An enumerative formula for the spherical cap discrepancy

The spherical cap discrepancy is a widely used measure for how uniformly a sample of points on the sphere is distributed. Being hard to compute, this discrepancy measure is typically replaced by some lower or upper estimates when designing optimal sampling schemes for the uniform distribution on the...

Full description

Saved in:
Bibliographic Details
Published in:Journal of computational and applied mathematics 2021-07, Vol.390, p.113409, Article 113409
Main Authors: Heitsch, Holger, Henrion, René
Format: Article
Language:English
Subjects:
Optimality conditions
Spherical cap discrepancy
Uniform distribution on sphere
Citations: Items that this one cites
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
cited_by
cites cdi_FETCH-LOGICAL-c292t-c26c2f7377147dd126ea9ce0002f3a76f91d968e495c1c609899c3ef5bc4c4113
container_end_page
container_issue
container_start_page 113409
container_title Journal of computational and applied mathematics
container_volume 390
creator Heitsch, Holger
Henrion, René
description The spherical cap discrepancy is a widely used measure for how uniformly a sample of points on the sphere is distributed. Being hard to compute, this discrepancy measure is typically replaced by some lower or upper estimates when designing optimal sampling schemes for the uniform distribution on the sphere. In this paper, we provide a fully explicit, easy to implement enumerative formula for the spherical cap discrepancy. Not surprisingly, this formula is of combinatorial nature and, thus, its application is limited to spheres of small dimension and moderate sample sizes. Nonetheless, it may serve as a useful calibrating tool for testing the efficiency of sampling schemes and its explicit character might be useful also to establish necessary optimality conditions when minimizing the discrepancy with respect to a sample of given size.
doi_str_mv 10.1016/j.cam.2021.113409
format article
fullrecord <record><control><sourceid>elsevier_cross</sourceid><recordid>TN_cdi_crossref_primary_10_1016_j_cam_2021_113409</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><els_id>S0377042721000285</els_id><sourcerecordid>S0377042721000285</sourcerecordid><originalsourceid>FETCH-LOGICAL-c292t-c26c2f7377147dd126ea9ce0002f3a76f91d968e495c1c609899c3ef5bc4c4113</originalsourceid><addsrcrecordid>eNp9j81KAzEUhYMoWKsP4G5eYMbcTJpMcFWKf1Bwo-sQ79zQlPkjmRb69qbUtZt7Vt8592PsEXgFHNTTvkLXV4ILqABqyc0VW0CjTQlaN9dswWutSy6FvmV3Ke0558qAXLBmPRQ0HHqKbg5HKvwY-0PnzlnMOyrStKMY0HUFuqloQ8JIkxvwdM9uvOsSPfzlkn2_vnxt3svt59vHZr0tURgx56tQeJ3XQeq2BaHIGaS8L3zttPIGWqMakmaFgIqbxhisya9-UKLMKksGl16MY0qRvJ1i6F08WeD2rG73Nqvbs7q9qGfm-cJQfuwYKNqEgQakNkTC2bZj-If-BUDzYJI</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype></control><display><type>article</type><title>An enumerative formula for the spherical cap discrepancy</title><source>ScienceDirect Freedom Collection</source><creator>Heitsch, Holger ; Henrion, René</creator><creatorcontrib>Heitsch, Holger ; Henrion, René</creatorcontrib><description>The spherical cap discrepancy is a widely used measure for how uniformly a sample of points on the sphere is distributed. Being hard to compute, this discrepancy measure is typically replaced by some lower or upper estimates when designing optimal sampling schemes for the uniform distribution on the sphere. In this paper, we provide a fully explicit, easy to implement enumerative formula for the spherical cap discrepancy. Not surprisingly, this formula is of combinatorial nature and, thus, its application is limited to spheres of small dimension and moderate sample sizes. Nonetheless, it may serve as a useful calibrating tool for testing the efficiency of sampling schemes and its explicit character might be useful also to establish necessary optimality conditions when minimizing the discrepancy with respect to a sample of given size.</description><identifier>ISSN: 0377-0427</identifier><identifier>EISSN: 1879-1778</identifier><identifier>DOI: 10.1016/j.cam.2021.113409</identifier><language>eng</language><publisher>Elsevier B.V</publisher><subject>Optimality conditions ; Spherical cap discrepancy ; Uniform distribution on sphere</subject><ispartof>Journal of computational and applied mathematics, 2021-07, Vol.390, p.113409, Article 113409</ispartof><rights>2021 Elsevier B.V.