[petsc-users] Eigensolution with matrices from mixed formulation problems
Jose E. Roman
jroman at dsic.upv.es
Thu Dec 5 01:54:17 CST 2019
In symmetric problems (GHEP) you might have large residuals due to B being singular. The explanation is the following. By default, in SLEPc GHEPs are solved via a B-Lanczos recurrence, that is, using the inner product induced by B. If B is singular this is not a true inner product and numerical problems may arise, in particular, the computed eigenvectors may be corrupted with components in the null space of B. This is easily solved by a small computation called 'eigenvector purification' which is on by default in SLEPc, see EPSSetPurify(). This is described with a bit more detail in section 3.4.4 of the manual.
In non-symmetric problems (GNHEP) you should not see this problem. The only precaution is not to solve systems with matrix B, e.g., using shift-and-invert.
Jose
> El 5 dic 2019, a las 5:04, Manav Bhatia <bhatiamanav at gmail.com> escribió:
>
> Hi,
>
> I am working on mixed form finite element discretization which leads to eigenvalues of the form
>
> A x = lambda B x
>
> With the matrices defined in a block structure as
>
> A = [ K D^T ]
> [ D 0 ]
>
> B = [ M 0 ]
> [ 0 0 ]
>
> The second row of equations come from Lagrange multipliers in our discretization scheme. A system with m Lagrange multiplier is expected to have m Inf eigenvalues. We are testing the standard eigensolvers in Matlab and as the system size increases the eigensolves are stopping with larger residuals, || r_i ||, of the eigensystem:
>
> r_i = A x_i - lambda_i B x_i
>
> I am working towards setting this up in SLEPc. In the meantime I am curious about the following:
>
> 1. Is the eigensolution of such systems known to be problematic?
> 2. Are there standard tricks in SLEPc or elsewhere that are geared towards more robust solutions of such systems?
>
> I would appreciate guidance on this.
>
> Regards,
> Manav
>
>
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