Using Radial Basis Functions for Control Variate Integration of High-Dimensional Functions Defined by Sparse Sample Data

Using Radial Basis Functions for Control Variate Integration of High-Dimensional Functions Defined by Sparse Sample Data | 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 Using Radial Basis Functions for Control Variate Integration of High-Dimensional Functions Defined by Sparse Sample Data Lin Li, James M. Hyman This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6720522/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 04 Oct, 2025 Read the published version in Numerical Algorithms → Version 1 posted 7 You are reading this latest preprint version Abstract We introduce a control variate integration (CVI) method using radial basis functions (RBFs) for high-dimensional numerical integration with sparse sample data. Unlike polynomial-based CVI, which effectively captures large-scale variations but suffers from the curse of dimensionality, our RBF approach inherently captures both large- and small-scale data variations. For problems with moderately high dimensions (d ≤ 20), numerical experiments demonstrate that the RBF-based CVI significantly outperforms the traditional Monte Carlo and Quasi-Monte Carlo methods. Although the polynomial CVI typically maintains the same convergence rate as the underlying sampling method, our RBF CVI achieves substantially faster convergence rates, empirically observed between O(N ^{−1.5}) and O(N ^{−2}). This performance advantage holds for both independent and low-discrepancy sampling scenarios, offering a practical solution for high-dimensional integration problems with sparse data. Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 04 Oct, 2025 Read the published version in Numerical Algorithms → Version 1 posted Editorial decision: Revision requested 08 Aug, 2025 Reviews received at journal 07 Aug, 2025 Reviewers agreed at journal 24 Jun, 2025 Reviewers invited by journal 23 Jun, 2025 Editor assigned by journal 04 Jun, 2025 Submission checks completed at journal 04 Jun, 2025 First submitted to journal 21 May, 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-6720522","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":475851131,"identity":"9bfd862c-6567-41b7-8743-6ad133d41007","order_by":0,"name":"Lin Li","email":"","orcid":"","institution":"Alphatec Spine","correspondingAuthor":false,"prefix":"","firstName":"Lin","middleName":"","lastName":"Li","suffix":""},{"id":475851132,"identity":"b2a93695-2a5f-47bd-8574-a18356c057aa","order_by":1,"name":"James M. Hyman","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAwklEQVRIiWNgGAWjYDACCTBpw8DADuEzNhCpJU2CgZlELYdJ0KI7u8d0w88d5+sMDjM/e/CBwUZ2wwECWszunDG72XvmtoTBYTZzwxkMacaEtdzIMbvB2wbSwmAmzcNwOJEoLTf/tp0DamH/Jv2H4T9xWm7zth0AauExk2ZgOECMlrSy27JtyZIzD/OUSfYYJBvPJKwledvNt212/HzH27dJ/Kiwk+0jpIWBgcMAiWOAUxkyYH9AlLJRMApGwSgYwQAAhyBFv0gx+RoAAAAASUVORK5CYII=","orcid":"","institution":"Tulane University","correspondingAuthor":true,"prefix":"","firstName":"James","middleName":"M.","lastName":"Hyman","suffix":""}],"badges":[],"createdAt":"2025-05-22 02:53:18","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-6720522/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-6720522/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s11075-025-02223-5","type":"published","date":"2025-10-04T15:57:40+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":92884427,"identity":"425a1462-da25-4f52-af1a-0bf1409fe483","added_by":"auto","created_at":"2025-10-06 16:12:49","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":623805,"visible":true,"origin":"","legend":"","description":"","filename":"HymanLiCVI.pdf","url":"https://assets-eu.researchsquare.com/files/rs-6720522/v1_covered_3214c49f-06a8-4ff2-823a-4596cc735bb1.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Using Radial Basis Functions for Control Variate Integration of High-Dimensional Functions Defined by Sparse Sample Data","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":"numerical-algorithms","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"numa","sideBox":"Learn more about [Numerical Algorithms](http://link.springer.com/journal/11075)","snPcode":"11075","submissionUrl":"https://submission.nature.com/new-submission/11075/3","title":"Numerical Algorithms","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"","lastPublishedDoi":"10.21203/rs.3.rs-6720522/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-6720522/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"We introduce a control variate integration (CVI) method using radial basis functions (RBFs) for high-dimensional numerical integration with sparse sample data. Unlike polynomial-based CVI, which effectively captures large-scale variations but suffers from the curse of dimensionality, our RBF approach inherently captures both large- and small-scale data variations. For problems with moderately high dimensions (d ≤ 20), numerical experiments demonstrate that the RBF-based CVI significantly outperforms the traditional Monte Carlo and Quasi-Monte Carlo methods. Although the polynomial CVI typically maintains the same convergence rate as the underlying sampling method, our RBF CVI achieves substantially faster convergence rates, empirically observed between O(N ^{−1.5}) and O(N ^{−2}). This performance advantage holds for both independent and low-discrepancy sampling scenarios, offering a practical solution for high-dimensional integration problems with sparse data.","manuscriptTitle":"Using Radial Basis Functions for Control Variate Integration of High-Dimensional Functions Defined by Sparse Sample Data","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-06-25 02:51:49","doi":"10.21203/rs.3.rs-6720522/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2025-08-08T10:04:03+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-08-08T03:47:09+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"30204996695414689996130902265850701395","date":"2025-06-24T15:09:17+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2025-06-23T13:50:17+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2025-06-04T09:17:41+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2025-06-04T07:53:26+00:00","index":"","fulltext":""},{"type":"submitted","content":"Numerical Algorithms","date":"2025-05-22T02:48:51+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"numerical-algorithms","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"numa","sideBox":"Learn more about [Numerical Algorithms](http://link.springer.com/journal/11075)","snPcode":"11075","submissionUrl":"https://submission.nature.com/new-submission/11075/3","title":"Numerical Algorithms","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"80ef7eb8-6fee-449e-8fbe-d11dc05f55e5","owner":[],"postedDate":"June 25th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[],"tags":[],"updatedAt":"2025-10-06T16:09:19+00:00","versionOfRecord":{"articleIdentity":"rs-6720522","link":"https://doi.org/10.1007/s11075-025-02223-5","journal":{"identity":"numerical-algorithms","isVorOnly":false,"title":"Numerical Algorithms"},"publishedOn":"2025-10-04 15:57:40","publishedOnDateReadable":"October 4th, 2025"},"versionCreatedAt":"2025-06-25 02:51:49","video":"","vorDoi":"10.1007/s11075-025-02223-5","vorDoiUrl":"https://doi.org/10.1007/s11075-025-02223-5","workflowStages":[]},"version":"v1","identity":"rs-6720522","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-6720522","identity":"rs-6720522","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}