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.
pip install pytreegrav or clone the repo and run python setup.py install from the repo directory.
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, PotentialN = 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 functionsNow 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
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:
theta |
RMS error | max error | solve time |
|---|---|---|---|
| 0.1 | 1.4e-05 | 1.3e-04 | 1.07 s |
| 0.4 | 4.8e-04 | 6.4e-03 | 0.13 s |
| 0.7 (default) | 1.8e-03 | 1.9e-02 | 0.07 s |
| 1.0 | 4.8e-03 | 6.2e-02 | 0.06 s |
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
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
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^-2This 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.