Sparse recovery of elliptic solvers from matrix-vector products

Abstract: 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 $$\mathcal{O}\left(\log^{1 + d}\left(\epsilon<sup>{-1}\right)\right)$$ 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.