## Berichte der Arbeitsgruppe Technomathematik (AGTM Report)

### Refine

#### Year of publication

- 1997 (14) (remove)

#### Keywords

- Anisotropic smoothness classes (1)
- Dense gas (1)
- Elliptic-parabolic equation (1)
- Enskog equation (1)
- Kohonen's SOM (1)
- Locally stationary processes (1)
- Neural networks (1)
- Panel clustering (1)
- adaptive estimation (1)
- asymptotic analysis (1)

- 184
- The mathematical simulation of the liquid transport in a multilayered nonwoven (1997)
- In this report we treat an optimization task, which should make the choice of nonwoven for making diapers faster. A mathematical model for the liquid transport in nonwoven is developed. The main attention is focussed on the handling of fully and partially saturated zones, which leads to a parabolic-elliptic problem. Finite-difference schemes are proposed for numerical solving of the differential problem. Paralle algorithms are considered and results of numerical experiments are given.

- 183
- Spherical panel clustering and its numerical aspects (1997)
- In modern approximation methods linear combinations in terms of (space localizing) radial basis functions play an essential role. Areas of application are numerical integration formulas on the uni sphere omega corresponding to prescribed nodes, spherical spline interpolation, and spherical wavelet approximation. the evaluation of such a linear combination is a time consuming task, since a certain number of summations, multiplications and the calculation of scalar products are required. This paper presents a generalization of the panel clustering method in a spherical setup. The economy and efficiency of panel clustering is demonstrated for three fields of interest, namely upward continuation of the earth's gravitational potential, geoid computation by spherical splines and wavelet reconstruction of the gravitational potential.

- 174
- Regularization Wavelets and Multiresolution (1997)
- Many problems arising in (geo)physics and technology can be formulated as compact operator equations of the first kind \(A F = G\). Due to the ill-posedness of the equation a variety of regularization methods are in discussion for an approximate solution, where particular emphasize must be put on balancing the data and the approximation error. In doing so one is interested in optimal parameter choice strategies. In this paper our interest lies in an efficient algorithmic realization of a special class of regularization methods. More precisely, we implement regularization methods based on filtered singular value decomposition as a wavelet analysis. This enables us to perform, e.g., Tikhonov-Philips regularization as multiresolution. In other words, we are able to pass over from one regularized solution to another one by adding or subtracting so-called detail information in terms of wavelets. It is shown that regularization wavelets as proposed here are efficiently applicable to a future problem in satellite geodesy, viz. satellite gravity gradiometry.

- 176
- Runge-Walsh Wavelet Approximation for (1997)
- Metaharmonic wavelets are introduced for constructing the solution of theHelmholtz equation (reduced wave equation) corresponding to Dirichlet's orNeumann's boundary values on a closed surface approach leading to exactreconstruction formulas is considered in more detail. A scale discrete version ofmultiresolution is described for potential functions metaharmonic outside theclosed surface and satisfying the radiation condition at infinity. Moreover, wediscuss fully discrete wavelet representations of band-limited metaharmonicpotentials. Finally, a decomposition and reconstruction (pyramid) scheme foreconomical numerical implementation is presented for Runge-Walsh waveletapproximation.

- 175
- An Asymptotic-Induced Scheme for Nonstationary Transport Equations in the Diffusive Limit (1997)
- An asymptotic-induced scheme for nonstationary transport equations with thediffusion scaling is developed. The scheme works uniformly for all ranges ofmean free paths. It is based on the asymptotic analysis of the diffusion limit ofthe transport equation. A theoretical investigation of the behaviour of thescheme in the diffusion limit is given and an approximation property is proven.Moreover, numerical results for different physical situations are shown and atheuniform convergence of the scheme is established numerically.

- 187
- A Numerical Method for Kinetic Semiconductor Equations in the Drift Diffusion limit (1997)
- An asymptotic-induced scheme for kinetic semiconductor equations with the diffusion scaling is developed. The scheme is based on the asymptotic analysis of the kinetic semiconductor equation. It works uniformly for all ranges of mean free paths. The velocity discretization is done using quadrature points equivalent to a moment expansion method. Numerical results for different physical situations are presented.

- 132
- Wavelet Thresholding in Anisotropic Function Classes and Application to Adaptive Estimation of Evolutionary Spectra (1997)
- We derive minimax rates for estimation in anisotropic smoothness classes. This rate is attained by a coordinatewise thresholded wavelet estimator based on a tensor product basis with separate scale parameter for every dimension. It is shown that this basis is superior to its one-scale multiresolution analog, if different degrees of smoothness in different directions are present.; As an important application we introduce a new adaptive wavelet estimator of the time-dependent spectrum of a locally stationary time series. Using this model which was resently developed by Dahlhaus, we show that the resulting estimator attains nearly the rate, which is optimal in Gaussian white noise, simultaneously over a wide range of smoothness classes. Moreover, by our new approach we overcome the difficulty of how to choose the right amount of smoothing, i.e. how to adapt to the appropriate resolution, for reconstructing the local structure of the evolutionary spectrum in the time-frequency plane.

- 177
- Industrial Mathematics - Ideas and Examples (1997)

- 186
- Industrial Mathematics - Ideas and Examples (1997)

- 180
- Grid-Free Particle Method for the Inhomogeneous Enskog Equation and its Application to a Riemann-Problem (1997)
- Starting from the mollified version of the Enskog equation for a hard-sphere fluid, a grid-free algorithm to obtain the solution is proposed. The algorithm is based on the finite pointset method. For illustration, it is applied to a Riemann problem. The shock-wave solution is compared to the results of Frezzotti and Sgarra where a good agreement is found.