Introduction to hierarchical matrices with applications

We give a short introduction to methods for the data-sparse approximation of matrices resulting from the discretisation of non-local operators occurring in boundary integral methods, as the inverses of partial differential operators or as solutions of control problems. The result of the approximation will be so-called hierarchical matrices (or short H-matrices). These matrices form a subset of the set of all matrices and have a data-sparse representation. The essential operations for these matrices (matrix-vector and matrix-matrix multiplication, addition and inversion) can be performed in, up to logarithmic factors, optimal complexity. We give a review of specialised variants of H-matrices, especially of H-2-matrices, and finally consider applications of the different methods to problems from integral equations, partial differential equations and control theory. (C) 2003 Elsevier Science Ltd. All rights reserved.