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，基于精心选择的向量（右端项），仅需 $\mathcal{O}\left(\log(N)\log^{d}(N / \epsilon)\right)$ 次矩阵-向量乘积，即可近似 $d$ 维椭圆边值问题的求解器，精度达到 $\epsilon$。该求解器仅作为黑箱访问，且底层算子可能未知且具有任意高阶性质。我们的算法（1）具备 $\mathcal{O}\left(N\log^2(N)\log^{2d}(N / \epsilon)\right)$ 复杂度，并将解算子表示为含 $\mathcal{O}\left(N\log(N)\log^{d}(N / \epsilon)\right$ 个非零元的稀疏Cholesky分解；（2）支持解算子的惊人数量的并行求值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 $\mathcal{O}\left(N\log^{d}(N / \epsilon)\right)$ 复杂度表示。作为副产品，我们的压缩方案生成了具有近最优逼近精度的均匀化解算子。通过多项式逼近，还可基于 $\mathcal{O}\left(\log^{1 + d}\left(\epsilon^{-1}\right)\right)$ 个偏微分方程的解，以精度 $\epsilon$ 逼近连续格林函数（在算子范数和Hilbert-Schmidt范数下）。我们给出了这些结果的严格证明。据我们所知，我们的算法在精度 $\epsilon$ 与所需矩阵-向量乘积次数之间实现了当前已知的最佳权衡。