raptop changed the topic of #kspacademia to: https://gist.github.com/pdn4kd/164b9b85435d87afbec0c3a7e69d3e6d | Dogs are cats. Spiders are cat interferometers. | Космизм сегодня! | Document well, for tomorrow you may get mauled by a ネコバス. | <UmbralRaptor> egg|nomz|egg: generally if your eyes are dewing over, that's not the weather. | <ferram4> I shall beat my problems to death with an engineer. | We can haz pdf
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<egg>
mofh: underlying one-step method
<egg>
mofh: but can it be computed, at least for a simple ODE
<egg>
mofh: flyswat
<mofh>
egg:
<mofh>
okay let me look at the paper
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<egg>
mofh: flyswat
<mofh>
egg: was helping someone with a condensed matter problem, featuring me taking an hour to realize that the integral of 1 over 3-space is a volume (literally figured out the rest of the highly technical argument immediately, then got stuck on how to convert \int 1 dV into V).
<mofh>
literally just got that done, will look at paper now.
<egg>
...
<egg>
mofh: s/\\int/∫/
<galois>
egg thinks mofh meant to say: egg: was helping someone with a condensed matter problem, featuring me taking an hour to realize that the integral of 1 over 3-space is a volume (literally figured out the rest of the highly technical argument immediately, then got stuck on how to convert ∫ 1 dV into V).
<kmath>
<ThePatanoiac> The messenger raven drops the thumb drive onto your desk, then squawks out a carefully-rehearsed encryption key.
<mofh>
egg: that... is not a nice functional equation at all.
<egg>
mofh: if it were nice i would not be summoning a moth
<mofh>
egg: like first off, quick sanity check: what is F as related to f?
<mofh>
(i.e. equation 2)
<egg>
mofh: depends on the multistep method
<egg>
for a linear multistep method it's a linear combination of the f(x_i)
<egg>
mofh: for a multistep method for a 2nd order ODE, it's that combined with a difference formula to get the derivatives, I guess?
<egg>
or something
<egg>
not sure what happens in the 2nd order case really
<mofh>
okay first off this paper is atrociously written, the author could stand to, like, make it a bloody page longer and not leave half the notation up to guesstimation or having to work it out oneself.
* mofh
squints at the functional equation again
<mofh>
Like I'm actually uncertain under what conditions this can be computed, and phrases like "the dependence of S on f is nonlinear even when F is linear in f" (which makes obvious sense when you look at it) aren't comforting.
<egg>
mofh: related literature: Hairer (2006), Conjugate-symplecticity of linear multistep methods, which explicitly gives the 1-step method for the harmonic oscillator, but in a weird complex 1st order formulation so I'm not sure how to apply that to methods for 2nd order ODEs; Chartier, Faou, and Murua (2006), An algebraic approach to invariant preserving integators: The case of quadratic and Hamiltonian invariants; Hairer and
<egg>
Lubich (2004) Symmetric multistep methods over long times
<egg>
Current Challenges in Stability Issues for Numerical Differential Equations seems interesting
<_whitenotifier-3d18>
[Principia] pleroy closed issue #2111: Do not crash when the target prediction cannot be computed until the desired time - https://git.io/fjkmE
<_whitenotifier-3d18>
[Principia] pleroy closed issue #2112: Extending the target vessel prediction should be asynchronous - https://git.io/fjkmH
<_whitenotifier-3d18>
[Principia] pleroy closed pull request #2116: Asynchronous computation of target trajectory and frame - https://git.io/fjkoA
<_whitenotifier-3d18>
[Principia] pleroy pushed 3 commits to master [+0/-0/±17] https://git.io/fjkK6
<_whitenotifier-3d18>
[Principia] pleroy 69abe25 - Remove FlowPrediction and add a 1-parameter RefreshPrediction.
<_whitenotifier-3d18>
[Principia] pleroy dad9459 - All tests passing.
<_whitenotifier-3d18>
[Principia] pleroy a4c8034 - Merge pull request #2116 from pleroy/Targetting Asynchronous computation of target trajectory and frame
<mofh>
egg: I mean this is what you expect for the harmonic oscillator, just trying to generalize it to a higher-order ODE is being tricky
<egg|cell|egg>
Yes
<egg>
mofh: but it's not really interesting in the harmonic oscillator case, since there the underlying one-step method is symplectic rather than just conjugate symplectic, right?
