New Algebraic Fast Algorithms for $N$-body Problems in Two and Three Dimensions

TL;DR

The paper advances fast algebraic MVP for $N$-body kernel matrices in $d$ dimensions by introducing two new nested hierarchical representations based on a higher-dimensional weak admissibility: $\mathcal{H}^2_{*}$ (fully nested) and $(\mathcal{H}^2 + \mathcal{H})_{*}$ (semi-nested). It achieves kernel-independence via ACA/NCA and partitions interactions into far-field and vertex-sharing components, enabling quasi-linear MVP and reduced memory in 2D and 3D. A key novelty is adapting NCAs to the weak admissibility regime in higher dimensions and partitioning the interaction lists to maintain compact pivot spaces, with detailed complexity analyses and extensive numerical validation against standard $\mathcal{H}^2$ methods across multiple kernels. The authors provide two fully implemented algorithms (and a semi-nested variant), demonstrate competitive performance with established algebraic FMM/H2 methods, and release the C++ code for public use, highlighting practical impact for fast solvers in integral equations and RBF interpolation. Overall, the work broadens kernel-independent hierarchical methods to higher dimensions with improved efficiency and scalability for dense kernel matrices.

Abstract

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/.

Figures (23)

• Figure 1: Hierarchical subdivision of the computational domain $C$ using $2^d$ uniform tree and the numbering convention followed in this article at different levels $(d=2)$.
• Figure 1: Illustration of search spaces of the pivots in B$2$T NCA to construct a nested bases algorithm based on our weak admissibility condition in higher dimensions, which leads to poor approximation. The green and cyan-colored regions represent the interaction list of $X$, i.e., $\mathcal{IL}_{*} \left(X\right)$.
• Figure 1: Partitioning of the interaction list corresponding to cluster $X$ at level $3$ of the quad tree in $2$D. The interaction list of $X$$\left(\mathcal{IL}_{*}(X)\right)$ consists of green and cyan colored clusters enclosed by a noticeable black border. The lighter shade colors represent the interaction list at the coarser levels of the tree (parent or grandparent of $X$).
• Figure 1: Plots of Memory, Initialization time and Solution time of the fast MVP algorithms.
• Figure 1: Illustration of search spaces of the pivots in B$2$T NCA for $\mathcal{H}^2_{\sqrt{d}}$(b) construction.

Theorems & Definitions (10)

• Remark 3.1
• Remark 3.2
• Remark 3.3
• Remark 4.1
• Remark 5.1
• Remark 5.2
• Remark 5.3
• Remark 5.4
• Remark 5.5
• Remark 5.6