[petsc-users] Is there efficeint method for matrix with one extremely small eigen value?

[Jed Brown]

On Wed, 6 Apr 2011 22:24:50 +0800 (CST), "Gong Ding" <gdiso at ustc.edu> wrote:
> Hi,
> Can some one gives me advise on how to solve the ill conditioned problem
> efficiently with iterative method (since the problem size is big). 
> 
> I calculated the smallest eigen values as well as the largest eigen values.
> There exist one extremely small eigen value, which made the system ill conditioned.
> I guess method such as Tikhonov regularization may work? 
> Or there are some cheaper method works, if I can endure some inaccuracy in the solution.
>   
> 
> Smallest 0 eigen value: -2.112144e-15 with error 9.452618e-14

Your problem has a null space of dimension 1. Determine the eigenvector associated with this eigenvalue. That is the null space, it might just be a constant. Create a MatNullSpace and use KSPSetNullSpace(). (If it is the constant, you can just use -ksp_constant_null_space.) See the section in the users manual on solving singular systems.