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<egg>
mofh: "For symmetric linear multistep methods, which cannot be strictly stable, such an underlying one-step method exists as a formal series in powers of h (see [6, p. 274] and [12, Sect. XV.2.2]). Despite its non-convergence, it can give much insight into the long-time behavior of the method."
<egg>
hm
<mofh>
egg: ...I believe so, yes.
<egg>
yeah it even says so in (16) of that book
<mofh>
And yeah, formal series was my first thought as well, but since I couldn't confirm convergence I didn't mention it.
<egg>
mofh: is the formal series 1. computable decently 2. usable in any way?
<egg>
mofh: say for the simple pendulum or something like that that's not quite as trivial as the harmonic oscillator
<egg>
mofh: otherwise if the underlying 1-step method is symplectic rather than conjugate symplectic, and also is a rotation of the phase space, not much will happen to the cats
<egg>
they will be bored
<mofh>
1. yes, 2. dunno, I need to check those references which claim that it *is*, somehow.
<mofh>
hmm
<egg>
mofh: 1. how
<egg>
mofh: also 2. if ys
<mofh>
egg: moment
<egg>
mofh: tbh already with the harmonic oscillator I can see interesting things wrt to exact vs. judicious startup
<mofh>
egg: oh?
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<egg>
mofh: unit harmonic oscillator, h = 1.35, method: ρ(ζ) = (ζ − 1)^2(ζ^2 + 1), σ(ζ) = 1/6(7ζ − 2ζ^2 + 7ζ^3), startup values in black + 6 steps fading grey; left: startup from exact values, right: startup from judicious values (values resulting from the underlying one-step method)
<egg>
(yoinked the word judicious from Kirchgraber)
<egg>
"Unfortunately the judicious supplementary initial values are not easily available." << It would be fun to get them for a pendulum to see something actually conjugate-symplectic, here the right-hand-side cats are symplectic
<egg>
mofh: also this is a picture of atlas,
<egg>
mofh: flyswat
<mofh>
egg: trying to figue out how to obtain those "judicious supplementary initial values" >_< (also rereading notes on dynamical systems theory, it's been awhile)
* UmbralRaptop
🔪 the alcubierre metric
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<egg>
mofh: how do you like the catplot though
<UmbralRaptor>
pip install catterplot
<egg>
mofh: flyswat
<mofh>
egg: pretty sure the catplot is giving me an impression of the cats being trapped in a demonstration of the Coriolis Force,
<mofh>
(it's good)
<egg>
tbh the cats are a bit dishonest because the velocities are evolved with the underlying one-step method
<egg>
I should try getting the velocities solely from the positions by a difference formula (or maybe a cohen-hubbard oesterwinter formula)
<mofh>
Would that actually make much of a meaningful difference? The two methods of time-evolution you just described *should* result in very similar results...
<egg>
mofh: well, the velocities aren't evolved in a 2nd order multistep method
<egg>
they are derived from the positions, so they don't accumulate their own errors
<egg>
mofh: so it turns out that using a modified cohen-hubbard oesterwinter method for the velocities does change the area of the right-hand-side cats a bit
<egg>
mofh: but it remains constant at 0.95686 after startup, with no further changes
<egg>
mofh: basically the cats are slightly squished vertically but nothing else happens, no parasitic solution in the position evolution and thus no net change in area
<egg>
whereas the left-hand-side cats go down to 0.274129 because of the parasitic solution
<mofh>
Huh. So you *do* get a parasitic solution if you don't use cohen-hubbard oesterwinter?
<egg>
mofh: ? the parasitic solution is from nonjudicious startup, not tied to the velocity computation; if I don't use CHO I'm not sure what I'm doing, I'm using a 1-step method to propagate the velocities next to (and with) positions computed multistep, so it's just silly and I wanted to avoid that
<mofh>
Ohh, I see what you mean now.
<mofh>
Right, that's what would make sense.
<_whitenotifier-3d18>
[Principia] RocketSquid opened issue #2118: "Apocalypse has occurred" message never goes away - https://git.io/fjk14
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<_whitenotifier-3d18>
[Principia] eggrobin commented on issue #2118: "Apocalypse has occurred" message never goes away - https://git.io/fjkMR
<_whitenotifier-3d18>
[Principia] eggrobin commented on issue #2118: "Apocalypse has occurred" message never goes away - https://git.io/fjkM0
<egg>
mofh: UmbralRaptor: help users want to cancel the apocalypse
<UmbralRaptor>
Yay?
<egg>
UmbralRaptor: but you can't cancel the apocalypse
<egg>
mofh: anyway, how do i do the same catplots with a simple pendulum,