy is the handling of fill-in patterns for those parameters related to uncertain components. These patterns are used in linear circuit analysis for efficient equation setup. Such systems can efficiently be solved by successive application of the Sherman-Morrison formula. The approach can also be extended to complex-valued systems from frequency domain analysis of more general linear circuits. Complex values result here from a Laplace transform of frequency-dependent components like capacitances and inductances. In order to apply interval techniques, a real representation of the linear system of equations can be used for separate treatment of real and imaginary part of the variables. In this representation each parameter corresponds to the superposition of two fill-in patterns. Crude bounds – obtained by treating both patterns independently – can be improved by consideration of the correlations to tighter enclosures of the solution. The techniques described above have been implemented as an extension to the toolbox Analog Insydes, an add-on package to the computer algebra system Mathematica for modeling, analysis, and design of analog circuits.

In this paper we propose a general approach solution method for the single facility ordered median problem in the plane. All types of weights (non-negative, non-positive, and mixed) are considered. The big triangle small triangle approach is used for the solution. Rigorous and heuristic algorithms are proposed and extensively tested on eight different problems with excellent results.

Safety and reliability requirements on the one side and short development cycles, low costs and lightweight design on the other side are two competing aspects of truck engineering. For safety critical components essentially no failures can be tolerated within the target mileage of a truck. For other components the goals are to stay below certain predefined failure rates. Reducing weight or cost of structures often also reduces strength and reliability. The requirements on the strength, however, strongly depend on the loads in actual customer usage. Without sufficient knowledge of these loads one needs large safety factors, limiting possible weight or cost reduction potentials. There are a lot of different quantities influencing the loads acting on the vehicle in actual usage. These ‘influencing quantities’ are, for example, the road quality, the driver, traffic conditions, the mission (long haulage, distribution or construction site), and the geographic region. Thus there is a need for statistical methods to model the load distribution with all its variability, which in turn can be used for the derivation of testing specifications.

We present a unified approach of several boundary conditions for lattice Boltzmann models. Its general framework is a generalization of previously introduced schemes such as the bounce-back rule, linear or quadratic interpolations, etc. The objectives are two fold: first to give theoretical tools to study the existing boundary conditions and their corresponding accuracy; secondly to design formally third- order accurate boundary conditions for general flows. Using these boundary conditions, Couette and Poiseuille flows are exact solution of the lattice Boltzmann models for a Reynolds number Re = 0 (Stokes limit). Numerical comparisons are given for Stokes flows in periodic arrays of spheres and cylinders, linear periodic array of cylinders between moving plates and for Navier-Stokes flows in periodic arrays of cylinders for Re < 200. These results show a significant improvement of the overall accuracy when using the linear interpolations instead of the bounce-back reflection (up to an order of magnitude on the hydrodynamics fields). Further improvement is achieved with the new multi-reflection boundary conditions, reaching a level of accuracy close to the quasi-analytical reference solutions, even for rather modest grid resolutions and few points in the narrowest channels. More important, the pressure and velocity fields in the vicinity of the obstacles are much smoother with multi-reflection than with the other boundary conditions. Finally the good stability of these schemes is highlighted by some simulations of moving obstacles: a cylinder between flat walls and a sphere in a cylinder.

On a Multigrid Adaptive Refinement Solver for Saturated Non-Newtonian Flow in Porous Media A multigrid adaptive refinement algorithm for non-Newtonian flow in porous media is presented. The saturated flow of a non-Newtonian fluid is described by the continuity equation and the generalized Darcy law. The resulting second order nonlinear elliptic equation is discretized by a finite volume method on a cell-centered grid. A nonlinear full-multigrid, full-approximation-storage algorithm is implemented. As a smoother, a single grid solver based on Picard linearization and Gauss-Seidel relaxation is used. Further, a local refinement multigrid algorithm on a composite grid is developed. A residual based error indicator is used in the adaptive refinement criterion. A special implementation approach is used, which allows us to perform unstructured local refinement in conjunction with the finite volume discretization. Several results from numerical experiments are presented in order to examine the performance of the solver.

Flow of non-Newtonian fluid in saturated porous media can be described by the continuity equation and the generalized Darcy law. Efficient solution of the resulting second order nonlinear elliptic equation is discussed here. The equation is discretized by a finite volume method on a cell-centered grid. Local adaptive refinement of the grid is introduced in order to reduce the number of unknowns. A special implementation approach is used, which allows us to perform unstructured local refinement in conjunction with the finite volume discretization. Two residual based error indicators are exploited in the adaptive refinement criterion. Second order accurate discretization of the fluxes on the interfaces between refined and non-refined subdomains, as well as on the boundaries with Dirichlet boundary condition, are presented here, as an essential part of the accurate and efficient algorithm. A nonlinear full approximation storage multigrid algorithm is developed especially for the above described composite (coarse plus locally refined) grid approach. In particular, second order approximation of the fluxes around interfaces is a result of a quadratic approximation of slave nodes in the multigrid - adaptive refinement (MG-AR) algorithm. Results from numerical solution of various academic and practice-induced problems are presented and the performance of the solver is discussed.

