In this work, we show that solvers of elliptic boundary value problems in $d$ dimensions can be approximated to accuracy $\epsilon$ from only $\mathcal{O}\left(\log(N)\log^{d}(N / \epsilon)\right)$ matrix-vector products with carefully chosen vectors (right-hand sides). The solver is only accessed as a black box, and the underlying operator may be unknown and of an arbitrarily high order. Our algorithm (1) has complexity $\mathcal{O}\left(N\log^2(N)\log^{2d}(N / \epsilon)\right)$ and represents the solution operator as a sparse Cholesky factorization with $\mathcal{O}\left(N\log(N)\log^{d}(N / \epsilon)\right)$ nonzero entries, (2) allows for embarrassingly parallel evaluation of the solution operator and the computation of its log-determinant, (3) allows for $\mathcal{O}\left(\log(N)\log^{d}(N / \epsilon)\right)$ complexity computation of individual entries of the matrix representation of the solver that, in turn, enables its recompression to an $\mathcal{O}\left(N\log^{d}(N / \epsilon)\right)$ complexity representation. As a byproduct, our compression scheme produces a homogenized solution operator with near-optimal approximation accuracy. By polynomial approximation, we can also approximate the continuous Green's function (in operator and Hilbert-Schmidt norm) to accuracy $\epsilon$ from $\mathcal{O}\left(\log^{1 + d}\left(\epsilon^{-1}\right)\right)$ solutions of the PDE. We include rigorous proofs of these results. To the best of our knowledge, our algorithm achieves the best known trade-off between accuracy $\epsilon$ and the number of required matrix-vector products.

翻译：本文证明，通过仅使用精心选取的向量（右端项）进行$\mathcal{O}\left(\log(N)\log^{d}(N / \epsilon)\right)$次矩阵-向量乘积，即可将$d$维椭圆边值问题的求解器近似到精度$\epsilon$。该求解器仅作为黑箱访问，底层算子可能未知且具有任意高阶数。我们的算法：(1) 复杂度为$\mathcal{O}\left(N\log^2(N)\log^{2d}(N / \epsilon)\right)$，并将求解算子表示为具有$\mathcal{O}\left(N\log(N)\log^{d}(N / \epsilon)\right)$个非零元的稀疏乔列斯基分解；(2) 支持求解算子的高度并行化评估及其对数行列式的计算；(3) 允许以$\mathcal{O}\left(\log(N)\log^{d}(N / \epsilon)\right)$复杂度计算求解器矩阵表示中的单个元素，进而可将其重压缩为$\mathcal{O}\left(N\log^{d}(N / \epsilon)\right)$复杂度的表示。作为副产品，我们的压缩方案产生了具有近最优逼近精度的均匀化求解算子。通过多项式逼近，我们还可以从$\mathcal{O}\left(\log^{1 + d}\left(\epsilon^{-1}\right)\right)$个PDE解中，以$\epsilon$精度逼近连续格林函数（在算子和希尔伯特-施密特范数下）。我们给出了这些结果的严格证明。据我们所知，本算法在精度$\epsilon$与所需矩阵-向量乘积数量之间实现了当前已知的最优权衡。