Definition and Properties of Hierarchical Matrices

Abstract

The set $\mathcal{H}(r,P)$ of hierarchical matrices ($\mathcal{H}$-matrices) is defined in Section 6.1. Section 6.2 mentions elementary properties; e.g., the H-matrix structure is invariant with respect to transformations by diagonal matrices and transposition. The first essential property of $\mathcal{H}$-matrices is data-sparsity proved in Section 6.3. The storage cost of an n × n matrix is O(n log* n). The precise estimate together with a description of the constants is given in §6.3.2 using the quantity C _sp from (6.5b). In Section 6.4 we prove that matrices arising from a finite element discretisation lead to a constant C _sp depending only on the shape regularity of the finite elements. In Section 6.5 we analyse how approximation errors of the submatrices affect the whole matrix. In the definition of $\mathcal{H}(r,P)$, the parameter r can be understood as a fixed local rank. In practice, an adaptive computation of the ranks is more interesting, as described in Section 6.6. The construction of the partition yields an a priori choice of the local ranks. These may too large. Therefore, a subsequent reduction of the rank (‘recompression’) is advisable, as explained in Section 6.7. In Section 6.8 we discuss how additional side conditions can be taken into consideration.