Assume I know all eigenvalues of a matrix $A$ fall into a certain set $\Omega \subset \mathbb{C}$. Is there any way I can exploit this knowledge to design a preconditioner for $A$?

Some further remarks:

  • The eigenvalues fill $\Omega$ uniformly, i.e. there are no isolated eigenvalues.
  • I know the eigenvalues but not the eigenfunctions.
  • $\begingroup$ What do you mean with "fill uniformly"? A matrix $A$ has finitely many eigenvalues, so they are all isolated up to multiplicity. $\endgroup$ – Wolfgang Bangerth Aug 14 '16 at 18:44
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    $\begingroup$ More precisely, I mean that as I let the size of $A$ go to infinity, it's spectrum $\sigma(A)$ becomes dense in $\Omega$. The main point however is that the largest/smallest eigenvalues aren't very far from the rest of the spectrum such that deflation techniques won't be effective. $\endgroup$ – gTcV Aug 15 '16 at 7:56
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    $\begingroup$ It's really not my area, but my gut feeling is that if (i) you don't know anything about the eigenfunctions, and (ii) the eigenvalues have no clustering properties at all, that there is not very much you can do. On the other hand, since you do have some kind of hierarchy in the problem it might be interesting to see whether there are ways to use smaller matrices from your family as preconditioners for bigger matrices. $\endgroup$ – Wolfgang Bangerth Aug 18 '16 at 14:25

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