Introduction

pytreegrav is a package for computing the gravitational potential and/or field of a set of particles. It includes methods for brute-force direction summation and for the fast, approximate Barnes-Hut treecode method. For the Barnes-Hut method we implement an oct-tree as a numba jitclass to achieve much higher peformance than the equivalent pure Python implementation, without writing a single line of C or Cython. Full documentation is available here.

Installation

pip install pytreegrav, or clone the repo and run pip install . (or pip install -e .) from the repo directory.

Walkthrough

First let's import the stuff we want and generate some particle positions and masses - these would be your particle data for whatever your problem is.

import numpy as np
from pytreegrav import Accel, Potential

N = 10**5 # number of particles
x = np.random.rand(N,3) # positions randomly sampled in the unit cube
m = np.repeat(1./N,N) # masses - let the system have unit mass
h = np.repeat(0.01,N) # softening radii - these are optional, assumed 0 if not provided to the frontend functions

Now we can use the Accel and Potential functions to compute the gravitational field and potential at each particle position:

print(Accel(x,m,h))
print(Potential(x,m,h))

[[-0.1521787   0.2958852  -0.30109005]
 [-0.50678204 -0.37489886 -1.0558666 ]
 [-0.24650087  0.95423467 -0.175074  ]
 ...
 [ 0.87868472 -1.28332176 -0.22718531]
 [-0.41962742  0.32372245 -1.31829084]
 [ 2.45127054  0.38292881  0.05820412]]
[-2.35518057 -2.19299372 -2.28494218 ... -2.11783337 -2.1653377
 -1.80464695]

By default, pytreegrav will try to make the optimal choice between brute-force and tree methods for speed, but we can also force it to use one method or another. Let's try both and compare their runtimes (all timings quoted in this walkthrough are single-core, on an otherwise-idle Intel Xeon Gold 6244):

from time import time
t = time()
# tree gravitational acceleration
accel_tree = Accel(x,m,h,method='tree')
print("Tree accel runtime: %gs"%(time() - t)); t = time()

accel_bruteforce = Accel(x,m,h,method='bruteforce')
print("Brute force accel runtime: %gs"%(time() - t)); t = time()

phi_tree = Potential(x,m,h,method='tree')
print("Tree potential runtime: %gs"%(time() - t)); t = time()

phi_bruteforce = Potential(x,m,h,method='bruteforce')
print("Brute force potential runtime: %gs"%(time() - t)); t = time()

Tree accel runtime: 0.556318s
Brute force accel runtime: 40.9757s
Tree potential runtime: 0.326653s
Brute force potential runtime: 18.9015s

As you can see, the tree-based methods can be much faster than the brute-force methods, especially for particle counts exceeding a few thousand. Here's an example of how much faster the treecode is when run on a Plummer sphere with a variable number of particles, on a single core of an Intel Xeon Gold 6244 workstation:

But there's no free lunch here: the tree methods are approximate. Let's quantify the RMS errors of the stuff we just computed, compared to the exact brute-force solutions:

acc_error = np.sqrt(np.mean(np.sum((accel_tree-accel_bruteforce)**2,axis=1))) # RMS force error
print("RMS force error: ", acc_error)
phi_error = np.std(phi_tree - phi_bruteforce)
print("RMS potential error: ", phi_error)

RMS force error:  0.00390130
RMS potential error:  0.00025342

The above errors are typical for default settings: ~0.2% RMS force error and ~0.1% RMS potential error (relative to the RMS field strength). The error in the tree approximation is controlled by the Barnes-Hut opening angle theta, set to 0.7 by default. Smaller theta gives higher accuracy, but also runs slower:

thetas = 0.1,0.2,0.4,0.8 # different thetas to try
for theta in thetas:
    t = time()    
    accel_tree = Accel(x,m,h,method='tree',theta=theta)
    acc_error = np.sqrt(np.mean(np.sum((accel_tree-accel_bruteforce)**2,axis=1)))
    print("theta=%g Runtime: %gs RMS force error: %g"%(theta, time()-t, acc_error))

theta=0.1 Runtime: 19.4092s RMS force error: 2.62033e-05
theta=0.2 Runtime: 5.70894s RMS force error: 0.000161552
theta=0.4 Runtime: 1.46086s RMS force error: 0.00087864
theta=0.8 Runtime: 0.430208s RMS force error: 0.00618697

