{"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/multidimensional-phase-recovery-and","title":"Multidimensional Phase Recovery and Interpolative Decomposition Butterfly Factorization","arxiv_id":"1908.09376","date":"2019-08-25","proceeding":null,"authors":["Ze Chen","Juan Zhang","Kenneth L. Ho","Haizhao Yang"],"abstract":"This paper focuses on the fast evaluation of the matvec $g=Kf$ for $K\\in \\mathbb{C}^{N\\times N}$, which is the discretization of a multidimensional oscillatory integral transform $g(x) = \\int K(x,\\xi) f(\\xi)d\\xi$ with a kernel function $K(x,\\xi)=e^{2\\pi\\i \\Phi(x,\\xi)}$, where $\\Phi(x,\\xi)$ is a piecewise smooth phase function with $x$ and $\\xi$ in $\\mathbb{R}^d$ for $d=2$ or $3$. A new framework is introduced to compute $Kf$ with $O(N\\log N)$ time and memory complexity in the case that only indirect access to the phase function $\\Phi$ is available. This framework consists of two main steps: 1) an $O(N\\log N)$ algorithm for recovering the multidimensional phase function $\\Phi$ from indirect access is proposed; 2) a multidimensional interpolative decomposition butterfly factorization (MIDBF) is designed to evaluate the matvec $Kf$ with an $O(N\\log N)$ complexity once $\\Phi$ is available. Numerical results are provided to demonstrate the effectiveness of the proposed framework.","url_abs":"http://arxiv.org/abs/1908.09376v2","url_pdf":"http://arxiv.org/pdf/1908.09376v2.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":"multidimensional-phase-recovery-and","repo_url":"https://github.com/ButterflyLab/ButterflyLab","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"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}