Optimize many variables 1

[MoNotts]

Hi,

I've got a quick question: Imagine you have a field of 100x100x3 points with known spacings. Each individual point can be moved by a known range in x, y and z. Your task it to maximize the volume enclosed by the points. Now easiest thing might be to use an optimization algorithm however, due to having so many data points it is fairly impossible due to computing power restrictions. So could I just optimize it for a lets say 5x5x2 field and then scale the results? Is there any academic background supporting this so I can justify the method? So far it seems to work but I'd prefer to have something backing it up.

Thank you very much!

 

[GregLocock]

Well, unless I've misunderstood totally your big task is to find the volume enclosed by your set of points. That is non trivial. A sensitivity analysis thereafter seems like small beer.

This depends entirely on whether your point cloud is a reasonably well defined shape or just random x y z. and whether dx dy dz is small in comparison.

So it is a good interesting question but the answer is ...it depends.





Cheers

Greg Locock


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[MoNotts]

Hi Greg,

thank you for your reply. I have just read back what I have written and I have not expressed myself clearly enough I believe. The x y z points are not random. The spacing between the individual points remains constant on a x, y and z basis. E.g. the x spacing is 2mm for all, y spacing is 3.5mm for all and z spacing is 1mm for all. Then a movement range of +- 0.5mm is assigned to x, +-1 to y and -0.25 to z. This is just an example of course.

I hope this is a better explanation. I will have a look into sensitivity studies. I have to admit the last time I have used them was back at Uni while I was doing my mechanical engineering degree but let's see whether my education pays off.

Thank you!

 

[IRstuff]

?? I don't understand what there is to do with the volume if the points are essentially fixed. If the points are essentially fixed, other than jiggling around their nominal locations, the mean volume isn't that different than the volume of the mean point locations. Am I misreading this?

TTFN
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[MoNotts]

No you are not, it is not that different but it is different. E.g. if you have four points in a square and you move the opposite points in the opposite direction by their maximum movement range you can increase the volume (by how much of course depends on the range of movement).

 

[IRstuff]

?? But with a large array, what's to optimize? You draw a ellipsoid around each point and draw a volume around all the ellipsoids. What else is there to do?

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[IRstuff]

Sounds like you're trying to find a convex hull:

http://en.wikipedia.org/wiki/Convex_hull

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[MoNotts]

Hi IRstuff,

That's a great solution. The only question I'm asking myself is how could I miss the obvious. I need to adapt it a bit because the z axis is iterative, i.e. the distance between the points is dependent on the previous point placement but a simple additional term should take care of this. Thank you so much!