On the Performance Prediction of BLAS-based Tensor Contractions

Tensor operations are surging as the computational building blocks for a variety of scientific simulations and the development of high-performance kernels for such operations is known to be a challenging task. While for operations on one- and two-dimensional tensors there exist standardized interfaces and highly-optimized libraries (BLAS), for higher dimensional tensors neither standards nor highly-tuned implementations exist yet. In this paper, we consider contractions between two tensors of arbitrary dimensionality and take on the challenge of generating high-performance implementations by resorting to sequences of BLAS kernels. The approach consists in breaking the contraction down into operations that only involve matrices or vectors. Since in general there are many alternative ways of decomposing a contraction, we are able to methodically derive a large family of algorithms. The main contribution of this paper is a systematic methodology to accurately identify the fastest algorithms in the bunch, without executing them. The goal is instead accomplished with the help of a set of cache-aware micro-benchmarks for the underlying BLAS kernels. The predictions we construct from such benchmarks allow us to reliably single out the best-performing algorithms in a tiny fraction of the time taken by the direct execution of the algorithms.Comment: Submitted to PMBS1

Periastron Advance in Spinning Black Hole Binaries: Gravitational Self-Force from Numerical Relativity

We study the general relativistic periastron advance in spinning black hole binaries on quasi-circular orbits, with spins aligned or anti-aligned with the orbital angular momentum, using numerical-relativity simulations, the post-Newtonian approximation, and black hole perturbation theory. By imposing a symmetry by exchange of the bodies' labels, we devise an improved version of the perturbative result, and use it as the leading term of a new type of expansion in powers of the symmetric mass ratio. This allows us to measure, for the first time, the gravitational self-force effect on the periastron advance of a non-spinning particle orbiting a Kerr black hole of mass M and spin S = -0.5 M^2, down to separations of order 9M. Comparing the predictions of our improved perturbative expansion with the exact results from numerical simulations of equal-mass and equal-spin binaries, we find a remarkable agreement over a wide range of spins and orbital separations.Comment: 18 pages, 12 figures; matches version to appear in Phys. Rev.

2.5PN kick from black-hole binaries in circular orbit: Nonspinning case

Using the Multipolar post-Minskowskian formalism, we compute the linear momentum flux from black-hole binaries in circular orbits and having no spins. The total linear momentum flux contains various types of instantaneous (which are functions of the retarded time) and hereditary (which depends on the dynamics of the binary in the past) terms both of which are analytically computed. In addition to the inspiral contribution, we use a simple model of plunge to compute the kick or recoil accumulated during this phase.Comment: To appear in Proceedings of "Relativity and Gravitation - 100 Years after Einstein in Prague" Ed. J. Bicak (2013

The Statistical Mechanics of Horizons and Black Hole Thermodynamics

Although we know that black holes are characterized by a temperature and an entropy, we do not yet have a satisfactory microscopic ``statistical mechanical'' explanation for black hole thermodynamics. I describe a new approach that attributes the thermodynamic properties to ``would-be gauge'' degrees of freedom that become dynamical on the horizon. For the (2+1)-dimensional black hole, this approach gives the correct entropy. (Talk given at the Pacific Conference on Gravitation and Cosmology, Seoul, February 1996.)Comment: 11 pages, LaTe

Gravitational Radiation from Post-Newtonian Sources and Inspiralling Compact Binaries

The article reviews the current status of a theoretical approach to the problem of the emission of gravitational waves by isolated systems in the context of general relativity. Part A of the article deals with general post-Newtonian sources. The exterior field of the source is investigated by means of a combination of analytic post-Minkowskian and multipolar approximations. The physical observables in the far-zone of the source are described by a specific set of radiative multipole moments. By matching the exterior solution to the metric of the post-Newtonian source in the near-zone we obtain the explicit expressions of the source multipole moments. The relationships between the radiative and source moments involve many non-linear multipole interactions, among them those associated with the tails (and tails-of-tails) of gravitational waves. Part B of the article is devoted to the application to compact binary systems. We present the equations of binary motion, and the associated Lagrangian and Hamiltonian, at the third post-Newtonian (3PN) order beyond the Newtonian acceleration. The gravitational-wave energy flux, taking consistently into account the relativistic corrections in the binary moments as well as the various tail effects, is derived through 3.5PN order with respect to the quadrupole formalism. The binary's orbital phase, whose prior knowledge is crucial for searching and analyzing the signals from inspiralling compact binaries, is deduced from an energy balance argument.Comment: 109 pages, 1 figure; this version is an update of the Living Review article originally published in 2002; available on-line at http://www.livingreviews.org

Two binary stars gravitational waves - homotopy perturbation method

Homotopy perturbation is one of the newest methods for numerical analysis of deferential equations. We have used for solving wave equation around a black hole. Our conclusions have this method far reaching consequences for comparison of theoritical physics and experimental physics.Comment: The manuscript considers the important problem of solve equation wave around a black hole. We have solved that by using Homotopy perturbation methods. Homotopy perturbation is one of the newest methods for numerical analysis of deferential equations. Our conclusions have far reaching consequences for comparison of theoritical physics and experimental physic

Developing an inverted Barrovian sequence; insights from monazite petrochronology

