H- and H²-matrices
by
Steffen Börm
Max-Planck-Institut für Mathematik in den Naturwissenschaften

H- and H²-matrices

Compared to other techniques like multipole or wavelet expansions, this representation offers the advantage that not only the matrix-vector multiplication, but also more complicated operations like the matrix-matrix multiplication, the inversion or the LU factorization can be performed in almost linear complexity. Due to this property, hierarchical matrices can be used to construct efficient preconditioners for integral or partial differential equations, evaluate matrix functions or even solve certain matrix equations.

The talk gives an introduction to the basic concepts of H-matrix techniques and demonstrates their usefulness in practical applications.

H²-matrices combine the basic ideas of H-matrices with those of multilevel methods in order to improve the efficiency of the data-sparse representation. In an H²-matrix, each submatrix is not only of low rank, but its range and the range of its adjoint are contained in prescribed spaces. This means that only the coefficients with respect to bases of these spaces have to be stored, therefore the storage requirements compared to H-matrices are reduced by a significant amount.

Suitable recursive algorithms can perform the matrix-matrix multiplication of H²-matrices efficiently, and based on these algorithms the matrix inversion and LU factorization can be implemented.

Numerical experiments illustrate the performance of H- and H²-matrix techniques in the context of integral and partial differential equations.

Date received: July 6, 2007

Copyright © 2007 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cavl-10.