Markus
Seidel
(Chemnitz University of Technology, Germany)
" Non-strongly
converging approximation methods and the approximation of pseudospectra "
Abstract. Classically,
for a given equation Ax=b and a sequence of compact projections P=(P_n)
which converges strongly to the identity one studies the sequence of
(truncated) equations P_n A P_n x_n=P_n b in order to find approximate
solutions for the initial problem. The theory behind that idea is
heavily based on the interactions between compactness, Fredholmness and
strong convergence.
In the first part of this talk we now turn the table in a sense, and we
take a sequence P=(P_n) (called approximate projection) as a starting
point for the definition of appropriate substitutes which we call
P-compactness, P-Fredholmness and P-strong convergence. On the one
hand, this adapted framework permits to develop a theory that mimics
the classical one and that provides very similar results on the
applicability of the projection method, the stability, and on the
asymptotics of norms, condition numbers or pseudospectra. On the other
hand, it can be applied to much more general settings since it is
detached from the fixed classical notions.
The second part picks up the approximation of pseudospectra in more
detail. We particular demonstrate how Hansens concept of (N,
ɛ)-pseudospectra can be generalized to the Banach space case and how
this may help to deal with the phenomenon of jumping pseudospectra.
