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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2405.15573 |
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| _version_ | 1866916747025580032 |
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| author | Bruyninckx, Kobe Huybrechs, Daan Meerbergen, Karl |
| author_facet | Bruyninckx, Kobe Huybrechs, Daan Meerbergen, Karl |
| contents | Boundary integral equations lead to dense system matrices when discretized, yet they are data-sparse. Using the $\mathcal{H}$-matrix format, this sparsity is exploited to achieve $\mathcal{O}(N\log N)$ complexity for storage and multiplication by a vector. This is achieved purely algebraically, based on low-rank approximations of subblocks, and hence the format is also applicable to a wider range of problems. The $\mathcal{H}^2$-matrix format improves the complexity to $\mathcal{O}(N)$ by introducing a recursive structure onto subblocks on multiple levels. However, in many cases this comes with a large proportionality constant, making the $\mathcal{H}^2$-matrix format advantageous mostly for large problems. In this paper we investigate the usefulness of a matrix format that lies in between these two: Uniform $\mathcal{H}$-matrices. An algebraic compression algorithm is introduced to transform a regular $\mathcal{H}$-matrix into a uniform $\mathcal{H}$-matrix, which maintains the asymptotic complexity. Using examples of the BEM formulation of the Helmholtz equation, we show that this scheme lowers the storage requirement and execution time of the matrix-vector product without significantly impacting the construction time. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_15573 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Uniform H-matrix Compression with Applications to Boundary Integral Equations Bruyninckx, Kobe Huybrechs, Daan Meerbergen, Karl Numerical Analysis Mathematical Software 35J05, 65F30, 65N38 Boundary integral equations lead to dense system matrices when discretized, yet they are data-sparse. Using the $\mathcal{H}$-matrix format, this sparsity is exploited to achieve $\mathcal{O}(N\log N)$ complexity for storage and multiplication by a vector. This is achieved purely algebraically, based on low-rank approximations of subblocks, and hence the format is also applicable to a wider range of problems. The $\mathcal{H}^2$-matrix format improves the complexity to $\mathcal{O}(N)$ by introducing a recursive structure onto subblocks on multiple levels. However, in many cases this comes with a large proportionality constant, making the $\mathcal{H}^2$-matrix format advantageous mostly for large problems. In this paper we investigate the usefulness of a matrix format that lies in between these two: Uniform $\mathcal{H}$-matrices. An algebraic compression algorithm is introduced to transform a regular $\mathcal{H}$-matrix into a uniform $\mathcal{H}$-matrix, which maintains the asymptotic complexity. Using examples of the BEM formulation of the Helmholtz equation, we show that this scheme lowers the storage requirement and execution time of the matrix-vector product without significantly impacting the construction time. |
| title | Uniform H-matrix Compression with Applications to Boundary Integral Equations |
| topic | Numerical Analysis Mathematical Software 35J05, 65F30, 65N38 |
| url | https://arxiv.org/abs/2405.15573 |