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 ǫ from only O log(N ) log d (N/ǫ) 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 O N log 2 (N ) log 2d (N/ǫ) and represents the solution operator as a sparse Cholesky factorization with O N log(N ) log d (N/ǫ) nonzero entries, (2) allows for embarrassingly parallel evaluation of the solution operator and the computation of its log-determinant, (3) allows for O log(N ) log d (N/ǫ) complexity computation of individual entries of the matrix representation of the solver that in turn enables its recompression to an O N log d (N/ǫ) 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 ǫ and the number of required matrix-vector products of the original solver.