Uniform H-matrix Compression with Applications to Boundary Integral Equations

TL;DR

The paper addresses the challenge of storing and operating on dense yet data-sparse matrices arising from boundary integral equations by introducing Uniform $\mathcal{H}$-matrices as a practical middle ground between $\mathcal{H}$- and $\mathcal{H}^2$-matrices.It develops an algebraic compression pipeline that transforms a regular $\mathcal{H}$-matrix into a uniform $\mathcal{H}$-matrix with shared cluster bases, while preserving $\mathcal{O}(N\log N)$ storage and mat-vec complexity and enabling direct construction from entries.The authors provide a log-linear algorithm for computing optimal cluster bases, present error bounds that relate local per-cluster accuracy to global matrix error, and demonstrate substantial memory reductions (roughly 2–4x) with competitive construction and parallel mat-vec performance on Helmholtz BEM problems.These findings indicate that the uniform approach offers a simpler, easier-to-implement alternative to $\mathcal{H}^2$-based methods while delivering meaningful performance gains for large-scale boundary integral computations.

Abstract

Boundary integral equations lead to dense system matrices when discretized, yet they are data-sparse. Using the $\mathcal{H}$-matrix format, this sparsity is exploited to achieve $\mathcal{O}(N\log N)$ complexity for storage and multiplication by a vector. This is achieved purely algebraically, based on low-rank approximations of subblocks, and hence the format is also applicable to a wider range of problems. The $\mathcal{H}^2$-matrix format improves the complexity to $\mathcal{O}(N)$ by introducing a recursive structure onto subblocks on multiple levels. However, in many cases this comes with a large proportionality constant, making the $\mathcal{H}^2$-matrix format advantageous mostly for large problems. In this paper we investigate the usefulness of a matrix format that lies in between these two: Uniform $\mathcal{H}$-matrices. An algebraic compression algorithm is introduced to transform a regular $\mathcal{H}$-matrix into a uniform $\mathcal{H}$-matrix, which maintains the asymptotic complexity. Using examples of the BEM formulation of the Helmholtz equation, we show that this scheme lowers the storage requirement and execution time of the matrix-vector product without significantly impacting the construction time.

Figures (5)

• Figure 1: Shapes used in the numerical experiments. From left to right and from top to bottom: trefoil knot (548870 triangles), submarine (400886 triangles), crankshaft (672128 triangles), frame (451664 triangles), falcon (609104 triangles) and lathe part (535054 triangles).
• Figure 2: Memory usage as a function of $\eta$ for the trefoil knot ($\kappa h = 0.1$, left) and crankshaft ($\kappa h = 0.4$, right) in regular and uniform $\mathcal{H}$-matrix format.
• Figure 3: Memory usage and construction time as a function of $N$ for the trefoil knot in regular and uniform $\mathcal{H}$-matrix format at $\kappa h=0$ (blue) and $\kappa h=0.3$ (orange).
• Figure 4: Memory usage and parallel construction time as a function of the tolerance $\epsilon$ for the crankshaft and submarine at $\kappa h=0.1$ in regular and uniform $\mathcal{H}$-matrix format. Left: Absolute values for the crankshaft. Middle: Absolute values for the submarine. Right: Relative values $\mathcal{H}/\mathcal{UH}$.
• Figure 5: Total memory usage as a function of the relative spectral error $\|A-A^\bullet\|_2/\|A\|_2$ ($\bullet=\mathcal{H},\mathcal{UH}$) for the crankshaft (left) and submarine (right) at $\kappa h=0.1$. Lines indicate the mean error while the light bands indicate the minimum and maximum errors.

Theorems & Definitions (18)

• Definition 2.1: cluster tree
• Definition 2.2: block cluster tree
• Definition 2.3: hierarchical matrix
• Definition 2.4: uniform hierarchical matrix
• Definition 2.5: sparsity constant
• Lemma 2.6: Storage cost of hierarchical matrices
• Proof 1
• Remark 2.7
• Remark 4.1: Difference with $\mathcal{H}^2$-matrix compression
• Lemma 4.2: Complexity of compression