In this paper, we propose multi-level Monte Carlo(MLMC) methods that use ensemble level mixed multiscale methods in the simulations of multi-phase flow and transport. The main idea of ensemble level multiscale methods is to construct local multiscale basis functions that can be used for any member of the ensemble. We consider two types of ensemble level mixed multiscale finite element methods, (1) the no-local-solve-online ensemble level method (NLSO) and (2) the local-solve-online ensemble level method (LSO). Both mixed multiscale methods use a number of snapshots of the permeability media to generate a multiscale basis.
As a result, in the offline stage, we construct multiple basis functions for
each coarse region where basis functions correspond to different realizations.
In the no-local-solve-online ensemble level method one uses the whole set of pre-computed basis functions to approximate the solution for an arbitrary realization. In the local-solve-online ensemble level method one uses the pre-computed functions to construct a multiscale basis for a particular realization. With this basis the solution corresponding to this
particular realization is approximated in LSO mixed MsFEM. In both approaches
the accuracy of the method is related to the number of snapshots computed based on different realizations that one uses to pre-compute a
multiscale basis. We note that LSO approaches share similarities with reduced basis methods [11, 21, 22].
In multi-level Monte Carlo methods ([14, 13]), more accurate (and expensive) forward simulations are run with fewer samples while less accurate(and inexpensive) forward simulations are run with a larger number of samples. Selecting the number of expensive and inexpensive simulations carefully, one can show that MLMC methods can provide better accuracy
at the same cost as MC methods. In our simulations, our goal is twofold. First, we would like to compare NLSO and LSO mixed MsFEMs. In particular, we show that NLSO
mixed MsFEM is more accurate compared to LSO mixed MsFEM. Further, we use both approaches in the context of MLMC to speed-up MC
calculations. We present basic aspects of the algorithm and numerical
results for coupled flow and transport in heterogeneous porous media.

Simulation of multibody systems (mbs) is an inherent part in developing and design of complex mechanical systems. Moreover, simulation during operation gained in importance in the recent years, e.g. for HIL-, MIL- or monitoring applications. In this paper we discuss the numerical simulation of multibody systems on different platforms. The main section of this paper deals with the simulation of an established truck model [9] on different platforms, one microcontroller and two real-time processor boards. Additional to numerical C-code the latter platforms provide the possibility to build the model with a commercial mbs tool, which is also investigated. A survey of different ways of generating code and equations of mbs models is given and discussed concerning handling, possible limitations as well as performance. The presented benchmarks are processed under terms of on-board real time applications. A further important restriction, caused by the real-time requirement, is a fixed integration step size. Whence, carefully chosen numerical integration algorithms are necessary, especially in the case of closed loops in the model. We investigate linearly-implicit time integration methods with fixed step size, so-called Rosenbrock methods, and compare them with respect to their accuracy and performance on the tested processors.

The modelling of hedge funds poses a difficult problem since the available reported data sets are often small and incomplete. We propose a switching regression model for hedge funds, in which the coefficients are able to switch between different regimes. The coefficients are governed by a Markov chain in discrete time. The different states of the Markov chain represent different states of the economy, which influence the performance of the independent variables. Hedge fund indices are chosen as regressors. The parameter estimation for the switching parameter as well as for the switching error term is done through a filtering technique for hidden Markov models developed by Elliott (1994). Recursive parameter estimates are calculated through a filter-based EM-algorithm, which uses the hidden information of the underlying Markov chain. Our switching regression model is applied on hedge fund series and hedge fund indices from the HFR database.

Traditional methods fail for the purpose of simulating the complete flow process in urban areas as a consequence of heavy rainfall and as required by the European Standard EN-752 since the bi-directional coupling between sewer and surface is not properly handled. The methodology, developed in the BMBF/ EUREKA-project RisUrSim, solves this problem by carrying out the runoff on the basis of shallow water equations solved on high-resolution surface grids. Exchange nodes between the sewer and the surface, like inlets and manholes, are located in the computational grid and water leaving the sewer in case of surcharge is further distributed on the surface. So far, it has been a problem to get the dense topographical information needed to build models suitable for hydrodynamic runoff calculation in urban areas. Recent airborne data collection methods like laser scanning, however, offer a great chance to economically gather densely sampled input data. This paper studies the potential of such laser-scan data sets for urban water hydrodynamics.

