2015 | OriginalPaper | Chapter
Definition and Properties of Hierarchical Matrices
Author : Wolfgang Hackbusch
Published in: Hierarchical Matrices: Algorithms and Analysis
Publisher: Springer Berlin Heidelberg
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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 Csp from (6.5b). In Section 6.4 we prove that matrices arising from a finite element discretisation lead to a constant Csp 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.