Adaptive BDDC in Three Dimensions

The adaptive BDDC method is extended to the selection of face constraints in three dimensions. A new implementation of the BDDC method is presented based on a global formulation without an explicit coarse problem, with massive parallelism provided by a multifrontal solver. Constraints are implemented by a projection and sparsity of the projected operator is preserved by a generalized change of variables. The effectiveness of the method is illustrated on several engineering problems.Comment: 28 pages, 9 figures, 9 table

Remarks on "Resolving isospectral `drums' by counting nodal domains"

In [3] the authors studied the 4-parameter family of isospectral flat 4-tori T^\pm(a,b,c,d) discovered by Conway and Sloane. With a particular method of counting nodal domains they were able to distinguish these tori (numerically) by computing the corresponding nodal sequences relative to a few explicit tuples (a,b,c,d). In this note we confirm the expectation expressed in [3] by proving analytically that their nodal count distinguishes any 4-tuple of distinct positive real numbers.Comment: 5 page

Fuzzy clustering: insights and a new approach

Fuzzy clustering extends crisp clustering in the sense that objects can belong to various clusters with different membership degrees at the same time, whereas crisp or deterministic clustering assigns each object to a unique cluster. The standard approach to fuzzy clustering introduces the so-called fuzzifier which controls how much clusters may overlap. In this paper we illustrate, how this fuzzifier can help to reduce the number of undesired local minima of the objective function that is associated with fuzzy clustering. Apart from this advantage, the fuzzifier has also some drawbacks that are discussed in this paper. A deeper analysis of the fuzzifier concept leads us to a more general approach to fuzzy clustering that can overcome the problems caused by the fuzzifier

BDDC and FETI-DP under Minimalist Assumptions

The FETI-DP, BDDC and P-FETI-DP preconditioners are derived in a particulary simple abstract form. It is shown that their properties can be obtained from only on a very small set of algebraic assumptions. The presentation is purely algebraic and it does not use any particular definition of method components, such as substructures and coarse degrees of freedom. It is then shown that P-FETI-DP and BDDC are in fact the same. The FETI-DP and the BDDC preconditioned operators are of the same algebraic form, and the standard condition number bound carries over to arbitrary abstract operators of this form. The equality of eigenvalues of BDDC and FETI-DP also holds in the minimalist abstract setting. The abstract framework is explained on a standard substructuring example.Comment: 11 pages, 1 figure, also available at http://www-math.cudenver.edu/ccm/reports

Robust rank correlation coefficients on the basis of fuzzy

The goal of this paper is to demonstrate that established rank correlation measures are not ideally suited for measuring rank correlation for numerical data that are perturbed by noise. We propose to use robust rank correlation measures based on fuzzy orderings. We demonstrate that the new measures overcome the robustness problems of existing rank correlation coe cients. As a rst step, this is accomplished by illustrative examples. The paper closes with an outlook on future research and applicationsPeer Reviewe

Learning fuzzy systems: an ojective function-approach

One of the most important aspects of fuzzy systems is that they are easily understandable and interpretable. This property, however, does not come for free but poses some essential constraints on the parameters of a fuzzy system (like the linguistic terms), which are sometimes overlooked when learning fuzzy system automatically from data. In this paper, an objective function-based approach to learn fuzzy systems is developed, taking these constraints explicitly into account. Starting from fuzzy c-means clustering, several modifications of the basic algorithm are proposed, affecting the shape of the membership functions, the partition of individual variables and the coupling of input space partitioning and local function approximation

Efficient Adaptive Elimination Strategies in Nonlinear FETI-DP Methods in Combination with Adaptive Spectral Coarse Spaces

Nonlinear FETI-DP (Finite Element Tearing and Interconnecting - Dual Primal) is a nonlinear nonoverlapping domain decomposition method (DDM) which has a superior nonlinear convergence behavior compared with classical Newton-Krylov-DDMs - at least for many problems. Its fast and robust nonlinear convergence is strongly influenced by the choice of the second level or, in other words, the choice of the coarse constraints. Additionally, the convergence is also affected by the choice of an elimination set, that is, a set of degrees of freedom which are eliminated nonlinearly before linearization. In this article, an adaptive coarse space is combined with a problem-dependent and residual-based choice of the elimination set. An efficient implementation exploiting sparse local saddle point problems instead of an explicit transformation of basis is used. Unfortunately, this approach makes a further adaption of the elimination sets necessary, that is, edges and faces with coarse constraints have to be either included in the elimination set completely or not at all. Different strategies to fulfill this additional constraint are discussed and compared with a solely residual-based approach. The latter approach has to be implemented with an explicit transformation of basis. In general, the residual which is used to choose the elimination set has to be transformed to a space which basis functions explicitly contain the coarse constraints. This is computationally expensive. Here, for the first time, it is suggested to use an approximation of the transformed residual instead to compute the elimination set

Dynamic data assigning assessment clustering of streaming data

Discovering interesting patterns or substructures in data streams is an important challenge in data mining. Clustering algorithm are very often applied to identify substructures, although they are designed to partition a data set. Another problem of clustering algorithms is that most of them are not designed for data streams. They assume that the data set to be analysed is already complete and will not be extended by new data. This paper discusses an extension of an algorithm that uses ideas from cluster analysis, but was designed to identify single clusters in large data sets without the necessity to partition the whole data set into clusters. The new extended version of this algorithm can applied to stream data and is able to identify new clusters in an incoming data stream. As a case study weather data are use