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

Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps

Coarse spaces are instrumental in obtaining scalability for domain decomposition methods for partial differential equations (PDEs). However, it is known that most popular choices of coarse spaces perform rather weakly in the presence of heterogeneities in the PDE coefficients, especially for systems of PDEs. Here, we introduce in a variational setting a new coarse space that is robust even when there are such heterogeneities. We achieve this by solving local generalized eigenvalue problems in the overlaps of subdomains that isolate the terms responsible for slow convergence. We prove a general theoretical result that rigorously establishes the robustness of the new coarse space and give some numerical examples on two and three dimensional heterogeneous PDEs and systems of PDEs that confirm this property

An integrative systematic framework helps to reconstruct skeletal evolution of glass sponges (Porifera, Hexactinellida)

BACKGROUND: Glass sponges (Class Hexactinellida) are important components of deep-sea ecosystems and are of interest from geological and materials science perspectives. The reconstruction of their phylogeny with molecular data has only recently begun and shows a better agreement with morphology-based systematics than is typical for other sponge groups, likely because of a greater number of informative morphological characters. However, inconsistencies remain that have far-reaching implications for hypotheses about the evolution of their major skeletal construction types (body plans). Furthermore, less than half of all described extant genera have been sampled for molecular systematics, and several taxa important for understanding skeletal evolution are still missing. Increased taxon sampling for molecular phylogenetics of this group is therefore urgently needed. However, due to their remote habitat and often poorly preserved museum material, sequencing all 126 currently recognized extant genera will be difficult to achieve. Utilizing morphological data to incorporate unsequenced taxa into an integrative systematics framework therefore holds great promise, but it is unclear which methodological approach best suits this task. RESULTS: Here, we increase the taxon sampling of four previously established molecular markers (18S, 28S, and 16S ribosomal DNA, as well as cytochrome oxidase subunit I) by 12 genera, for the first time including representatives of the order Aulocalycoida and the type genus of Dactylocalycidae, taxa that are key to understanding hexactinellid body plan evolution. Phylogenetic analyses suggest that Aulocalycoida is diphyletic and provide further support for the paraphyly of order Hexactinosida; hence these orders are abolished from the Linnean classification. We further assembled morphological character matrices to integrate so far unsequenced genera into phylogenetic analyses in maximum parsimony (MP), maximum likelihood (ML), Bayesian, and morphology-based binning frameworks. We find that of these four approaches, total-evidence analysis using MP gave the most plausible results concerning congruence with existing phylogenetic and taxonomic hypotheses, whereas the other methods, especially ML and binning, performed more poorly. We use our total-evidence phylogeny of all extant glass sponge genera for ancestral state reconstruction of morphological characters in MP and ML frameworks, gaining new insights into the evolution of major hexactinellid body plans and other characters such as different spicule types. CONCLUSIONS: Our study demonstrates how a comprehensive, albeit in some parts provisional, phylogeny of a larger taxon can be achieved with an integrative approach utilizing molecular and morphological data, and how this can be used as a basis for understanding phenotypic evolution. The datasets and associated trees presented here are intended as a resource and starting point for future work on glass sponge evolution

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

A Precision Measurement of pp Elastic Scattering Cross Sections at Intermediate Energies

We have measured differential cross sections for \pp elastic scattering with internal fiber targets in the recirculating beam of the proton synchrotron COSY. Measurements were made continuously during acceleration for projectile kinetic energies between 0.23 and 2.59 GeV in the angular range $30 \leq \theta_{c.m.} \leq 90$ deg. Details of the apparatus and the data analysis are given and the resulting excitation functions and angular distributions presented. The precision of each data point is typically better than 4%, and a relative normalization uncertainty of only 2.5% within an excitation function has been reached. The impact on phase shift analysis as well as upper bounds on possible resonant contributions in lower partial waves are discussed.Comment: 23 pages 29 figure

First Measurement of Antikaon Phase-Space Distributions in Nucleus-Nucleus Collisions at Subthreshold Beam Energies

Differential production cross sections of K$^-$ and K$^+$ mesons have been measured as function of the polar emission angle in Ni+Ni collisions at a beam energy of 1.93 AGeV. In near-central collisions, the spectral shapes and the widths of the rapidity distributions of K$^-$ and K$^+$ mesons are in agreement with the assumption of isotropic emission. In non-central collisions, the K$^-$ and K$^+$ rapidity distributions are broader than expected for a single thermal source. In this case, the polar angle distributions are strongly forward-backward peaked and the nonisotropic contribution to the total yield is about one third both for K$^+$ and K$^-$ mesons. The K$^-$/K$^+$ ratio is found to be about 0.03 independent of the centrality of the reaction. This value is significantly larger than predicted by microscopic transport calculations if in-medium modifications of K mesons are neglected.Comment: 16 pages, 3 figures, accepted for publication in Physics Letters

Refined saddle-point preconditioners for discretized Stokes problems

This paper is concerned with the implementation of efficient solution algorithms for elliptic problems with constraints. We establish theory which shows that including a simple scaling within well-established block diagonal preconditioners for Stokes problems can result in significantly faster convergence when applying the preconditioned MINRES method. The codes used in the numerical studies are available online