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On the best approximation of the hierarchical matrix product

Dölz, Jürgen and Harbrecht, Helmut and Multerer, Michael D.. (2018) On the best approximation of the hierarchical matrix product. Preprints Fachbereich Mathematik, 2018 (10).

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Official URL: https://edoc.unibas.ch/70167/

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Abstract

The multiplication of matrices is an important arithmetic operation in computational mathematics. In the context of hierarchical matrices, this operation can be realized by the multiplication of structured block-wise low-rank matrices, resulting in an almost linear cost. However, the computational efficiency of the algorithm is based on a recursive scheme which makes the error analysis quite involved. In this article, we propose a new algorithmic framework for the multiplication of hierarchical matrices. It improves currently known implementations by reducing the multiplication of hierarchical matrices towards finding a suitable low-rank approximation of sums of matrix products. We propose several compression schemes to address this task. As a consequence, we are able to compute the best-approximation of hierarchical matrix products. A cost analysis shows that, under reasonable assumptions on the low-rank approximation method, the cost of the framework is almost linear with respect to the size of the matrix. Numerical experiments show that the new approach produces indeed the best-approximation of the product of hierarchical matrices for a given tolerance. They also show that the new multiplication can accomplish this task in less computation time than the established multiplication algorithm without error control.
Faculties and Departments:05 Faculty of Science > Departement Mathematik und Informatik > Mathematik > Computational Mathematics (Harbrecht)
12 Special Collections > Preprints Fachbereich Mathematik
UniBasel Contributors:Dölz, Jürgen and Harbrecht, Helmut
Item Type:Preprint
Publisher:Universität Basel
Language:English
Last Modified:11 Apr 2019 15:41
Deposited On:11 Apr 2019 15:41

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