Nonlinear Preconditioning: How to use a Nonlinear Schwarz Method to Precondition Newton's Method

For linear problems, domain decomposition methods can be used directly as iterative solvers, but also as preconditioners for Krylov methods. In practice, Krylov acceleration is almost always used, since the Krylov method finds a much better residual polynomial than the stationary iteration, and thus converges much faster. We show in this paper that also for non-linear problems, domain decomposition methods can either be used directly as iterative solvers, or one can use them as preconditioners for Newton's method. For the concrete case of the parallel Schwarz method, we show that we obtain a preconditioner we call RASPEN (Restricted Additive Schwarz Preconditioned Exact Newton) which is similar to ASPIN (Additive Schwarz Preconditioned Inexact Newton), but with all components directly defined by the iterative method. This has the advantage that RASPEN already converges when used as an iterative solver, in contrast to ASPIN, and we thus get a substantially better preconditioner for Newton's method. The iterative construction also allows us to naturally define a coarse correction using the multigrid full approximation scheme, which leads to a convergent two level non-linear iterative domain decomposition method and a two level RASPEN non-linear preconditioner. We illustrate our findings with numerical results on the Forchheimer equation and a non-linear diffusion problem

Toward the assessment of the susceptibility of a digital system to lightning upset

Accomplishments and directions for further research aimed at developing methods for assessing a candidate design of an avionic computer with respect to susceptability to lightning upset are reported. Emphasis is on fault tolerant computers. Both lightning stress and shielding are covered in a review of the electromagnetic environment. Stress characterization, system characterization, upset detection, and positive and negative design features are considered. A first cut theory of comparing candidate designs is presented including tests of comparative susceptability as well as its analysis and simulation. An approach to lightning induced transient fault effects is included

Magnetostrictive hysteresis of TbCo/CoFe multilayers and magnetic domains

Magnetic and magnetostrictive hysteresis loops of TbCo/CoFe multilayers under field applied along the hard magnetization axis are studied using vectorial magnetization measurements, optical deflectometry and magneto optical Kerr microscopy. Even a very small angle misalignment between hard axis and magnetic field direction is shown to drastically change the shape of magnetization and magnetostrictive torsion hysteresis loops. Two kinds of magnetic domains are revealed during the magnetization: big regions with opposite rotation of spontaneous magnetization vector and spontaneous magnetic domains which appear in a narrow field interval and provide an inversion of this rotation. We show that the details of the hysteresis loops of our exchange-coupled films can be described using the classical model of homogeneous magnetization rotation of single uniaxial films and the configuration of observed domains. The understanding of these features is crucial for applications (for MEMS or microactuators) which benefit from the greatly enhanced sensitivity near the point of magnetic saturation at the transverse applied field.Comment: 10 pages, 11 figure

Efficient C(sp³)−H Carbonylation of Light and Heavy Hydrocarbons with Carbon Monoxide via Hydrogen Atom Transfer Photocatalysis in Flow

Despite their abundance in organic molecules, considerable limitations still exist in synthetic methods that target the direct C−H functionalization at sp3-hybridized carbon atoms. This is even more the case for light alkanes, which bear some of the strongest C−H bonds known in Nature, requiring extreme activation conditions that are not tolerant to most organic molecules. To bypass these issues, synthetic chemists rely on prefunctionalized alkyl halides or organometallic coupling partners. However, new synthetic methods that target regioselectively C−H bonds in a variety of different organic scaffolds would be of great added value, not only for the late-stage functionalization of biologically active molecules but also for the catalytic upgrading of cheap and abundant hydrocarbon feedstocks. Here, we describe a general, mild and scalable protocol which enables the direct C(sp3)−H carbonylation of saturated hydrocarbons, including natural products and light alkanes, using photocatalytic hydrogen atom transfer (HAT) and gaseous carbon monoxide (CO). Flow technology was deemed crucial to enable high gas-liquid mass transfer rates and fast reaction kinetics, needed to outpace deleterious reaction pathways, but also to leverage a scalable and safe process.</p

Synchronous Behavior of Two Coupled Electronic Neurons

We report on experimental studies of synchronization phenomena in a pair of analog electronic neurons (ENs). The ENs were designed to reproduce the observed membrane voltage oscillations of isolated biological neurons from the stomatogastric ganglion of the California spiny lobster Panulirus interruptus. The ENs are simple analog circuits which integrate four dimensional differential equations representing fast and slow subcellular mechanisms that produce the characteristic regular/chaotic spiking-bursting behavior of these cells. In this paper we study their dynamical behavior as we couple them in the same configurations as we have done for their counterpart biological neurons. The interconnections we use for these neural oscillators are both direct electrical connections and excitatory and inhibitory chemical connections: each realized by analog circuitry and suggested by biological examples. We provide here quantitative evidence that the ENs and the biological neurons behave similarly when coupled in the same manner. They each display well defined bifurcations in their mutual synchronization and regularization. We report briefly on an experiment on coupled biological neurons and four dimensional ENs which provides further ground for testing the validity of our numerical and electronic models of individual neural behavior. Our experiments as a whole present interesting new examples of regularization and synchronization in coupled nonlinear oscillators.Comment: 26 pages, 10 figure