In the Himalayan region of Sikkim, the well-developed inverted metamorphic sequence of the Main Central Thrust (MCT) zone is folded, thus exposing several transects through the structure that reached similar metamorphic grades at different times. In-situ LA-ICP-MS U–Th–Pb monazite ages, linked to pressure–temperature conditions via trace-element reaction fingerprints, allow key aspects of the evolution of the thrust zone to be understood for the first time. The ages show that peak metamorphic conditions were reached earliest in the structurally highest part of the inverted metamorphic sequence, in the Greater Himalayan Sequence (GHS) in the hanging wall of the MCT. Monazite in this unit grew over a prolonged period between ~37 and 16 Ma in the southerly leading-edge of the thrust zone and between ~37 and 14.5 Ma in the northern rear-edge of the thrust zone, at peak metamorphic conditions of ~790 ◦C and 10 kbar. Monazite ages in Lesser Himalayan Sequence (LHS) footwall rocks show that identical metamorphic conditions were reached ~4–6 Ma apart along the ~60 km separating samples along the MCT transport direction. Upper LHS footwall rocks reached peak metamorphic conditions of ~655 ◦C and 9 kbar between ~21 and 16 Ma in the more southerly-exposed transect and ~14.5–12 Ma in the northern transect. Similarly, lower LHS footwall rocks reached peak metamorphic conditions of ~580 ◦C and 8.5 kbar at ~16 Ma in the south, and 9–10 Ma in the north. In the southern transect, the timing of partial melting in the GHS hanging wall (~23–19.5 Ma) overlaps with the timing of prograde metamorphism (~21 Ma) in the LHS footwall, confirming that the hanging wall may have provided the heat necessary for the metamorphism of the footwall. Overall, the data provide robust evidence for progressively downwards-penetrating deformation and accretion of original LHS footwall material to the GHS hanging wall over a period of ~5 Ma. These processes appear to have occurred several times during the prolonged ductile evolution of the thrust. The preserved inverted metamorphic sequence therefore documents the formation of sequential ‘paleothrusts’ through time, cutting down from the original locus of MCT movement at the LHS–GHS protolith boundary and forming at successively lower pressure and temperature conditions. The petrochronologic methods applied here constrain a complex temporal and thermal deformation history, and demonstrate that inverted metamorphic sequences can preserve a rich record of the duration of progressive ductile thrusting

Global Chromatin Architecture Reflects Pluripotency and Lineage Commitment in the Early Mouse Embryo

An open chromatin architecture devoid of compact chromatin is thought to be associated with pluripotency in embryonic stem cells. Establishing this distinct epigenetic state may also be required for somatic cell reprogramming. However, there has been little direct examination of global structural domains of chromatin during the founding and loss of pluripotency that occurs in preimplantation mouse development. Here, we used electron spectroscopic imaging to examine large-scale chromatin structural changes during the transition from one-cell to early postimplantation stage embryos. In one-cell embryos chromatin was extensively dispersed with no noticeable accumulation at the nuclear envelope. Major changes were observed from one-cell to two-cell stage embryos, where chromatin became confined to discrete blocks of compaction and with an increased concentration at the nuclear envelope. In eight-cell embryos and pluripotent epiblast cells, chromatin was primarily distributed as an extended meshwork of uncompacted fibres and was indistinguishable from chromatin organization in embryonic stem cells. In contrast, lineage-committed trophectoderm and primitive endoderm cells, and the stem cell lines derived from these tissues, displayed higher levels of chromatin compaction, suggesting an association between developmental potential and chromatin organisation. We examined this association in vivo and found that deletion of Oct4, a factor required for pluripotency, caused the formation of large blocks of compact chromatin in putative epiblast cells. Together, these studies show that an open chromatin architecture is established in the embryonic lineages during development and is sufficient to distinguish pluripotent cells from tissue-restricted progenitor cells

Parameter estimation method that directly compares gravitational wave observations to numerical relativity

We present and assess a Bayesian method to interpret gravitational wave signals from binary black holes. Our method directly compares gravitational wave data to numerical relativity (NR) simulations. In this study, we present a detailed investigation of the systematic and statistical parameter estimation errors of this method. This procedure bypasses approximations used in semianalytical models for compact binary coalescence. In this work, we use the full posterior parameter distribution for only generic nonprecessing binaries, drawing inferences away from the set of NR simulations used, via interpolation of a single scalar quantity (the marginalized log likelihood, ln L ) evaluated by comparing data to nonprecessing binary black hole simulations. We also compare the data to generic simulations, and discuss the effectiveness of this procedure for generic sources. We specifically assess the impact of higher order modes, repeating our interpretation with both l ≤ 2 as well as l ≤ 3 harmonic modes. Using the l ≤ 3 higher modes, we gain more information from the signal and can better constrain the parameters of the gravitational wave signal. We assess and quantify several sources of systematic error that our procedure could introduce, including simulation resolution and duration; most are negligible. We show through examples that our method can recover the parameters for equal mass, zero spin, GW150914-like, and unequal mass, precessing spin sources. Our study of this new parameter estimation method demonstrates that we can quantify and understand the systematic and statistical error. This method allows us to use higher order modes from numerical relativity simulations to better constrain the black hole binary parameters