6 ms·

I wonder how can authors claim such great certainty in their results, after all, it is all based on number crunching. Some floating point error, and similar is

I wonder how can authors claim such great certainty in their results, after all, it is all based on number crunching. Some floating point error, and similar is bound to creep in...

privong 9y ago
In the text they say they use high-precision floating point calculations to reduce this type of error: > At first, using the obtained initial conditions, we checked the 137 periodic orbits by means of the high-order Taylor series method in the 100-digit precision with truncation errors less than 10^−70 , and guaranteed that they are indeed periodic orbits. > Besides, we use the CNS with even smaller round-off error (in 120-digit precision) and truncation error (less than 10^−90 ) to guarantee the reliability of these 27 families. They do reference some software (e.g., "dop853"), but I'm not familiar with the details of those ODE solvers.
- MichailP 9y ago
  That sounds really impressive. I jumped to comment before reading the paper. I found a nice gem in David Tongs lectures [1] in chapter about Dimensional analysis. He mentions Planck legnth Lp ~ 10^-35m, for which "All indications are that this is the shortest distance scale possible; at distances shorter than Lp, space itself is likely to have no meaning". So 10^-70 sounds like a nice margin. [1] http://www.damtp.cam.ac.uk/user/tong/relativity/dynrel.pdf http://www.damtp.cam.ac.uk/user/tong/relativity/dynrel.pdf
  - DamonHD 9y ago
    Lovely: a completely 'practical' bound on a totally theoretical numeric problem.
  - privong 9y ago
    > He mentions Planck legnth Lp ~ 10^-35m, for which "All indications are that this is the shortest distance scale possible; at distances shorter than Lp, space itself is likely to have no meaning". So 10^-70 sounds like a nice margin. That's an apples-to-oranges comparison, though. The 10^-70 number is a relative error while the planck length is just a length. The numerical computations in the paper were likely done in a system of "simulation units" where the actual lengths, velocities, forces, etc. are normalized to values of order unity. This has advantages for the numerical aspect, in terms of preserving floating point accuracy. But it also means that to translate it to a physical system (e.g., a triple system of stars), the simulation needs to be scaled to physical units. The 10^-70 that's quoted just means that the values should be numerically accurate to 1 part in 10^70. If you wanted to compare the (floating point) accuracy of the simulation with the Planck length, you would need to scale the simulation to a physical size and see what the 10^-70 fractional error would translate to in physical units.
    - azernik 9y ago
      10^-70 would get you below the Planck length for any length scales lower than 10^35m, which is 10^9 times the diameter of the observable universe.
exDM69 9y ago
The numerical methods used for this kind of calculations are engineered to compensate numerical errors (floating point and errors inherent to integrators). They take advantage of a priori knowledge about the laws of physics, in particular the conservation of mechanical energy and angular momentum. Predictor-corrector is one family of methods. Other methods rely on convergence as timesteps change. There is a lot of literature on numerical integration applied to celestial mechanics and new methods being released every year. These methods are tailored to this problem space.
- nonbel 9y ago
  When I looked into making a solar system simulation awhile back, I actually found it shocking how much goes into monitoring and ensuring conservation of energy/momentum in these models. Once I was aware of this issue though, even more shocking to me was that many papers on climate models I came across do not even mention they monitored adherence to conservation laws. There is a disconnect there, I would think this adherence either is a big deal (as would be suggested by the practice in astronomy) or not (as would be suggested by climate research), not that it would change by subfield.
  - DamonHD 9y ago
    It would surely depended on how numerically (un)stable the calculations were. Some of the stuff we did in fixed income derivatives was rock solid robust, and some would go off the rails at the slightest provocation, so I don't think that your thesis is valid as it stands.
    - nonbel 9y ago
      I'm not sure I follow. Both responses to my post seem suggest that "following conservation laws is not an important issue for climate models". Ok, I am open to that but I don't really follow the argument that the stability of the simulation can be used to tell us that violation of conservation laws will not lead to inaccurate results.
  - strainer 9y ago
psi-squared 9y ago
The paper has a section on this, around the end of page 4, which is really interesting. The short version is: They compared their double-precision results to extremely high-precision Taylor expansions (with theoretical 70+ digit accuracy, and calculated in 100+ digit accuracy), and found that they matched to the accuracy you'd expect. That doesn't guarantee that the orbits are perfectly periodic, I suppose, but it does suggest that the orbits are stable with respect to rounding errors up to those you get from using doubles.
- deleted 9y ago
  [deleted]
kmill 9y ago
Well, they can't, because the system might still fly apart at some future point. It is possible that these are close to periodic solutions, but to make it rigorous what they need to do is show that the principle of least action is satisfied for some configuration close to the approximations. This was done by Greg Minton well before the posted paper (abstract [1]). [1] http://jointmathematicsmeetings.org/amsmtgs/2141_abstracts/1086-65-1626.pdf http://jointmathematicsmeetings.org/amsmtgs/2141_abstracts/1...

