🤖 AI Summary
Discretizing boundary integral equations yields dense matrices; a key challenge is reducing memory consumption and computational time while preserving $mathcal{O}(Nlog N)$ storage and matrix–vector multiplication complexity.
Method: We propose the algebraic Uniform $mathcal{H}$-matrix format—a novel sparse representation framework situated between standard $mathcal{H}$- and $mathcal{H}^2$-matrices—enabling fully algebraic, geometry-free compression via hierarchical clustering, low-rank subblock approximation, and boundary element discretization. The method is applicable to oscillatory kernel problems such as the Helmholtz equation.
Contribution/Results: To our knowledge, this is the first purely algebraic $mathcal{H}$-matrix variant achieving uniform complexity without geometric input. Experiments on medium-scale problems show ~30% reduction in matrix storage, 1.8× speedup in matrix–vector multiplication, and negligible overhead in construction time—outperforming state-of-the-art $mathcal{H}$- and $mathcal{H}^2$-matrix methods in practice.
📝 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 $mathcal{O}(Nlog N)$ 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.