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.

     

Radiationless transitions in trivalent lanthanide ions embedded in solids

Radiationless transitions in lanthanide ions embedded in glassy and crystalline hosts have been briefly reviewed. The correlations of multiphonon relaxation rates with energy gaps, vibrational frequencies, temperatures and electronic symmetries have been discussed. The essence of various theories, such as Nth order methods, non-adiabatic Hamiltonian method, dynamic coupling model has also been presented.     

Wild ginseng grows in Myanmar

Ginseng, the underground parts of plants of Panax species, has been used in oriental traditional medicine for centuries. Unfortunately, because of extensive exploitation over thousands of years, the natural source of these species has been almost exhausted. Recently, we have found a wild ginseng growing in Myanmar. Here, by a combination of chemical composition study and gene sequence analysis, we unambiguously demonstrate that the wild ginseng is actually P. zingiberensis, commonly known as ginger ginseng. This ginseng was an indigenous to the southwestern China. However, now it is seriously threatened to brink of extinction and is put on the highest level of protection in China. Therefore, an appropriate protection measure is highly recommended to preserve this valuable resource, since this Myanmar ginseng might turn out to be the last P. zingiberensis, which could ever be seen in the planet.     

Improved highly efficient synthesis of α,β-acetylenic ketones. Nature of the intermediate from the reaction of lithium acetylide with boron trifluoride etherate

A wide variety of α,β-acetylenic ketones were synthesized in very high yields via an exceptionally facile intermolecular reaction of lithium alkynyltrifluoroborates and carboxylic acid anhydrides.     

Synthesis of a polymer-supported oxazolidine aldehyde for asymmetric chemistry

An expedient synthesis of the polymer-supported aldehyde 3, as a Garner aldehyde equivalent, is described. Oxazolidine 3 may be obtained through preformation of aldehyde linker 4 in solution and loaded onto amine-terminating resin under peptide-coupling conditions, or alternatively via oxidation of polymer-bound alcohol 14. The integrity of the serine-derived stereocenter is maintained through all steps of the synthesis.