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Author Topic: Philip Moriarty on the mandelbrot set  (Read 8499 times)
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TheRedshiftRider
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« Reply #15 on: August 29, 2016, 02:38:09 PM »

Please explain, how did I say you can't advertise perturbation methods?
Circular argument here: you can't completely automate if the machine can't detect something authomatically, therefore you can never completely automate.

The original algorithm doesn't need checking, which is the goal, which is why Knighty and others are working on it. The UltraFractal author claimed to have a bulletproof method at the 2014 Fractal Symposium, and explained to us how it works, but I haven't seen any results yet (and it's obviously not something I can share).
Thanks for the reminder, it's almost as if I didn't say this in my very first post lips are sealed
Scared of the Mandel-mob...
You are being silly like you always do. I am out of this discussion.
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cKleinhuis
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« Reply #16 on: August 29, 2016, 03:49:24 PM »

unlocked it, it is a valid discussion here

in my opinion it is is way over my head to argue the theoretical foundations of pertubation theory, what lycium wants is a value for the error, i believe this is somewhere stated in the proofs of the theory, but in no way known to me .... at least physicians use pertubation theory to build atom bombs ... or to simulate atom bombs and then build these fluffing crap .... so that it is too wrong cant be possible, for me they are both apples, one grown naturally and the other grown in the artifical apple building lab ....
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lycium
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« Reply #17 on: August 29, 2016, 04:06:09 PM »

Thank you for unlocking the topic (it seems when he leaves the discussion, no one else may continue); it is frankly unbelievable that a "moderator" resorts to ad hominem when he runs out of substantial replies (after getting his last word, of course, not about the points raised, of course). As for being "silly", you can see why I was hesitant to write an aside with my opinion...

Again, all I was doing is advocating clear thinking / distinctions, and once again I regret doing so on this forum. I'm out of this "discussion" too, serves me right for trying to raise a point about furthering the field.
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kram1032
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« Reply #18 on: August 29, 2016, 10:01:09 PM »

I'm pretty sure Dr. Moriarty actually knows quite a bit about perturbation theory since it's heavily applied in physics and particularly in those kinds of physics he studies. Indeed it's a rather large problem that you almost can't calculate anything meaningful in the full standard model of physics (as well as things like super symmetry or SUSY) without making use of perturbation theory, and to my knowledge (I might be wrong: It's not like I have worked with that stuff yet) it's a huge challenge to get all those calculations to meet all conditions for them to actually be valid.
I can only assume that the random artifacts you tend to see appear for the same reason. It's a challenge for rendering fractals because it's a challenge everywhere. It's a topic where you have to step forward with extreme caution, as per the motto: "Garbage in -> Garbage out", but the garbage in this case looks almost indistinguishable from the real thing.
And many people are actually calling for non-perturbative (i.e. exact) rewrites of those same theories in hopes to get rid of all the mess that it entails. Many of the tools we currently have available just aren't up to snuff for now. It's being worked on, at the very edge of today's math and as such it'll probably take years if not decades for the hopefully eventually successful tools to trickle down into niche communities such as fractal art creators.

Of course that doesn't mean people shouldn't be trying anyway. smiley
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quaz0r
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« Reply #19 on: August 29, 2016, 11:35:09 PM »

lycium has taken this stance before about perturbation.  i think (?) he is more saying, how can we Prove that perturbation will always produce the exact same output as the standard method?  I don't think one can offer such a proof.  Anecdotally speaking after a few years of using perturbation rendering with automatic glitch detection and correction, it appears to be solid enough to use.  One would also have to decide what exactly they mean by "proof" and "exact."  Maybe if you used full arbitrary precision for all variables in the perturbation method it could be "proven" (?) to produce "exactly the same" results as the standard method, but the whole reason we use perturbation is to use less precision to gain speed and hope it all works out in the end.  How much does this affect the outcome and in what ways?  Are the numbers maybe a little bit off, but the iterations still come out the same, or maybe are the number of iterations sometimes off by a few in the end?  I dont think anyone has really investigated this much.

At least practically speaking, perturbation itself appears to produce identical or virtually identical images.  What is really left to nail down 100% is the utilization of series approximation to initialize points - more specifically, the stop condition, which is what is currently being further investigated, and looks to have a promising outcome thanks to the likes of knighty and claude  A Beer Cup
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cKleinhuis
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« Reply #20 on: August 30, 2016, 01:03:53 AM »

at least i dont understand why the visual proof is not beeing accepted, stuff like the shape stacking and the obvious behaviour that is quite regularly e.g. doubling spirals, real minibrots and - in my point of view - are concrete hints to that the general approach seems to be intact using the method ... i just mean, if it looks like a mandelbrot, if it smells like a mandelbrot, sounds like a mandelbrot, feels like a mandelbrot and behaves like a mandelbrot ... it has to be THE mandelbrot

in my youth i really was thinking there is something new down there, today we now it is "just" endles repetition of previous behaviour, spiral doubling every n iterations and so on

recent competition nevertheless produced some stunning views from down there
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kram1032
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« Reply #21 on: August 30, 2016, 01:20:43 AM »

exactness in this case is rather easy to define.
If you go into the limit of n->inf for the perturbation method, does it converge against the exact same thing as if you were to use a standard iteration method? - Rounding errors and the like not withstanding.
If the answer is yes, then it's exact.

