$\mathcal{H}$-矩阵近似反BEM矩阵的存在性：超奇异积分算子
Existence of $\mathcal{H}$-matrix approximants to the inverse of BEM matrices: the hyper-singular integral operator
AI总结：
研究超奇异积分算子在闭合曲面上的离散化，证明对应系统矩阵的逆可被分块低秩矩阵以指数速率近似，涵盖$\mathcal{H}$-矩阵的数据稀疏格式，针对两种离散化方法证明近似结果。
摘要：
我们考虑闭合曲面上超奇异积分算子的离散化，并证明相应系统矩阵的逆可被分块低秩矩阵以指数速率近似。我们特别涵盖了$\mathcal{H}$-矩阵的数据稀疏格式。我们针对两种类型的离散化方法证明了近似结果。第一种是鞍点公式，它包含了解的均值为零的约束。第二种离散化基于稳定化的超奇异算子，导致对称正定矩阵。在后者设置中，我们还证明了分层Cholesky分解可被以块秩的指数速率近似。
英文摘要：
We consider discretizations of the hyper-singular integral operator on closed surfaces and show that the inverses of the corresponding system matrices can be approximated by blockwise low-rank matrices at an exponential rate in the block rank. We cover in particular the data-sparse format of $\mathcal{H}$-matrices. We show the approximability result for two types of discretizations. The first one is a saddle point formulation, which incorporates the constraint of vanishing mean of the solution. The second discretization is based on a stabilized hyper-singular operator, which leads to symmetric positive definite matrices. In this latter setting, we also show that the hierarchical Cholesky factorization can be approximated at an exponential rate in the block rank.
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