On Thu, Jun 23, 2011 at 6:13 AM, Juha Jäykkä <span dir="ltr"><<a href="mailto:juhaj@iki.fi">juhaj@iki.fi</a>></span> wrote:<br><div class="gmail_quote"><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex;">
> What physical system does it represent and what sort of discretization are<br>
> you using?<br>
<br>
Please see arXiv:0809.4303 for details. The equation is obtained from the<br>
Lagrangian (4) by imposing cylindrically symmetric u with z and t appearing in<br>
complex exponential in a certain way, which decouples z and t from the planar<br>
variables. Furthermore, the angular variable vanishes (the whole point of the<br>
ansatz), leaving one with just the equation for the radial profile of u. (This<br>
is all similar to what is done in the article at Eq. (17), but the article has<br>
further constraints imposed, which eventually gives exact solutions.)<br>
<br>
After some rescaling of the domain and the codomain, one ends up with unknown<br>
g:[0,1] -> [0,1], which is what I am solving.<br>
<br>
> Do you know that the equations have a solution for all values of the<br>
> parameter? Even simple problems may not have solutions for all values of a<br>
<br>
Given the origin of the equation - a well defined Hamiltonian/Lagrangian, I<br>
would be very surprised if there were no solutions. It is hard to prove,<br>
either way, though. If I treat the problem as 3D energy minimisation one, I do<br>
find solutions (of course I do: the energy is bouded from below!), which look<br>
very much like what the diverged SNES line searches end up with, but not<br>
quite. Therefore I believe there are solutions and end up with the theory I<br>
explained in the original post.<br>
<br>
Oh, now that the equations are there, the parameters I am scanning are the<br>
product \beta e^2 and n, where n is comparable to the n in Eq (17). Obviously,<br>
n is an integer so cannot be continued as such, but \beta e^2 is real, so I<br>
start with \beta e^2 = 1, n=1, where the solution is g(y) = y.<br></blockquote><div><br></div><div>You could relax integrality.</div><div> </div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex;">
Which reminds me of another oddity: if I start with the exact solution, my<br>
function value is ~ 1e-11, so I know for sure to keep -snes_atol ~1e-10<br>
because that's the best my function evaluation can do. Could this be the<br>
problem? Too little accuracy in function? I did try up to 6th order central<br>
differences, but it does not help.<br></blockquote><div><br></div><div>It is possible for the discrete equation to have no real solutions, while the</div><div>continuous equation does. Even if it is expensive, I suggest continuing in the</div>
<div>nonlinearity to try to get to a solution. If you find one, it could give you insight</div><div>into designing a search strategy that will work for your equation.</div><div><br></div><div> Matt</div><div> </div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex;">
Cheers,<br>
-Juha<br>
<font color="#888888"><br>
--<br>
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| Juha Jäykkä, <a href="mailto:juhaj@iki.fi">juhaj@iki.fi</a> |<br>
| <a href="http://www.maths.leeds.ac.uk/~juhaj" target="_blank">http://www.maths.leeds.ac.uk/~juhaj</a> |<br>
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</font></blockquote></div><br><br clear="all"><br>-- <br>What most experimenters take for granted before they begin their experiments is infinitely more interesting than any results to which their experiments lead.<br>-- Norbert Wiener<br>