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20th GAMM-Seminar Leipzig on
Numerical Methods for Non-Local Operators

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  20th GAMM-Seminar
January, 22th-24th, 2004
 
     
  Winterschool on hierarchical matrices  
     
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  Abstract Wolfgang Hackbusch, Sat, 10.30-11.00 Previous Contents Next  
  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 tex2html_wrap_inline564, which is necessary to obtain an accuracy tex2html_wrap_inline566 on tex2html_wrap_inline568 is given by tex2html_wrap_inline570 The block structure for BEM matrices corresponding to intervals (or curve) are of the type depicted on the left side:

tabular24       tabular346

The simpler format on the right side follows, if we allow blocks tex2html_wrap_inline572 where the clusters tex2html_wrap_inline574 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 tex2html_wrap_inline568 can be obtained by a similar rank tex2html_wrap_inline578 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 tex2html_wrap_inline580 is the integral operator

displaymath560            
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 one-dimensional manifolds) and discuss consequences for the arithmetic.
 

 
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