3 ms·

The "CNS" is as far as I can tell just the use of a high order Taylor series together with multiple precision arithmetic for solving ODEs. This is a well known

by fdej 9y ago

The "CNS" is as far as I can tell just the use of a high order Taylor series together with multiple precision arithmetic for solving ODEs. This is a well known method for long term simulation of dynamical systems that has been around for decades. The authors basically imply that they invented this method and were the first to be able to study long term evolution of dynamical systems reliably because of it. Fair enough if they just weren't aware of previous work (which happens all the time), but it's an oversight that shouldn't have passed peer review. That said, using this method to discover new periodic solutions of the three-body system is a very neat application, which deserves applause!

spuz 9y ago
Are you sure that their numerical method isn't new? The abstract says that the author created it in 2009 and this paper further develops it to apply to the three body problem. If it already existed, can you say why it had not already been applied to the three body problem?
- fdej 9y ago
  In fact the more general N-body problem is perhaps one of the most classical uses of high order Taylor methods, in particular for studying the long term stability of the solar system. Here is a paper from 1993: http://adsabs.harvard.edu/full/1993A%26A...272..687L http://adsabs.harvard.edu/full/1993A%26A...272..687L. Rigorous error analysis of Taylor methods for ODEs goes all the way back to Moore's original work on interval arithmetic in the 1960s; in the early 2000s Makino and Berz developed so-called Taylor models which combine rigorous error bounds with accurate long-term propagation of changes due to perturbations in the initial values (see http://bt.pa.msu.edu/index_TaylorModels.htm); http://bt.pa.msu.edu/index_TaylorModels.htm); they used it to study the dynamics of the solar system among other things. R. Barrio and others have published several papers about reliable solution of ODEs using Taylor methods; leading to the development of the TIDES software. See several references listed on http://cody.unizar.es/tides.html http://cody.unizar.es/tides.html. In particular the paper https://doi.org/10.1016/j.amc.2004.02.015 https://doi.org/10.1016/j.amc.2004.02.015 from 2005 investigates using variable order Taylor series with extended precision for the solutions of dynamical systems. The paper http://dx.doi.org/10.1155/2012/716024 http://dx.doi.org/10.1155/2012/716024 from 2012 is specifically about obtaining periodic solutions of dynamical systems using a high order Taylor method with multiple precision arithmetic. Actually, I made the first comment after I checked the second author's earlier paper https://arxiv.org/abs/1109.0130 https://arxiv.org/abs/1109.0130 where they essentially made the claim about inventing "CNS" as the first-ever reliable technique for long-term solutions of dynamical systems, without referencing any of the earlier work I mentioned above. However, I just checked the actual text of this new article on the three-body problem (through sci-hub) and there the authors do cite the earlier work I mentioned above, giving proper attribution to others for the basic ideas behind what they call "CNS"! So all is actually well, and the peer review presumably did work (or the authors found out about the earlier work even before writing the new paper). It is just the press release that is misleading, as usual. To answer your question, I'm not really familiar with the research on the N-body problem, so I can't say why or whether no one tried looking for periodic solutions in this way before. Perhaps no one actually thought of it, or they didn't try since they didn't expect to find anything, or they didn't have the computational resources, or they just couldn't figure out the details of how to do it (which the present authors did, and deserve credit for). Again, I was not trying to downplay the significance of this work, and finding new applications of existing methods (and making even tiny improvements along the way) is how science progresses. It also happens all the time that methods get rediscovered/reinvented independently.
  - jcoffland 9y ago
    The novel part is not the use of Taylor series for simulation. CNS stands for Clean Numerical Solution. The method of correcting the numerical error in the simulation is what is claimed to be novel. From the CSN paper: "the residual and round-off errors are verified and estimated carefully by means of different time-step Δt, different precision of data, and different order M of Taylor expansion....for the considered problem, the truncation and round-off errors of the CNS can be reduced even to the level of 10^−1244 and 10^−1000, respectively, so that the micro-level inherent physical uncertainty of the initial condition (in the level of 10^−60) of the H\'{e}non-Heiles system can be investigated accurately."
    - fdej 9y ago
      This is not a new idea. If you have a variable-order, variable-precision implementation, it's the most obvious way to estimate the error of a numerical solution.

