A posteriori error estimation and basis adaptivity for reduced-basis approximation of nonaffine-parametrized linear elliptic partial differential equations

In this paper, we extend the earlier work [M. Barrault, Y. Maday, N. C. Nguyen, A.T. Patera, An “empirical interpolation” method: application to efficient reduced-basis discretization of partial differential equations, C.R. Acad. Sci. Paris, Série I 339 (2004) 667–672; M.A. Grepl, Y. Maday, N.C. Nguyen, A.T. Patera, Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations, M2AN Math. Model. Numer. Anal. 41 (3) (2007) 575–605.] to provide a posteriori error estimation and basis adaptivity for reduced-basis approximation of linear elliptic partial differential equations with nonaffine parameter dependence. The essential components are (i) rapidly convergent reduced-basis approximations – (Galerkin) projection onto a space spanned by N global hierarchical basis functions which are constructed from solutions of the governing partial differential equation at judiciously selected points in parameter space; (ii) stable and inexpensive interpolation procedures – methods which allow us to replace nonaffine parameter functions with a coefficient-function expansion as a sum of M products of parameter-dependent coefficients and parameter-independent functions; (iii) a posteriori error estimation – relaxations of the error-residual equation that provide inexpensive yet sharp error bounds for the error in the outputs of interest; (iv) optimal basis construction – processes which make use of the error bounds as an inexpensive surrogate for the expensive true error to explore the parameter space in the quest for an optimal sampling set; and (v) offline/online computational procedures – methods which decouple the generation and projection stages of the approximation process. The operation count for the online stage – in which, given a new parameter value, we calculate the output of interest and associated error bounds – depends only on N, M, and the affine parametric complexity of the problem; the method is thus ideally suited for repeated and reliable evaluation of input–output relationships in the many-query or real-time contexts.     

Interfacial reactions: mixed order kinetics and segregation effects

We study A-B reaction kinetics at a fixed interface separating A and B bulks. Initially, the number of reactions R(t) approximately tn(infinity)(A)n(infinity)(B) is second order in the far-field densities n(infinity)(A), n(infinity)(B). First order kinetics, governed by diffusion from the dilute bulk, onset at long times: R(t) approximately x(t)n(infinity)(A), where x(t) approximately t(1/z) is the rms molecular displacement. Below a critical dimension, d0) leads to anomalous decay of interfacial densities. Numerical simulations for z = 2 support the theory.     

Bounds in the Yukawa2 quantum field theory: Upper bound on the pressure,Hamiltonian bound and linear lower bound

We prove bounds of the formZ e ^a|| and (SZ)e ^a|| in theY ₂ Euclidean field theory and from this obtain Glimm's Hamiltonian bound and Schrader's linear lower bound.This work partially supported by USNSF grants GP-40768X and GP-39048.On leave from Max-Planck-Institut für Physik und Astrophysik, München, Federal Republic of Germany.A. Sloan Foundation Fellow.     

Two-dimensional matter: order,curvature and defects

Many systems in nature and the synthetic world involve ordered arrangements of units on two-dimensional surfaces. We review here the fundamental role payed by both the topology of the underlying surface and its Gaussian curvature. Topology dictates certain broad features of the defect structure of the ground state but curvature-driven energetics control the detailed structure of the ordered phases. Among the surprises are the appearance in the ground state of structures that would normally be thermal excitations and thus prohibited at zero temperature. Examples include excess dislocations in the form of grain boundary scars for spherical crystals above a minimal system size, dislocation unbinding for toroidal hexatics, interstitial fractionalization in spherical crystals and the appearance of well-separated disclinations for toroidal crystals. Much of the analysis leads to universal predictions that do not depend on the details of the microscopic interactions that lead to order in the first place. These predictions are subject to test by the many experimental soft- and hard-matter systems that lead to curved ordered structures such as colloidal particles self-assembling on droplets of one liquid in a second liquid. The defects themselves may be functionalized to create ligands with directional bonding. Thus, nano- to meso-scale superatoms may be designed with specific valency for use in building supermolecules and novel bulk materials. Parameters such as particle number, geometrical aspect ratios and anisotropy of elastic moduli permit the tuning of the precise architecture of the superatoms and associated supermolecules. Thus, the field has tremendous potential from both a fundamental and materials science/supramolecular chemistry viewpoint.     

Valley views: instantons,large order behaviors,and supersymmetry

The elucidation of the properties of the instantons in the topologically trivial sector has been a long-standing puzzle. Here we claim that the properties can be summarized in terms of the geometrical structure in the configuration space, the valley. Evidence for this claim is presented in various ways. The conventional perturbation theory and the non-perturbatioe calculation are unified, and the ambiguity of the Borel transform of the perturbation series is removed. A ‘proof’ of Bogomolny's “trick” is presented, which enables us to go beyond the dilute-gas approximation. The prediction of the large order behavior of perturbation theory is confirmed by explicit calculations, in some cases to the 478th order. A new type of supersymmetry is found as a by-product, and our result is shown to be consistent with the non-renormalization theorem. The prediction of the energy levels is confirmed with numerical solutions of the Schrödinger equation.     

