## Berichte des Fraunhofer-Instituts für Techno- und Wirtschaftsmathematik (ITWM Report)

179

Classical geometrically exact Kirchhoff and Cosserat models are used to study the nonlinear deformation of rods. Extension, bending and torsion of the rod may be represented by the Kirchhoff model. The Cosserat model additionally takes into account shearing effects. Second order finite differences on a staggered grid define discrete viscoelastic versions of these classical models. Since the rotations are parametrised by unit quaternions, the space discretisation results in differential-algebraic equations that are solved numerically by standard techniques like index reduction and projection methods. Using absolute coordinates, the mass and constraint matrices are sparse and this sparsity may be exploited to speed-up time integration. Further improvements are possible in the Cosserat model, because the constraints are just the normalisation conditions for unit quaternions such that the null space of the constraint matrix can be given analytically. The results of the theoretical investigations are illustrated by numerical tests.

174

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.

173

Home Health Care (HHC) services are becoming increasingly important in Europe’s aging societies. Elderly people have varying degrees of need for assistance and medical treatment. It is advantageous to allow them to live in their own homes as long as possible, since a long-term stay in a nursing home can be much more costly for the social insurance system than a treatment at home providing assistance to the required level. Therefore, HHC services are a cost-effective and flexible instrument in the social system. In Germany, organizations providing HHC services are generally either larger charities with countrywide operations or small private companies offering services only in a city or a rural area. While the former have a hierarchical organizational structure and a large number of employees, the latter typically only have some ten to twenty nurses under contract. The relationship to the patients (“customers”) is often long-term and can last for several years. Therefore acquiring and keeping satisfied customers is crucial for HHC service providers and intensive competition among them is observed.

172

The understanding of the motion of long slender elastic fibers in turbulent flows is of great interest to research, development and production in technical textiles manufacturing. The fiber dynamics depend on the drag forces that are imposed on the fiber by the fluid. Their computation requires in principle a coupling of fiber and flow with no-slip interface conditions. However, theneeded high resolution and adaptive grid refinement make the direct numerical simulation of the three-dimensional fluid-solid-problem for slender fibers and turbulent flows not only extremely costly and complex, but also still impossible for practically relevant applications. Embedded in a slender body theory, an aerodynamic force concept for a general drag model was therefore derived on basis of a stochastic k-o; description for a turbulent flow field in [23]. The turbulence effects on the fiber dynamics were modeled by a correlated random Gaussian force and its asymptotic limit on a macroscopic fiber scale by Gaussian white noise with flow-dependent amplitude. The concept was numerically studied under the conditions of a melt-spinning process for nonwoven materials in [24] – for the specific choice of a non-linear Taylor drag model. Taylor [35] suggested the heuristic model for high Reynolds number flows, Re in [20, 3 · 105], around inclined slender objects under an angle of attack of alpha in (pi/36, pi/2] between flow and object tangent. Since the Reynolds number is considered with respect to the relative velocity between flow and fiber, the numerical results lackaccuracy evidently for small Re that occur in cases of flexible light fibers moving occasionally with the flow velocity. In such a regime (Re << 1), linear Stokes drag forces were successfully applied for the prediction of small particles immersed in turbulent flows, see e.g. [25, 26, 32, 39], a modifiedStokes force taking also into account the particle oscillations was presented in [14]. The linear drag relation was also conferred to longer filaments by imposing free-draining assumptions [29, 8]. Apart from this, the Taylor drag suffers from its non-applicability to tangential incident flow situations (alpha = 0) that often occur in fiber and nonwoven production processes.

171

We present a parsimonious multi-asset Heston model. All single-asset submodels follow the well-known Heston dynamics and their parameters are typically calibrated on implied market volatilities. We focus on the calibration of the correlation structure between the single-asset marginals in the absence of sucient liquid cross-asset option price data. The presented model is parsimonious in the sense that d(d􀀀1)=2 asset-asset cross-correlations are required for a d-asset Heston model. In order to calibrate the model, we present two general setups corresponding to relevant practical situations: (1) when the empirical cross-asset correlations in the risk neutral world are given by the user and we need to calibrate the correlations between the driving Brownian motions or (2) when they have to be estimated from the historical time series. The theoretical background, including the ergodicity of the multidimensional CIR process, for the proposed estimators is also studied.

170

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.

169

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.

168

A general multi-period network redesign problem arising in the context of strategic supply chain planning (SCP) is studied. Several aspects of practical relevance in SCP are captured namely, multiple facility layers with different types of facilities, flows between facilities in the same layer, direct shipments to customers, and facility relocation. An efficient two-phase heuristic approach is proposed for obtaining feasible solutions to the problem, which is initially modeled as a large-scale mixed-integer linear program. In the first stage of the heuristic, a linear programming rounding strategy is applied to second initial values for the binary location variables in the model. The second phase of the heuristic uses local search to correct the initial solution when feasibility is not reached or to improve the solution when its quality does not meet given criteria. The results of an extensive computational study performed on randomly generated instances are reported.

167

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].

166

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.