Convergence of equation-free methods in the case of finite time scale separation with application to deterministic and stochastic systems

This is the author accepted manuscript. The final version is available from SIAM via the DOI in this record.41 pages of supplementary material available at https://doi.org/10.6084/m9.figshare.6166421A common approach to studying high-dimensional systems with emergent low-dimensional behavior is based on lift-evolve-restrict maps (called equation-free methods): first, a user-defined lifting operator maps a set of low-dimensional coordinates into the high-dimensional phase space, then the high-dimensional (microscopic) evolution is applied for some time, and finally a user-defined restriction operator maps down into a low-dimensional space again. We prove convergence of equation-free methods for finite time-scale separation with respect to a method parameter, the so-called healing time. Our convergence result justifies equation-free methods as a tool for performing high-level tasks such as bifurcation analysis on high-dimensional systems. More precisely, if the high-dimensional system has an attracting invariant manifold with smaller expansion and attraction rates in the tangential direction than in the transversal direction (normal hyperbolicity), and restriction and lifting satisfy some generic transversality conditions, then an implicit formulation of the lift-evolve-restrict procedure generates an approximate map that converges to the flow on the invariant manifold for healing time going to infinity. In contrast to all previous results, our result does not require the time scale separation to be large. A demonstration with Michaelis-Menten kinetics shows that the error estimates of our theorem are sharp. The ability to achieve convergence even for finite time scale separation is especially important for applications involving stochastic systems, where the evolution occurs at the level of distributions, governed by the Fokker-Planck equation. In these applications the spectral gap is typically finite. We investigate a low-dimensional stochastic differential equation where the ratio between the decay rates of fast and slow variables is 2.J. Sieber’s research was supported by funding from the European Union’s Horizon 2020 research and innovation programme under Grant Agreement number 643073, by the EPSRC Centre for Predictive Modelling in Healthcare (Grant Number EP/N014391/1) and by the EPSRC Fellowship EP/N023544/1. C. Marschler and J. Starke would like to thank Civilingeniør Frederik Christiansens Almennyttige Fond for financial support. J. Starke would also like to thank the Villum Fonden (VKR-Centre of Excellence Ocean Life), the Technical University of Denmark and Queen Mary University of London for financial support

Temperature-induced reversal of magnetic interlayer exchange coupling

For epitaxial trilayers of the magnetic rare-earth metals Gd and Tb, exchange coupled through a non-magnetic Y spacer layer, element-specific hysteresis loops were recorded by the x-ray magneto-optical Kerr effect at the rare-earth $M_5$ thresholds. This allowed us to quantitatively determine the strength of interlayer exchange coupling (IEC). In addition to the expected oscillatory behavior as a function of spacer-layer thickness $d_Y$, a temperature-induced sign reversal of IEC was observed for constant $d_Y$, arising from magnetization-dependent electron reflectivities at the magnetic interfaces.Comment: 4 pages, 4 figures; accepted version; minor changes and new Figs. 2 and 4 containing more dat

LEED Holography applied to a complex superstructure: a direct view of the adatom cluster on SiC(111)-(3x3)

For the example of the SiC(111)-(3x3) reconstruction we show that a holographic interpretation of discrete Low Energy Electron Diffraction (LEED) spot intensities arising from ordered, large unit cell superstructures can give direct access to the local geometry of a cluster around an elevated atom, provided there is only one such prominent atom per surface unit cell. By comparing the holographic images obtained from experimental and calculated data we illuminate validity, current limits and possible shortcomings of the method. In particular, we show that periodic vacancies such as cornerholes may inhibit the correct detection of the atomic positions. By contrast, the extra diffraction intensity due to slight substrate reconstructions, as for example buckling, seems to have negligible influence on the images. Due to the spatial information depth of the method the stacking of the cluster can be imaged down to the fourth layer. Finally, it is demonstrated how this structural knowledge of the adcluster geometry can be used to guide the dynamical intensity analysis subsequent to the holographic reconstruction and necessary to retrieve the full unit cell structure.Comment: 11 pages RevTex, 6 figures, Phys. Rev. B in pres

Diagnosis of tuberculosis in children: increased need for better methods.

In the last decade tuberculosis (TB) has reemerged as a major worldwide public health hazard with increasing incidence among adults and children. Although cases among children represent a small percentage of all TB cases, infected children are a reservoir from which many adult cases will arise. TB diagnosis in children usually follows discovery of a case in an adult, and relies on tuberculin skin testing, chest radiograph, and clinical signs and symptoms. However, clinical symptoms are nonspecific, skin testing and chest radiographs can be difficult to interpret, and routine laboratory tests are not helpful. Although more rapid and sensitive laboratory testing, which takes into account recent advances in molecular biology, immunology, and chromatography, is being developed, the results for children have been disappointing. Better techniques would especially benefit children and infants in whom early diagnosis is imperative for preventing progressive TB

Prediction of huge X-ray Faraday rotation at the Gd N_4,5 threshold

X-ray absorption spectra in a wide energy range around the 4d-4f excitation threshold of Gd were recorded by total electron yield from in-plane magnetized Gd metal films. Matching the experimental spectra to tabulated absorption data reveals unprecedented short light absorption lengths down to 3 nm. The associated real parts of the refractive index for circularly polarized light propagating parallel or antiparallel to the Gd magnetization, determined through the Kramers-Kronig transformation, correspond to a magneto-optical Faraday rotation of 0.7 degrees per atomic layer. This finding shall allow the study of magnetic structure and magnetization dynamics of lanthanide elements in nanosize systems and dilute alloys.Comment: 4 pages, 2 figures, final version resubmitted to Phys. Rev. B, Brief Reports. Minor change

Missouri litter laws (1995)

Reviewed March 1995