H2-Matrices with Adaptive Bases
Steffen Börm (MPI Leipzig)
H2-matrices can be used to find data-sparse representations
of the densely populated matrices occurring, e.g., in boundary element methods.
We give a short introduction to the basic concepts of H2-matrices
and their application to simple boundary element methods. While these
techniques lead to fast algorithms, the corresponding memory requirements are
relatively high. In order to avoid this drawback, we present an algorithm
that adapts the function systems used in the kernel expansion in order to
compress the H2-matrices even further. The algorithm can work
``on the fly'', i.e., there is no need to store the uncompressed matrix.
The original approximation scheme can be shown to converge exponentially if
the kernel function satisfies the usual asymptotic smoothness condition.
Since the compression methods use only algebraic information, no additional
requirements have to be imposed on the kernel function or the geometry in
order to preserve exponential convergence.
Numerical experiments with discretizations of Poisson's and Maxwell's
equations show that the compression algorithm reduces the memory requirements
significantly and also improves the performance of the matrix-vector
multiplication that is crucial for most iterative solvers.