The discretization of non-local operators, e.g., solution operators of partial differential equations or integral operators, leads to large densely populated matrices. $\mathcal{H}^2$-matrices take advantage of local low-rank structures in these matrices to provide an efficient data-sparse approximation that allows us to handle large matrices efficiently, e.g., to reduce the storage requirements to $\mathcal{O}(n k)$ for $n$-dimensional matrices with local rank $k$, and to reduce the complexity of the matrix-vector multiplication to $\mathcal{O}(n k)$ operations. In order to perform more advanced operations, e.g., to construct efficient preconditioners or evaluate matrix functions, we require algorithms that take $\mathcal{H}^2$-matrices as input and approximate the result again by $\mathcal{H}^2$-matrices, ideally with controllable accuracy. In this manuscript, we introduce an algorithm that approximates the product of two $\mathcal{H}^2$-matrices and guarantees block-relative error estimates for the submatrices of the result. It uses specialized tree structures to represent the exact product in an intermediate step, thereby allowing us to apply mathematically rigorous error control strategies.

翻译：非局部算子（例如偏微分方程的解算子或积分算子）的离散化会产生大规模稠密矩阵。$\mathcal{H}^2$-矩阵利用这些矩阵中的局部低秩结构，提供高效的数据稀疏近似，使我们能够高效处理大型矩阵，例如将$n$维局部秩为$k$的矩阵存储需求降低至$\mathcal{O}(n k)$，并将矩阵-向量乘法的复杂度降低至$\mathcal{O}(n k)$次运算。为了实现更高级的操作（例如构造高效预处理器或计算矩阵函数），我们需要能够将$\mathcal{H}^2$-矩阵作为输入，并再次用$\mathcal{H}^2$-矩阵近似结果的算法，理想情况下需具备可控精度。本文提出一种算法，该算法近似两个$\mathcal{H}^2$-矩阵的乘积，并保证结果子矩阵的块相对误差估计。该算法利用专用树结构在中间步骤中表示精确乘积，从而允许我们应用数学上严格的误差控制策略。