Randomized weakly admissible meshes

Yiming Xu; Akil Narayan

doi:10.1016/j.jat.2022.105835

Randomized weakly admissible meshes

Yiming Xu, Akil Narayan

Mathematics

Research output: Contribution to journal › Article › peer-review

4 Scopus citations

Abstract

A weakly admissible mesh (WAM) on a continuum real-valued domain is a sequence of discrete grids such that the discrete maximum norm of polynomials on the grid is comparable to the supremum norm of polynomials on the domain. The asymptotic rate of growth of the grid sizes and of the comparability constants must grow in a controlled manner. In this paper, we recognize that the notion of a WAM can be generalized to a sequence of hierarchical subspaces not necessarily of polynomial functions, and we analyze particular strategies for random sampling as a technique for generating WAM's. Our main results show that WAM's and their stronger variant, admissible meshes (AM's), can be generated by random sampling, and our analysis provides concrete estimates for growth and rates of growth of both the meshes and the discrete–continuum comparability constants.

Original language	English
Article number	105835
Journal	Journal of Approximation Theory
Volume	285
DOIs	https://doi.org/10.1016/j.jat.2022.105835
State	Published - Jan 2023

Bibliographical note

Publisher Copyright:
© 2022 Elsevier Inc.

Funding

We would like to thank the anonymous referees for their very helpful comments which significantly improved the results and presentation of the paper. A. Narayan thanks Norm Levenberg and Sione Ma’u for a careful reading of an early draft, and for providing several comments that greatly improved the quality of the manuscript. Y. Xu thanks Piotr Hajlasz for a helpful conversation about the measure density condition. Y. Xu and A. Narayan are partially supported by NSF, USA DMS-1848508 . This material is based partially upon work supported by the National Science Foundation, USA under Grant No. DMS-1439786 and by the Simons Foundation, USA Grant No. 50736 while A. Narayan was in residence at the Institute for Computational and Experimental Research in Mathematics in Providence, RI, during the “Model and dimension reduction in uncertain and dynamic systems” program.

Funders	Funder number
National Science Foundation Arctic Social Science Program	DMS-1439786, USA DMS-1848508
National Science Foundation Arctic Social Science Program
Simons Foundation	50736
Simons Foundation

Keywords

Admissible meshes
Near-isometry
Norming sets
Random sampling
Weighted covering

ASJC Scopus subject areas

Analysis
Numerical Analysis
General Mathematics
Applied Mathematics

Access to Document

10.1016/j.jat.2022.105835

Cite this

@article{bbd63578683040f59802c60156240160,

title = "Randomized weakly admissible meshes",

abstract = "A weakly admissible mesh (WAM) on a continuum real-valued domain is a sequence of discrete grids such that the discrete maximum norm of polynomials on the grid is comparable to the supremum norm of polynomials on the domain. The asymptotic rate of growth of the grid sizes and of the comparability constants must grow in a controlled manner. In this paper, we recognize that the notion of a WAM can be generalized to a sequence of hierarchical subspaces not necessarily of polynomial functions, and we analyze particular strategies for random sampling as a technique for generating WAM's. Our main results show that WAM's and their stronger variant, admissible meshes (AM's), can be generated by random sampling, and our analysis provides concrete estimates for growth and rates of growth of both the meshes and the discrete–continuum comparability constants.",

keywords = "Admissible meshes, Near-isometry, Norming sets, Random sampling, Weighted covering",

author = "Yiming Xu and Akil Narayan",

note = "Publisher Copyright: {\textcopyright} 2022 Elsevier Inc.",

year = "2023",

month = jan,

doi = "10.1016/j.jat.2022.105835",

language = "English",

volume = "285",

}

TY - JOUR

T1 - Randomized weakly admissible meshes

AU - Xu, Yiming

AU - Narayan, Akil

PY - 2023/1

Y1 - 2023/1

N2 - A weakly admissible mesh (WAM) on a continuum real-valued domain is a sequence of discrete grids such that the discrete maximum norm of polynomials on the grid is comparable to the supremum norm of polynomials on the domain. The asymptotic rate of growth of the grid sizes and of the comparability constants must grow in a controlled manner. In this paper, we recognize that the notion of a WAM can be generalized to a sequence of hierarchical subspaces not necessarily of polynomial functions, and we analyze particular strategies for random sampling as a technique for generating WAM's. Our main results show that WAM's and their stronger variant, admissible meshes (AM's), can be generated by random sampling, and our analysis provides concrete estimates for growth and rates of growth of both the meshes and the discrete–continuum comparability constants.

AB - A weakly admissible mesh (WAM) on a continuum real-valued domain is a sequence of discrete grids such that the discrete maximum norm of polynomials on the grid is comparable to the supremum norm of polynomials on the domain. The asymptotic rate of growth of the grid sizes and of the comparability constants must grow in a controlled manner. In this paper, we recognize that the notion of a WAM can be generalized to a sequence of hierarchical subspaces not necessarily of polynomial functions, and we analyze particular strategies for random sampling as a technique for generating WAM's. Our main results show that WAM's and their stronger variant, admissible meshes (AM's), can be generated by random sampling, and our analysis provides concrete estimates for growth and rates of growth of both the meshes and the discrete–continuum comparability constants.

KW - Admissible meshes

KW - Near-isometry

KW - Norming sets

KW - Random sampling

KW - Weighted covering

UR - http://www.scopus.com/inward/record.url?scp=85140589994&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85140589994&partnerID=8YFLogxK

U2 - 10.1016/j.jat.2022.105835

DO - 10.1016/j.jat.2022.105835

M3 - Article

AN - SCOPUS:85140589994

SN - 0021-9045

VL - 285

JO - Journal of Approximation Theory

JF - Journal of Approximation Theory

M1 - 105835

ER -

Randomized weakly admissible meshes

Abstract

Bibliographical note

Funding

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this