发表机构
The Department of Mathematics, The Ohio State University（俄亥俄州立大学数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对$\mathcal{H}^2$矩阵，提出两种基于随机草图的恢复算法，在仅需矩阵-伴随乘积下实现拟最优误差，携带恢复以更少算子应用达到与重草图相当的精度。
AI 中文摘要
$\mathcal{H}^2$ 矩阵利用嵌套低秩基表示良好分离的相互作用，同时显式保留邻近相互作用。我们考虑仅利用矩阵及其伴随的乘积来构造显式、强可容许的 $\mathcal{H}^2$ 近似的问题，并在自适应几何层次结构上开发了两种随机恢复算法。携带恢复从一对稠密高斯测试矩阵开始，无需特定于层次结构的探测，并跨层级更新所得的前向-伴随草图。因此，所有算子样本可以一次性获取。重草图恢复在每一层使用新的高斯测试矩阵应用相同的恢复过程。对于该变体，我们证明了相对于具有规定层次结构、秩界和保留近邻模式的最近似，期望 Frobenius 误差的全局拟最优界。携带恢复仅需要 $O(k+p)$ 次矩阵及其伴随的应用，与树深度无关，而重草图恢复在对数深度树上需要 $O((k+p)\log(N/k))$ 次。这里 $N$ 是矩阵维度，$k$ 是最大规定盒秩，$p$ 是过采样参数。这些界假设有界近邻计数和 $O(k)$ 局部维度。在二维和三维中，对 Laplace、Dirichlet-to-Neumann 和 Helmholtz 算子进行的实验，未知量多达 $1.28$ 百万，表明在充分过采样下，携带恢复实现了几乎与细化无关的误差，并且精度与重草图恢复相当，同时使用的算子应用次数少得多。对于固定秩，其压缩应用成本线性扩展。
英文摘要
An $\mathcal H^2$ matrix represents well-separated interactions using nested low-rank bases while retaining nearby interactions explicitly. We consider the problem of constructing an explicit, strongly admissible $\mathcal H^2$ approximation using only products with the matrix and its adjoint, and develop two randomized recovery algorithms on an adaptive geometric hierarchy. Carried recovery begins with a single pair of dense Gaussian test matrices, without hierarchy-specific probing, and updates the resulting forward--adjoint sketches across levels. All operator samples can therefore be acquired in one batch. Resketched recovery applies the same recovery procedure using fresh Gaussian test matrices at each level. For this variant, we prove a global quasi-optimal bound on the expected Frobenius error relative to the best approximation with the prescribed hierarchy, rank bounds, and retained near-neighbor pattern. Carried recovery requires only $O(k+p)$ applications of the matrix and its adjoint, independent of tree depth, whereas resketched recovery requires $O((k+p)\log(N/k))$ on trees of logarithmic depth. Here $N$ is the matrix dimension, $k$ the maximum prescribed box rank, and $p$ the oversampling parameter. These bounds assume bounded near-neighbor counts and $O(k)$ local dimensions. Experiments on Laplace, Dirichlet-to-Neumann, and Helmholtz operators in two and three dimensions with up to $1.28$ million unknowns show that, with sufficient oversampling, carried recovery achieves nearly refinement-independent error and accuracy comparable to resketched recovery while using many fewer operator applications. Its compressed application cost scales linearly for a fixed rank.
