TY - GEN A1 - Galliat, Tobias A1 - Deuflhard, Peter T1 - Adaptive hierarchical cluster analysis by Self-Organizing Box Maps N2 - The present paper aims at an extension of {\sc Kohonen's} Self-Organizing Map (SOM) algorithm to be called Self-Organizing Box Map (SOBM) algorithm; it generates box codebooks in lieu of point codebooks. Box codebooks just like point codebooks indirectly define a Voronoi tessellation of the input space, so that each codebook vector represents a unique set of points. Each box codebook vector comprises a multi-dimensional interval that approximates the related partition of the Voronoi tessellation. Upon using the automated cluster identification method that has recently been developed by the authors, the codebook vectors can be grouped in such a way that each group represents a point cluster in the input space. Since the clustering usually depends on the size of the SOM, one cannot be sure, whether the clustering comes out to be optimal. Refinement of part of the identified clusters would often improve the results. This paper presents the concept of an adaptive multilevel cluster algorithm that performs such refinements automatically. Moreover the paper introduces a concept of essential dimensions and suggests a method for their identification based on our herein suggested box codebooks. Applications of the algorithm to molecular dynamics will be described in a forthcoming paper. T3 - ZIB-Report - 00-13 KW - Self-Organizing Maps KW - cluster analysis KW - Voronoi tessellation KW - feature extraction KW - essential dimensions KW - multilevel methods Y1 - 2000 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-5815 ER - TY - GEN A1 - Bornemann, Folkmar A. A1 - Yserentant, Harry T1 - A Basic Norm Equivalence for the Theory of Multilevel Methods. N2 - Subspace decompositions of finite element spaces based on $L2$-like orthogonal projections play an important role for the construction and analysis of multigrid like iterative methods. Recently several authors proved the equivalence of the associated discrete norms with the $H^1$-norm. The present report gives an elementary, self-contained derivation of this result which is based on the use of $ K$-functionals known from the theory of interpolation spaces. {\bf Keywords:} multilevel methods, nonuniform meshes, optimal convergence rates. {\bf AMS(MOS) Subject classifications:} 65N55, 65N30, 65N50. T3 - ZIB-Report - SC-92-01 KW - multilevel methods KW - nonuniform meshes KW - optimal convergence rates Y1 - 1992 U6 - https://doi.org/10.1007/BF01388699 N1 - No preprint available ER -