I wonder how can authors claim such great certainty in their results, after all, it is all based on number crunching. Some floating point error, and similar is bound to creep in...

privong 9y ago

In the text they say they use high-precision floating point calculations to reduce this type of error: > At first, using the obtained initial conditions, we checked the 137 periodic orbits by means of the high-order Taylor series method in the 100-digit precision with truncation errors less than 10^−70 , and guaranteed that they are indeed periodic orbits. > Besides, we use the CNS with even smaller round-off error (in 120-digit precision) and truncation error (less than 10^−90 ) to guarantee the reliability of these 27 families. They do reference some software (e.g., "dop853"), but I'm not familiar with the details of those ODE solvers.

MichailP 9y ago
That sounds really impressive. I jumped to comment before reading the paper. I found a nice gem in David Tongs lectures [1] in chapter about Dimensional analysis. He mentions Planck legnth Lp ~ 10^-35m, for which "All indications are that this is the shortest distance scale possible; at distances shorter than Lp, space itself is likely to have no meaning". So 10^-70 sounds like a nice margin. [1] http://www.damtp.cam.ac.uk/user/tong/relativity/dynrel.pdf http://www.damtp.cam.ac.uk/user/tong/relativity/dynrel.pdf
- DamonHD 9y ago
  Lovely: a completely 'practical' bound on a totally theoretical numeric problem.
- privong 9y ago
  > He mentions Planck legnth Lp ~ 10^-35m, for which "All indications are that this is the shortest distance scale possible; at distances shorter than Lp, space itself is likely to have no meaning". So 10^-70 sounds like a nice margin. That's an apples-to-oranges comparison, though. The 10^-70 number is a relative error while the planck length is just a length. The numerical computations in the paper were likely done in a system of "simulation units" where the actual lengths, velocities, forces, etc. are normalized to values of order unity. This has advantages for the numerical aspect, in terms of preserving floating point accuracy. But it also means that to translate it to a physical system (e.g., a triple system of stars), the simulation needs to be scaled to physical units. The 10^-70 that's quoted just means that the values should be numerically accurate to 1 part in 10^70. If you wanted to compare the (floating point) accuracy of the simulation with the Planck length, you would need to scale the simulation to a physical size and see what the 10^-70 fractional error would translate to in physical units.
  - azernik 9y ago
    10^-70 would get you below the Planck length for any length scales lower than 10^35m, which is 10^9 times the diameter of the observable universe.

exDM69 9y ago

The numerical methods used for this kind of calculations are engineered to compensate numerical errors (floating point and errors inherent to integrators). They take advantage of a priori knowledge about the laws of physics, in particular the conservation of mechanical energy and angular momentum. Predictor-corrector is one family of methods. Other methods rely on convergence as timesteps change. There is a lot of literature on numerical integration applied to celestial mechanics and new methods being released every year. These methods are tailored to this problem space.

nonbel 9y ago
When I looked into making a solar system simulation awhile back, I actually found it shocking how much goes into monitoring and ensuring conservation of energy/momentum in these models. Once I was aware of this issue though, even more shocking to me was that many papers on climate models I came across do not even mention they monitored adherence to conservation laws. There is a disconnect there, I would think this adherence either is a big deal (as would be suggested by the practice in astronomy) or not (as would be suggested by climate research), not that it would change by subfield.
- DamonHD 9y ago
  It would surely depended on how numerically (un)stable the calculations were. Some of the stuff we did in fixed income derivatives was rock solid robust, and some would go off the rails at the slightest provocation, so I don't think that your thesis is valid as it stands.
  - nonbel 9y ago
    I'm not sure I follow. Both responses to my post seem suggest that "following conservation laws is not an important issue for climate models". Ok, I am open to that but I don't really follow the argument that the stability of the simulation can be used to tell us that violation of conservation laws will not lead to inaccurate results.
- strainer 9y ago

psi-squared 9y ago

The paper has a section on this, around the end of page 4, which is really interesting. The short version is: They compared their double-precision results to extremely high-precision Taylor expansions (with theoretical 70+ digit accuracy, and calculated in 100+ digit accuracy), and found that they matched to the accuracy you'd expect. That doesn't guarantee that the orbits are perfectly periodic, I suppose, but it does suggest that the orbits are stable with respect to rounding errors up to those you get from using doubles.

deleted 9y ago
[deleted]

kmill 9y ago

Well, they can't, because the system might still fly apart at some future point. It is possible that these are close to periodic solutions, but to make it rigorous what they need to do is show that the principle of least action is satisfied for some configuration close to the approximations. This was done by Greg Minton well before the posted paper (abstract [1]). [1] http://jointmathematicsmeetings.org/amsmtgs/2141_abstracts/1086-65-1626.pdf http://jointmathematicsmeetings.org/amsmtgs/2141_abstracts/1...