Accuracy versus cost

The tree walk's cost scales roughly as theta^-3, so it is worth knowing what that buys. The sweep above is serial and quotes absolute error; the figure below is the same experiment run in parallel with errors normalised, so the two sets of timings are not directly comparable. Running examples/error_benchmark.py on a 10^5-particle Plummer sphere gives:

Errors are relative to the RMS field strength of the system (and to std(phi) for the potential, whose zero point is arbitrary). Some representative points for the acceleration:

Two things worth noting. First, the maximum error is consistently ~10x the RMS error across the whole range: the treecode error distribution has a long tail, so if your problem is sensitive to the worst-case error on any single particle, budget an order of magnitude above the RMS figure. Second, the returns are strongly diminishing in the direction of small theta -- going from theta=1.0 to theta=0.1 costs ~17x the runtime to buy ~340x the accuracy, but most of that accuracy gain is already available by theta=0.4 at a quarter of the cost. The right-hand panel plots error directly against solve time, which is usually the more decision-relevant view.

The potential is roughly 4-5x more accurate than the acceleration at fixed theta, because it converges faster in the multipole expansion.

Both brute-force and tree-based calculations can be parallelized across all available logical cores via OpenMP, by specifying parallel=True. This can speed things up considerably, with parallel scaling that will vary with your core and particle number:

from time import time
t = time()
# tree gravitational acceleration
accel_tree = Accel(x,m,h,method='tree',parallel=True)
print("Tree accel runtime in parallel: %gs"%(time() - t)); t = time()

accel_bruteforce = Accel(x,m,h,method='bruteforce',parallel=True)
print("Brute force accel runtime in parallel: %gs"%(time() - t)); t = time()

phi_tree = Potential(x,m,h,method='tree',parallel=True)
print("Tree potential runtime in parallel: %gs"%(time() - t)); t = time()

phi_bruteforce = Potential(x,m,h,method='bruteforce',parallel=True)
print("Brute force potential runtime in parallel: %gs"%(time() - t)); t = time()

Tree accel runtime in parallel: 0.222271s
Brute force accel runtime in parallel: 7.25576s
Tree potential runtime in parallel: 0.181393s
Brute force potential runtime in parallel: 5.72611s

For parallel brute force there are two kernels, and pytreegrav picks between them for you. The straightforward one gives each thread a single target particle, so it can only ever write that particle's own result and must therefore evaluate all N² pairs — twice the work of the serial upper-triangular loop. Above SYMMETRIC_NMIN particles (1000) it instead uses a symmetrized kernel that evaluates each pair once and writes both sides, worth close to the expected 2× (e.g. 145 ms → 74 ms at N=32768 on 16 threads). The price is per-thread scratch — nthreads*N doubles for the potential, 3× that for the acceleration, so 13/38 MB at N=10⁵ on 16 threads. Below the crossover the simpler kernel wins anyway, because the symmetrized one runs two parallel regions to its one and a prange costs a full thread-team barrier however few iterations it has.

You can call Potential_bruteforce_symmetric / Accel_bruteforce_symmetric directly if you want to bypass the dispatch, but there is rarely a reason to.

What if I want to evaluate the fields at different points than where the particles are?

We got you covered. The Target methods do exactly this: you specify separate sets of points for the particle positions and the field evaluation, and everything otherwise works exactly the same (including optional parallelization and choice of solver):

from pytreegrav import AccelTarget, PotentialTarget

# generate a separate set of "target" positions where we want to know the potential and field
N_target = 10**4
x_target = np.random.rand(N_target,3)
h_target = np.repeat(0.01,N_target) # optional "target" softening: this sets a floor on the softening length of all forces/potentials computed

accel_tree = AccelTarget(x_target, x,m, softening_target=h_target, softening_source=h,method='tree') # we provide the points/masses/softenings we generated before as the "source" particles
accel_bruteforce = AccelTarget(x_target,x,m,softening_source=h,method='bruteforce')

acc_error = np.sqrt(np.mean(np.sum((accel_tree-accel_bruteforce)**2,axis=1))) # RMS force error
print("RMS force error: ", acc_error)

phi_tree = PotentialTarget(x_target, x,m, softening_target=h_target, softening_source=h,method='tree') # we provide the points/masses/softenings we generated before as the "source" particles
phi_bruteforce = PotentialTarget(x_target,x,m,softening_target=h_target, softening_source=h,method='bruteforce')

phi_error = np.std(phi_tree - phi_bruteforce)
print("RMS potential error: ", phi_error)

