meeting 2024 02 05 n315 - JacobPilawa/TriaxSchwarzschild_wiki_5 GitHub Wiki

Context

  • I've run a few tests on the convex hull sampling routine, given that my previous attempt at sampling points from inside the convex hall seemed to place many more proposed models away from the minimum rather than near it, especially at high Tmin.

  • I've sliced the set of input points we're feeding into the convex hall into a variety of sizes and re-run our routine. I was hoping to gain some insight into the geometry of the hull and understand why we were missing proposed points near the minimum.

  • Takeaway: It seems like the well fitting points at Tmin>0.7 or so occupy an exceptionally small volume of the 6D space, and so the resulting distribution of points from the convex hull is skewed toward the larger volume region (Tmin<0.7 or so). I don't think this is an issue in the convex hull fitting or sampling, but simply from the geometry of our surface at the present.

    • I think it will be a bit difficult to make this into a set of useable points, so I'm inclined at this point to submit a uniform hypercube over the range of parameters we want to explore, and use the convex-hull samping on the next batch of models.
First, here are convex hulls buily from slicing Tmaj into chunks of size 0.1; note that I'm using only the best (models within 500 of the minimum) to build these convex hulls
Slice Plot
0.0-0.1
0.1-0.2
0.2-0.3
0.3-0.4
0.4-0.5
0.5-0.6
0.6-0.7
0.7-0.8
0.8-0.9
0.9-1.0
Taking the resulting proposed points (blue histograms in the above panels) and plotting them next to one another, we can see that the geometry of the hull is a bit strange. In particular, the slices in Tmin for Tmin > 0.7 or so contain a much, much smaller volume in the remaining 5 dimensions than the slices for smaller Tmin. I think this largely explains why the resulting point distributions were so skewed away from the high Tmin points -- there is just not a lot of volume there to begin with relative to the rest of the space.
Plot
Same idea as above but in Tmin bins of 0.2 --
Slice Plot
0.0-0.2
0.2-0.4
0.4-0.6
0.6-0.8
0.8-1.0
Here's the same comparison of the hulls as we did for the case of 0.1 stepsize.
Plot
Same idea as above but in Tmin bins of 0.33 --
Slice Plot
0.00-0.33
0.33-0.66
0.66-1.00
Here's the same comparison of the hulls as we did for the case of 0.1 and 0.2 stepsizes.
Plot
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