1. A novel approach to visualizing dark matter simulations
By Kähler et al. Presentation by Gonzalo J. Carracedo 1

2. Introduction
Computing power growth ⇒ accuracy + speed in physical simulations
Astrophysics (cosmology) as one of benefited fields
Smoothed Particle Hydrodinamics (SPH)
N-body problem (some of you already know what I am talking about) 2

3. The N-body problem (reminder for those not having P3)
Goal: simulate space-time evolution of large distributions of mass within our computational domain, central forces considered.
Idea: tracer particles representing centers around which fixed amounts of mass are distributed.
N particles ⇒ N² - N interactions!
Trick: use FFT to compute Poisson equation.
Nowadays: N ≈ 134 · 10⁶ particles! 3

4. The representation problem
Big computational problem at this point too, just because the size of the data (whose storage format says very little about the underlying structure)
Importance of structure in several scale levels (this is, we need to zoom in as much as we can)
This is not the final step: representation can be the input for further post-processing stages (we need to be accurate) 4

5. The classical approach
PCA analysis to hierarchical clustering of information (close relation with K-means) => points of big density
Resampling methods over a regular grid (usually with GPU- assisted algorithms) => 3-D density map
Some work in resampling over tetrahedral grids, tetrahedral decomposition depending of line-of-sight. 5

6. The new proposed method
Non line-of-sight dependant tetrahedral decomposition.
Assumption: first time step has all particles distributed along a regular grid, where d0 · d1 · d2 = N
Connectivity information derived from initial distribution, kept for all time steps.
Tetrahedra made of neighbor vertices representing equal mass => evolution modifies volume (density)
Reusing GPU structures to store this information 6

7. The new proposed method (II)
We do not work over tracer particles anymore, but over tetrahedra!
We can derive a density distribution computing the volumes of each tetrahedron.
Density in one point: sum of densities of tetrahedra containing that point.
Method can be extended for any physical quantity considered constant (and linear) along every tetrahedron. 7

8. Using the tetrahedra representation
Having this internal representation, we have three ways to display it:
a) Centroid approach
b) Resampling method
c) Cell projection 8

9. Implementation details: data storing
The whole method is a «hack» in the meaning of it uses GPU structures in a non conventional way
At first: store the “initial prism” of particles within a 3-D structure. Store positions in the RGB components (we can do this as they're seen as floating point numbers by the GPU). Connectivity given by point indices. 9

10. Implementation details: tetrahedra
Vertex shader called for every tracer, adds connectivity information defining cubical cells.
Geometry shaders generate 6 tetrahedra per cell, adding density information.
In order to stay within GPU resource limit, we work over separate blocks fitting the graphics memory.
Once we have generated all tetrahedra, we'll need to render them, and that takes us to the following point 10

11. Implementation details: Centroids
Compute barycenters of each tetrahedron.
Put a kernel centered on the barycenter, scale it according to its density.
Project all kernels against a 2-D texture buffer using an additive blending equation.
Results are noisy, but it's rather fast and relatively easy to implement. 11

12. Implementation details: Centroids (II)12

13. Implementation details: Resampling
Render-to-texture: we're going to render against a 3-D texture first
For each tetrahedron: we determine a “slab” perpendicular to the z-direction, containing the whole tetrahedron.
Measuring how many forward-facing and backward-facing triangles we cross from the background to each point in the z- direction, we can measure the density in each point. 13

14. Implementation details: Cell-projection
Best display method so-far (according to the requirements described before)
Integrates density along all lines-of-sight (one per tetrahedron)
Additive terms (from back-facing triangles) are computed separately from substractive terms (from front-facing triangles)
Both terms are stored in different color channels. 14

15. Implementation details: Cell projection (II)15

16. Comparative results 16 a) Constant kernel smoothing
b) Adaptive kernel smoothing
c) Voronoi tessellation (Voro++)
d) Centroids
e) Resampling
f) Cell projection 16

17. Comparative results (II)17

18. Future directions
High quality images let us us use them as direct input for further models.
Gravitational lensing
Convergence maps / shear maps are straightforward to compute.
Idea: why not use this for plasma physics? 18

19. Conclusions
It works in a reasonable amount of time, and the results fit with what they promised.
Strong points: massive use of GPU parallelism, hq images and detail in all scale levels.
Weak points: we need a homogeneous (and prismatic) initial configuration!
Personal opinion.
P3 soutenance is in a couple of weeks, just sayin' 19

20. Thank you

21. Bibliography http://ranger.uta.edu/~chqding/papers/KmeansPCA1.pdf