Band 5, Heft 2

In this paper we show that space-mapping optimization can be understood in the framework of defect correction. Then, space-mapping algorithms can be seen as special cases of defect correction iteration. In order to analyze the properties of space mapping and the space-mapping function, we introduce the new concept of flexibility of the underlying models. The best space-mapping results are obtained for so-called equally flexible models. By introducing an a±ne operator as a left preconditioner, two models can be made equally flexible, at least in the neighborhood of a solution. This motivates an improved space-mapping (or manifold-mapping) algorithm. The left preconditioner complements traditional space mapping where only a right preconditioner is used. In the last section a few simple examples illustrate some of the phenomena analyzed in this paper.

We study numerical approximations of nonnegative solutions of the p-Laplacian equation with a nonlinear source. We describe when solutions of a semidiscretization in space exist globally in time and when they blow up in a finite time. We also find the blow-up rates and the blow-up sets by means of the discrete self-similar profiles.

In this paper we present a general derivation of kinetic models for traffic flows including different kinds of interaction rules. We show that most kinetic previously derived models can be cast in the actual formulation. The development of Monte-Carlo methods for direct simulation of kinetic models is considered as an initial step towards realistic and effcient computations of traffic phenomena. Monte-Carlo methods are developed for these kinetic models. Several numerical examples are computed and compared to the previously obtained solutions of the stationary equation.

The Perona|Malik equation for nonlinear di®usion is the well-known equation widely used in image processing. This equation can be applied with great success for selective smoothing of images, where the noise is represented by small gradients and the spurious edges are represented by the large ones. For a numerical solution of this equation we use the ¯nite volume method. We analyze convergence of a corresponding explicit finite volume scheme to its weak solution and show some numerical experiments. We suggest also some modi¯cations of this scheme to obtain better CPU time performance of the algorithm.

We apply a functional-discrete method with convergence rate not worse than that of a geometric series for an eigenvalue transmission problem with periodic boundary conditions. It is shown that the convergence rate increases with an increase in the ordinal number of trial eigenvalue. Based on asymptotic behavior of eigenvalues of the basic problem and the functional-discrete method, qualitative results concerning the arrangement of the eigenvalues of the original eigenvalue transmission problem with periodic boundary conditions are proved. A number of numerical examples are given to support the theory.