Efficient numerical methods for non-local operators. \(\mathcal H^2\)-matrix compression, algorithms and analysis.

*(English)*Zbl 1208.65037
EMS Tracts in Mathematics 14. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-091-3/hbk). ix, 432 p. (2010).

Hierarchical matrices (\(\mathcal H\)-matrices) are data-sparse representations which approximate very well matrices arising in many applications and offer matrix arithmetic operations like evaluation, multiplication, factorization and inversion that can be used to construct efficient preconditioners and solve matrix equations. By using a multilevel basis of submatrices, \({\mathcal H}^2\)-matrices introduce an additional hierarchical structure to reduce the storage requirements and computational complexity of \(\mathcal H\)-matrices, particularly for large problems.

The author presents an overview of theoretical results and practical algorithms for working with \({\mathcal H}^2\)-matrices. The following questions are addressed: (a) which kinds of matrices can be compressed; (b) which kinds of operations can be performed efficiently; (c) which problems can be solved efficiently.

The Bibliography contains 108 entries, most of them being quite recent. The HLib software package used for the numerical experiments described in Chapters 4–10 is provided for free at http://www.hlib.org for research purposes.

The author presents an overview of theoretical results and practical algorithms for working with \({\mathcal H}^2\)-matrices. The following questions are addressed: (a) which kinds of matrices can be compressed; (b) which kinds of operations can be performed efficiently; (c) which problems can be solved efficiently.

The Bibliography contains 108 entries, most of them being quite recent. The HLib software package used for the numerical experiments described in Chapters 4–10 is provided for free at http://www.hlib.org for research purposes.

Reviewer: Rémi Vaillancourt (Ottawa)

##### MSC:

65F05 | Direct numerical methods for linear systems and matrix inversion |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |

65F30 | Other matrix algorithms (MSC2010) |

65N22 | Numerical solution of discretized equations for boundary value problems involving PDEs |

65N38 | Boundary element methods for boundary value problems involving PDEs |

65R20 | Numerical methods for integral equations |