Hierarchical Matrices based on a Weak Admissibility Condition
Wolfgang Hackbusch (MPI Leipzig)
The usual (strong) admissibility condition requires that
the diameters of the clusters and their distance have a comparable size. Under
this condition, the remainder terms of interpolation formulae or of the Taylor
expansion prove the exponential decay of the singular values in an admissible
block and justify the truncation to a rank k. The dependence of the rank
, which is necessary to obtain an accuracy
on is given by The block structure for
BEM matrices corresponding to intervals (or curve) are of the type depicted on
the left side:
The simpler format on the right side follows, if we allow blocks
where the clusters may be neighboured, but
do not intersect. This is the ``weak admissibility''. Obviously, the overhead
costs are lower for the second format. The crucial question is however,
whether an accuracy can be obtained by a similar rank
as before. Since the clusters are neighboured,
interpolations or Taylor expansions cannot be used to produce a reasonable
separable expansion. The continuous counterpart in the case of the kernel
is the integral operator
and the question whether the eigenvalues of K decay exponentially.
We will give an almost optimal answer to these questions (restricted to the 1D
case, i.e. BEM matrices corresponding to onedimensional manifolds) and
discuss consequences for the arithmetic.
