{"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/the-inverse-fast-multipole-method","title":"The Inverse Fast Multipole Method","arxiv_id":"1407.1572","date":"2014-07-07","proceeding":null,"authors":["Sivaram Ambikasaran","Eric Darve"],"abstract":"This article introduces a new fast direct solver for linear systems arising out of wide range of applications, integral equations, multivariate statistics, radial basis interpolation, etc., to name a few. \\emph{The highlight of this new fast direct solver is that the solver scales linearly in the number of unknowns in all dimensions.} The solver, termed as Inverse Fast Multipole Method (abbreviated as IFMM), works on the same data-structure as the Fast Multipole Method (abbreviated as FMM). More generally, the solver can be immediately extended to the class of hierarchical matrices, denoted as $\\mathcal{H}^2$ matrices with strong admissibility criteria (weak low-rank structure), i.e., \\emph{the interaction between neighboring cluster of particles is full-rank whereas the interaction between particles corresponding to well-separated clusters can be efficiently represented as a low-rank matrix}. The algorithm departs from existing approaches in the fact that throughout the algorithm the interaction corresponding to neighboring clusters are always treated as full-rank interactions. Our approach relies on two major ideas: (i) The $N \\times N$ matrix arising out of FMM (from now on termed as FMM matrix) can be represented as an extended sparser matrix of size $M \\times M$, where $M \\approx 3N$. (ii) While solving the larger extended sparser matrix, \\emph{the fill-in's that arise in the matrix blocks corresponding to well-separated clusters are hierarchically compressed}. The ordering of the equations and the unknowns in the extended sparser matrix is strongly related to the local and multipole coefficients in the FMM~\\cite{greengard1987fast} and \\emph{the order of elimination is different from the usual nested dissection approach}. Numerical benchmarks on $2$D manifold confirm the linear scaling of the algorithm.","url_abs":"https://arxiv.org/abs/1407.1572v1","url_pdf":"https://arxiv.org/pdf/1407.1572v1.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":"the-inverse-fast-multipole-method","repo_url":"https://github.com/takahashitoru/IFMM","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}