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.
Keywords
- Property-based Clustering
- Leaf Clusters
- Hierarchical Matrices
- Inadmissible Blocks
- Efficient Error Control Scheme
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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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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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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DOI: https://doi.org/10.1007/978-3-642-27343-8_18
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