Adaptive Cross Approximation with a Geometrical Pivot Choice: ACA-GP Method

TL;DR

Addresses the high storage and compute cost of dense admissible blocks in $\mathcal{H}$-matrices arising from boundary integral methods. The method ACA-GP blends algebraic Adaptive Cross Approximation with geometry-guided pivot selection based on central subsets and error structure (extreme curves). The paper develops pivot strategies for ranks $k=2$, $k=3$, and $k\ge 4$, culminating in a full algorithm that uses a single tunable parameter $\varepsilon_r$ and a matrix-free implementation with $A'_k=U_k V_k^T$. Results on two interacting clouds with kernel $\kappa(x,y)=1/\|x-y\|$ show ACA-GP substantially improves over classical ACA and can approach the truncated SVD accuracy for small ranks, while maintaining low cost. The work demonstrates geometry-aware low-rank approximation as a practical enhancement for BIM/H-matrix solvers and outlines directions for extensions to collocation BIM and more complex domains.

Abstract

The Adaptive Cross Approximation (ACA) method is widely used to approximate admissible blocks of hierarchical matrices, or H-matrices, from discretized operators in the boundary integral method. These matrices are fully populated, making their storage and manipulation resource-intensive. ACA constructs a low-rank approximation by evaluating only a few rows and columns of the original operator, significantly reducing computational costs. A key aspect of ACA's effectiveness is the selection of pivots, which are entries common to the evaluated row and column of the original matrix. This paper proposes combining the classical, purely algebraic ACA method with a geometrical pivot selection based on the central subsets and extreme property subsets. The method is named ACA-GP, GP stands for Geometrical Pivots. The superiority of the ACA-GP compared to the classical ACA is demonstrated using a classical Green operator for two clouds of interacting points.

Figures (10)

• Figure 1: Illustration of the optimal choice of the first pivot $\{i_1^*,j_1^*\}$ for the rank $k=1$ for two clouds of points ($x_{i_1^*},y_{j_1^*}$ are marked with stars). The optimal pivot points are located near the centers of the clouds. Left column: relative error above the SVD approximation: $\tilde{E}_1 = (E_1 - E_1^{\text{\tiny{SVD}}})/E_1^{\text{\tiny{SVD}}}$. Central column: interaction matrix's column $a_{i_1,j}$ colored according to their value. Right column: the location of the optimal column $i_1$ in the cloud $X$. Black crosses correspond to the barycenters of the clouds.
• Figure 1: Set-up of the experiment: two clouds of points $X$ and $Y$ are generated, each in a rectangle $a\times b$ ($\xi = a/b$).
• Figure 2: Illustration of the error structure for ranks $k=\{2,3,4,5\}$ starting from the top and a randomly selected $\{i_2,i_3,i_4,i_5\}$ (note that all pivots $i_l^*,j_l^*$ with $l<k$ were assumed to be selected in an optimal way and are shown with black circles) in cloud $X$ shown with a cross on the right column; the left column represents the relative difference in genetic ACA's error (relative error of the Frobenius norm of the matrix approximation) with respect to SVD $\tilde{E}_k = (E_g - E_{\text{SVD}})/E_{\text{SVD}}$; the central column shows the absolute value of the residual column corresponding to selected $i_k$; the clouds have $n=m=400$ points.
• Figure 2: The effect of the relative central subset fraction $\varepsilon_r$ on the accuracy of the low-rank approximation: the gain factor $\tilde{E}_k^{\textrm{ACA-GP}}/\tilde{E}_k^{\textrm{ACA}}$ for all ranks are plotted.
• Figure 3: Illustration of the error structure for the rank $k=2$ for two clouds of structured (upper panel) and random (lower panel) points within square boxes. A randomly selected point $x_{i_2}$ in $X$ cloud is marked with a green cross and the corresponding optimal $y_{j'_2}$ is shown with a star, the corresponding circle $\mathcal{C}_2(i_2)$ is shown with a red dashed line, the conjugate circle $\mathcal{C}_2^{\perp}$ is shown with a green dotted line; black circles highlight the first optimal pivot $x(i_1^*)$ and $y(j_1^*)$.