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Survey on the Technique of Hierarchical Matrices

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Abstract

Usually, one avoids numerical algorithms involving operations with large, fully populated matrices. Instead, one tries to reduce all algorithms to matrix-vector multiplications involving only sparse matrices. The reason is the large number of floating point operations; e.g., \(\mathcal {O}(n^{3})\) for the multiplication of two general n × n matrices. The hierarchical matrix (\(\mathcal {H}\)-matrix) technique provides tools to perform the matrix operations approximately in almost linear work \(\mathcal {O}(n\log ^{\ast }n)\). The approximation errors are nevertheless acceptable, since large-scale matrices are usually obtained from discretisations which anyway contain a discretisation error. Adjusting the approximation error to the discretisation error yields the factor \(\mathcal {O}(\log ^{\ast }n).\) The operations enabled by the \(\mathcal {H}\)-matrix technique are not only the matrix addition and multiplication but also the matrix inversion and the LU or Cholesky decomposition. The positive statements from above do not hold for all matrices, but they are valid for the important class of matrices originating from standard discretisations of elliptic partial differential equations or the related integral equations. An important aspect is the fact that the algorithms can be applied in a black-box fashion. Having all matrix operations available, a much larger class of problems can be treated than by the restriction to matrix-vector multiplications. The LU decomposition can be used to construct fast iterations for solving linear systems. Also eigenvalue problems can be treated. The computation of matrix-valued functions is possible (e.g., the matrix exponential function) as well as the solution of matrix equations (e.g., of the Riccati equation).

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Notes

  1. The \(\mathcal {H}\)-matrix algorithms will implicitly produce an ordering (cf. (13).

  2. It would be more precise to consider triples (M, A, B) with M = AB T, but this would complicate the later notations.

  3. The support of a function \(f:X\rightarrow Y\) is defined by \(\operatorname {supp}(f):=\overline {\{x:f(x)\neq 0\}}\).

  4. In practice, interpolation is preferred.

  5. We use the subsets \(\tau \subset I\) as identifiers of the vertices in T(I),since in the usual cases different vertices correspond to different subsets. If one allows vertices τ with exactly one son vertex, one may introduce a more involved notations by labels.

  6. For a precise notation, one has to change the identifier of the vertices of the tree to distinguish the father τ from the son τ 1.

  7. For simplicity we assume that the nodal points satisfy ξ i ≠ξ j for i≠j.

  8. For more general trees see [26, Sections 5.5.2–5.5.3].

  9. Even if I = J,the Petrov–Galerkin discretisation may use different ansatz functions ϕ i (i∈I) and test functions ψ j (j∈J) so that X i ≠Y i .

  10. For variants of the admissibility condition see [26, (5.7a–c) and Section 5.2.3].

  11. This statement shows that the partition might be finer than necessary. Therefore it makes sense to coarsen the partition. This technique checks whether a coarser partition with the same accuracy exists without increasing the storage size (cf. [26, Section 6.7.2]).

  12. This includes the zero blocks which are considered as low-rank blocks of rank zero: r(b)=0.

  13. Of course, the arising matrix-matrix multiplications and additions can be parallelised.

  14. Without loss of generality, A can be scaled so that the smallest eigenvalue is ≥1.

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Acknowledgments

I thank the anonymous referees for their valuable comments.

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  1. Max-Planck-Institut Mathematik in den Naturwissenschaften, Inselstr. 22, 04103, Leipzig, Germany

    Wolfgang Hackbusch

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Correspondence to Wolfgang Hackbusch.

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Dedicated to Eberhard Zeidler’s 75th birthday.

The asterix in \(\protect \qopname \relax o{log}^{\ast }\) replaces some exponent.

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Hackbusch, W. Survey on the Technique of Hierarchical Matrices. Vietnam J. Math. 44, 71–101 (2016). https://doi.org/10.1007/s10013-015-0168-5

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