{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/sparse-recovery-of-elliptic-solvers-from","title":"Sparse recovery of elliptic solvers from matrix-vector products","arxiv_id":"2110.05351","date":"2021-10-11","proceeding":null,"authors":["Florian Schäfer","Houman Owhadi"],"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 $\\mathcal{O}\\left(\\log^{1 + d}\\left(\\epsilon^{-1}\\right)\\right)$ 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.","url_abs":"https://arxiv.org/abs/2110.05351v4","url_pdf":"https://arxiv.org/pdf/2110.05351v4.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"sparse-recovery-of-elliptic-solvers-from","repo_url":"https://github.com/f-t-s/sparse_recovery_of_elliptic_solution_operators_from_matrix-vector_products","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok","spdx":"MIT"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":"https://syntology.ai/paper/2110.05351","atlas_url":"https://app.syntology.ai/?focus=2110.05351","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}