Kaiserslautern - Fachbereich Mathematik
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- 1993 (28) (remove)
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- Families of Probability Measures (1)
- Information Theory (1)
- Low-discrepancy sequences (1)
- Monte Carlo method (1)
- Optimal Prior Distribution (1)
- Shannon-Capacity (1)
- Statistical Experiments (1)
- Structure Theory (1)
- Van Neumann-Kakutani transformation (1)
- discrete measure (1)
- distribution (1)
- low discrepancy (1)
- particle method (1)
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Questions arising from Statistical Decision Theory, Bayes Methods and other probability theoretic fields lead to concepts of orthogonality of a family of probability measures. In this paper we therefore give a sketch of a generalized information theory which is very helpful in considering and answering those questions. In this adapted information theory Shannon's classical transition channels modelled by finite stochastic matrices are replaced by compact families of probability measures that are uniformly integrable. These channels are characterized by concepts such as information rate and capacity and by optimal priors and the optimal mixture distribution. For practical studies we introduce an algorithm to calculate the capacity of the whole probability family which is appli cable even for general output space. We then explain how the algorithm works and compare its numerical costs with those of the classical Arimoto-Blahut-algorithm.
The system of shallow water waves is one of the classical examples for nonlinear, twodimensional conservation laws. The paper investigates a simple kinetic equation depending on a parameter e which leads for e to 0 to the system of shallow water waves. The corresponding equilibrium distribution function has a compact support which depends on the eigenvalues of the hyperbolic system. It is shown that this kind of kinetic approach is restricted to a special class of nonlinear conservation laws. The kinetic model is used to develop a simple particle method for the numerical solution of shallow water waves. The particle method can be implemented in a straightforward way and produces in test examples sufficiently accurate results.
The paper presents a fast implementation of a constructive method to generate a special class of low-discrepancy sequences which are based on Van Neumann-Kakutani tranformations. Such sequences can be used in various simulation codes where it is necessary to generate a certain number of uniformly distributed random numbers on the unit interval.; From a theoretical point of view the uniformity of a sequence is measured in terms of the discrepancy which is a special distance between a finite set of points and the uniform distribution on the unit interval.; Numerical results are given on the cost efficiency of different generators on different hardware architectures as well as on the corresponding uniformity of the sequences. As an example for the efficient use of low-discrepancy sequences in a complex simulation code results are presented for the simulation of a hypersonic rarefied gas flow.
This paper considers the numerical solution of a transmission boundary-value problem for the time-harmonic Maxwell equations with the help of a special finite volume discretization. Applying this technique to several three-dimensional test problems, we obtain large, sparse, complex linear systems, which are solved by using BiCG, CGS, BiCGSTAB resp., GMRES. We combine these methods with suitably chosen preconditioning matrices and compare the speed of convergence.
In these lectures we will mainly treat a billard game. Our particles will be hard spheres. Not always: We will also touch cases, where particles have interior energies due to rotation or vibration, which they exchange in a collision, and we will talk about chemical reactions happening during a collision. But many essential aspects occur already in the billard case which will be therefore paradigmatic. I do not know enough about semiconductors to handle collisions there - the Boltzmann case is certainly different but may give some idea even for the other cases.
Discrete families of functions with the property that every function in a certain space can be represented by its formal Fourier series expansion are developed on the sphere. A Fourier series type expansion is obviously true if the family is an orthonormal basis of a Hilbert space, but it also can hold in situations where the family is not orthogonal and is overcomplete. Furthermore, all functions in our approach are axisymmetric (depending only on the spherical distance) so that they can be used adequately in (rotation) invariant pseudodifferential equations on the frames (ii) Gauss- Weierstrass frames, and (iii) frames consisting of locally supported kernel functions. Abel-Poisson frames form families of harmonic functions and provide us with powerful approximation tools in potential theory. Gauss-Weierstrass frames are intimately related to the diffusion equation on the sphere and play an important role in multiscale descriptions of image processing on the sphere. The third class enables us to discuss spherical Fourier expansions by means of axisymmetric finite elements.
Spline functions that interpolate data given on the sphere are developed in a weighted Sobolev space setting. The flexibility of the weights makes possible the choice of the approximating function in a way which emphasizes attributes desirable for the particular application area. Examples show that certain choices of the weight sequences yield known methods. A pointwise convergence theorem containing explicit constants yields a useable error bound.