On Thu, Jun 23, 2011 at 6:13 AM, Juha J?ykk? <juhaj at iki.fi> wrote:
> > What physical system does it represent and what sort of discretization > are > > you using? > > Please see arXiv:0809.4303 for details. The equation is obtained from the > Lagrangian (4) by imposing cylindrically symmetric u with z and t appearing > in > complex exponential in a certain way, which decouples z and t from the > planar > variables. Furthermore, the angular variable vanishes (the whole point of > the > ansatz), leaving one with just the equation for the radial profile of u. > (This > is all similar to what is done in the article at Eq. (17), but the article > has > further constraints imposed, which eventually gives exact solutions.) > > After some rescaling of the domain and the codomain, one ends up with > unknown > g:[0,1] -> [0,1], which is what I am solving. > > > Do you know that the equations have a solution for all values of the > > parameter? Even simple problems may not have solutions for all values of > a > > Given the origin of the equation - a well defined Hamiltonian/Lagrangian, I > would be very surprised if there were no solutions. It is hard to prove, > either way, though. If I treat the problem as 3D energy minimisation one, I > do > find solutions (of course I do: the energy is bouded from below!), which > look > very much like what the diverged SNES line searches end up with, but not > quite. Therefore I believe there are solutions and end up with the theory I > explained in the original post. > > Oh, now that the equations are there, the parameters I am scanning are the > product \beta e^2 and n, where n is comparable to the n in Eq (17). > Obviously, > n is an integer so cannot be continued as such, but \beta e^2 is real, so I > start with \beta e^2 = 1, n=1, where the solution is g(y) = y. > You could relax integrality. > Which reminds me of another oddity: if I start with the exact solution, my > function value is ~ 1e-11, so I know for sure to keep -snes_atol ~1e-10 > because that's the best my function evaluation can do. Could this be the > problem? Too little accuracy in function? I did try up to 6th order central > differences, but it does not help. > It is possible for the discrete equation to have no real solutions, while the continuous equation does. Even if it is expensive, I suggest continuing in the nonlinearity to try to get to a solution. If you find one, it could give you insight into designing a search strategy that will work for your equation. Matt > Cheers, > -Juha > > -- > ----------------------------------------------- > | Juha J?ykk?, juhaj at iki.fi | > | http://www.maths.leeds.ac.uk/~juhaj | > ----------------------------------------------- > -- What most experimenters take for granted before they begin their experiments is infinitely more interesting than any results to which their experiments lead. -- Norbert Wiener -------------- next part -------------- An HTML attachment was scrubbed... URL: <http://lists.mcs.anl.gov/pipermail/petsc-users/attachments/20110623/c7120d58/attachment-0001.htm>
