A smoothing-regularization approach to mathematical programs with vanishing constraints

We consider a numerical approach for the solution of a difficult class of optimization problems called mathematical programs with vanishing constraints. The basic idea is to reformulate the characteristic constraints of the program via a nonsmooth function and to eventually smooth it and regularize the feasible set with the aid of a certain smoothing- and regularization parameter t>0 such that the resulting problem is more tractable and coincides with the original program for t=0. We investigate the convergence behavior of a sequence of stationary points of the smooth and regularized problems by letting t tend to zero. Numerical results illustrating the performance of the approach are given. In particular, a large-scale example from topology optimization of mechanical structures with local stress constraints is investigated.     

On hedging in finite security markets

A market is considered where trading can take place only at discrete time points, the trading frequency cannot grow without bound, and the number of states of nature is finite. The main objectives of the paper are to show that the market can be completed also with highly correlated risky assets, and to describe an efficient algorithm to compute a self-financing hedging strategy. The algorithm consists off-line of a backwards recursion and on-line of the solution, in each period, of a system of linear equations; it is a consequence of a proof where, using a well-known mathematical property, it is shown that uniqueness of the martingale measure implies completeness also in our setting. The significance of ‘multistate’ models versus the familiar binomial model is discussed and it is shown how the evolution of prices of the (correlated) risky assets can be chosen so that a given probability measure is already the unique equivalent martingale measure.     

Multiresolution schemes for conservation laws

Summary. In recent years a variety of high–order schemes for the numerical solution of conservation laws has been developed. In general, these numerical methods involve expensive flux evaluations in order to resolve discontinuities accurately. But in large parts of the flow domain the solution is smooth. Hence in these regions an unexpensive finite difference scheme suffices. In order to reduce the number of expensive flux evaluations we employ a multiresolution strategy which is similar in spirit to an approach that has been proposed by A. Harten several years ago. Concrete ingredients of this methodology have been described so far essentially for problems in a single space dimension. In order to realize such concepts for problems with several spatial dimensions and boundary fitted meshes essential deviations from previous investigations appear to be necessary though. This concerns handling the more complex interrelations of fluxes across cell interfaces, the derivation of appropriate evolution equations for multiscale representations of cell averages, stability and convergence, quantifying the compression effects by suitable adapted multiscale transformations and last but not least laying grounds for ultimately avoiding the storage of data corresponding to a full global mesh for the highest level of resolution. The objective of this paper is to develop such ingredients for any spatial dimension and block structured meshes obtained as parametric images of Cartesian grids. We conclude with some numerical results for the two–dimensional Euler equations modeling hypersonic flow around a blunt body. Received June 24, 1998 / Revised version received February 21, 2000 / Published online November 8, 2000     

Elementare Eigenschaften lokaler–Polynom–Schätzer: ein Streifzug durch die lineare Algebra

Zusammenfassung. Der lokale–Polynom–Sch?tzer besteht darin, an eine Sequenz von aufeinanderfolgenden Datenpunkten lokal ein Polynom anzupassen und damit zuf?llige Schwankungen in den Daten wegzugl?tten. Dieser Ansatz gewinnt in der nichtparametrischen Kurvensch?tzung st?ndig an Bedeutung. Es wird gezeigt, wie die Herleitung einiger elementarer Eigenschaften dieses Handwerkszeuges des Datenanalytikers einen Streifzug durch die lineare Algebra erforderlich macht. Eingegangen am 27.03.1998 / Angenommen am 15.06.1998     

Manifolds with S 1-category 2 have cyclic fundamental groups

A closed topological n-manifold M ⁿ is of S ¹-category 2 if it can be covered by two open subsets W ₁, W ₂ such that the inclusions W _i → M ⁿ factor homotopically through maps W _i → S ¹. We show that for n > 3 the fundamental group of such an n-manifold is either trivial or infinite cyclic.