</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c292t-c26c2f7377147dd126ea9ce0002f3a76f91d968e495c1c609899c3ef5bc4c4113</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,780,784,27924,27925</link.rule.ids></links><search><creatorcontrib>Heitsch, Holger</creatorcontrib><creatorcontrib>Henrion, René</creatorcontrib><title>An enumerative formula for the spherical cap discrepancy</title><title>Journal of computational and applied mathematics</title><description>The spherical cap discrepancy is a widely used measure for how uniformly a sample of points on the sphere is distributed. Being hard to compute, this discrepancy measure is typically replaced by some lower or upper estimates when designing optimal sampling schemes for the uniform distribution on the sphere. In this paper, we provide a fully explicit, easy to implement enumerative formula for the spherical cap discrepancy. Not surprisingly, this formula is of combinatorial nature and, thus, its application is limited to spheres of small dimension and moderate sample sizes. Nonetheless, it may serve as a useful calibrating tool for testing the efficiency of sampling schemes and its explicit character might be useful also to establish necessary optimality conditions when minimizing the discrepancy with respect to a sample of given size.</description><subject>Optimality conditions</subject><subject>Spherical cap discrepancy</subject><subject>Uniform distribution on sphere</subject><issn>0377-0427</issn><issn>1879-1778</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2021</creationdate><recordtype>article</recordtype><recordid>eNp9j81KAzEUhYMoWKsP4G5eYMbcTJpMcFWKf1Bwo-sQ79zQlPkjmRb69qbUtZt7Vt8592PsEXgFHNTTvkLXV4ILqABqyc0VW0CjTQlaN9dswWutSy6FvmV3Ke0558qAXLBmPRQ0HHqKbg5HKvwY-0PnzlnMOyrStKMY0HUFuqloQ8JIkxvwdM9uvOsSPfzlkn2_vnxt3svt59vHZr0tURgx56tQeJ3XQeq2BaHIGaS8L3zttPIGWqMakmaFgIqbxhisya9-UKLMKksGl16MY0qRvJ1i6F08WeD2rG73Nqvbs7q9qGfm-cJQfuwYKNqEgQakNkTC2bZj-If-BUDzYJI</recordid><startdate>202107</startdate><enddate>202107</enddate><creator>Heitsch, Holger</creator><creator>Henrion, René</creator><general>Elsevier B.V</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>202107</creationdate><title>An enumerative formula for the spherical cap discrepancy</title><author>Heitsch, Holger ; Henrion, René</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c292t-c26c2f7377147dd126ea9ce0002f3a76f91d968e495c1c609899c3ef5bc4c4113</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2021</creationdate><topic>Optimality conditions</topic><topic>Spherical cap discrepancy</topic><topic>Uniform distribution on sphere</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Heitsch, Holger</creatorcontrib><creatorcontrib>Henrion, René</creatorcontrib><collection>CrossRef</collection><jtitle>Journal of computational and applied mathematics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Heitsch, Holger</au><au>Henrion, René</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>An enumerative formula for the spherical cap discrepancy</atitle><jtitle>Journal of computational and applied mathematics</jtitle><date>2021-07</date><risdate>2021</risdate><volume>390</volume><spage>113409</spage><pages>113409-</pages><artnum>113409</artnum><issn>0377-0427</issn><eissn>1879-1778</eissn><abstract>The spherical cap discrepancy is a widely used measure for how uniformly a sample of points on the sphere is distributed. Being hard to compute, this discrepancy measure is typically replaced by some lower or upper estimates when designing optimal sampling schemes for the uniform distribution on the sphere. In this paper, we provide a fully explicit, easy to implement enumerative formula for the spherical cap discrepancy. Not surprisingly, this formula is of combinatorial nature and, thus, its application is limited to spheres of small dimension and moderate sample sizes. Nonetheless, it may serve as a useful calibrating tool for testing the efficiency of sampling schemes and its explicit character might be useful also to establish necessary optimality conditions when minimizing the discrepancy with respect to a sample of given size.</abstract><pub>Elsevier B.V</pub><doi>10.1016/j.cam.2021.113409</doi><oa>free_for_read</oa></addata></record>
fulltext fulltext
identifier ISSN: 0377-0427
ispartof Journal of computational and applied mathematics, 2021-07, Vol.390, p.113409, Article 113409
issn 0377-0427
1879-1778
language eng
recordid cdi_crossref_primary_10_1016_j_cam_2021_113409
source ScienceDirect Freedom Collection
subjects Optimality conditions
Spherical cap discrepancy
Uniform distribution on sphere
title An enumerative formula for the spherical cap discrepancy
url http://sfxeu10.hosted.exlibrisgroup.com/loughborough?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2024-12-29T06%3A09%3A29IST&url_ver=Z39.88-2004&url_ctx_fmt=infofi/fmt:kev:mtx:ctx&rfr_id=info:sid/primo.exlibrisgroup.com:primo3-Article-elsevier_cross&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.genre=article&rft.atitle=An%20enumerative%20formula%20for%20the%20spherical%20cap%20discrepancy&rft.jtitle=Journal%20of%20computational%20and%20applied%20mathematics&rft.au=Heitsch,%20Holger&rft.date=2021-07&rft.volume=390&rft.spage=113409&rft.pages=113409-&rft.artnum=113409&rft.issn=0377-0427&rft.eissn=1879-1778&rft_id=info:doi/10.1016/j.cam.2021.113409&rft_dat=%3Celsevier_cross%3ES0377042721000285%3C/elsevier_cross%3E%3Cgrp_id%3Ecdi_FETCH-LOGICAL-c292t-c26c2f7377147dd126ea9ce0002f3a76f91d968e495c1c609899c3ef5bc4c4113%3C/grp_id%3E%3Coa%3E%3C/oa%3E%3Curl%3E%3C/url%3E&rft_id=info:oai/&rft_id=info:pmid/&rfr_iscdi=true