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

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

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

Global Diversity of Sponges (Porifera)

With the completion of a single unified classification, the Systema Porifera (SP) and subsequent development of an online species database, the World Porifera Database (WPD), we are now equipped to provide a first comprehensive picture of the global biodiversity of the Porifera. An introductory overview of the four classes of the Porifera is followed by a description of the structure of our main source of data for this paper, the WPD. From this we extracted numbers of all ‘known’ sponges to date: the number of valid Recent sponges is established at 8,553, with the vast majority, 83%, belonging to the class Demospongiae. We also mapped for the first time the species richness of a comprehensive set of marine ecoregions of the world, data also extracted from the WPD. Perhaps not surprisingly, these distributions appear to show a strong bias towards collection and taxonomy efforts. Only when species richness is accumulated into large marine realms does a pattern emerge that is also recognized in many other marine animal groups: high numbers in tropical regions, lesser numbers in the colder parts of the world oceans. Preliminary similarity analysis of a matrix of species and marine ecoregions extracted from the WPD failed to yield a consistent hierarchical pattern of ecoregions into marine provinces. Global sponge diversity information is mostly generated in regional projects and resources: results obtained demonstrate that regional approaches to analytical biogeography are at present more likely to achieve insights into the biogeographic history of sponges than a global perspective, which appears currently too ambitious. We also review information on invasive sponges that might well have some influence on distribution patterns of the future

Lampreys, the jawless vertebrates, contain three Pax6 genes with distinct expression in eye, brain and pancreas

10.1038/s41598-019-56085-8Scientific Reports911955

Eukaryotic Protein Kinases (ePKs) of the Helminth Parasite Schistosoma mansoni

<p>Abstract</p> <p>Background</p> <p>Schistosomiasis remains an important parasitic disease and a major economic problem in many countries. The <it>Schistosoma mansoni </it>genome and predicted proteome sequences were recently published providing the opportunity to identify new drug candidates. Eukaryotic protein kinases (ePKs) play a central role in mediating signal transduction through complex networks and are considered druggable targets from the medical and chemical viewpoints. Our work aimed at analyzing the <it>S. mansoni </it>predicted proteome in order to identify and classify all ePKs of this parasite through combined computational approaches. Functional annotation was performed mainly to yield insights into the parasite signaling processes relevant to its complex lifestyle and to select some ePKs as potential drug targets.</p> <p>Results</p> <p>We have identified 252 ePKs, which corresponds to 1.9% of the <it>S. mansoni </it>predicted proteome, through sequence similarity searches using HMMs (Hidden Markov Models). Amino acid sequences corresponding to the conserved catalytic domain of ePKs were aligned by MAFFT and further used in distance-based phylogenetic analysis as implemented in PHYLIP. Our analysis also included the ePK homologs from six other eukaryotes. The results show that <it>S. mansoni </it>has proteins in all ePK groups. Most of them are clearly clustered with known ePKs in other eukaryotes according to the phylogenetic analysis. None of the ePKs are exclusively found in <it>S. mansoni </it>or belong to an expanded family in this parasite. Only 16 <it>S. mansoni </it>ePKs were experimentally studied, 12 proteins are predicted to be catalytically inactive and approximately 2% of the parasite ePKs remain unclassified. Some proteins were mentioned as good target for drug development since they have a predicted essential function for the parasite.</p> <p>Conclusions</p> <p>Our approach has improved the functional annotation of 40% of <it>S. mansoni </it>ePKs through combined similarity and phylogenetic-based approaches. As we continue this work, we will highlight the biochemical and physiological adaptations of <it>S. mansoni </it>in response to diverse environments during the parasite development, vector interaction, and host infection.</p

Coupling of ocean redox and animal evolution during the Ediacaran-Cambrian transition

The late Ediacaran to early Cambrian interval witnessed extraordinary radiations of metazoan life. The role of the physical environment in this biological revolution, such as changes to oxygen levels and nutrient availability, has been the focus of longstanding debate. Seemingly contradictory data from geochemical redox proxies help to fuel this controversy. As an essential nutrient, nitrogen can help to resolve this impasse by establishing linkages between nutrient supply, ocean redox, and biological changes. Here we present a comprehensive N-isotope dataset from the Yangtze Basin that reveals remarkable coupling between δ¹⁵N, δ¹³C, and evolutionary events from circa 551 to 515 Ma. The results indicate that increased fixed nitrogen supply may have facilitated episodic animal radiations by reinforcing ocean oxygenation, and restricting anoxia to near, or even at the sediment–water interface. Conversely, sporadic ocean anoxic events interrupted ocean oxygenation, and may have led to extinctions of the Ediacaran biota and small shelly animals