Increasing Parallelism and Reducing Thread Contentions in Mapping Localized N-Body Simulations to GPUs

Abstract

The use of graphics processors to accelerate N-body simulations has been widely studied. The primary focus of most of these studies has been a class of problems that model the entire system of bodies interacting under Newtonian dynamic laws. A separate class of N-body problems, referred to herein as localized N-body simulations, focus on simulating only a small region of the system, with random state updates in order to find a local optimum. Due to the differences in the problem geometries, the widely applied algorithms and data structures for accelerating N-body simulations are less effective when applied to localized N-body problems. In this chapter, we present techniques for effective parallelization and acceleration of such localized N-body simulations on GPUs. Using energy minimization simulations as a case study, we show the challenges in using the existing data structures in accelerating localized N-body simulations and propose modified data structures and algorithms that enable better parallelism, achieving 7× to 27× speedup over serial code.

Appendices

Appendix A: Energy Expressions

The total energy of a system of atoms is given as a sum of various bonded and non-bonded energies for all the atoms:

$$ {E}^{total}=\underset{non- bonded}{\underbrace{E^{vdw}+{E}^{elec}}}+\underset{ bond ed}{\underbrace{E^{bond}+{E}^{angle}+{E}^{torsion}+{E}^{improper}}} $$

(18.1)

Energy minimization involves repeated evaluation of this expression, once during each minimization iteration. In the current discussion, only the non-bonded terms are mapped to the GPU and evaluated using the neighbor lists.

The non-bonded energy of each atom is the sum of the contributions due to the neighboring atoms within a cutoff distance. Non-bonded energy is a sum of the electrostatic and van der Waals energy terms. A variant of the Lennard-Jones 6–12 potential representing the van der Waals energy term is shown in Eq. (18.2)

$$ {E}_{ik}^{vdw}= ep{s}_{ik}\left(\frac{r{m}_{ik}^6}{r_{ik}^{12}}-\frac{{8 r{m}_{ik}^6}\!\left/ \!{{r}_c^6}\right.}{r_{ik}^6}+\frac{r{m}_{ik}^6}{r_c^{12}}\left(1+\frac{2{r}_{ik}^6}{r_c^6}\right)\right) $$

(18.2)

$$ ep{s}_{i k}= ep{s}_i. ep{s}_k $$

$$ r{m}_{i k}={\left( r{m}_i+ r{m}_k\right)}^2 $$

where eps _i and rm _i represent the van der Waals parameters of atom ‘i’, r _ik is the distance between the atoms ‘i’ and ‘k’ and r _c is the cut-off distance.

The electrostatic energy of a solute can be decomposed into two components; a sum of the self energies E ^self _i of all the charges and a sum of pair-wise interaction energies E ^int _ij [9] [Eq. (18.3)]

$$ {E}^{elec}={\displaystyle \sum_i{E}_i^{\mathit{self}}+{\displaystyle \sum_{i< j}{E}_{i j}^{\mathrm{int}}}} $$

(18.3)

Using the Analytic Continuum Electrostatics (ACE) model [9], the self-energy of an atom can be represented as a sum of its Born self-energy in the solvent and the sum of effective pair-wise interactions, E ^self _ik , due to all the other solute atoms [see Eqs. (18.4) and (18.5)].

$$ {E}_i^{s elf}=\frac{q_i^2}{2{\varepsilon}_s{R}_i}+{\displaystyle \sum_{k\ne i}{E}_{i k}^{\mathit{self}}}$$

(18.4)

$$ {E}_{i k}^{\mathit{self}}=\tau {q}_i^2\left(\frac{e^{-\left({{r}_{i k}^2}\!\left/ \!{{\sigma}_{i k}^2}\right.\right)}}{\omega_{i k}}+\frac{{\tilde{V}}_k}{8\pi}{\left(\frac{r_{i k}^3}{r_{i k}^4+{\mu}_{i k}^4}\right)}^4\right) $$

(18.5)

where q _i represents the charge on atom ‘i’, r _ik is the distance between the two atoms, ${\tilde{V}}_k$ is the size of the solute volume associated with atom ‘k’, ω _ik and σ _ik determine the height and width of the Gaussian that approximates E ^self _ik , and μ _ik is an atom-atom parameter.

The pair-wise interaction term of Eq. (18.3) is given by the generalized Born (GB) equation shown in Eq. (18.6), which is the sum of Coulomb’s law in a dielectric and the Born equation [10]

$$ {E}_{i j}^{\mathrm{int}}=332{\displaystyle \sum_{j\ne i}\frac{q_i{q}_j}{r_{i j}}}-166\tau {\displaystyle \sum_{j\ne i}\frac{q_i{q}_j}{\sqrt{r_{i j}^2+{\alpha}_i{\alpha}_j{e}^{-\left({{r}_{i j}^2}\!\left/ \!{4{\alpha}_i{\alpha}_j}\right.\right)}}}} $$

(18.6)

where α _i and α _j represent the Born radii for atoms ‘i’ and ‘j’, respectively. These in turn depend on the self energies of the atoms. The self energy of each individual atom thus needs to be computed before computing the pair-wise interactions.

Appendix B: Source Code