An intrinsic Proper Generalized Decomposition for parametric symmetric elliptic problems

We introduce in this paper a technique for the reduced order approximation of parametric symmetric elliptic partial differential equations. For any given dimension, we prove the existence of an optimal subspace of at most that dimension which realizes the best approximation in mean of the error with respect to the parameter in the quadratic norm associated to the elliptic operator, between the exact solution and the Galerkin solution calculated on the subspace. This is analogous to the best approximation property of the Proper Orthogonal Decomposition (POD) subspaces, excepting that in our case the norm is parameter-depending, and then the POD optimal sub-spaces cannot be characterized by means of a spectral problem. We apply a deflation technique to build a series of approximating solutions on finite-dimensional optimal subspaces, directly in the on-line step. We prove that the partial sums converge to the continuous solutions, in mean quadratic elliptic norm.Comment: 18 page

The Influence of Quadrature Errors on Isogeometric Mortar Methods

Mortar methods have recently been shown to be well suited for isogeometric analysis. We review the recent mathematical analysis and then investigate the variational crime introduced by quadrature formulas for the coupling integrals. Motivated by finite element observations, we consider a quadrature rule purely based on the slave mesh as well as a method using quadrature rules based on the slave mesh and on the master mesh, resulting in a non-symmetric saddle point problem. While in the first case reduced convergence rates can be observed, in the second case the influence of the variational crime is less significant

Fluctuating Dark Energy and the Luminosity Distance

The origin of dark energy driving the accelerated expansion of the universe is still mysterious. We explore the possibility that dark energy fluctuates, resulting in spatial correlations. Due to these fluctuations, the Hubble rate itself becomes a fluctuating quantity. We discuss the effect this has on measurements of type Ia supernovae, which are used to constrain the luminosity distance. We show that the luminosity distance is affected by spatial correlations in several ways. First, the luminosity distance becomes dressed by the fluctuations, thereby differing from standard $\Lambda$CDM. Second, angular correlations become visible in the two-point correlation function of the luminosity distance. To investigate the latter we construct the angular power spectrum of luminosity distance fluctuations. We then perform a forecast for two supernova surveys, the ongoing Dark Energy Survey (DES) and the upcoming Legacy Survey of Space and Time (LSST), and compare this effect with relativistic lensing effects from perturbed $\Lambda$CDM. We find that the signal can rise above the lensing effects and that LSST could test this effect for a large part of the parameter space. As an example, a specific realisation of such a scenario is that quantum fluctuations of some field in the early universe imprint spatial correlations with a predictable form in the dark energy density today. In this case, the Hubble rate fluctuates due to the intrinsic quantum nature of the dark energy density field. We study whether the signal of this specific model would be measurable, and conclude that testing this model with LSST would be challenging. However, taking into account a speed of sound $c_s<1$ of the dark energy fluid can make this model observable.Comment: 38 pages, 9 figure

Optical Properties of Wide Band Gap Indium Sulphide Thin Films Obtained by Physical Vapor Deposition

Thin films of indium sulphide containing oxygen have been synthesized following a dry physical process. The constituents are deposited by thermal evaporation on glass substrates and then annealed under argon flow. Polycrystalline β-In2S3 containing oxygen thin films are obtained as soon as the temperature of annealing is between 623 and 723 K. In this paper, these β-In2S3 thin films have optically been studied. The optical band gap is direct. Its value is not dependent on the temperature of annealing. It is about 2.8 eV, which is higher than that of β-In2S3 single crystal. This high value is related to the presence of oxygen in the films. The extinction coefficient k and the refractive index n of the films have also been found independent of the annealing temperature. These optical properties make the films studied good candidates to be substituted to CdS in Cu(In,Ga)Se2 based solar cells

Reduced basis isogeometric mortar approximations for eigenvalue problems in vibroacoustics

We simulate the vibration of a violin bridge in a multi-query context using reduced basis techniques. The mathematical model is based on an eigenvalue problem for the orthotropic linear elasticity equation. In addition to the nine material parameters, a geometrical thickness parameter is considered. This parameter enters as a 10th material parameter into the system by a mapping onto a parameter independent reference domain. The detailed simulation is carried out by isogeometric mortar methods. Weakly coupled patch-wise tensorial structured isogeometric elements are of special interest for complex geometries with piecewise smooth but curvilinear boundaries. To obtain locality in the detailed system, we use the saddle point approach and do not apply static condensation techniques. However within the reduced basis context, it is natural to eliminate the Lagrange multiplier and formulate a reduced eigenvalue problem for a symmetric positive definite matrix. The selection of the snapshots is controlled by a multi-query greedy strategy taking into account an error indicator allowing for multiple eigenvalues

