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. q

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The efficient treatment of dense matrices arising, e.g., from the finite element discretisation of integral operators requires special compression techniques. In this article we use the H-matrix representation that approximates the dense stiffness matrix in admissible blocks (corresponding to subdomains where the underlying kernel function is smooth) by low-rank matrices. The low-rank matrices are assembled by a new hybrid algorithm (HCA) that has the same proven convergence as standard interpolation but also the same efficiency as the (heuristic) adaptive cross approximation (ACA). Mathematics Subject Classification (2000)45B05 · 65N38 · 68P05

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