On Mon, Aug 20, 2012 at 8:10 AM, Barry Smith <bsmith at mcs.anl.gov> wrote:
> > I think it is even simpler than this. Let your coupled matrix look > like > > ( A C12 ) > ( C21 B ) > > and the uncoupled be > > ( A 0 ) > ( 0 B ) > > > Now solving the uncoupled with GMRES, CG etc is actually solving at > the same time two independent systems and the relative size of the > eigenvalues of A and B don't really matter. > > Jed is right that the first thing you need to do is > > > 1. non-dimensionalize: scale your model to use some representative units > so that the solution and residual are well-scaled > > > you want the A part and the B part in the Jacobian and function > evaluations to have similar scales of values. > > > BTW: the FAQ has one sentence "Change the units (nondimensionalization), > boundary condition scaling, or formulation so that the Jacobian is better > conditioned." > How about putting some references there and some good examples on how to > do this? What this means? This is hard shit and we dismiss it in one > sentence like a kindergartener can do it. http://en.wikipedia.org/wiki/Buckingham_%CF%80_theorem Matt > > Barry > > > On Aug 20, 2012, at 7:39 AM, Jed Brown <jedbrown at mcs.anl.gov> wrote: > > > On Mon, Aug 20, 2012 at 7:07 AM, w_ang_temp <w_ang_temp at 163.com> wrote: > > Hello, > > I have a problem about the ill-conditioned matrix. I deal with the > soil-water coupled problem in > > geotechnical engineering. It can be demonstrated that the stiffness > matrix of soil-water coupled problem is > > ill-conditioned due to the coupled effects of large relative differences > in stiffnesses and permeabilities > > of materials. But it should not be ill-conditioned for soil-water > uncoupled problem, although there is no derivation > > in papers. > > In my three models, the condition numbers are 3.5595E+05, > 1.8849E+06, 9.0206E+07 for soil-water > > coupled models and 5.2885E+04, 3.3826 E+05, 3.4565E+05 for soil-water > uncoupled ones, respectively. > > Therefore, in my opinion, the uncoupled models are also ill-conditioned > due to the large condition > > numbers. > > However, when using GMRES, CG and CGS, the three models of uncoupled > ones show rapid convergent rate > > while coupled models have bad convergence performance. > > > > This indicates that the spectrum of the simpler model has a small number > of outliers. It could be as simple as boundary condition scaling or it > could be more meaningful. The coupled model does not have that nice > property, so convergence is slower. You should start with: > > > > 1. non-dimensionalize: scale your model to use some representative units > so that the solution and residual are well-scaled > > > > 2. find an effective preconditioner. PCFIELDSPLIT might work well, but > you have to understand methods to know how to drive it. Look at the > examples and read through the literature in your field to find something > you want to implement, then ask here if you need suggestions for how to > implement. (It can often be done using only command line options.) > > -- 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/20120820/bee8cba1/attachment-0001.html>