RMS force error:  0.0029070938409950310
RMS potential error:  0.00018373931733379673

Tidal fields

TidalTensor returns the tidal tensor $T_{ij} = \partial g_i / \partial x_j = -\partial^2 \Phi / \partial x_i \partial x_j$, as a shape (N,3,3) array. In this convention $T_{ij} \delta x_j$ is the relative tidal acceleration of a neighbour at small separation $\delta x$, so positive eigenvalues are stretching and negative ones compressive, and $\mathrm{tr},T = -4\pi G \rho$ (exactly zero wherever no softening kernel overlaps the evaluation point). All the usual options apply — method, parallel, theta, quadrupole, tree, and a TidalTensorTarget for evaluating at separate points:

from pytreegrav import TidalTensor, TidalTensorTarget

T = TidalTensor(x, m, h, theta=0.4, quadrupole=True, parallel=True)  # shape (N,3,3)
T_target = TidalTensorTarget(x_target, x, m, softening_source=h, parallel=True)

eigenvalues = np.linalg.eigvalsh(T)  # ascending; use eigh if you want the principal axes too
compressive = (eigenvalues < 0).all(axis=1)  # fully compressive tides

Note that the tidal tensor is one derivative higher than the acceleration, so at a given theta its fractional error is correspondingly larger. Use a smaller theta than you would for forces and turn on quadrupole; on a uniform box of $10^5$ particles that costs 0.14 s against 0.04 s for the monopole walk at theta=0.7.

Ray-tracing

pytreegrav's octree implementation can be used for efficient tree-based searches for ray-tracing of unstructured data. Currently implemented is the method ColumnDensity, which calculates the integral of the density field to infinity along a grid of rays originating at each particle (defaulting to 6 rays). For example:

columns = ColumnDensity(x, m, h, parallel=True) # shape (N,6) array of column densities in 6 angular bins - this is fastest but least accurate
columns_10 = ColumnDensity(x, m, h, rays=10, parallel=True) # shape (N, 10) array column densities along 10 random rays
columns_random = ColumnDensity(x, m, h, randomize_rays=True, parallel=True) # can randomize the ray grid for each particle so that there are no correlated errors due to the angular discretization
columns_custom = ColumnDensity(x, m, h, rays=np.random.normal(size=(100,3)), parallel=True)  # can also pass an arbitrary set of rays for the raygrid; these need not be normalized
κ = 0.02 # example opacity, in code units
σ = m * κ # total cross-section in each particle is product of mass and opacity
𝛕 = ColumnDensity(x, σ, h, parallel=True) # can pass cross-section instead of mass to get optical depth
𝛕_eff = -np.log(np.exp(-𝛕.clip(-300,300)).mean(axis=1)) # effective optical depth that would give the same radiation flux from a background; note clipping because overflow is not uncommon here
Σ_eff = 𝛕_eff / κ # effective column density *for this opacity* in code mass/code length^2
NH_eff = Σ_eff X_H / m_p  # column density in H nuclei code length^-2

GPU acceleration (optional)

Column density, monopole tree gravity and brute-force gravity all have an optional CUDA backend, reached with the same device="cuda" flag. Measured on an RTX A6000 against 32 Xeon Gold 6244 threads, on a real astrophysical snapshot (22.3M gas particles):

The two GPU columns differ by what each call has to redo. device="cuda" is a complete one-shot solve: Morton-order the targets, narrow them to float32, pack and upload the tree, launch, copy back. The tree column is a pytreegrav.cuda.CudaPotential/CudaAccel/CudaColumnDensity context, which pays a one-time 0.3-0.4 s pack-and-upload and then only launches. Gravity is where the distinction bites: the walk itself is under a second, so a single call is dominated by everything around it, and holding a context is worth 8x more than the flag. Column density does enough work per call that it barely matters.

pip install pytreegrav[cuda]     # adds numba-cuda; nothing changes for CPU-only users

pytreegrav.cuda is never imported by the package itself, so a CPU-only install is unaffected. All of it is single precision — see the per-quantity accuracy notes below. Below N ≈ 5e3 the GPU is slower than 32 CPU threads for any of these: a kernel launch plus a tree upload is not worth amortizing over that little work. See examples/benchmark_scaling.py --cuda and examples/benchmark_scaling_cuda.png.

