We propose a technique for a training set approximation and its usage in kernel methods. The approach aims to represent data in a low dimensional space with possibly minimal representation error which is similar to the Principal Component Analysis (PCA). In contrast to the PCA, the basis vectors of the low dimensional space used for data representation are properly selected vectors from the tra...

We discuss various analytical approximation methods for following the evolution of cosmological density perturbations into the strong (i.e. nonlinear) clustering regime. We start by giving a thorough treatment of linear gravitational instability in cosmological models and discussing the statistics of primordial density fluctuations produced in various scenarios of structure formation, and the r...

In this paper we propose a cross-validation selection criterion to determine asymptotically the correct model among the family of all possible partially linear models when the underlying model is a partially linear model. We establish the asymptotic consistency of the criterion. In addition, the criterion is illustrated using two real sets of data. © 2001 Elsevier Science

Anisotropic triangulations are utilized in recent methods for sparse representation and adaptive approximation of image data. This article first addresses selected computational aspects concerning image approximation on triangular meshes, before four recent image approximation algorithms, each relying on anisotropic triangulations, are discussed. The discussion includes generic triangulations o...

The numerical result provided by an approximation method is affected by a global error, which consists of both a truncation error and a round-off error. Let us consider the converging sequence generated by successively dividing by two the step size used. If computations are performed until, in the convergence zone, the difference between two successive approximations is only due to round-off er...

Acknowledgements The organizers would like to thank the Hausdorff Center for Mathematics at the University of Bonn and the Institute of Mathematics at the University of Lübeck for the generous financial support, which made this workshop possible. In addition, they also acknowledge the support by the German Research Foundation (DFG) within the Emmy-Noether program and their kind assistance in va...

The stretching energy functional and the bending energy functional are widely used for approximating the solution of the Plateau-Béizer Problem. This paper presents another two simple methods by using the extended stretching energy functional and the extended bending energy functional. The resulting surface obtained by the new methods will have a smaller area. Comparisons are made with both the...

Approximation theory and functional analysis share many common problems and points of contact. One of the areas of mutual interest is that of density results. In this paper we briefly survey various methods and results in this area starting from work of Weierstrass and Riesz, and extending to more recent times.

Reaction–diffusion equations are commonly used in different science and engineering fields to describe spatial patterns arising from the interaction of chemical or biochemical reactions and diffusive transport mechanisms. The aim of this work is to show that a Green’s function formulation of reaction–diffusion PDEs is a suitable framework to derive FD schemes incorporating both O(h2) accuracy a...