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A wider perspective: Why shadowing?

With an introductory level of understanding of shadowing behind us, it may be instructive to devote some time to ``devil's advocate'' questions and answers.

tex2html_wrap_inline3057 : Science is usually interested in generally applicable results. Why should we care about finding an exact solution to a specific initial condition?

tex2html_wrap_inline3251 : We don't. What shadowing tries to do, empirically for now, is show that given any particular initial condition and cheaply integrated solution, that a solution of a more expensive integration exists nearby. Ultimately, using GHYS's containment algorithm, we aim to show that any particular noisy solution has a true solution nearby. If we cannot find such solutions, then we may have good reason to suspect the reliability of our simulations.

tex2html_wrap_inline3057 : OK, but if the system is chaotic, then it is supposed to have sensitive dependence on initial conditions. Shouldn't it be fine that there is noise in our orbits? i.e.,  isn't our theory robust enough to withstand small perturbations? Isn't computational noise roughly equivalent to chaos anyway?

tex2html_wrap_inline3251 : We (or at least I) do not know enough about chaotic systems to say, so for now we should play it safe. In fact, one way of looking at shadowing is that we are actually trying to discover if the theory is robust enough!

tex2html_wrap_inline3057 : There is a large body of theory behind numerical integration of ODE's. In particular, the theory of defect-based backwards error analysis (DBBEA) shows that, for any particular ODE y'=f(t,y), a numerical method will give the exact solution to a slightly perturbed problem tex2html_wrap_inline3261 , where tex2html_wrap_inline3263 is a function whose magnitude is small, roughly the size of the tolerance requested of the numerical ODE method. Furthermore, there exists a theorem of ODEs that states equivalence between the problem of how solutions change with changing initial conditions, and how they change with changing parameters. Isn't this good enough when we realize that tex2html_wrap_inline3263 can be considered a parameter with tex2html_wrap_inline3267 giving the true solution?

tex2html_wrap_inline3251 : Again, we don't know enough about chaotic systems to say. If I want to integrate the gravitational N-body problem, DBBEA tells me that I will get the exact solution to the problem


But this is not the N-body problem. Generally in science, one tries to change as few things as possible between the real world and the model of the world. It seems to me that asking for an exact solution to the same ODE with sligthly different initial conditions is changing less than asking for an exact solution to a slightly different ODE. It is certainly not clear that the latter is good enough, especially considering that we are dealing specifically with problems that we know have sensitive dependence on small changes. Furthermore, Hamiltonian systems have many special properties that will be preserved if we change the initial conditions, but may not be preserved by equation 1.11.

tex2html_wrap_inline3275 What is the ultimate goal of shadowing?

tex2html_wrap_inline3251 : That's a tough question. For one, we are answering question 0 above. A possible ultimate goal of shadowing could be to identify, learn about, recognize, and finally eliminate glitches from our simulations, but it is unclear if all of these are feasible, especially the last. I suspect the GHYS containment procedure may allow us to eliminate glitches when we detect them, but this line of reasoning requires more thought.

next up previous
Next: High Dimensional Shadowing & Up: Shadowing Previous: A new shadowing procedure:

Wayne Hayes
Sun Dec 29 23:43:59 EST 1996

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