Linear and nonlinear substructured Restricted Additive Schwarz iterations and preconditioning

Iterative substructuring Domain Decomposition (DD) methods have been extensively studied, and they are usually associated with nonoverlapping decompositions. It is less known that classical overlapping DD methods can also be formulated in substructured form, i.e., as iterative methods acting on variables defined exclusively on the interfaces of the overlapping domain decomposition. We call such formulations substructured domain decomposition methods. We introduce here a substructured version of Restricted Additive Schwarz (RAS) which we call SRAS. We show that RAS and SRAS are equivalent when used as iterative solvers, as they produce the same iterates, while they are substantially different when used as preconditioners for GMRES. We link the volume and substructured Krylov spaces and show that the iterates are different by deriving the least squares problems solved at each GMRES iteration. When used as iterative solvers, SRAS presents computational advantages over RAS, as it avoids computations with matrices and vectors at the volume level. When used as preconditioners, SRAS has the further advantage of allowing GMRES to store smaller vectors and perform orthogonalization in a lower dimensional space. We then consider nonlinear problems, and we introduce SRASPEN (Substructured Restricted Additive Schwarz Preconditioned Exact Newton), where SRAS is used as a preconditioner for Newton’s method. In contrast to the linear case, we prove that Newton’s method applied to the preconditioned volume and substructured formulation produces the same iterates in the nonlinear case. Next, we introduce two-level versions of nonlinear SRAS and SRASPEN. Finally, we validate our theoretical results with numerical experiments.

     

Exploring AuRh Nanoalloys: A Computational Perspective on the Formation and Physical Properties

We studied the formation of AuRh nanoalloys (between 20–150 atoms) in the gas phase by means of Molecular Dynamics (MD) calculations, exploring three possible formation processes: one-by-one growth, coalescence, and nanodroplets annealing. As a general trend, we recover a predominance of Rh@Au core-shell ordering over other chemical configurations. We identify new structural motifs with enhanced thermal stabilities. The physical features of those selected systems were studied at the Density Functional Theory (DFT) level, revealing profound correlations between the nanoalloys morphology and properties. Surprisingly, the arrangement of the inner Rh core seems to play a dominant role on nanoclusters’ physical features like the HOMO-LUMO gap and magnetic moment. Strong charge separations are recovered within the nanoalloys suggesting the existence of charge-transfer transitions.     

EFFECT OF EXPOSURE TO LIGHT ON ENZYME ACTIVITIES and TRYPTOPHAN METABOLITES OF THE KYNURENINE PATHWAY IN WISTAR, IN HETEROZYGOUS and HOMOZYGOUS ADULT and NEWBORN GUNN RATS

Abstract— The purpose of this study was to determine the influence of phototherapy used in the treatment of hyperbilirubinemia in infants, on tryptophan metabolism and on enzyme activities involved along the kynurenine pathway in Wistar, icteric homozygous and nonicteric heterozygous Gunn rats after an intraperitoneal loading of 1.0 g/kg body weight L-tryptophan before and after exposure to visible light. The total mean 24-h excretion of the tryptophan metabolites in the groups of the Wistar and heterozygous Gunn rats was higher in females than in males and higher than in homozygous groups before phototherapy. Only after exposure to light did the groups of Wistar and heterozygous Gunn rats of both sexes show a decrease in the total excretion of the metabolites. The activities of tryptophan pyrrolase and kynureninase in liver and kynurenine aminotransferase in liver and kidneys were assayed in each group of rats. Male and female Wistar and heterozygous Gunn rats showed a higher activity of tryptophan pyrrolase than the groups of homozygotes in agreement with the data of the metabolites excreted. No difference in enzyme activities was found between the groups of heterozygous and homozygous neonates before and after phototherapy. Phototherapy did not seem to influence these enzyme activities.     

Substructured two-grid and multi-grid domain decomposition methods

Two-level Schwarz domain decomposition methods are very powerful techniques for the efficient numerical solution of partial differential equations (PDEs). A two-level domain decomposition method requires two main components: a one-level preconditioner (or its corresponding smoothing iterative method), which is based on domain decomposition techniques, and a coarse correction step, which relies on a coarse space. The coarse space must properly represent the error components that the chosen one-level method is not capable to deal with. In the literature, most of the works introduced efficient coarse spaces obtained as the span of functions defined on the entire space domain of the considered PDE. Therefore, the corresponding two-level preconditioners and iterative methods are defined in volume. In this paper, we use the excellent smoothing properties of Schwarz domain decomposition methods to define, for general elliptic problems, a new class of substructured two-level methods, for which both Schwarz smoothers and coarse correction steps are defined on the interfaces (except for the application of the smoother that requires volumetric subdomain solves). This approach has several advantages. On the one hand, the required computational effort is cheaper than the one required by classical volumetric two-level methods. On the other hand, our approach does not require, like classical multi-grid methods, the explicit construction of coarse spaces, and it permits a multilevel extension, which is desirable when the high dimension of the problem or the scarce quality of the coarse space prevents the efficient numerical solution. Numerical experiments demonstrate the effectiveness of the proposed new numerical framework.