The convergence of Krylov subspace methods like conjugate gradients
depends on the eigenvalues of the underlying matrix. In many cases
the exact location of the eigenvalues is unknown, but one has some
information about the distribution of eigenvalues in an asymptotic
sense. This could be the case for linear systems arising from a
discretization of a PDE. The asymptotic behavior then takes place
when the meshsize tends to zero.
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We discuss two possible approaches to study the convergence of
conjugate gradients based on such information.
The first approach is based on a straightforward idea to estimate
the condition number. This method is illustrated by means of a
comparison of preconditioning techniques.
The second approach takes into account the full asymptotic
spectrum. It gives a bound on the asymptotic convergence factor
which explains the superlinear convergence observed in many situations.
This method is mathematically more involved since it deals with
potential theory. I will explain the basic ideas.