Acceleration function benchmarks

Sebastian Micluța-Câmpeanu, Mikhail Vaganov

Solving the equations of notions for an N-body problem implies solving a (large) system of differential equations. In DifferentialEquations.jl these are represented through ODE or SDE problems. To build the problem we need a function that describe the equations. In the case of N-body problems, this function gives the accelerations for the particles in the system.

Here we will test the performance of several acceleration functions used in N-body simulations. The systems that will be used are not necessarily realistic as we are not solving the problem, we just time how fast is an acceleration function call.

using BenchmarkTools, NBodySimulator
using NBodySimulator: gather_bodies_initial_coordinates, gather_accelerations_for_potentials,
    gather_simultaneous_acceleration, gather_group_accelerations
using StaticArrays

const SUITE = BenchmarkGroup();

function acceleration(simulation)

    (u0, v0, n) = gather_bodies_initial_coordinates(simulation)

    acceleration_functions = gather_accelerations_for_potentials(simulation)
    simultaneous_acceleration = gather_simultaneous_acceleration(simulation)

    function soode_system!(dv, v, u, p, t)
        @inbounds for i = 1:n
            a = MVector(0.0, 0.0, 0.0)
            for acceleration! in acceleration_functions
                acceleration!(a, u, v, t, i);
            end
            dv[:, i] .= a
        end
        for acceleration! in simultaneous_acceleration
            acceleration!(dv, u, v, t);
        end
    end

    return soode_system!
end
acceleration (generic function with 1 method)

Gravitational potential

let SUITE=SUITE
    G = 6.67e-11 # m^3/kg/s^2
    N = 200 # number of bodies/particles
    m = 1.0 # mass of each of them
    v = 10.0 # mean velocity
    L = 20.0 # size of the cell side

    bodies = generate_bodies_in_cell_nodes(N, m, v, L)
    g_parameters = GravitationalParameters(G)
    system = PotentialNBodySystem(bodies, Dict(:gravitational => g_parameters))
    tspan = (0.0, 1.0)
    simulation = NBodySimulation(system, tspan)

    f = acceleration(simulation)
    u0, v0, n = gather_bodies_initial_coordinates(simulation)
    dv = zero(v0)

    b = @benchmarkable $f(dv, $v0, $u0, $g_parameters, 0.) setup=(dv=zero($v0)) evals=1

    SUITE["gravitational"] = b
end
Benchmark(evals=1, seconds=5.0, samples=10000)

Coulomb potential

let SUITE=SUITE
    n = 200
    bodies = ChargedParticle[]
    L = 20.0
    m = 1.0
    q = 1.0
    count = 1
    dL = L / (ceil(n^(1 / 3)) + 1)
    for x = dL / 2:dL:L, y = dL / 2:dL:L, z = dL / 2:dL:L
        if count > n
            break
        end
        r = SVector(x, y, z)
        v = SVector(.0, .0, .0)
        body = ChargedParticle(r, v, m, q)
        push!(bodies, body)
        count += 1
    end

    k = 9e9
    τ = 0.01 * dL / sqrt(2 * k * q * q / (dL * m))
    t1 = 0.0
    t2 = 1000 * τ

    potential = ElectrostaticParameters(k, 0.45 * L)
    system = PotentialNBodySystem(bodies, Dict(:electrostatic => potential))
    pbc = CubicPeriodicBoundaryConditions(L)
    simulation = NBodySimulation(system, (t1, t2), pbc)

    f = acceleration(simulation)
    u0, v0, n = gather_bodies_initial_coordinates(simulation)
    dv = zero(v0)

    b = @benchmarkable $f(dv, $v0, $u0, $potential, 0.) setup=(dv=zero($v0)) evals=1

    SUITE["coulomb"] = b
end
Benchmark(evals=1, seconds=5.0, samples=10000)

Magnetic dipole potential

let SUITE=SUITE
    n = 200
    bodies = MagneticParticle[]
    L = 20.0
    m = 1.0
    count = 1
    dL = L / (ceil(n^(1 / 3)) + 1)
    for x = dL / 2:dL:L, y = dL / 2:dL:L, z = dL / 2:dL:L
        if count > n
            break
        end
        r = SVector(x, y, z)
        v = SVector(.0, .0, .0)
        mm = rand(SVector{3})
        body = MagneticParticle(r, v, m, mm)
        push!(bodies, body)
        count += 1
    end

    μ_4π = 1e-7
    t1 = 0.0  # s
    t2 = 1.0 # s
    τ = (t2 - t1) / 100

    parameters = MagnetostaticParameters(μ_4π)
    system = PotentialNBodySystem(bodies, Dict(:magnetic => parameters))
    simulation = NBodySimulation(system, (t1, t2))

    f = acceleration(simulation)
    u0, v0, n = gather_bodies_initial_coordinates(simulation)
    dv = zero(v0)

    b = @benchmarkable $f(dv, $v0, $u0, $parameters, 0.) setup=(dv=zero($v0)) evals=1

    SUITE["magnetic_dipole"] = b
end
Benchmark(evals=1, seconds=5.0, samples=10000)

