The Beylkin-Cramer Summation Rule and A New Fast Algorithm of Cosmic Statistics for Large Data Sets

Based on the Beylkin-Cramer summation rule, we introduce a new fast algorithm that enable us to explore the high order statistics efficiently in large data sets. Central to this technique is to make decomposition both of fields and operators within the framework of multi-resolution analysis (MRA), and realize theirs discrete representations. Accordingly, a homogenous point process could be equivalently described by a operation of a Toeplitz matrix on a vector, which is accomplished by making use of fast Fourier transformation. The algorithm could be applied widely in the cosmic statistics to tackle large data sets. Especially, we demonstrate this novel technique using the spherical, cubic and cylinder counts in cells respectively. The numerical test shows that the algorithm produces an excellent agreement with the expected results. Moreover, the algorithm introduces naturally a sharp-filter, which is capable of suppressing shot noise in weak signals. In the numerical procedures, the algorithm is somewhat similar to particle-mesh (PM) methods in N-body simulations. As scaled with $O(N\log N)$, it is significantly faster than the current particle-based methods, and its computational cost does not relies on shape or size of sampling cells. In addition, based on this technique, we propose further a simple fast scheme to compute the second statistics for cosmic density fields and justify it using simulation samples. Hopefully, the technique developed here allows us to make a comprehensive study of non-Guassianity of the cosmic fields in high precision cosmology. A specific implementation of the algorithm is publicly available upon request to the author.Comment: 27 pages, 9 figures included. revised version, changes include (a) adding a new fast algorithm for 2nd statistics (b) more numerical tests including counts in asymmetric cells, the two-point correlation functions and 2nd variances (c) more discussions on technic

Efficient Cosmological Parameter Estimation from Microwave Background Anisotropies

We revisit the issue of cosmological parameter estimation in light of current and upcoming high-precision measurements of the cosmic microwave background power spectrum. Physical quantities which determine the power spectrum are reviewed, and their connection to familiar cosmological parameters is explicated. We present a set of physical parameters, analytic functions of the usual cosmological parameters, upon which the microwave background power spectrum depends linearly (or with some other simple dependence) over a wide range of parameter values. With such a set of parameters, microwave background power spectra can be estimated with high accuracy and negligible computational effort, vastly increasing the efficiency of cosmological parameter error determination. The techniques presented here allow calculation of microwave background power spectra $10^5$ times faster than comparably accurate direct codes (after precomputing a handful of power spectra). We discuss various issues of parameter estimation, including parameter degeneracies, numerical precision, mapping between physical and cosmological parameters, and systematic errors, and illustrate these considerations with an idealized model of the MAP experiment.Comment: 22 pages, 12 figure

ASCR/HEP Exascale Requirements Review Report

This draft report summarizes and details the findings, results, and recommendations derived from the ASCR/HEP Exascale Requirements Review meeting held in June, 2015. The main conclusions are as follows. 1) Larger, more capable computing and data facilities are needed to support HEP science goals in all three frontiers: Energy, Intensity, and Cosmic. The expected scale of the demand at the 2025 timescale is at least two orders of magnitude -- and in some cases greater -- than that available currently. 2) The growth rate of data produced by simulations is overwhelming the current ability, of both facilities and researchers, to store and analyze it. Additional resources and new techniques for data analysis are urgently needed. 3) Data rates and volumes from HEP experimental facilities are also straining the ability to store and analyze large and complex data volumes. Appropriately configured leadership-class facilities can play a transformational role in enabling scientific discovery from these datasets. 4) A close integration of HPC simulation and data analysis will aid greatly in interpreting results from HEP experiments. Such an integration will minimize data movement and facilitate interdependent workflows. 5) Long-range planning between HEP and ASCR will be required to meet HEP's research needs. To best use ASCR HPC resources the experimental HEP program needs a) an established long-term plan for access to ASCR computational and data resources, b) an ability to map workflows onto HPC resources, c) the ability for ASCR facilities to accommodate workflows run by collaborations that can have thousands of individual members, d) to transition codes to the next-generation HPC platforms that will be available at ASCR facilities, e) to build up and train a workforce capable of developing and using simulations and analysis to support HEP scientific research on next-generation systems.Comment: 77 pages, 13 Figures; draft report, subject to further revisio

The New Horizon Run Cosmological N-Body Simulations

We present two large cosmological N-body simulations, called Horizon Run 2 (HR2) and Horizon Run 3 (HR3), made using 6000^3 = 216 billions and 7210^3 = 374 billion particles, spanning a volume of (7.200 Gpc/h)^3 and (10.815 Gpc/h)^3, respectively. These simulations improve on our previous Horizon Run 1 (HR1) up to a factor of 4.4 in volume, and range from 2600 to over 8800 times the volume of the Millennium Run. In addition, they achieve a considerably finer mass resolution, down to 1.25x10^11 M_sun/h, allowing to resolve galaxy-size halos with mean particle separations of 1.2 Mpc/h and 1.5 Mpc/h, respectively. We have measured the power spectrum, correlation function, mass function and basic halo properties with percent level accuracy, and verified that they correctly reproduce the LCDM theoretical expectations, in excellent agreement with linear perturbation theory. Our unprecedentedly large-volume N-body simulations can be used for a variety of studies in cosmology and astrophysics, ranging from large-scale structure topology, baryon acoustic oscillations, dark energy and the characterization of the expansion history of the Universe, till galaxy formation science - in connection with the new SDSS-III. To this end, we made a total of 35 all-sky mock surveys along the past light cone out to z=0.7 (8 from the HR2 and 27 from the HR3), to simulate the BOSS geometry. The simulations and mock surveys are already publicly available at http://astro.kias.re.kr/Horizon-Run23/.Comment: 18 pages, 10 figures. Added clarification on Fig 6. Published in the Journal of the Korean Astronomical Society (JKAS). The paper with high-resolution figures is available at http://jkas.kas.org/journals/2011v44n6/v44n6.ht

Including parameter dependence in the data and covariance for cosmological inference

The final step of most large-scale structure analyses involves the comparison of power spectra or correlation functions to theoretical models. It is clear that the theoretical models have parameter dependence, but frequently the measurements and the covariance matrix depend upon some of the parameters as well. We show that a very simple interpolation scheme from an unstructured mesh allows for an efficient way to include this parameter dependence self-consistently in the analysis at modest computational expense. We describe two schemes for covariance matrices. The scheme which uses the geometric structure of such matrices performs roughly twice as well as the simplest scheme, though both perform very well.Comment: 17 pages, 4 figures, matches version published in JCA