CMA, University of Oslo, Norway
(Seminar room B1036, N. H. Abels building)
March 8-9, 2005.
We invite all who are interested to a two-day intensive workshop
- or lecture series -. Keynote speaker will be Professor
Klaus Neymeyr from University of Rostock, Germany who willl
give four lectures on the topic. Also Simen Kvaal
of CMA and Torgeir Rusten from Veritas Research
will contribute on related topics. The remaining time will be left
Multigrid preconditioned conjugate gradient schemes belong to the
most efficient solvers for boundary value problems
for elliptic partial differential operators. In contrast to this,
the subject of efficient solvers for eigenvalue problems
for self-adjoint elliptic partial differential operators has still
not gained a similar attention, though direct multigrid solvers
have been devised.
The topic of this lecture are preconditioned iterative solvers
for self-adjoint elliptic eigenproblems. First a comparison of boundary
value problems and eigenvalue problems for these differential operators
is given and similarities and dissimilarities are worked out. On
the basis of this background it is shown that several ideas underlying
solvers for boundary value problems can be transferred to those
for the eigenvalue problem.
A central point of the lecture is a geometric interpretation of
preconditioned eigensolvers which allows to derive a new theoretical
framework for a larger class of preconditioned eigensolvers. This
framework includes new ideas for the convergence analysis of such
preconditioned eigensolvers. We discuss a hierarchy of eigensolvers
which includes not only simple preconditioned gradient type eigensolvers,
but also preconditioned CG-like eigensolvers and subspace iterations.
All these iterative eigensolvers allow the solution of mesh eigenproblems
(deriving from finite element discretizations) by using present-day
personal computers with optimal complexity. Standard multigrid or
multilevel preconditioners, as designed for the solution of boundary
value problems, can be applied as "black-box'' solvers. The
resulting schemes are conceptually simple, easy to implement, computationally
cheap and robust with respect to the initial guess.