Variations of Infinite Derivative Modified Gravity

We consider nonlocal modified Einstein gravity without matter, where nonlocal term has the form $P(R) F(\Box) Q(R)$. For this model, in this paper we give the derivation of the equations of motion in detail. This is not an easy task and presented derivation should be useful to a researcher who wants to investigate nonlocal gravity. Also, we present the second variation of the related Einstein-Hilbert modified action and basics of gravity perturbations.Comment: 22 page

A FETI method with a mesh independent condition number for the iteration matrix

We introduce a framework for FETI methods using ideas from the decomposition via Lagrange multipliers of H1 0 (Ω) derived by Raviart-Thomas [22] P.-A. Raviart, J.-M. Thomas, Primal Hybrid Finite Element Metho and complemented with the detailed work on polygonal domains developed by Grisvard [17] P. Grisvard, Singularities in Boundary value problems. Recherches en Mathématiques Appliquées, 22. Masson, 1992.. We compute the action of the Lagrange multipliers using the natural H 1/2 00 scalar product, therefore no consistency error appears. As a byproduct, we obtain that the condition number for the iteration matrix is independent of the mesh size and there is no need for preconditioning. This result improves the standard asymptotic bound for this condition number shown by Mandel-Tezaur in [19] J. Mandel, R. Tezaur, Convergence of a substructuring method with Lagrange multipliers. Numer. Math., 73 (1996), 473–487. Numerical results that confirm our theoretical analysis are presented.Nous proposons une nouvelle approche des méthodes FETI: la décomposition de domaine fait appel aux multiplicateurs de Lagrange tels qu’introduits par Raviart-Thomas [22] P.-A. Raviart, J.-M. Thomas, Primal Hybrid Finite Element Methods for second order eliptic equations. Math. Comp., 31 (1977), 391-413 et au traitement des domaines polygonaux dù á Grisvard [17] P. Grisvard, Singularities in Boundary value problems. Recherches en Mathématiques Appliquées, 22. Masson, 1992. Ces multiplicateurs utilisent le produit scalaire de H 1/2 00, de sorte qu’aucune erreur de consistance n’apparaît. En outre, nous prouvons que le nombre de condition de la matrice liée à chaque itération est indépendant de la taille du maillage, ce qui améliore le résultat de Mandel-Tezaur [19] J. Mandel, R. Tezaur, Convergence of a substructuring method with Lagrange multipliers. Numer. Math., 73 (1996), 473–487; par suite, aucun préconditionnement n’est nécessaire. Nous présentons des expériences numériques qui confirment notre analyse.Ministerio de Educación y Cienci

A guided tour of asynchronous cellular automata

Research on asynchronous cellular automata has received a great amount of attention these last years and has turned to a thriving field. We survey the recent research that has been carried out on this topic and present a wide state of the art where computing and modelling issues are both represented.Comment: To appear in the Journal of Cellular Automat

Prior Mating Experience Modulates the Dispersal of Drosophila in Males More Than in Females

Cues from both an animal’s internal physiological state and its local environment may influence its decision to disperse. However, identifying and quantifying the causative factors underlying the initiation of dispersal is difficult in uncontrolled natural settings. In this study, we automatically monitored the movement of fruit flies and examined the influence of food availability, sex, and reproductive status on their dispersal between laboratory environments. In general, flies with mating experience behave as if they are hungrier than virgin flies, leaving at a greater rate when food is unavailable and staying longer when it is available. Males dispersed at a higher rate and were more active than females when food was unavailable, but tended to stay longer in environments containing food than did females. We found no significant relationship between weight and activity, suggesting the behavioral differences between males and females are caused by an intrinsic factor relating to the sex of a fly and not simply its body size. Finally, we observed a significant difference between the dispersal of the natural isolate used throughout this study and the widely-used laboratory strain, Canton-S, and show that the difference cannot be explained by allelic differences in the foraging gene