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

Analysis of contingency tables based on generalised median polish with power transformations and non-additive models

Contingency tables are a very common basis for the investigation of effects of different treatments or influences on a disease or the health state of patients. Many journals put a strong emphasis on p-values to support the validity of results. Therefore, even small contingency tables are analysed by techniques like t-test or ANOVA. Both these concepts are based on normality assumptions for the underlying data. For larger data sets, this assumption is not so critical, since the underlying statistics are based on sums of (independent) random variables which can be assumed to follow approximately a normal distribution, at least for a larger number of summands. But for smaller data sets, the normality assumption can often not be justified. Robust methods like the Wilcoxon-Mann-Whitney-U test or the Kruskal-Wallis test do not lead to statistically significant p-values for small samples. Median polish is a robust alternative to analyse contingency tables providing much more insight than just a p-value. Median polish is a technique that provides more information than just a p-value. It explains the contingency table in terms of an overall effect, row and columns effects and residuals. The underlying model for median polish is an additive model which is sometimes too restrictive. In this paper, we propose two related approach to generalise median polish. A power transformation can be applied to the values in the table, so that better results for median polish can be achieved. We propose a graphical method how to find a suitable power transformation. If the original data should be preserved, one can apply other transformations – based on so-called additive generators – that have an inverse transformation. In this way, median polish can be applied to the original data, but based on a non-additive model. The non-linearity of such a model can also be visualised to better understand the joint effects of rows and columns in a contingency table

Infection- and procedure-dependent effects on pulmonary gene expression in the early phase of influenza A virus infection in mice

BACKGROUND: Investigating the host response in the early stage of influenza A virus (IAV) infection is of considerable interest. However, it is conceivable that effects due to the anesthesia and/or intranasal infection procedure might introduce artifacts. We therefore aimed to evaluate the effects of anesthesia and/or intranasal infection on transcription of selected pulmonary mRNAs in two inbred mouse strains with differential susceptibility to IAV infection. RESULTS: DBA/2J and C57BL/6J mice were evaluated in a time course experiment in which lung tissue was sampled after 6, 12, 18, 24, 48 and 120 h. After anesthesia with ketamine and xylazine, a suspension of mouse-adapted IAV strain PR8_Mun in 20 μl sterile buffer, or 20 μl sterile buffer only, was instilled intranasally. The mice receiving anesthesia and PBS only were designated the “mock treatment” group. Pulmonary expression of 10 host mRNAs (Fos, Retnla, Irg1, Il6, Il1b, Cxcl10, Stat1, Ifng, Ifnl2, and Mx1) and viral hemagglutinin (HA) mRNA were determined at the designated time points. As expected, weight loss and viral replication were greater in the DBA/2J strain (which is more susceptible to IAV infection). Four mRNAs (Retnla, Irg1, Il6, and Cxcl10) were procedure-dependently regulated in DBA/2J mice between 6 and 24 h, and two (Retnla and Il6) in C57BL/6J mice, although to a lesser extent. All 10 mRNAs rose after infection, but one (Fos) only in DBA/2J mice. These infection-dependent effects could be separated from procedure-dependent effects beginning around 12 h in DBA/2J and 18 h in C57BL/6J mice. The interferon-related mRNAs Stat1, Ifng, Infl2, and Mx1 were unaffected by mock treatment in either mouse strain. Mx1 and Infl2 correlated best with HA mRNA expression (r = 0.97 and 0.93, respectively, in DBA/2J). CONCLUSIONS: These results demonstrate effects of the anesthesia and/or intranasal infection procedure on pulmonary gene expression, which are detectable between approximately 6 and 24 h post procedure and vary in intensity and temporal evolution depending on the mouse strain used. Mock infection controls should be included in all studies on pulmonary gene expression in the early phase of infection with IAV and, likely, other respiratory pathogens

Parallel implementation of Multilevel BDDC

In application of the Balancing Domain Decomposition by Constraints (BDDC) to a case with many substructures, solving the coarse problem exactly becomes the bottleneck which spoils scalability of the solver. However, it is straightforward for BDDC to substitute the exact solution of the coarse problem by another step of BDDC method with subdomains playing the role of elements. In this way, the algorithm of three-level BDDC method is obtained. If this approach is applied recursively, multilevel BDDC method is derived. We present a detailed description of a recently developed parallel implementation of this algorithm. The implementation is applied to an engineering problem of linear elasticity and a benchmark problem of Stokes flow in a cavity. Results by the multilevel approach are compared to those by the standard (two-level) BDDC method.Comment: 9 pages, 2 figures, 3 table

On dual Schur domain decomposition method for linear first-order transient problems

