Boundary integral equation formulations of elliptic partial differential 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 practice 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.

翻译：椭圆型偏微分方程的边界积分方程形式在离散化后会产生稠密的系统矩阵，但这些矩阵具有数据稀疏性。利用$\mathcal{H}$矩阵格式，可通过子块的低秩近似来利用这种稀疏性，从而实现存储和向量乘法的$\mathcal{O}(N\log N)$复杂度。该方法完全基于代数操作，因此该格式也适用于更广泛的问题类别。$\mathcal{H}^2$矩阵格式通过在多层级上引入子块的递归结构，将复杂度改进为$\mathcal{O}(N)$。然而在实际应用中，这会带来较大的比例常数，使得$\mathcal{H}^2$矩阵格式主要对大规模问题具有优势。本文研究介于这两种格式之间的矩阵格式——一致化$\mathcal{H}$矩阵——的实用性。我们引入了一种代数压缩算法，可将常规$\mathcal{H}$矩阵转换为保持渐近复杂度的一致化$\mathcal{H}$矩阵。