Theorie und Anwendungen Hierarchischer Matrizen
The modeling of physical properties often leads to the task of solving partial differential equations or integral equations. The results of some discretisation and linearisation process are matrix equations or linear systems of equations with special features. In the case of partial differential equations one exploits the local character of the differentiation by using some finite element method or finite difference scheme and gains a sparse system matrix. In the case of (nonlocal) integral operators low rank approximations seem to be the method of choice. These are either given explicitly by some multipole method or panel clustering technique or implicitly by rank revealing decompositions. Both types of matrices can be represented as so-called H-matrices. In this thesis we investigate algorithms that perform the addition, multiplication and inversion of H-matrices approximately. Under moderate assumptions the complexity of these new arithmetics is almost linear (linear up to logarithmic terms of order 1 to 3). The arithmetic operations can be performed adaptively, that is up to some given accuracy epsilon the relative error of the operations is zero. The question arises under which circumstances the inverse of an H-matrix can be approximated by an H-matrix. For the techniques used in this thesis we need very restrictive assumptions, but the numerical examples in the last part indicate that the approximability does not depend on these assumptions.