The Barnes-Hut Algorithm

J. Barnes, P. Hut: A hierarchical O(N log N) force calculation algorithm. Nature 324, 1986, p. 446.

# Barnes Hut Method – Key Ideas

## Short-range Forces:

strong and weak nuclear force
fast decay of force contributions with increasing distance
- dense force matrix with $\mathcal{O}(n^2)$, but mostly very small entries
- with cut-off: force matrix is sparse
- complexity of entire force calculation can be reduced to $\mathcal{O}(N)$

Barnes Hut Method – Key Ideas

Long-range Forces:

gravity, electromagnetic force
slow decay of force contributions with increasing distance
- cut-off of force matrix not possible, even far away particles need to be taken into account

Is it possible to reduce the complexity for the computation of long-range forces?

Barnes Hut Method – Key Ideas

Consider Astrophysics:

force w.r.t. a far-away individual star might be neglected
but not the force w.r.t. a far-away galaxy
- approximate forces on a individual star by grouping far-away stars, galaxies, etc. into clusters
- represent clusters by accumulated mass located at its centre-of-mass

Barnes Hut Method – Key Ideas

developed 1986 for applications in Astrophysics
for gravitational force: $\vec{F}_{ij} = \vec{F}(\vec{r}_i, \vec{r}_j) = -\gamma_\text{grav}\frac{m_i m_j (\vec{r}_i - \vec{r}_j)}{\|\vec{r}_i - \vec{r}_j \|^3}$
uses octree with 0 or 1 particles per cell
inner nodes corresp. to clusters of particles (pseudo particle)
idea: gravity force of particle cluster approximated (sum of masses, localised in centre of mass)
same can be done for electromagnetic force (sum charges)

Octrees and Quadtrees for Domain Decomposition

clustering of particles required, where size of clusters depends on the distance to each individual particle
solved by multi-level tree-based domain decomposition
to be done for every particles position (in practice via hierarchical domain decomposition)

Octrees and Quadtrees for Domain Decomposition

Barnes-Hut Algorithm

### Create point cloud - Visualisation by Jeffrey Heer - https://jheer.github.io/barnes-hut/

### Subdivide domain into quadtree - distribute long-range region into subdomains: $$ \Omega^{\rm{far}} = \bigcup_i \Omega^{\rm{far}}_i$$ For each particle (position $x\in\Omega$): - start in root node - descent into subdomains and subdivide until every domain contains 0 or 1 particles Add points

### Compute centers of mass or charge - assign a point $y_0^i$ to each $\Omega^{\rm{far}}_i$ corresponding to the center of mass - for each subdomain compute total mass by summing particles in that subdomain - decomposition depending on size of subdomains: $$ \text{diam} := \sup_{y\in\Omega^{\rm{far}}_i} \|y-y_0^i\| $$

### Force computation For each particle (position $x\in\Omega$): - start in root node - descent into subdomains, until $\theta$-rule satisfied: $$ \frac{diam}{r} \leq \theta,$$ $r$ the distance of pseudo particle from $x$ - accumulate corresp. partial force to current particle Change theta

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Barnes-Hut: Computation of Forces

Implicit separation of short- and longe-range forces:

short-range: all leaf nodes that are reached (containing 1 particle)
long-range: all inner nodes, where descent is stopped (force caused by pseudo particle)

Barnes-Hut: Accuracy and Complexity

Accuracy of Barnes-Hut:

depends on choice of $\theta$
the smaller $\theta$, the more accurate the long-range forces
the smaller $\theta$, the larger the short-range (i.e., the costs)
slow convergence w.r.t. $\theta$ (low-order method)

Complexity:

grows for small $\theta$
for $\theta\rightarrow 0$: algorithm degenrates to ``all-to-all’’ $\to$ $\mathcal{O}(N^2)$
for more or less homogeneously distributed particles:
- number of active cells: $\mathcal{O}( \log N / \theta^3)$
- total effort therefore $\mathcal{O}(\theta^{-3}N\log N)$

Barnes-Hut: Implementation

computation of pseudo particles:
- bottom-up-traversal (post-order)
- sum up masses, weighted average for centre-of-mass
computation of forces:
- traversal of entire tree (outer loop on all particles)
- top-down traversal (pre-order) until $\theta$-rule satisfied (inner loop)
further traversals for time integration
re-build (or update) octree structure after each time step $\to$ requires efficient data structures and algorithms