Finite difference discretizations of 1D poroelasticity equations with discontinuous coefficients are analyzed. A recently suggested FD discretization of poroelasticity equations with constant coefficients on staggered grid, [5], is used as a basis. A careful treatment of the interfaces leads to harmonic averaging of the discontinuous coefficients. Here, convergence for the pressure and for the displacement is proven in certain norms for the scheme with harmonic averaging (HA). Order of convergence 1.5 is proven for arbitrary located interface, and second order convergence is proven for the case when the interface coincides with a grid node. Furthermore, following the ideas from [3], modified HA discretization are suggested for particular cases. The velocity and the stress are approximated with second order on the interface in this case. It is shown that for wide class of problems, the modified discretization provides better accuracy. Second order convergence for modified scheme is proven for the case when the interface coincides with a displacement grid node. Numerical experiments are presented in order to illustrate our considerations.

Two-level domain decomposition preconditioner for 3D flows in anisotropic highly heterogeneous porous media is presented. Accurate finite volume discretization based on multipoint flux approximation (MPFA) for 3D pressure equation is employed to account for the jump discontinuities of full permeability tensors. DD/MG type preconditioner for above mentioned problem is developed. Coarse scale operator is obtained from a homogenization type procedure. The influence of the overlapping as well as the influence of the smoother and cell problem formulation is studied. Results from numerical experiments are presented and discussed.

An efficient approach for calculating the effective heat conductivity for a class of industrial composite materials, such as metal foams, fibrous glass materials, and the like, is discussed. These materials, used in insulation or in advanced heat exchangers, are characterized by a low volume fraction of the highly conductive material (glass or metal) having a complex, network-like structure and by a large volume fraction of the insulator (air). We assume that the composite materials have constant macroscopic thermal conductivity tensors, which in principle can be obtained by standard up-scaling techniques, that use the concept of representative elementary volumes (REV), i.e. the effective heat conductivities of composite media can be computed by post-processing the solutions of some special cell problems for REVs. We propose, theoretically justify, and numerically study an efficient approach for calculating the effective conductivity for media for which the ratio of low and high conductivities satisfies 1. In this case one essentially only needs to solve the heat equation in the region occupied by the highly conductive media. For a class of problems we show, that under certain conditions on the microscale geometry, the proposed approach produces an upscaled conductivity that is O() close to the exact upscaled permeability. A number of numerical experiments are presented in order to illustrate the accuracy and the limitations of the proposed method. Applicability of the presented approach to upscaling other similar problems, e.g. flow in fractured porous media, is also discussed.

A number of water flow problems in porous media are modelled by Richards’ equation [1]. There exist a lot of different applications of this model. We are concerned with the simulation of the pressing section of a paper machine. This part of the industrial process provides the dewatering of the paper layer by the use of clothings, i.e. press felts, which absorb the water during pressing [2]. A system of nips are formed in the simplest case by rolls, which increase sheet dryness by pressing against each other (see Figure 1). A lot of theoretical studies were done for Richards’ equation (see [3], [4] and references therein). Most articles consider the case of x-independent coefficients. This simplifies the system considerably since, after Kirchhoff’s transformation of the problem, the elliptic operator becomes linear. In our case this condition is not satisfied and we have to consider nonlinear operator of second order. Moreover, all these articles are concerned with the nonstationary problem, while we are interested in the stationary case. Due to complexity of the physical process our problem has a specific feature. An additional convective term appears in our model because the porous media moves with the constant velocity through the pressing rolls. This term is zero in immobile porous media. We are not aware of papers, which deal with such kind of modified steady Richards’ problem. The goal of this paper is to obtain the stability results, to show the existence of a solution to the discrete problem, to prove the convergence of the approximate solution to the weak solution of the modified steady Richards’ equation, which describes the transport processes in the pressing section. In Section 2 we present the model which we consider. In Section 3 a numerical scheme obtained by the finite volume method is given. The main part of this paper is theoretical studies, which are given in Section 4. Section 5 presents a numerical experiment. The conclusion of this work is given in Section 6.

A new stability preserving model reduction algorithm for discrete linear SISO-systems based on their impulse response is proposed. Similar to the Padé approximation, an equation system for the Markov parameters involving the Hankel matrix is considered, that here however is chosen to be of very high dimension. Although this equation system therefore in general cannot be solved exactly, it is proved that the approximate solution, computed via the Moore-Penrose inverse, gives rise to a stability preserving reduction scheme, a property that cannot be guaranteed for the Padé approach. Furthermore, the proposed algorithm is compared to another stability preserving reduction approach, namely the balanced truncation method, showing comparable performance of the reduced systems. The balanced truncation method however starts from a state space description of the systems and in general is expected to be more computational demanding.