Fast N-body Research Articles

Improving the accuracy of halo mass based statistics for fast approximate N-body simulations

ABSTRACT Approximate N-body methods, such as fastpm and cola, have been successful in modelling halo and galaxy clustering statistics, but their low resolution on small scales is a limitation for applications that require high precision. Full N-body simulations can provide better accuracy but are too computationally expensive for a quick exploration of cosmological parameters. This paper presents a method for correcting distinct haloes identified in fast N-body simulations, so that various halo statistics improve to a percent level accuracy. The scheme seeks to find empirical corrections to halo properties such that the virial mass is the same as that of a corresponding halo in a full N-body simulation. The modified outer density contour of the corrected halo is determined on the basis of the fastpm settings and the number of particles inside the halo. This method only changes some parameters of the halo finder, and does not require any extra CPU-cost. We demonstrate that the adjusted halo catalogues of fastpm simulations significantly improve the precision of halo mass-based statistics from redshifts $z=0.0$ to 1.0, and that our calibration can be applied to different cosmologies without needing to be recalibrated.

Revisiting Vainshtein screening for fast N-body simulations

We revisit a method to incorporate the Vainshtein screening mechanism in N-body simulations proposed by R. Scoccimarro in [1]. We further extend this method to cover a subset of Horndeski theories that evade the bound on the speed of gravitational waves set by the binary neutron star merger GW170817. The procedure consists of the computation of an effective gravitational coupling that is time and scale dependent, G eff (k,z), where the scale dependence will incorporate the screening of the fifth-force. This is a fast procedure that when contrasted to the alternative of solving the full equation of motion for the scalar field inside N-body codes, reduces considerably the computational time and complexity required to run simulations. To test the validity of this approach in the non-linear regime, we have implemented it in a COmoving Lagrangian Approximation (COLA) N-body code, and ran simulations for two gravity models that have full N-body simulation outputs available in the literature, nDGP and Cubic Galileon. We validate the combination of the COLA method with this implementation of the Vainshtein mechanism with full N-body simulations for predicting the boost function: the ratio between the modified gravity non-linear matter power spectrum and its General Relativity counterpart. This quantity is of great importance for building emulators in beyond-ΛCDM models, and we find that the method described in this work has an agreement of below 2% for scales down to k ≈ 3h/Mpc with respect to full N-body simulations.

SCF-FDPS: A Fast N-body Code for Simulating Disk–Halo Systems

A fast N-body code has been developed for simulating a stellar disk embedded in a live dark matter halo. In generating its Poisson solver, a self-consistent field (SCF) code that inherently possesses perfect scalability is incorporated into a tree code that is parallelized using a library termed Framework for Developing Particle Simulators (FDPS). Thus, the code developed here is called SCF-FDPS. This code has realized the speedup of a conventional tree code by applying an SCF method not only to the calculation of the self-gravity of the halo but also to that of the gravitational interactions between the disk and halo particles. Consequently, in the SCF-FDPS code, a tree algorithm is applied only to calculate the self-gravity of the disk. On a many-core parallel computer, the SCF-FDPS code has performed at least 3 times (in one case, nearly an order of magnitude) faster than an extremely tuned tree code on it, if the numbers of disk and halo particles are, respectively, fixed for both codes. In addition, the SCF-FDPS code shows that the central processing unit cost scales almost linearly with the total number of particles and almost inversely with the number of cores. We find that the time evolution of a disk–halo system simulated with the SCF-FDPS code is, in large measure, similar to that obtained using the tree code. We suggest how the present code can be extended to cope with a wide variety of disk-galaxy simulations.

Projection-tree reduced-order modeling for fast N-body computations

This work presents a data-driven reduced-order modeling framework to accelerate the computations of nonlocal and N-body methods that model dynamical systems. The proposed framework differs from traditional acceleration methods, like the Barnes–Hut method, which requires online tree building of the state space, or the fast-multipole method, which requires rigorous a priori analysis of governing kernels and online tree building. Our approach combines Barnes–Hut hierarchical decomposition, projection-based reduced-order modeling via the least-squares Petrov–Galerkin (LSPG) projection, and hyper-reduction by way of the Gauss–Newton with approximated tensor (GNAT) approach. The resulting projection-tree reduced-order model (PTROM) enables a drastic reduction in operational count complexity by constructing sparse hyper-reduced pairwise interactions of the non-compact N-body dynamical system. As a result, the presented framework is capable of achieving an operational count complexity that is independent of N, the number of bodies in the numerical domain. Capabilities of the PTROM method are demonstrated on the two-dimensional fluid-dynamic Biot–Savart kernel within a parametric and reproductive setting. Results show the PTROM is capable of achieving over 2000× wall-time speed-up with respect to the full-order model, where the speed-up increases with N. The resulting solution delivers quantities of interest with errors that are less than 0.1% with respect to full-order model.

A gradient based method for modeling baryons and matter in halos of fast simulations

Fast N-body PM simulations with a small number of time steps such as FastPM or COLA have been remarkably successful in modeling the galaxy statistics, but their lack of small scale force resolution and long time steps cannot give accurate halo matter profiles or matter power spectrum. High resolution N-body simulations can improve on this, but lack baryonic effects, which can only be properly included in hydro simulations. Here we present a scheme to calibrate the fast simulations to mimic the precision of the hydrodynamic simulations or high resolution N-body simulations. The scheme is based on a gradient descent of either effective gravitational potential, which mimics the short range force, or of effective enthalpy, which mimics gas hydrodynamics and feedback. The scheme is fast and differentiable, and can be incorporated as a post-processing step into any simulation. It gives very good results for the matter power spectrum for several of the baryonic feedback and dark matter simulations, and also gives improved dark matter halo profiles. The scheme is even able to find the large subhalos, and increase the correlation coefficient between the fast simulations and the high resolution N-body or hydro simulations. It can also be used to add baryonic effects to the high resolution N-body simulations. While the method has free parameters that can be calibrated on various simulations, they can also be viewed as astrophysical nuisance parameters describing baryonic effects that can be marginalized over during the data analysis. In this view these parameters can be viewed as an efficient parametrization of baryonic effects.

