On Clustering of Non-stationary Meteorological Time Series

A method for clustering of multidimensional non-stationary meteorological time series is presented. The approach is based on optimization of the regularized averaged clustering functional describing the quality of data representation in terms of several regression models and a metastable hidden process switching between them. Proposed numerical clustering algorithm is based on application of the finite element method (FEM) to the problem of non-stationary time series analysis. The main advantage of the presented algorithm compared to Hidden Markov Models (HMMs) and to finite mixture models is that no a priori assumptions about the probability model for the hidden and observed processes (e.g., Markovianity or stationarity) are necessary for the proposed method. Another attractive numerical feature of the discussed algorithm is the possibility to choose the optimal number of metastable clusters and a natural opportunity to control the fuzziness of the resulting decomposition a posteriory, based on the statistical distinguishability of the resulting persistent cluster states. The resulting FEM-K-trends algorithm is compared with some standard fuzzy clustering methods on toy model examples and on analysis of multidimensional historical temperature data locally in Europe and on the global temperature data set

Dual variable methods for mixed-hybrid finite element approximation of the potential fluid flow problem in porous media

Mixed-hybrid finite element discretization of Darcy's law and the continuity equation that describe the potential fluid flow problem in porous media leads to symmetric indefinite saddle-point problems. In this paper we consider solution techniques based on the computation of a null-space basis of the whole or of a part of the left lower off-diagonal block in the system matrix and on the subsequent iterative solution of a projected system. This approach is mainly motivated by the need to solve a sequence of such systems with the same mesh but different material properties. A fundamental cycle null-space basis of the whole off-diagonal block is constructed using the spanning tree of an associated graph. It is shown that such a basis may be theoretically rather ill-conditioned. Alternatively, the orthogonal null-space basis of the sub-block used to enforce continuity over faces can be easily constructed. In the former case, the resulting projected system is symmetric positive definite and so the conjugate gradient method can be applied. The projected system in the latter case remains indefinite and the preconditioned minimal residual method (or the smoothed conjugate gradient method) should be used. The theoretical rate of convergence for both algorithms is discussed and their efficiency is compared in numerical experiments. Copyright © 2006, Kent State University