A Short Survey on Data Clustering Algorithms

With rapidly increasing data, clustering algorithms are important tools for data analytics in modern research. They have been successfully applied to a wide range of domains; for instance, bioinformatics, speech recognition, and financial analysis. Formally speaking, given a set of data instances, a clustering algorithm is expected to divide the set of data instances into the subsets which maximize the intra-subset similarity and inter-subset dissimilarity, where a similarity measure is defined beforehand. In this work, the state-of-the-arts clustering algorithms are reviewed from design concept to methodology; Different clustering paradigms are discussed. Advanced clustering algorithms are also discussed. After that, the existing clustering evaluation metrics are reviewed. A summary with future insights is provided at the end

Theory and implementation of H\mathcal{H}-matrix based iterative and direct solvers for Helmholtz and elastodynamic oscillatory kernels

In this work, we study the accuracy and efficiency of hierarchical matrix ($\mathcal{H}$-matrix) based fast methods for solving dense linear systems arising from the discretization of the 3D elastodynamic Green's tensors. It is well known in the literature that standard $\mathcal{H}$-matrix based methods, although very efficient tools for asymptotically smooth kernels, are not optimal for oscillatory kernels. $\mathcal{H}^2$-matrix and directional approaches have been proposed to overcome this problem. However the implementation of such methods is much more involved than the standard $\mathcal{H}$-matrix representation. The central questions we address are twofold. (i) What is the frequency-range in which the $\mathcal{H}$-matrix format is an efficient representation for 3D elastodynamic problems? (ii) What can be expected of such an approach to model problems in mechanical engineering? We show that even though the method is not optimal (in the sense that more involved representations can lead to faster algorithms) an efficient solver can be easily developed. The capabilities of the method are illustrated on numerical examples using the Boundary Element Method