{"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/distributed-memory-mathcal-h-matrix-algebra-i","title":"Distributed-memory $\\mathcal{H}$-matrix Algebra I: Data Distribution and Matrix-vector Multiplication","arxiv_id":"2008.12441","date":"2020-08-28","proceeding":null,"authors":["Yingzhou Li","Jack Poulson","Lexing Ying"],"abstract":"We introduce a data distribution scheme for $\\mathcal{H}$-matrices and a distributed-memory algorithm for $\\mathcal{H}$-matrix-vector multiplication. Our data distribution scheme avoids an expensive $\\Omega(P^2)$ scheduling procedure used in previous work, where $P$ is the number of processes, while data balancing is well-preserved. Based on the data distribution, our distributed-memory algorithm evenly distributes all computations among $P$ processes and adopts a novel tree-communication algorithm to reduce the latency cost. The overall complexity of our algorithm is $O\\Big(\\frac{N \\log N}{P} + \\alpha \\log P + \\beta \\log^2 P \\Big)$ for $\\mathcal{H}$-matrices under weak admissibility condition, where $N$ is the matrix size, $\\alpha$ denotes the latency, and $\\beta$ denotes the inverse bandwidth. Numerically, our algorithm is applied to address both two- and three-dimensional problems of various sizes among various numbers of processes. On thousands of processes, good parallel efficiency is still observed.","url_abs":"https://arxiv.org/abs/2008.12441v2","url_pdf":"https://arxiv.org/pdf/2008.12441v2.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":"distributed-memory-mathcal-h-matrix-algebra-i","repo_url":"https://github.com/YingzhouLi/dmhm","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}