Lennard Jones potential

let SUITE=SUITE
    T = 120.0 # K
    T0 = 90.0 # K
    kb = 8.3144598e-3 # kJ/(K*mol)
    ϵ = T * kb
    σ = 0.34 # nm
    ρ = 1374/1.6747# Da/nm^3
    N = 200
    m = 39.95# Da = 216 # number of bodies/particles
    L = (m*N/ρ)^(1/3)#10.229σ
    R = 0.5*L
    v_dev = sqrt(kb * T / m)
    bodies = generate_bodies_in_cell_nodes(N, m, v_dev, L)

    τ = 0.5e-3 # ps or 1e-12 s
    t1 = 0.0
    t2 = 2000τ

    lj_parameters = LennardJonesParameters(ϵ, σ, R)
    lj_system = PotentialNBodySystem(bodies, Dict(:lennard_jones => lj_parameters));

    pbc = CubicPeriodicBoundaryConditions(L)
    simulation = NBodySimulation(lj_system, (t1, t2), pbc, kb)

    f = acceleration(simulation)
    u0, v0, n = gather_bodies_initial_coordinates(simulation)
    dv = zero(v0)

    b = @benchmarkable $f(dv, $v0, $u0, $lj_parameters, 0.) setup=(dv=zero($v0)) evals=1

    SUITE["lennard_jones"] = b
end
Benchmark(evals=1, seconds=5.0, samples=10000)

WaterSPCFw model

function acceleration(simulation::NBodySimulation{<:WaterSPCFw})

    (u0, v0, n) = gather_bodies_initial_coordinates(simulation)

    (o_acelerations, h_acelerations) = gather_accelerations_for_potentials(simulation)
    group_accelerations = gather_group_accelerations(simulation)
    simultaneous_acceleration = gather_simultaneous_acceleration(simulation)

    function soode_system!(dv, v, u, p, t)
        @inbounds for i = 1:n
            a = MVector(0.0, 0.0, 0.0)
            for acceleration! in o_acelerations
                acceleration!(a, u, v, t, 3 * (i - 1) + 1);
            end
            dv[:, 3 * (i - 1) + 1]  .= a
        end
        @inbounds for i in 1:n, j in (2, 3)
            a = MVector(0.0, 0.0, 0.0)
            for acceleration! in h_acelerations
                acceleration!(a, u, v, t, 3 * (i - 1) + j);
            end
            dv[:, 3 * (i - 1) + j]   .= a
        end
        @inbounds for i = 1:n
            for acceleration! in group_accelerations
                acceleration!(dv, u, v, t, i);
            end
        end
        for acceleration! in simultaneous_acceleration
            acceleration!(dv, u, v, t);
        end
    end

    return soode_system!
end

let SUITE=SUITE
    T = 370 # K
    T0 = 275 # K
    kb = 8.3144598e-3 # kJ/(K*mol)
    ϵOO = 0.1554253*4.184 # kJ
    σOO = 0.3165492 # nm
    ρ = 997/1.6747# Da/nm^3
    mO = 15.999 # Da
    mH = 1.00794 # Da
    mH2O = mO+2*mH
    N = 200
    L = (mH2O*N/ρ)^(1/3)
    R = 0.9 # ~3*σOO
    Rel = 0.49*L
    v_dev = sqrt(kb * T /mH2O)
    τ = 0.5e-3 # ps
    t1 = 0τ
    t2 = 5τ # ps
    k_bond = 1059.162*4.184*1e2 # kJ/(mol*nm^2)
    k_angle = 75.90*4.184 # kJ/(mol*rad^2)
    rOH = 0.1012 # nm
    ∠HOH = 113.24*pi/180 # rad
    qH = 0.41
    qO = -0.82
    k = 138.935458 #
    bodies = generate_bodies_in_cell_nodes(N, mH2O, v_dev, L)
    jl_parameters = LennardJonesParameters(ϵOO, σOO, R)
    e_parameters = ElectrostaticParameters(k, Rel)
    spc_parameters = SPCFwParameters(rOH, ∠HOH, k_bond, k_angle)
    pbc = CubicPeriodicBoundaryConditions(L)
    water = WaterSPCFw(bodies, mH, mO, qH, qO,  jl_parameters, e_parameters, spc_parameters);
    simulation = NBodySimulation(water, (t1, t2), pbc, kb);

    f = acceleration(simulation)
    u0, v0, n = gather_bodies_initial_coordinates(simulation)
    dv = zero(v0)

    b = @benchmarkable $f(dv, $v0, $u0, $spc_parameters, 0.) setup=(dv=zero($v0)) evals=1

    SUITE["water_spcfw"] = b
end
Benchmark(evals=1, seconds=5.0, samples=10000)

Here are the results of the benchmarks

r = run(SUITE)

minimum(r)
5-element BenchmarkTools.BenchmarkGroup:
  tags: []
  "gravitational" => TrialEstimate(4.941 ms)
  "coulomb" => TrialEstimate(593.856 μs)
  "lennard_jones" => TrialEstimate(485.927 μs)
  "water_spcfw" => TrialEstimate(6.865 ms)
  "magnetic_dipole" => TrialEstimate(21.712 ms)

and

memory(r)
5-element BenchmarkTools.BenchmarkGroup:
  tags: []
  "gravitational" => 7670400
  "coulomb" => 9600
  "lennard_jones" => 9600
  "water_spcfw" => 124912
  "magnetic_dipole" => 25564800

Appendix

These benchmarks are a part of the SciMLBenchmarks.jl repository, found at: https://github.com/SciML/SciMLBenchmarks.jl. For more information on high-performance scientific machine learning, check out the SciML Open Source Software Organization https://sciml.ai.

