Uniform $\mathcalH$-matrix Compression with Applications to Boundary Integral Equations
<|reference_start|>Uniform $\mathcalH$-matrix Compression with Applications to Boundary Integral Equations: 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.<|reference_end|>
@article{bruyninckx2024uniform,
    title={Uniform $\mathcal{H}$-matrix Compression with Applications to Boundary
  Integral Equations},
    author={Kobe Bruyninckx, Daan Huybrechs, Karl Meerbergen},
    journal={arXiv preprint arXiv:2405.15573},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.15573},
    primaryClass={math.NA cs.MS cs.NA}
}