Another possible way to accelerate MB set deep zooming

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Offline gerrit

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« Reply #15 on: January 28, 2018, 12:24:43 AM »
For warped mini doesn't look so good. I just bailout the fp iterations for |z|> R = 0.09-0.01 with attached results. (K,M)=(6,5), increasing makes no difference.

I think the stopping condition |z|>R for the approximation should be replaced by something more sophisticated, maybe \( \frac{\partial fp}{ \partial z} >1 \) or something similar? (That doesn't seem to work...)

Offline knighty

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« Reply #16 on: January 28, 2018, 01:16:41 PM »
Found the bug, see attached for various (K,M). Iterating till 1e5, except (4,3)b uses bailout rad = 0.003, (4,3) also and then continues normal PT.
Where did you get the number 0.003 from? Seems pretty good now. But (K,M)=(2,1) (normal renormalization approx) seems as good as using higher order.
Let me try a warped mini next.
:horsie: :clapping:
I guess it doesn't need high order approximation because the minibrot's deformation is very small.
the 0.003 is approximately the square root of the size of the embedded julia set of that minibrot. It is just an estimation, I don't know a good method that gives the correct escape radius. Maybe it is related somehow to the size of the atom domain or maybe something that happen in the dynamic plane...

For warped mini doesn't look so good. I just bailout the fp iterations for |z|> R = 0.09-0.01 with attached results. (K,M)=(6,5), increasing makes no difference.

I think the stopping condition |z|>R for the approximation should be replaced by something more sophisticated, maybe \( \frac{\partial fp}{ \partial z} >1 \) or something similar? (That doesn't seem to work...)
The escape radius to use is obviously greater than 0.09, maybe around 0.25. It is strange that the minibrot that is obtained doesn't seem to be deformed. Does it need a higher order approximation or aren't you using only the very first terms of fp instead of all of them?

Here are two pictures using orbit trap in dynamic and parameter planes. The trap is a circle (here centred at 0 and of radius 0.28..).

Offline gerrit

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« Reply #17 on: January 28, 2018, 02:22:20 PM »
The escape radius to use is obviously greater than 0.09, maybe around 0.25. It is strange that the minibrot that is obtained doesn't seem to be deformed. Does it need a higher order approximation or aren't you using only the very first terms of fp instead of all of them?
Strange indeed. I checked that all terms are actually contributing, not underflowing etc.
Perhaps the approximation breaks down already inside the warped mini?
I'll try adding some probes to see what's going on.

Offline knighty

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« Reply #18 on: January 30, 2018, 10:11:49 PM »
Ok! here is my attempt. ;D Very simple implementation: no optimization and no scaling.
The location below needs a high order approximation to look right.

Offline gerrit

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« Reply #19 on: January 31, 2018, 12:15:13 AM »
Which pixels are computed with the accelerated method?

Offline pauldelbrot

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« Reply #20 on: January 31, 2018, 01:56:48 AM »
You can see the dwell bands. So the outer red points got no acceleration, the green inside the outer circular border got the minibrot's period iterations, the next red band in got twice the minibrot's period, and so forth.

Offline gerrit

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« Reply #21 on: January 31, 2018, 02:40:21 AM »
I found my off-by-one error. I'm getting the same now  :thumbs:

Offline pauldelbrot

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« Reply #22 on: January 31, 2018, 02:58:13 AM »
I'd like to see zooms of an embedded Julia near that mini, both with and without this method, to see if it's rendering even those accurately. A shallowish one in the third or fourth renormalization-dwell-band would suffice for this.

Offline gerrit

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« Reply #23 on: January 31, 2018, 03:14:57 AM »
If I set escape rad = 0.1 I see no bands or distortion at all even with only (4,4).

Offline gerrit

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« Reply #24 on: January 31, 2018, 03:28:02 AM »
I'd like to see zooms of an embedded Julia near that mini, both with and without this method, to see if it's rendering even those accurately. A shallowish one in the third or fourth renormalization-dwell-band would suffice for this.
I tried but I get all pixels escape after 1-2 super-iteration (= p normal ones) in a small ball around the nucleus of the Julia, so there seems no saving there.
Maybe knighty can get it to work. I'm not clear on how to chose the escape radius.

Offline gerrit

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« Reply #25 on: January 31, 2018, 08:04:27 AM »
With DE I can't get rid of the band artifacts even at high order and low bailout. Doing something wrong again?

Offline knighty

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« Reply #26 on: January 31, 2018, 11:01:13 AM »
Maybe another "off by one" error; or initialization or bailout.

I'd like to see zooms of an embedded Julia near that mini, both with and without this method, to see if it's rendering even those accurately. A shallowish one in the third or fourth renormalization-dwell-band would suffice for this.
I tried but I get all pixels escape after 1-2 super-iteration (= p normal ones) in a small ball around the nucleus of the Julia, so there seems no saving there.
Maybe knighty can get it to work. I'm not clear on how to chose the escape radius.
That (^^^) location with distorted minibrot is not a very good example because it at a zoom of just 100x or so. The embedded julia set is already visible in those (^^^) pictures.
I'm also not clear but it seems to work! If smb finds the root at some zoom level, which usually happens first at the first embedded Julia set, the escape radius can be set to that zoom.
(better) Example:
(Of course, for the first zoom level, we merely do one "super iteration". the escape radius here is 1e-13)

« Last Edit: January 31, 2018, 11:39:27 AM by knighty »

Offline knighty

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« Reply #27 on: January 31, 2018, 11:02:08 AM »
Continuing...

Offline knighty

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« Reply #28 on: January 31, 2018, 12:12:35 PM »
For the escape radius of the "super iteration" I think the size of the atom domain is the way to go. Just make it a little bigger (not too much). It is good that in most cases order 4 approximation is sufficient. Also:
...the first coefficient is 0 (or very small value) and that the corfficient in odd powers of b are also very small (ideally 0). The first is obvisously because c0 have period n and the second comes from the fact that z0=0 (I don't have a formal proof, just a verification on a small example).
That may explain the results.
is true.

I did a little mistake in the code posted yesterday. in iteratePt() function it should be:
Code: [Select]
if(abs(d0)<Bout){
while(i<maxiter && norm(d)<Bout){
d = fp.eval(d,d0);
i+=period;
si++;
}
}
instead of:
Code: [Select]
if(abs(d0)<Bout){
while(i<maxiter && abs(d)<Bout){
d = fp.eval(d,d0);
i+=period;
si++;
}
}
There are still things to be verified...

Offline gerrit

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« Reply #29 on: January 31, 2018, 06:02:17 PM »
Maybe another "off by one" error; or initialization or bailout.
Can you try DE with this method and with a more revealing color scheme than just b/w? My iterations look perfect, but the DE not, but my code is not very good, maybe I should stick to my math :)

Instead of a "bailout radius" maybe it's better to think of it as a "stop criterion" like the methods for usual SA?

If you write the expansion as \( \sum_{ij}a_{ij}z^i (c-c_0)^j \), the \( a_{0j} \) is just the good old SA. Could that observation be used to integrate this method seamlessly with conventional SA? Sort of like far away from the mini it becomes usual SA but closer you can skip more if you go into the higher "super" iteration bands?

This has the potential to become the next big advance in speedup!



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