To locally run this benchmark, do the following commands:

using SciMLBenchmarks
SciMLBenchmarks.weave_file("benchmarks/NBodySimulator","acceleration_functions.jmd")

Computer Information:

Julia Version 1.6.5
Commit 9058264a69 (2021-12-19 12:30 UTC)
Platform Info:
  OS: Linux (x86_64-pc-linux-gnu)
  CPU: AMD EPYC 7502 32-Core Processor
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-11.0.1 (ORCJIT, znver2)
Environment:
  BUILDKITE_PLUGIN_JULIA_CACHE_DIR = /cache/julia-buildkite-plugin
  JULIA_DEPOT_PATH = /cache/julia-buildkite-plugin/depots/5b300254-1738-4989-ae0a-f4d2d937f953

Package Information:

      Status `/cache/build/exclusive-amdci3-0/julialang/scimlbenchmarks-dot-jl/benchmarks/NBodySimulator/Project.toml`
  [6e4b80f9] BenchmarkTools v0.7.0
  [a93c6f00] DataFrames v1.1.1
  [0e6f8da7] NBodySimulator v1.6.1
  [1dea7af3] OrdinaryDiffEq v5.53.1
  [91a5bcdd] Plots v1.13.2
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  [31c91b34] SciMLBenchmarks v0.1.0
  [90137ffa] StaticArrays v1.1.3
  [f3b207a7] StatsPlots v0.14.20

And the full manifest:

      Status `/cache/build/exclusive-amdci3-0/julialang/scimlbenchmarks-dot-jl/benchmarks/NBodySimulator/Manifest.toml`
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  [975044d2] Xorg_xcb_util_keysyms_jll v0.4.0+1
  [0d47668e] Xorg_xcb_util_renderutil_jll v0.3.9+1
  [c22f9ab0] Xorg_xcb_util_wm_jll v0.4.1+1
  [35661453] Xorg_xkbcomp_jll v1.4.2+4
  [33bec58e] Xorg_xkeyboard_config_jll v2.27.0+4
  [c5fb5394] Xorg_xtrans_jll v1.4.0+3
  [8f1865be] ZeroMQ_jll v4.3.2+6
  [3161d3a3] Zstd_jll v1.4.8+0
  [0ac62f75] libass_jll v0.14.0+4
  [f638f0a6] libfdk_aac_jll v0.1.6+4
  [b53b4c65] libpng_jll v1.6.37+6
  [a9144af2] libsodium_jll v1.0.20+0
  [f27f6e37] libvorbis_jll v1.3.6+6
  [1270edf5] x264_jll v2020.7.14+2
  [dfaa095f] x265_jll v3.0.0+3
  [d8fb68d0] xkbcommon_jll v0.9.1+5
  [0dad84c5] ArgTools
  [56f22d72] Artifacts
  [2a0f44e3] Base64
  [ade2ca70] Dates
  [8bb1440f] DelimitedFiles
  [8ba89e20] Distributed
  [f43a241f] Downloads
  [7b1f6079] FileWatching
  [9fa8497b] Future
  [b77e0a4c] InteractiveUtils
  [4af54fe1] LazyArtifacts
  [b27032c2] LibCURL
  [76f85450] LibGit2
  [8f399da3] Libdl
  [37e2e46d] LinearAlgebra
  [56ddb016] Logging
  [d6f4376e] Markdown
  [a63ad114] Mmap
  [ca575930] NetworkOptions
  [44cfe95a] Pkg
  [de0858da] Printf
  [3fa0cd96] REPL
  [9a3f8284] Random
  [ea8e919c] SHA
  [9e88b42a] Serialization
  [1a1011a3] SharedArrays
  [6462fe0b] Sockets
  [2f01184e] SparseArrays
  [10745b16] Statistics
  [4607b0f0] SuiteSparse
  [fa267f1f] TOML
  [a4e569a6] Tar
  [8dfed614] Test
  [cf7118a7] UUIDs
  [4ec0a83e] Unicode
  [e66e0078] CompilerSupportLibraries_jll
  [deac9b47] LibCURL_jll
  [29816b5a] LibSSH2_jll
  [c8ffd9c3] MbedTLS_jll
  [14a3606d] MozillaCACerts_jll
  [4536629a] OpenBLAS_jll
  [efcefdf7] PCRE2_jll
  [83775a58] Zlib_jll
  [8e850ede] nghttp2_jll
  [3f19e933] p7zip_jll