It's kinda silly that things like "automatic error detection" are even necessary. If it really is correct, and there should be a way to make it fully correct, there will be no error (larger than what ever the error is for a given n) - certainly no additional structure that persists no matter how high you push n. And as a result there will be no need for any kind of error detection.

And that should be provable. - In a formal mathematical sense.
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quaz0r
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« Reply #22 on: August 30, 2016, 02:04:01 AM »

Quote from: kram
It's kinda silly that things like "automatic error detection" are even necessary. If it really is correct, and there should be a way to make it fully correct, there will be no error

well, again, it seems the perturbation method itself is correct and will result in no errors if enough precision is used.  it is the utilization of not enough precision which introduces the errors.  hapf mentioned once conducting some experiments using more precision when necessary to avoid having to recalculate points.  i suspect this must be what the ultrafractal author meant if he in fact made some statement about "avoiding" glitches.  as it is a precision issue, there would really be no other way.
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kaludix
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« Reply #23 on: September 04, 2016, 06:39:37 AM »

Quote from: quaz0r
well, again, it seems the perturbation method itself is correct and will result in no errors if enough precision is used.  it is the utilization of not enough precision which introduces the errors.

I agree with the statement above.  I can't speak to the perturbation method in general but when applied to fractals, as discussed here, there are no approximations beyond numerical precision.  When deriving the perturbation expressions (for example equation (1) in the K.I. Martin technical note) nowhere is an approximation made.  Rather the equations are recast in terms of quantities that, for a suitable reference point, can be calculated using lower precision.  These quantities are the orbit relative to a known reference orbit which is usually calculated using high precision math.  Glitches occur when the reference orbit is not close enough to the orbit of interest such that their difference can be adequately expressed in terms of standard precision.

This isn't the case when the perturbation method is used with series approximation.  When using series approximation you are truncating the expression for the relative orbit which is an approximation beyond just numerical precision.
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kram1032
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« Reply #24 on: September 05, 2016, 09:52:18 PM »

Ok, that sounds fine. Though it ought to be possible to get bounds on when higher precision or a closer orbit are necessary without any sophisticated method to check that all is fine. What does this error correction actually entail? How does it decide whether an error has occurred and we need a boost? If that also is a simple check - akin to checking whether a point falls outside the circle of radius 2 to decide that it isn't in the Mset - then I really don't see a problem with the technique, based on what you guys are saying.
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kaludix
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« Reply #25 on: September 06, 2016, 08:34:50 AM »

The celebrated method discovered by Pauldelbrot here is as follows:
http://www.fractalforums.com/announcements-and-news/pertubation-theory-glitches-improvement/msg73027/#msg73027
Directly under his last formula, he briefly notes the implementation which is not difficult to code for.

Note that a simpler check of verifying the size of the relative orbit remains small is also needed.  Depending on how one codes the remaining details of the perturbation method, further checking may also be required.
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kram1032
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« Reply #26 on: September 06, 2016, 11:54:01 AM »

Ok, I see... Well, the solution Pauldelbrot came up with is brilliant but it still has a slight ad-hoc-ness smell to it. It's probably fine but I bet there are some further improvements to be made. For instance, perhaps we could figure out why 10^{-3} is a sensible threshold. Though the most convoluted part is where he starts hybridizing the two techniques.

Mind you, that's perfectly fine for image generation. If they look correct, that's probably good enough from an artistic standpoint. But there probably still are good improvements to be had. - And even if the numbers all turn out exactly correct (within the error bounds), the scheme could probably be simplified while yielding the same thing.

That post is over 2 years old, however. I didn't follow all the perturbation theory stuff too closely (and clearly missed out on a big deal) so I can only assume that improvements on that idea have been made since?
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quaz0r
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« Reply #27 on: September 06, 2016, 12:13:11 PM »

http://www.fractalforums.com/announcements-and-news/pertubation-theory-glitches-improvement/msg91934/#msg91934
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TheRedshiftRider
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« Reply #28 on: September 06, 2016, 12:16:10 PM »

In a previous post I said I left the topic. I regret saying that. I will join the discussion again as I have changed my mind and I have something to contribute.

Anyway, I have sent an e-mail to Sir Moriarity. And he replied, see the screenshots below:


Edit: Hey quasor!


* Screenshot_2016-09-06-12-07-36.png (90.1 KB, 480x800 - viewed 97 times.)

* Screenshot_2016-09-06-12-07-43.png (114.68 KB, 480x800 - viewed 99 times.)
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kram1032
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« Reply #29 on: September 06, 2016, 12:58:11 PM »

I see, so this discussion is already happening elsewhere smiley

Meanwhile, Dr. Moriarty's reply is more relevant to this thread. Btw, TheRedshiftRider, he was kind enough not to point it out but you might have an i to many in his name there wink
Keep us posted if that correspondence develops any further though. He probably wouldn't have the time but perhaps he could have a look at the perturbation work and give a pointer towards improvements.
The perturbation theory link he gave is sure to be a good start:
http://fizika.unios.hr/~ilukacevic/dokumenti/materijali_za_studente/qm2/Lecture_2_Perturbation_theory.pdf for your convenience.
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