{"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/multivariate-newton-interpolation-in-downward","title":"Multivariate Newton Interpolation in Downward Closed Spaces Reaches the Optimal Geometric Approximation Rates for Bos--Levenberg--Trefethen Functions","arxiv_id":"2504.17899","date":"2025-04-24","proceeding":null,"authors":["Michael Hecht","Phil-Alexander Hofmann","Damar Wicaksono","Uwe Hernandez Acosta","Krzysztof Gonciarz","Jannik Kissinger","Vladimir Sivkin","Ivo F. Sbalzarini"],"abstract":"We extend the univariate Newton interpolation algorithm to arbitrary spatial dimensions and for any choice of downward-closed polynomial space, while preserving its quadratic runtime and linear storage cost. The generalisation supports any choice of the provided notion of non-tensorial unisolvent interpolation nodes, whose number coincides with the dimension of the chosen-downward closed space. Specifically, we prove that by selecting Leja-ordered Chebyshev-Lobatto or Leja nodes, the optimal geometric approximation rates for a class of analytic functions -- termed Bos--Levenberg--Trefethen functions -- are achieved and extend to the derivatives of the interpolants. In particular, choosing Euclidean degree results in downward-closed spaces whose dimension only grows sub-exponentially with spatial dimension, while delivering approximation rates close to, or even matching those of the tensorial maximum-degree case, mitigating the curse of dimensionality. Several numerical experiments demonstrate the performance of the resulting multivariate Newton interpolation compared to state-of-the-art alternatives and validate our theoretical results.","url_abs":"https://arxiv.org/abs/2504.17899v1","url_pdf":"https://arxiv.org/pdf/2504.17899v1.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":"multivariate-newton-interpolation-in-downward","repo_url":"https://github.com/phil-hofmann/lpfun","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}