New Algebraic Fast Algorithms for $N$-body Problems in Two and Three Dimensions
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2023
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866910117362925568 |
|---|---|
| author | Khan, Ritesh Ambikasaran, Sivaram |
| author_facet | Khan, Ritesh Ambikasaran, Sivaram |
| contents | We present two new algebraic multilevel hierarchical matrix algorithms to perform fast matrix-vector product (MVP) for $N$-body problems in $d$ dimensions, namely efficient $\mathcal{H}^2_{*}$ (fully nested algorithm, i.e., $\mathcal{H}^2$ matrix-like algorithm) and $(\mathcal{H}^2 + \mathcal{H})_{*}$ (semi-nested algorithm, i.e., cross of $\mathcal{H}^2$ and $\mathcal{H}$ matrix-like algorithms). The efficient $\mathcal{H}^2_{*}$ and $(\mathcal{H}^2 + \mathcal{H})_{*}$ hierarchical representations are based on our recently introduced weak admissibility condition in higher dimensions, where the admissible clusters are the far-field and the vertex-sharing clusters. Due to the use of nested form of the bases, the proposed hierarchical matrix algorithms are more efficient than the non-nested algorithms ($\mathcal{H}$ matrix algorithms). We rely on purely algebraic low-rank approximation techniques (e.g., ACA and NCA) and develop both algorithms in a black-box fashion. Another noteworthy contribution of this article is that we perform a comparative study of the proposed algorithms with different algebraic (NCA or ACA-based compression) fast MVP algorithms in $2$D and $3$D. The fast algorithms are tested on various kernel matrices and applied to get fast iterative solutions of a dense linear system arising from the discretized integral equations and radial basis function interpolation. Notably, all the algorithms are developed in a similar fashion in $\texttt{C++}$ and tested within the same environment, allowing for meaningful comparisons. The numerical results demonstrate that the proposed algorithms are competitive to the NCA-based standard $\mathcal{H}^2$ matrix algorithm with respect to the memory and time. The C++ implementation of the proposed algorithms is available at https://github.com/riteshkhan/H2weak/. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_14085 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | New Algebraic Fast Algorithms for $N$-body Problems in Two and Three Dimensions Khan, Ritesh Ambikasaran, Sivaram Numerical Analysis Mathematical Physics 65F55, 65D12, 65R20, 65D05, 65R10 We present two new algebraic multilevel hierarchical matrix algorithms to perform fast matrix-vector product (MVP) for $N$-body problems in $d$ dimensions, namely efficient $\mathcal{H}^2_{*}$ (fully nested algorithm, i.e., $\mathcal{H}^2$ matrix-like algorithm) and $(\mathcal{H}^2 + \mathcal{H})_{*}$ (semi-nested algorithm, i.e., cross of $\mathcal{H}^2$ and $\mathcal{H}$ matrix-like algorithms). The efficient $\mathcal{H}^2_{*}$ and $(\mathcal{H}^2 + \mathcal{H})_{*}$ hierarchical representations are based on our recently introduced weak admissibility condition in higher dimensions, where the admissible clusters are the far-field and the vertex-sharing clusters. Due to the use of nested form of the bases, the proposed hierarchical matrix algorithms are more efficient than the non-nested algorithms ($\mathcal{H}$ matrix algorithms). We rely on purely algebraic low-rank approximation techniques (e.g., ACA and NCA) and develop both algorithms in a black-box fashion. Another noteworthy contribution of this article is that we perform a comparative study of the proposed algorithms with different algebraic (NCA or ACA-based compression) fast MVP algorithms in $2$D and $3$D. The fast algorithms are tested on various kernel matrices and applied to get fast iterative solutions of a dense linear system arising from the discretized integral equations and radial basis function interpolation. Notably, all the algorithms are developed in a similar fashion in $\texttt{C++}$ and tested within the same environment, allowing for meaningful comparisons. The numerical results demonstrate that the proposed algorithms are competitive to the NCA-based standard $\mathcal{H}^2$ matrix algorithm with respect to the memory and time. The C++ implementation of the proposed algorithms is available at https://github.com/riteshkhan/H2weak/. |
| title | New Algebraic Fast Algorithms for $N$-body Problems in Two and Three Dimensions |
| topic | Numerical Analysis Mathematical Physics 65F55, 65D12, 65R20, 65D05, 65R10 |
| url | https://arxiv.org/abs/2309.14085 |