Application of Fast N-Body Algorithm to Option Pricing under CGMY Model

The N-body problem is an active research topic in physics for which there are two major algorithms for efficient computation, the fast multipole method and treecode, but these algorithms are not popular in financial engineering. In this article, we apply a fast N-body algorithm called the Cartesian treecode to the computation of the integral operator of integro-partial differential equations to compute option prices under the CGMY model, a generalization of a jump-diffusion model. We present numerical examples to illustrate the accuracy and effectiveness of the method and thereby demonstrate its suitability for application in financial engineering.

An FMM Based on Dual Tree Traversal for Many-Core Architectures

The present work attempts to integrate the independent efforts in the fast N-body community to create the fastest N-body library for many-core and heterogenous architectures. Focus is placed on low accuracy optimizations, in response to the recent interest to use FMM as a preconditioner for sparse linear solvers. A direct comparison with other state-of-the-art fast N-body codes demonstrates that orders of magnitude increase in performance can be achieved by careful selection of the optimal algorithm and low-level optimization of the code. The current N-body solver uses a fast multipole method with an efficient strategy for finding the list of cell-cell interactions by a dual tree traversal. A task-based threading model is used to maximize thread-level parallelism and intra-node load-balancing. In order to extract the full potential of the SIMD units on the latest CPUs, the inner kernels are optimized using AVX instructions.

Hierarchical N-body Simulations with Autotuning for Heterogeneous Systems

With the current hybridization of treecodes and FMMs, combined with auto-tuning capabilities on heterogeneous architectures, the flexibility of fast N-body methods has been greatly enhanced. These features are a requirement to developing a black-box software library for fast N-body algorithms on heterogeneous systems, which is our immediate goal.

Modeling Interacting Galaxies: NGC 4449 revisited

Observing nearby interacting galaxies is a key to understanding galactic physics provided that we know the spatial and temporal perturbations acting on these galaxies. Thus, we have to know the orbits and the gross internal properties of the galaxies. In order to cope with the related extended parameter space, we developed the code MINGA which combines a genetic algorithm with a fast N-body method. As an example for this method, we present a re-analysis of the prototypical system NGC 4449 which is now based on both, the full HI data cube of the NGC 4449 system and on improved determinations of the galactic orbits within a restricted N-body calculation.

Fast multipole methods on a cluster of GPUs for the meshless simulation of turbulence

Recent advances in the parallelizability of fast N-body algorithms, and the programmability of graphics processing units (GPUs) have opened a new path for particle based simulations. For the simulation of turbulence, vortex methods can now be considered as an interesting alternative to finite difference and spectral methods. The present study focuses on the efficient implementation of the fast multipole method and pseudo-particle method on a cluster of NVIDIA GeForce 8800 GT GPUs, and applies this to a vortex method calculation of homogeneous isotropic turbulence. The results of the present vortex method agree quantitatively with that of the reference calculation using a spectral method. We achieved a maximum speed of 7.48 TFlops using 64 GPUs, and the cost performance was near $9.4/GFlops. The calculation of the present vortex method on 64 GPUs took 4120 s, while the spectral method on 32 CPUs took 4910 s.

M51 revisited: a genetic algorithm approach of its interaction history

Detailed models of observed interacting galaxies suffer from the extended parameter space. Here, we present results from our code MINGA which couples an evolutionary optimization strategy (a genetic algorithm) with a fast N-body method. MINGA allows for an automatic search of the optimal region(s) in parameter space within a few hours to a few days of CPU time on a modern PC by investigating of the order of 10^5 models. We demonstrate its applicability by modelling the HI intensity and velocity maps of the interacting system M51 and NGC 5195. We get a good fit for the HI intensity map and we can reproduce the counter-rotation feature of the HI arm. Our result corroborates the results of Salo & Laurikainen (2000) who favour a model with multiple passages through M51's disk.

Symplectic integration with floating-point arithmetic and other approximations

There is significant theoretical support for the use of symplectic integrators for the numerical solution of Hamiltonian systems. However, the theory does not apply to practical computations because of the failure to take into account the effects of roundoff errors, and other approximations such as the use of fast N-body solvers and the use of “not fully converged iterations” in implicit or semi-implicit integrators. Very often these effects grow exponentially with time and completely overwhelm the numerical results well before the integration is complete. By means of a simple and inexpensive modification of the integrator, we show that it is possible to maintain a symplectic integration with floating-point arithmetic and other approximations.

Periodic boundary conditions and the fast multipole method

The seminal work of Nijboer and De Wette [Physica 23, 309 (1957)] enables the calculation of lattice sums of spherical harmonics, but has long been overlooked. In this article, their central result is recast in a simplified form suitable for modern multipole algorithms that employ the solid harmonics. This formulation makes possible the imposition of periodic boundary conditions within modern versions of the fast multipole method, and other fast N-body methods. The distinction between the extrinsic values obtained with the lattice sums ℳ of the multipole interaction tensors, and the intrinsic values associated with Taylor’s expansion of the Ewald formulas, is made. The central constants, ℳ, are computed to 32 digit accuracy using extended precision arithmetic. Timings and corresponding errors obtained with a periodic version of the fast multipole method are presented for particle numbers spanning [103,106], and a range of expansion orders. A qualitative comparison is made between the present implementation, other periodic versions of the fast multipole method, and fast Ewald methods.