No abstract

Abstract. The classical Schwarz method is a domain decomposition method to solve elliptic partial differential equations in parallel. Convergence is achieved through overlap of the subdomains. We study in this paper a variant of the Schwarz method which converges without overlap for the Helmholtz equation. We show that the key ingredients for such an algorithm are the transmission conditions. We derive optimal transmission conditions which lead to convergence of the algorithm in a finite number of steps. These conditions are, however, nonlocal in nature, and we introduce local approximations which we optimize for performance of the Schwarz method. This leads to an algorithm in the class of optimized Schwarz methods. We present an asymptotic analysis of the optimized Schwarz method for two types of transmission conditions, Robin conditions and transmission conditions with second order tangential derivatives. Numerical results illustrate the effectiveness of the optimized Schwarz method on a model problem and on a problem from industry.Key words. optimized Schwarz methods, domain decomposition, preconditioner, iterative parallel methods, acoustics AMS subject classifications. 65F10, 65N22PII. S10648275013870121. Introduction. The classical Schwarz algorithm has a long history. It was invented by Schwarz more than a century ago [25] to prove existence and uniqueness of solutions to Laplace's equation on irregular domains. Schwarz decomposed the irregular domain into overlapping regular ones and formulated an iteration which used only solutions on regular domains and which converged to a unique solution on the irregular domain. A century later the Schwarz method was proposed as a computational method by Miller in [23], but it was only with the advent of parallel computers that the Schwarz method really gained popularity and was analyzed in depth both at the continuous level (see, for example, [17] [6]. We study in this paper the influence of the transmission conditions on the Schwarz algorithm for the Helmholtz equation. We derive optimal transmission conditions which lead to the best possible convergence of the Schwarz algorithm and which do not require overlap to be effective as in [12]. These optimal

SUMMARYOptimized Schwarz methods are working like classical Schwarz methods, but they are exchanging physically more valuable information between subdomains and hence have better convergence behavior. The new transmission conditions include also derivative information, not just function values, and optimized Schwarz methods can be used without overlap. In this paper, we present a new optimized Schwarz method without overlap in the 2D case, which uses a different Robin condition for neighboring subdomains at their common interface, and which we call two-sided Robin condition. We optimize the parameters in the Robin conditions and show that for a fixed frequency ω, an asymptotic convergence factor of 1 − O(h 1 4 ) in the mesh parameter h can be achieved. If the frequency is related to the mesh parameter h, h = O( 1 ω γ ) for γ ≥ 1, then the optimized asymptotic convergence factor is 1 − O(ω 1−2γ 8 ). We illustrate our analysis with 2D numerical experiments.

No abstract

Motivation Analysis toolkits for shotgun metagenomic data achieve strain-level characterization of complex microbial communities by capturing intra-species gene content variation. Yet, these tools are hampered by the extent of reference genomes that are far from covering all microbial variability, as many species are still not sequenced or have only few strains available. Binning co-abundant genes obtained from de novo assembly is a powerful reference-free technique to discover and reconstitute gene repertoire of microbial species. While current methods accurately identify species core parts, they miss many accessory genes or split them into small gene groups that remain unassociated to core clusters. Results We introduce MSPminer, a computationally efficient software tool that reconstitutes Metagenomic Species Pan-genomes (MSPs) by binning co-abundant genes across metagenomic samples. MSPminer relies on a new robust measure of proportionality coupled with an empirical classifier to group and distinguish not only species core genes but accessory genes also. Applied to a large scale metagenomic dataset, MSPminer successfully delineates in a few hours the gene repertoires of 1661 microbial species with similar specificity and higher sensitivity than existing tools. The taxonomic annotation of MSPs reveals microorganisms hitherto unknown and brings coherence in the nomenclature of the species of the human gut microbiota. The provided MSPs can be readily used for taxonomic profiling and biomarkers discovery in human gut metagenomic samples. In addition, MSPminer can be applied on gene count tables from other ecosystems to perform similar analyses. Availability and implementation The binary is freely available for non-commercial users at www.enterome.com/downloads . Supplementary information Supplementary data are available at Bioinformatics online.