Water and other tetrahedral liquids: order, anomalies and solvation

In order to understand the common features of tetrahedral liquids with water-like anomalies, the relationship between local order and anomalies has been studied using molecular dynamics simulations for three categories of such liquids: (a)?atomistic rigid-body models for water (TIP4P, TIP4P/2005, mTIP3P, SPC/E), (b)?ionic melts, BeF(2) (TRIM model) and SiO(2) (BKS potential) and (c)?Stillinger-Weber liquids parametrized to model water (mW) and silicon. Rigid-body, atomistic models for water and the Stillinger-Weber liquids show a strong correlation between tetrahedral and pair correlation order and the temperature for the onset of the density anomaly is close to the melting temperature. In contrast, the ionic melts show weaker and more variable degrees of correlation between tetrahedral and pair correlation metrics, and the onset temperature for the density anomaly is more than twice the melting temperature. In the case of water, the relationship between water-like anomalies and solvation is studied by examining the hydration of spherical solutes (Na(+), Cl(-), Ar) in water models with different temperature regimes of anomalies (SPC/E, TIP4P and mTIP3P). For both ionic and nonpolar solutes, the local structure and energy of water molecules is essentially the same as in bulk water beyond the second-neighbour shell. The local order and binding energy of water molecules are not perturbed by the presence of a hydrophobic solute. In the case of ionic solutes, the perturbation is largely localized within the first hydration shell. The binding energies for the ions are strongly dependent on the water models and clearly indicate that the geometry of the partial charge distributions, and the associated multipole moments, play an important role. However the anomalous behaviour of the water network has been found to be unimportant for polar solvation.     

Fractional order chaotic systems: history,achievements, applications,and future challenges

Motivated by the importance of study on the complex behaviors, which may be exhibited by fractional order differential equations, this review paper focuses on dynamical fractional order systems exhibiting chaotic behaviors. The review begins with a brief history on the first publications on the above-mentioned subject. Then, the review is continued by investigating the recent progresses relevant to fractional order chaotic systems. Furthermore, a summary on some applications for such systems, which have been reported in the literature, is presented. Finally, the paper is closed by discussing some open problems on the aforementioned research subject. These open problems, as future challenges for further study on fractional order chaotic systems, can specify some direction lines for continuing the research on that subject.

     

Improved radiative corrections for (e, e'p) experiments: Beyond the peaking approximation and implications of the soft-photon approximation

Analyzing (e, e'p) experimental data involves corrections for radiative effects which change the interaction kinematics and which have to be carefully considered in order to obtain the desired accuracy. Missing momentum and energy due to bremsstrahlung have so far often been incorporated into the simulations and the experimental analyses using the peaking approximation. It assumes that all bremsstrahlung is emitted in the direction of the radiating particle. In this article we introduce a full angular Monte Carlo simulation method which overcomes this approximation. As a test, the angular distribution of the bremsstrahlung photons is reconstructed from H(e, e'p) data. Its width is found to be underestimated by the peaking approximation and described much better by the approach developed in this work. The impact of the soft-photon approximation on the photon angular distribution is found to be minor as compared to the impact of the peaking approximation.     

Focus in production: tonal shape, intensity and word order

The effect of word order and prosodic focus on the tonal shape and intensity in the production of prosody was studied. The results show that the production of focus in Finnish follows a global pattern with regard to tonal features. The relative pitch height difference between contrasted words is the most important pitch-related factor in signaling narrow prosodic focus. Narrow focus is not localized to prosodically emphasized words only but relates to the utterance as a whole. It was also found that syntactic structure with respect to both intensity and tonal structure modulated relative prosodic prominence of individual words.     

Evolutionary drive: The effect of microscopic diversity,error making,and noise

In order to model any macroscopic system, it is necessary to aggregate both spatially and taxonomically. If average processes are assumed, then kinetic equations of population dynamics can be derived. Much effort has gone into showing the important effects introduced by non-average effects (fluctuations) in generating symmetry-breaking transitions and creating structure and form. However, the effects of microscopic diversity have been largely neglected. We show that evolution will select for populations which retain variability, even though this is, at any given time, loss-making, predicting that we shall not observe populations with optimal behavior, but populations which can learn. This lesser short-term efficiency may be why natural diversity is so great. Evolution is seen to be driven by the noise to which it leads.     

Introduction to focus issue: mixed mode oscillations: experiment, computation, and analysis

Mixed mode oscillations (MMOs) occur when a dynamical system switches between fast and slow motion and small and large amplitude. MMOs appear in a variety of systems in nature, and may be simple or complex. This focus issue presents a series of articles on theoretical, numerical, and experimental aspects of MMOs. The applications cover physical, chemical, and biological systems.     

Binary alloys of Ge and Te: order, voids, and the eutectic composition

The liquid and amorphous structures of Ge0.15Te0.85 and GeTe alloys are characterized using combined density functional/molecular dynamics simulations. Te is threefold coordinated, in contrast with predictions of the "8-N rule," and Ge atoms (fourfold coordinated) show octahedral and tetrahedral bonding angles. Cubic local environment occurs in both materials, and GeTe shows a pronounced alternation of atomic types. Tetrahedral Ge coordination is more common in the eutectic Ge0.15Te0.85, which comprises corner- and edge-sharing GeTe4 units surrounded by Te. There is no Te segregation, and the material resembles neither GeTe nor Te. The ubiquitous cavities (voids) have been overlooked in Ge0.15Te0.85, where they comprise over 25% of the volume.