[petsc-dev] Symmetry acceleration of the Jacobi-Davidson method (in SLEPc)

[Tobin Isaac]

On Fri, Feb 13, 2015 at 03:06:38PM +0100, Krzysztof Gawarecki wrote:
> Dear All,
> 
> I'm calculating eigenvalues and eigenvectors of the matrix which has
> specific kind of symmetry.
> Due to this symmetry I obtain the eigenvalues which are doubly degenerated.
> So eg. eigeinvalue 'e1' has eigenvectors 'a1' and 'b1'. These eigenvectors
> are related to each other by the relation a1 = T b1, where T is a matrix
> (given for my problem).
> So it is enough to calculate only one eigenvector for each eigenvalue (and
> the second one can be calculated by matvec operation). This situation has
> been described in http://dl.acm.org/citation.cfm?id=2494747.
> 
> How could I take advantage on this in EPSSolve in Jacobi-Davidson method?
> Could I add two vectors to the subspace (the second one would be calculated
> by multiplying the first one by matrix T) in every iteration? Should I
> modify function "dvd_updateV_update_gen" in dvd_updatev.c ?
> 
> I would be very grateful for any suggestion.

In that paper you're looking at a Hermitian operator, right?  In this
case, can't you use the symmetry to make the problem smaller? If you
run Jacobi-Davidson for the operator SAS, where S=0.5(I+T), you'll get
zero eigenvalues for the anti-symmetric vectors, but the other
eigenvalues should have multiplicity 1 and can be used to reconstruct
what you want.

  Toby

> 
> Krzysztof
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