Accurate estimators of power spectra in N-body simulations

abridged] A method to rapidly estimate the Fourier power spectrum of a point distribution is presented. This method relies on a Taylor expansion of the trigonometric functions. It yields the Fourier modes from a number of FFTs, which is controlled by the order N of the expansion and by the dimension D of the system. In three dimensions, for the practical value N=3, the number of FFTs required is 20. We apply the method to the measurement of the power spectrum of a periodic point distribution that is a local Poisson realization of an underlying stationary field. We derive explicit analytic expression for the spectrum, which allows us to quantify--and correct for--the biases induced by discreteness and by the truncation of the Taylor expansion, and to bound the unknown effects of aliasing of the power spectrum. We show that these aliasing effects decrease rapidly with the order N. The only remaining significant source of errors is reduced to the unavoidable cosmic/sample variance due to the finite size of the sample. The analytical calculations are successfully checked against a cosmological N-body experiment. We also consider the initial conditions of this simulation, which correspond to a perturbed grid. This allows us to test a case where the local Poisson assumption is incorrect. Even in that extreme situation, the third-order Fourier-Taylor estimator behaves well. We also show how to reach arbitrarily large dynamic range in Fourier space (i.e., high wavenumber), while keeping statistical errors in control, by appropriately "folding" the particle distribution.Comment: 18 Pages, 9 Figures. Accepted for publication in MNRAS. The Fourier-Taylor module as well as the associated power spectrum estimator tool we propose is available as an F90 package, POWMES, at http://www.projet-horizon.fr or on request from the author

Time shifted aliasing error upper bounds for truncated sampling cardinal series

AbstractTime shifted aliasing error upper bound extremals for the sampling reconstruction procedure are fully characterized. Sharp upper bounds are found on the aliasing error of truncated cardinal series and the corresponding extremals are described for entire functions from certain specific Lp, p>1, classes. Analogous results are obtained in multidimensional regular sampling. Truncation error analysis is provided in all cases considered. Moreover, sharpness of bounding inequalities, convergence rates and various sufficient conditions are discussed

Numerical approximation of poroelasticity with random coefficients using Polynomial Chaos and Hybrid High-Order methods

In this work, we consider the Biot problem with uncertain poroelastic coefficients. The uncertainty is modelled using a finite set of parameters with prescribed probability distribution. We present the variational formulation of the stochastic partial differential system and establish its well-posedness. We then discuss the approximation of the parameter-dependent problem by non-intrusive techniques based on Polynomial Chaos decompositions. We specifically focus on sparse spectral projection methods, which essentially amount to performing an ensemble of deterministic model simulations to estimate the expansion coefficients. The deterministic solver is based on a Hybrid High-Order discretization supporting general polyhedral meshes and arbitrary approximation orders. We numerically investigate the convergence of the probability error of the Polynomial Chaos approximation with respect to the level of the sparse grid. Finally, we assess the propagation of the input uncertainty onto the solution considering an injection-extraction problem.Comment: 30 pages, 15 Figure

A Neural Model of How the Brain Computes Heading from Optic Flow in Realistic Scenes

Animals avoid obstacles and approach goals in novel cluttered environments using visual information, notably optic flow, to compute heading, or direction of travel, with respect to objects in the environment. We present a neural model of how heading is computed that describes interactions among neurons in several visual areas of the primate magnocellular pathway, from retina through V1, MT+, and MSTd. The model produces outputs which are qualitatively and quantitatively similar to human heading estimation data in response to complex natural scenes. The model estimates heading to within 1.5° in random dot or photo-realistically rendered scenes and within 3° in video streams from driving in real-world environments. Simulated rotations of less than 1 degree per second do not affect model performance, but faster simulated rotation rates deteriorate performance, as in humans. The model is part of a larger navigational system that identifies and tracks objects while navigating in cluttered environments.National Science Foundation (SBE-0354378, BCS-0235398); Office of Naval Research (N00014-01-1-0624); National-Geospatial Intelligence Agency (NMA201-01-1-2016

Optical Identification of Cepheids in 19 Host Galaxies of Type Ia Supernovae and NGC 4258 with the Hubble Space Telescope

We present results of an optical search for Cepheid variable stars using the Hubble Space Telescope (HST) in 19 hosts of Type Ia supernovae (SNe Ia) and the maser-host galaxy NGC 4258, conducted as part of the SH0ES project (Supernovae and H0 for the Equation of State of dark energy). The targets include 9 newly imaged SN Ia hosts using a novel strategy based on a long-pass filter that minimizes the number of HST orbits required to detect and accurately determine Cepheid properties. We carried out a homogeneous reduction and analysis of all observations, including new universal variability searches in all SN Ia hosts, that yielded a total of 2200 variables with well-defined selection criteria -- the largest such sample identified outside the Local Group. These objects are used in a companion paper to determine the local value of H0 with a total uncertainty of 2.4%.Comment: ApJ, in press. v2 adds missing co-author to arXiv metadata and text in acknowledgment

Higher-order derivative correlations and the alignment of small-scale structures in isotropic numerical turbulence

In a three dimensional simulation higher order derivative correlations, including skewness and flatness factors, are calculated for velocity and passive scalar fields and are compared with structures in the flow. The equations are forced to maintain steady state turbulence and collect statistics. It is found that the scalar derivative flatness increases much faster with Reynolds number than the velocity derivative flatness, and the velocity and mixed derivative skewness do not increase with Reynolds number. Separate exponents are found for the various fourth order velocity derivative correlations, with the vorticity flatness exponent the largest. Three dimensional graphics show strong alignment between the vorticity, rate of strain, and scalar-gradient fields. The vorticity is concentrated in tubes with the scalar gradient and the largest principal rate of strain aligned perpendicular to the tubes. Velocity spectra, in Kolmogorov variables, collapse to a single curve and a short minus 5/3 spectral regime is observed

Generation of Vorticity and Velocity Dispersion by Orbit Crossing

We study the generation of vorticity and velocity dispersion by orbit crossing using cosmological numerical simulations, and calculate the backreaction of these effects on the evolution of large-scale density and velocity divergence power spectra. We use Delaunay tessellations to define the velocity field, showing that the power spectra of velocity divergence and vorticity measured in this way are unbiased and have better noise properties than for standard interpolation methods that deal with mass weighted velocities. We show that high resolution simulations are required to recover the correct large-scale vorticity power spectrum, while poor resolution can spuriously amplify its amplitude by more than one order of magnitude. We measure the scalar and vector modes of the stress tensor induced by orbit crossing using an adaptive technique, showing that its vector modes lead, when input into the vorticity evolution equation, to the same vorticity power spectrum obtained from the Delaunay method. We incorporate orbit crossing corrections to the evolution of large scale density and velocity fields in perturbation theory by using the measured stress tensor modes. We find that at large scales (k~0.1 h/Mpc) vector modes have very little effect in the density power spectrum, while scalar modes (velocity dispersion) can induce percent level corrections at z=0, particularly in the velocity divergence power spectrum. In addition, we show that the velocity power spectrum is smaller than predicted by linear theory until well into the nonlinear regime, with little contribution from virial velocities.Comment: 27 pages, 14 figures. v2: reorganization of the material, new appendix. Accepted by PR