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- Mehrskalenanalyse (3)
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- delay management problem (1)
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- multiscale approximation on regular telluroidal surfaces (1)
- never-meet property (1)
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In this paper we introduce a multiscale technique for the analysis of deformation phenomena of the Earth. Classically, the basis functions under use are globally defined and show polynomial character. In consequence, only a global analysis of deformations is possible such that, for example, the water load of an artificial reservoir is hardly to model in that way. Up till now, the alternative to realize a local analysis can only be established by assuming the investigated region to be flat. In what follows we propose a local analysis based on tools (Navier scaling functions and wavelets) taking the (spherical) surface of the Earth into account. Our approach, in particular, enables us to perform a zooming-in procedure. In fact, the concept of Navier wavelets is formulated in such a way that subregions with larger or smaller data density can accordingly be modelled with a higher or lower resolution of the model, respectively.

Caloric Restriction (CR) is the only intervention proven to retard aging and extend maximum lifespan in mammalians. A possible mechanism for the beneficial effects of CR is that the mild metabolic stress associated with CR induces cells to express stress proteins that increase their resistance to disease processes. In this article we therefore model the retardation of aging by dietary restriction within a mathematical framework. The resulting model comprises food intake, stress proteins, body growth and survival. We successfully applied our model to growth and survival data of mice exposed to different food levels.

Based on the well-known results of classical potential theory, viz. the limit and jump relations for layer integrals, a numerically viable and e±cient multiscale method of approximating the disturbing potential from gravity anomalies is established on regular surfaces, i.e., on telluroids of ellipsoidal or even more structured geometric shape. The essential idea is to use scale dependent regularizations of the layer potentials occurring in the integral formulation of the linearized Molodensky problem to introduce scaling functions and wavelets on the telluroid. As an application of our multiscale approach some numerical examples are presented on an ellipsoidal telluroid.

The mathematical formulation of many physical problems results in the task of inverting a compact operator. The only known sensible solution technique is regularization which poses a severe problem in itself. Classically one dealt with deterministic noise models and required both the knowledge of smoothness of the solution function and the overall error behavior. We will show that we can guarantee an asymptotically optimal regularization for a physically motivated noise model under no assumptions for the smoothness and rather weak assumptions on the noise behavior which can mostly obtained out of two input data sets. An application to the determination of the gravitational field out of satellite data will be shown.

In this paper we generalize the classical shortest path problem in two ways. We consider two objective functions and time-dependent data. The resulting problem, called the time-dependent bicriteria shortest path problem (TdBiSP), has several interesting practical applications, but has not gained much attention in the literature.

Porous media flow of polymers with Carreau law viscosities and their application to enhanced oil recovery (EOR) is considered. Applying the homogenization method leads to a nonlinear two-scale problem. In case of a small difference between the Carreau and the Newtonian case an asymptotic expansion based on the small deviation of the viscosity from the Newtonian case is introduced. For uni-directional pressure gradients, which is a reasonable assumption in applications like EOR, auxiliary problems to decouple the micro- from the macrovariables are derived. The microscopic flow field obtained by the proposed approach is compared to the solution of the two-scale problem. Finite element calculations for an isotropic and an anisotropic pore cell geometries are used to validate the accuracy and speed-up of the proposed approach. The order of accuracy has been studied by performing the simulations up to the third order expansion for the isotropic geometry.

The following two papers present recent developments in multiscale ocean circulation modeling and multiscale gravitational field modeling that have been presented at the 2nd International GOCE User Workshop 2004 in Frascati. Part A - Multiscale Modeling of Ocean Circulation In this paper the applicability of multiscale methods to oceanography is demonstrated. More precisely, we use convolutions with certain locally supported kernels to approximate the dynamic topography and the geostrophic flow. As data sets the French CLS01 data are used for the mean sea surface topography and are compared to the EGM96 geoid. Since those two data sets have very different levels of spatial resolutions the necessity of an interpolating or approximating tool is evident. Compared to the standard spherical harmonics approach, the strongly space localizing kernels improve the possibilities of local data analysis here. Part B - Multiscale Modeling from EIGEN-1S, EIGEN-2, EIGEN-GRACE01S, GGM01, UCPH2002_0.5, EGM96 Spherical wavelets have been developed by the Geomathematics Group Kaiserslautern for several years and have been successfully applied to georelevant problems. Wavelets can be considered as consecutive band-pass filters and allow local approximations. The wavelet transform can also be applied to spherical harmonic models of the Earth's gravitational field like the most up-to-date EIGEN-1S, EIGEN-2, EIGEN-GRACE01S, GGM01, UCPH2002_0.5, and the well-known EGM96. Thereby, wavelet coefficients arise. In this paper it is the aim of the Geomathematics Group to make these data available to other interested groups. These wavelet coefficients allow not only the reconstruction of the wavelet approximations of the gravitational potential but also of the geoid, of the gravity anomalies and other important functionals of the gravitational field. Different types of wavelets are considered: bandlimited wavelets (here: Shannon and Cubic Polynomial (CuP)) as well as non-bandlimited ones (in our case: Abel-Poisson). For these types wavelet coefficients are computed and wavelet variances are given. The data format of the wavelet coefficients is also included.

In the delay management problem we decide how to react in case of delays in public transportation. More specific, the question is if connecting vehicles should wait for delayed feeder vehicles or if it is better to depart in time. As objective we consider the convenience over all customers, expressed as the average delay of a customer when arriving at his destination.We present path-based and activity-based integer programming models for the delay management problem and show the equivalence of these formulations. Based on these, we present a simplification of the (cubic) activity-based model which results in an integer linear program. We identify cases in which this linearization is correct, namely if the so-called never-meet property holds. Fortunately, this property is often almost satisfied in our practical data. Finally, we show how to find an optimal solution in linear time in case of the never-meet property.