Adaptive multiplication of rank-structured matrices in linear complexity

TL;DR

This work tackles the challenge of multiplying $\mathcal{H}^2$-matrices in linear time by introducing a two-phase approach: first construct an intermediate, refined block-structure representation with adaptively induced bases, then re-compress to a user-specified coarse block tree while guaranteeing accuracy. Central to the method are compressed induced cluster bases, basis weights, and total weights, enabling efficient, nested, isometric bases and error-controlled compression. The authors provide a detailed procedure for condensation and a bottom-up/top-down scheme to obtain adaptive row and column bases, along with error bounds and coarsening strategies that keep complexity near $O(n)$ under standard assumptions. Numerical experiments on boundary element matrices demonstrate linear-time behavior per degree of freedom and high accuracy, suggesting the approach can significantly speed up large-scale energy- or boundary-integral computations and potentially extend to factorization-based solvers. The work thus offers a practical, adaptable framework for linear-complexity $\mathcal{H}^2$-matrix arithmetic with robust error control.

Abstract

Hierarchical matrices approximate a given matrix by a decomposition into low-rank submatrices that can be handled efficiently in factorized form. $\mathcal{H}^2$-matrices refine this representation following the ideas of fast multipole methods in order to achieve linear, i.e., optimal complexity for a variety of important algorithms. The matrix multiplication, a key component of many more advanced numerical algorithms, has so far proven tricky: the only linear-time algorithms known so far either require the very special structure of HSS-matrices or need to know a suitable basis for all submatrices in advance. In this article, a new and fairly general algorithm for multiplying $\mathcal{H}^2$-matrices in linear complexity with adaptively constructed bases is presented. The algorithm consists of two phases: first an intermediate representation with a generalized block structure is constructed, then this representation is re-compressed in order to match the structure prescribed by the application. The complexity and accuracy are analysed and numerical experiments indicate that the new algorithm can indeed be significantly faster than previous attempts.

Figures (9)

• Figure 1: Construction of the basis weights for the basis $W_Y$
• Figure 2: Construction of the total weights for the matrix $Y$
• Figure 3: Construction of a compressed induced row basis for an $\mathcal{H}^2$-matrix product
• Figure 4: Original block structure (left) and product block structure (right)
• Figure 5: Block structures and corresponding column trees

Theorems & Definitions (13)

• Definition 1: Cluster tree
• Definition 2: Block tree
• Definition 3: Cluster basis
• Definition 4: $\mathcal{H}^2$-matrix
• Example 5: Integral operator
• Definition 6: Induced row cluster basis
• Definition 7: Induced column cluster basis
• Definition 8: Isometric cluster basis
• Definition 9: Basis weight
• Definition 10: Total weights