Papers › New Algebraic Fast Algorithms for N-body Problems in Two and Three Dimensions
New Algebraic Fast Algorithms for N-body Problems in Two and Three Dimensions
Ritesh Khan, Sivaram Ambikasaran
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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 ℋ²_* (fully nested algorithm, i.e., ℋ² matrix-like algorithm) and (ℋ² + ℋ)_* (semi-nested algorithm, i.e., cross of ℋ² and ℋ matrix-like algorithms). The efficient ℋ²_* and (ℋ² + 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 (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 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 ℋ² matrix algorithm with respect to the memory and time. The C++ implementation of the proposed algorithms is available at https://github.com/riteshkhan/H2weak/.
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