Sparse recovery of elliptic solvers from matrix-vector products

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. We include rigorous proofs of these results, and to the best of our knowledge, the proposed algorithm achieves the best trade-off between accuracy $\epsilon$ and the number of required matrix-vector products of the original solver.