{"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/toeplitz-monte-carlo","title":"Toeplitz Monte Carlo","arxiv_id":"2003.03915","date":"2020-03-09","proceeding":null,"authors":["Josef Dick","Takashi Goda","Hiroya Murata"],"abstract":"Motivated mainly by applications to partial differential equations with random coefficients, we introduce a new class of Monte Carlo estimators, called Toeplitz Monte Carlo (TMC) estimator for approximating the integral of a multivariate function with respect to the direct product of an identical univariate probability measure. The TMC estimator generates a sequence $x_1,x_2,\\ldots$ of i.i.d. samples for one random variable, and then uses $(x_{n+s-1},x_{n+s-2}\\ldots,x_n)$ with $n=1,2,\\ldots$ as quadrature points, where $s$ denotes the dimension. Although consecutive points have some dependency, the concatenation of all quadrature nodes is represented by a Toeplitz matrix, which allows for a fast matrix-vector multiplication. In this paper we study the variance of the TMC estimator and its dependence on the dimension $s$. Numerical experiments confirm the considerable efficiency improvement over the standard Monte Carlo estimator for applications to partial differential equations with random coefficients, particularly when the dimension $s$ is large.","url_abs":"https://arxiv.org/abs/2003.03915v2","url_pdf":"https://arxiv.org/pdf/2003.03915v2.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":"toeplitz-monte-carlo","repo_url":"https://github.com/takashigoda/Toeplitz-Monte-Carlo","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"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}