This paper addresses some numerical and theoretical aspects of dual Schur domain decomposition methods for linear first-order transient partial differential equations. In this work, we consider the trapezoidal family of schemes for integrating the ordinary differential equations (ODEs) for each subdomain and present four different coupling methods, corresponding to different algebraic constraints, for enforcing kinematic continuity on the interface between the subdomains. Method 1 (d-continuity) is based on the conventional approach using continuity of the primary variable and we show that this method is unstable for a lot of commonly used time integrators including the mid-point rule. To alleviate this difficulty, we propose a new Method 2 (Modified d-continuity) and prove its stability for coupling all time integrators in the trapezoidal family (except the forward Euler). Method 3 (v-continuity) is based on enforcing the continuity of the time derivative of the primary variable. However, this constraint introduces a drift in the primary variable on the interface. We present Method 4 (Baumgarte stabilized) which uses Baumgarte stabilization to limit this drift and we derive bounds for the stabilization parameter to ensure stability. Our stability analysis is based on the ``energy'' method, and one of the main contributions of this paper is the extension of the energy method (which was previously introduced in the context of numerical methods for ODEs) to assess the stability of numerical formulations for index-2 differential-algebraic equations (DAEs).Comment: 22 Figures, 49 pages (double spacing using amsart

Clustering of nonstationary data streams: a survey of fuzzy partitional methods

YesData streams have arisen as a relevant research topic during the past decade. They are real‐time, incremental in nature, temporally ordered, massive, contain outliers, and the objects in a data stream may evolve over time (concept drift). Clustering is often one of the earliest and most important steps in the streaming data analysis workflow. A comprehensive literature is available about stream data clustering; however, less attention is devoted to the fuzzy clustering approach, even though the nonstationary nature of many data streams makes it especially appealing. This survey discusses relevant data stream clustering algorithms focusing mainly on fuzzy methods, including their treatment of outliers and concept drift and shift.Ministero dell‘Istruzione, dell‘Universitá e della Ricerca

Multispace and Multilevel BDDC

BDDC method is the most advanced method from the Balancing family of iterative substructuring methods for the solution of large systems of linear algebraic equations arising from discretization of elliptic boundary value problems. In the case of many substructures, solving the coarse problem exactly becomes a bottleneck. Since the coarse problem in BDDC has the same structure as the original problem, it is straightforward to apply the BDDC method recursively to solve the coarse problem only approximately. In this paper, we formulate a new family of abstract Multispace BDDC methods and give condition number bounds from the abstract additive Schwarz preconditioning theory. The Multilevel BDDC is then treated as a special case of the Multispace BDDC and abstract multilevel condition number bounds are given. The abstract bounds yield polylogarithmic condition number bounds for an arbitrary fixed number of levels and scalar elliptic problems discretized by finite elements in two and three spatial dimensions. Numerical experiments confirm the theory.Comment: 26 pages, 3 figures, 2 tables, 20 references. Formal changes onl

On the Nodal Count Statistics for Separable Systems in any Dimension

We consider the statistics of the number of nodal domains aka nodal counts for eigenfunctions of separable wave equations in arbitrary dimension. We give an explicit expression for the limiting distribution of normalised nodal counts and analyse some of its universal properties. Our results are illustrated by detailed discussion of simple examples and numerical nodal count distributions.Comment: 21 pages, 4 figure

Stability of nodal structures in graph eigenfunctions and its relation to the nodal domain count

The nodal domains of eigenvectors of the discrete Schrodinger operator on simple, finite and connected graphs are considered. Courant's well known nodal domain theorem applies in the present case, and sets an upper bound to the number of nodal domains of eigenvectors: Arranging the spectrum as a non decreasing sequence, and denoting by $\nu_n$ the number of nodal domains of the $n$'th eigenvector, Courant's theorem guarantees that the nodal deficiency $n-\nu_n$ is non negative. (The above applies for generic eigenvectors. Special care should be exercised for eigenvectors with vanishing components.) The main result of the present work is that the nodal deficiency for generic eigenvectors equals to a Morse index of an energy functional whose value at its relevant critical points coincides with the eigenvalue. The association of the nodal deficiency to the stability of an energy functional at its critical points was recently discussed in the context of quantum graphs [arXiv:1103.1423] and Dirichlet Laplacian in bounded domains in $R^d$ [arXiv:1107.3489]. The present work adapts this result to the discrete case. The definition of the energy functional in the discrete case requires a special setting, substantially different from the one used in [arXiv:1103.1423,arXiv:1107.3489] and it is presented here in detail.Comment: 15 pages, 1 figur