Re: questions on line-search stagnation for newton-raphson method


In article <1144858818.427869.218680@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"suhao" <hesuhao@xxxxxxxxx> writes:
Thanks for so perceptive post.
I'm using solver (newt+lnsrch) from "numerical receipes" with a
numerical Jacobian.
My equations are several PDEs. They can be rewritten as Mz'=F(t,z), if
? after applying the (vertical) method of lines?


M is singular, when I decrease h, isn't the singularity coming early
instead of getting delayed?
h = delta_t should have no influence on the singularity of M
you will apply an implicit dae solver, i assume bdf
(did you consider using DASSL for your system?
DASSL is one of the best (if not "the" best) solver for dae's
and is available from http://www.netlib.org/ode
your system then will read
M(z_new)*(z_new_z_old) -delta_t*F(t_new, c*z_new+ sum alpha_k z_old_k)
where the value inside F represents an interpolation formula for z(t_new)
and now your nonlinear system is in the variable z_new only.
the Jacobian of this one, i.e. essentially
M(...)-c*delta_t*F_z(...) + M_z(...)*(z_new-z_old)
where F_z is the Jacobian of F with respect to z and M_z is a tensor of order
three (the derivative of M(..) with respect to z)
and this one is becoming singular. if M itself is already singular
(? an incomplete parabolic system?) then playing with delta_t does not help,
a smaller delta_t will make things worse. maybe you succeed in getting
a system of index zero by further differentiating (part) of your system
with tepsect to t. clearly you must look into solving dae's

My Jacobian is not scaled well due to the physics (nonlinear
permeability in Richards Equation). But the solver works pretty well at
certain conditions. When I added some other nonlinear physical terms,
the results start to staganate.
this sounds indeed like a problem in scaling

I'll try to look at the matrix M and the scaling, also another
suggestion from jei.... Thank you for your suggestions.
Is there any existing solver using Levenberg-Marquardt method?
yes, it is in
http://www.netlib.org/minpack
the guide
http://plato.la.asu.edu/topics/problems/zero.html
has much more on solvers you might consider KINSOL
this is a system for doing exactly the kind of job you are involved in


I saw people suggest to use trust region method or pseudotransient
trust region = Levenberg Marquardt

continuation method if newton method failed. Should I test more or give
continuation will not help here: the natural continuation parameter is t
you have a solution for some t and want to proceed along the t axis, but
taking smaller t-steps already failed?

up present newton solver, assuming I didn't make mistake in coding and
physical eqns?
suhao


hth
peter
.

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