Clustered data is the harder case throughout: a warp can hold both dense-core and diffuse sightlines, so lanes wait on each other, and the tree no longer fits in cache. Smooth synthetic clouds do better, so the figures above are the ones to expect on production data.

Column density

columns = ColumnDensity(x, m, h, rays=6, device="cuda")  # single shot; repacks and uploads each call

Only the ray-traced path is supported: pass rays, and leave randomize_rays off. The 6-bin angular estimator (rays=None) has no GPU path.

On real data the error grows with the number of contributions summed along a sightline, so it is a distribution rather than one number. Measured over 1.2M sightlines of that snapshot, against the same walk in float64:

The worst cases are the densest sightlines — 0.0003% of entries exceed 1e-3, and their median column is ~2000x the overall median. Those are the ones where τ ≫ 1 and the answer is "opaque" regardless, so this is comfortably below the error of the uniform-sphere density model itself. But the CPU path agrees with direct summation to ~1e-15, so the two are not interchangeable if you need reproducible digits.

The pack-and-upload of the tree is a rounding error here (0.39 s against a 53 s pass), but for repeated evaluation — many ray grids, target subsets, or timesteps — hold a context and reuse it:

from pytreegrav.cuda import CudaColumnDensity
ctx = CudaColumnDensity(tree)                 # pack + upload once
columns = ctx(pos, rays)                      # per-call transfers are ~2% of runtime

Tree gravity

Monopole only:

phi = Potential(x, m, h, theta=0.7, device="cuda")   # or Accel(...)

Gravity walks are short — under 0.03 µs/particle on the device — so a stateless call spends most of its time reordering targets and uploading the tree rather than walking. On that snapshot, reusing a CudaPotential/CudaAccel context gives 31.7x and 31.6x (0.58 s and 0.65 s against 18.4 s and 20.7 s on 32 threads); the single-shot flag above gives 3.8x and 3.9x.

Because the walk is so short, holding a context matters much more here than it does for column density:

from pytreegrav.cuda import CudaPotential, CudaAccel
ctx = CudaPotential(tree)                     # pack + upload once
phi = ctx(pos, softening, G=1.0, theta=0.7)   # theta is per-call, so one upload serves any opening angle

Measured float32 error against the same, ungrouped walk in float64 — the algorithm the device actually runs — over all 2.2e7 particles of that snapshot, normalised as elsewhere here by rms|a| and by std(phi): potential 1.0e-6 median / 6.2e-6 p99, acceleration 4.3e-8 / 1.3e-5, three to five orders below the opening angle's own ~2e-3. Diffing against the default CPU path instead gives 1.4e-4 and 9.9e-5 medians, 100–2000x larger — but that is the CPU's target grouping opening a superset of nodes, not precision: compiling the device kernels in float64 barely moves either pair.

Quote the normalisation with the number, particularly for the acceleration. This snapshot has max|a| / rms|a| = 716, so dividing by max|a| reports that same 9.9e-5 median as 1.4e-7, and per-particle relative error reports it as 2.2e-3 — it diverges wherever |a| → 0 at force balance, which says nothing about the kernel. See the CUDA docs for the full table.

Brute force

This is where the GPU is at its best — no traversal, no divergence, pure arithmetic:

phi = Potential(x, m, h, method="bruteforce", device="cuda")

387 Gpair/s on an A6000 against roughly 10 on 32 CPU threads, about 40x. Because brute force is exact, the useful consequence is where the crossover moves: on the GPU it stays cheaper than the tree out to N ≈ 1e5, against N ≈ 7e3 on 32 CPU threads — a much wider range in which you can skip the approximation entirely. Its error is float32 accumulation and nothing else (no theta, so no tail), growing as √N: 2.4e-6 at N = 1e3, 5.1e-5 at N = 6e4.

Community

This code is actively developed and maintained by Mike Grudic.

If you would like help using pytreegrav, please ask a question on our Discussions page.

If you have found a bug or an issue using pytreegrav, please open an issue.