Sparse recovery of elliptic solvers from matrix-vector products
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In this work, we show that solvers of elliptic boundary value problems in d dimensions can be approximated to accuracy ϵ from only 𝒪(log(N)log^d(N / ϵ)) 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(Nlog^2(N)log^2d(N / ϵ)) and represents the solution operator as a sparse Cholesky factorization with 𝒪(Nlog(N)log^d(N / ϵ)) 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 / ϵ)) complexity computation of individual entries of the matrix representation of the solver that in turn enables its recompression to an 𝒪(Nlog^d(N / ϵ)) 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 ϵ and the number of required matrix-vector products of the original solver.
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