by fdej 9y ago

spuz 9y ago

Are you sure that their numerical method isn't new? The abstract says that the author created it in 2009 and this paper further develops it to apply to the three body problem. If it already existed, can you say why it had not already been applied to the three body problem?

fdej 9y ago
In fact the more general N-body problem is perhaps one of the most classical uses of high order Taylor methods, in particular for studying the long term stability of the solar system. Here is a paper from 1993: http://adsabs.harvard.edu/full/1993A%26A...272..687L http://adsabs.harvard.edu/full/1993A%26A...272..687L. Rigorous error analysis of Taylor methods for ODEs goes all the way back to Moore's original work on interval arithmetic in the 1960s; in the early 2000s Makino and Berz developed so-called Taylor models which combine rigorous error bounds with accurate long-term propagation of changes due to perturbations in the initial values (see http://bt.pa.msu.edu/index_TaylorModels.htm); http://bt.pa.msu.edu/index_TaylorModels.htm); they used it to study the dynamics of the solar system among other things. R. Barrio and others have published several papers about reliable solution of ODEs using Taylor methods; leading to the development of the TIDES software. See several references listed on http://cody.unizar.es/tides.html http://cody.unizar.es/tides.html. In particular the paper https://doi.org/10.1016/j.amc.2004.02.015 https://doi.org/10.1016/j.amc.2004.02.015 from 2005 investigates using variable order Taylor series with extended precision for the solutions of dynamical systems. The paper http://dx.doi.org/10.1155/2012/716024 http://dx.doi.org/10.1155/2012/716024 from 2012 is specifically about obtaining periodic solutions of dynamical systems using a high order Taylor method with multiple precision arithmetic. Actually, I made the first comment after I checked the second author's earlier paper https://arxiv.org/abs/1109.0130 https://arxiv.org/abs/1109.0130 where they essentially made the claim about inventing "CNS" as the first-ever reliable technique for long-term solutions of dynamical systems, without referencing any of the earlier work I mentioned above. However, I just checked the actual text of this new article on the three-body problem (through sci-hub) and there the authors do cite the earlier work I mentioned above, giving proper attribution to others for the basic ideas behind what they call "CNS"! So all is actually well, and the peer review presumably did work (or the authors found out about the earlier work even before writing the new paper). It is just the press release that is misleading, as usual. To answer your question, I'm not really familiar with the research on the N-body problem, so I can't say why or whether no one tried looking for periodic solutions in this way before. Perhaps no one actually thought of it, or they didn't try since they didn't expect to find anything, or they didn't have the computational resources, or they just couldn't figure out the details of how to do it (which the present authors did, and deserve credit for). Again, I was not trying to downplay the significance of this work, and finding new applications of existing methods (and making even tiny improvements along the way) is how science progresses. It also happens all the time that methods get rediscovered/reinvented independently.
- jcoffland 9y ago
  The novel part is not the use of Taylor series for simulation. CNS stands for Clean Numerical Solution. The method of correcting the numerical error in the simulation is what is claimed to be novel. From the CSN paper: "the residual and round-off errors are verified and estimated carefully by means of different time-step Δt, different precision of data, and different order M of Taylor expansion....for the considered problem, the truncation and round-off errors of the CNS can be reduced even to the level of 10^−1244 and 10^−1000, respectively, so that the micro-level inherent physical uncertainty of the initial condition (in the level of 10^−60) of the H\'{e}non-Heiles system can be investigated accurately."
  - fdej 9y ago
    This is not a new idea. If you have a variable-order, variable-precision implementation, it's the most obvious way to estimate the error of a numerical solution.