Systematic optimization methods for uniform designs

Systematic optimization methods for uniform designs | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Systematic optimization methods for uniform designs Liangwei Qi, Changxing Ma, Yongdao Zhou This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7572223/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 29 Apr, 2026 Read the published version in Statistics and Computing → Version 1 posted 11 You are reading this latest preprint version Abstract As a successful practice of the quasi-Monte Carlo method in computer experiments , uniform design aims to distribute points evenly on a restricted domain. Such a point set has low discrepancy and has enjoyed increasing popularity in applications. However, most designs obtained by the numerical optimization algorithms in literature are just nearly uniform, thus there is significant room for improvement. This paper reviews the existing work on uniform designs and then characterizes their structure. Deterministic construction methods for uniform designs with any number of levels are theoretically proposed, which break through a common limitation of setting the level number to be a prime or a prime power. Based on the above, we provide a systematic optimization algorithm to search for uniform designs under more general parameters for practical use. Numerical experiments show that the performance and runtime of the proposed algorithm are far superior to existing ones. Good lattice point set Optimization method Quasi-Monte Carlo method Wrap-around L2-discrepancy Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 29 Apr, 2026 Read the published version in Statistics and Computing → Version 1 posted Editorial decision: Revision requested 02 Dec, 2025 Reviews received at journal 02 Dec, 2025 Reviews received at journal 11 Oct, 2025 Reviewers agreed at journal 14 Sep, 2025 Reviewers agreed at journal 13 Sep, 2025 Reviewers agreed at journal 12 Sep, 2025 Reviewers agreed at journal 12 Sep, 2025 Reviewers invited by journal 12 Sep, 2025 Editor assigned by journal 12 Sep, 2025 Submission checks completed at journal 11 Sep, 2025 First submitted to journal 09 Sep, 2025 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-7572223","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":516319709,"identity":"6c3bcbb8-3e31-4985-abeb-8b029c34b5c5","order_by":0,"name":"Liangwei Qi","email":"","orcid":"","institution":"Donghua University","correspondingAuthor":false,"prefix":"","firstName":"Liangwei","middleName":"","lastName":"Qi","suffix":""},{"id":516319712,"identity":"9c8dd163-9e88-460d-83e6-bd7cc8925378","order_by":1,"name":"Changxing Ma","email":"","orcid":"","institution":"University at Buffalo","correspondingAuthor":false,"prefix":"","firstName":"Changxing","middleName":"","lastName":"Ma","suffix":""},{"id":516319713,"identity":"6cccd2ed-8587-41dc-af7c-e3f7c0da61fa","order_by":2,"name":"Yongdao Zhou","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA30lEQVRIiWNgGAWjYFCCBDYgYQPEQDqBgYGxgUgtaaRrOQzRwkCMFoPj6c8efGw7n8c/uy3xwwMGG9kNB5ifPcCr5cyDdMOZbbeLJe4cOyyRwJBmvOEAm7kBXi03Eo5J87bdTmy4kd4A1HI4ccMBHjYJ/FoS24BaziXOv5He/COB4T8xWpLZgFoOJG64kXYMaMsBwlokzzxjk5xxLjlx4420NIsEg2TjmYfZzPBq4QOGmMSHMrvEeTfSjG/+qLCT7Tve/AyvFoUDqO4EYmZ86oFAvoGAglEwCkbBKBgFDAC/TFAo5DqJVQAAAABJRU5ErkJggg==","orcid":"","institution":"NITFID, Nankai University","correspondingAuthor":true,"prefix":"","firstName":"Yongdao","middleName":"","lastName":"Zhou","suffix":""}],"badges":[],"createdAt":"2025-09-09 09:38:38","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7572223/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7572223/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s11222-026-10891-z","type":"published","date":"2026-04-29T15:57:20+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":108437588,"identity":"efa4aa02-ca49-4af2-b833-39c79dfc8a26","added_by":"auto","created_at":"2026-05-04 15:59:43","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":601594,"visible":true,"origin":"","legend":"","description":"","filename":"snarticle.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7572223/v1_covered_94227b8a-7c1a-45f5-9504-64123b5179c2.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Systematic optimization methods for uniform designs","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"statistics-and-computing","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"stco","sideBox":"Learn more about [Statistics and Computing](http://link.springer.com/journal/11222)","snPcode":"11222","submissionUrl":"https://submission.nature.com/new-submission/11222/3","title":"Statistics and Computing","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Good lattice point set, Optimization method, Quasi-Monte Carlo method, Wrap-around L2-discrepancy","lastPublishedDoi":"10.21203/rs.3.rs-7572223/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7572223/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"As a successful practice of the quasi-Monte Carlo method in computer experiments , uniform design aims to distribute points evenly on a restricted domain. Such a point set has low discrepancy and has enjoyed increasing popularity in applications. However, most designs obtained by the numerical optimization algorithms in literature are just nearly uniform, thus there is significant room for improvement. This paper reviews the existing work on uniform designs and then characterizes their structure. Deterministic construction methods for uniform designs with any number of levels are theoretically proposed, which break through a common limitation of setting the level number to be a prime or a prime power. Based on the above, we provide a systematic optimization algorithm to search for uniform designs under more general parameters for practical use. Numerical experiments show that the performance and runtime of the proposed algorithm are far superior to existing ones.","manuscriptTitle":"Systematic optimization methods for uniform designs","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-09-19 05:06:27","doi":"10.21203/rs.3.rs-7572223/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2025-12-02T05:59:52+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-12-02T05:50:35+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-10-12T03:01:09+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"24822055899051740615134579686385644282","date":"2025-09-14T07:10:11+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"148319497424502252664699736988484512099","date":"2025-09-13T13:36:57+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"133686271791838231412422987998375688397","date":"2025-09-12T09:01:15+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"267120229571272128398274962997431075862","date":"2025-09-12T07:28:02+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2025-09-12T06:28:32+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2025-09-12T06:27:40+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2025-09-11T05:48:32+00:00","index":"","fulltext":""},{"type":"submitted","content":"Statistics and Computing","date":"2025-09-09T09:35:34+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"statistics-and-computing","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"stco","sideBox":"Learn more about [Statistics and Computing](http://link.springer.com/journal/11222)","snPcode":"11222","submissionUrl":"https://submission.nature.com/new-submission/11222/3","title":"Statistics and Computing","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"89a4d1ae-fc59-4b91-9bae-eb9305bafb76","owner":[],"postedDate":"September 19th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[],"tags":[],"updatedAt":"2026-05-04T15:59:19+00:00","versionOfRecord":{"articleIdentity":"rs-7572223","link":"https://doi.org/10.1007/s11222-026-10891-z","journal":{"identity":"statistics-and-computing","isVorOnly":false,"title":"Statistics and Computing"},"publishedOn":"2026-04-29 15:57:20","publishedOnDateReadable":"April 29th, 2026"},"versionCreatedAt":"2025-09-19 05:06:27","video":"","vorDoi":"10.1007/s11222-026-10891-z","vorDoiUrl":"https://doi.org/10.1007/s11222-026-10891-z","workflowStages":[]},"version":"v1","identity":"rs-7572223","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7572223","identity":"rs-7572223","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}