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\({\mathcal{H}}^{2}\)-Matrix Compression

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New Developments in the Visualization and Processing of Tensor Fields

Part of the book series: Mathematics and Visualization ((MATHVISUAL))

Abstract

Representing a matrix in a hierarchical data structure instead of the standard two-dimensional array can offer significant advantages: submatrices can be compressed efficiently, different resolutions of a matrix can be handled easily, and even matrix operations like multiplication, factorization or inversion can be performed in the compressed representation, thus saving computation time and storage. \({\mathcal{H}}^{2}\)-matrices use a subdivision of the matrix into a hierarchy of submatrices in combination with a hierarchical basis, similar to a wavelet basis, to handle \(n \times n\) matrices in \(\mathcal{O}(nk)\) units of storage, where k is a parameter controlling the compression error. This chapters gives a short introduction into the basic concepts of the \({\mathcal{H}}^{2}\)-matrix method, particularly concerning the compression of arbitrary matrices.

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Acknowledgements

Most of the research results described in this chapter are the result of several years of work as a researcher at the Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany, and I particularly owe a debt of gratitude to Wolfgang Hackbusch and Lars Grasedyck for many insightful discussions.

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Authors and Affiliations

  1. Institute of Computer Science, Christian-Albrechts-Universität zu Kiel, D-24118, Kiel, Germany

    Steffen Börm

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  1. Steffen Börm
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Correspondence to Steffen Börm .

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Editors and Affiliations

  1. Dept. Computer Science, Brown University, Providence, Rhode Island, USA

    David H. Laidlaw

  2. Department of Biomedical Engineering, Einhoven University of Technology, Eindhoven, 5600 MB, Netherlands

    Anna Vilanova

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Börm, S. (2012). \({\mathcal{H}}^{2}\)-Matrix Compression. In: Laidlaw, D., Vilanova, A. (eds) New Developments in the Visualization and Processing of Tensor Fields. Mathematics and Visualization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27343-8_18

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