{"paper_id":"6fc4576f-c5d9-4ae7-9569-9161cd08f6fb","body_text":"A random-bound Chebyshev method for uncertainty propagation of nonlinear dynamics under imprecise probabilities | 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 A random-bound Chebyshev method for uncertainty propagation of nonlinear dynamics under imprecise probabilities Licong Zhang, Chunna Li, Hua Su, Xiaowei Wang, Sizhi Yang, Chunlin Gong This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3733444/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 4 You are reading this latest preprint version Abstract The distribution-free P-box is an effective quantification model for uncertainties with only imprecise probabilistic information. However, its application to nonlinear dynamical systems is limited due to a lack of efficient uncertainty propagation (UP) methods. To end this, this work develops a random-bound Chebyshev (RBC) UP method based on the framework of the interval Monte Carlo (IMC) method. First, the Chebyshev method is applied to solve the interval analysis in the IMC simulations. Here, the bounds of intervals can be regarded as random variables whose cumulative density functions (CDFs) are the CDF bounds of the P-box. Since the CDF bounds of distribution-free P-boxes are always arbitrary and non-parameterized, the data-driven polynomial chaos expansion (DD-PCE), which only requires the information of statistical moments, is introduced to solve the problem of random bounds. Then a sparse-regression strategy is utilized to deal with the ‘curse of dimensionality’ of the DD-PCE for high-dimensional problems. As a result, the RBC method efficiently achieves a non-intrusive UP of nonlinear dynamics with distribution-free P-boxes. The method is also effective for hybrid UP problems with random, interval, and P-box variables. Then the RBC method is validated based on test cases, including a duffing oscillator, a vehicle ride, and an engineering application of launch-vehicle trajectory. The results verify the ability of the method to deal with complex black-box problems. In comparison with the reference solutions based on the IMC simulations, with relative errors of less than 1%, the proposed method requires less than 0.0004% sample size and 0.015% calculation time. Nonlinear dynamics Uncertainty propagation Imprecise probabilities Distribution-free P-box Chebyshev method Data-driven polynomial chaos expansion Full Text Supplementary Files sourcecode.rar Cite Share Download PDF Status: Under Review Version 1 posted Reviewers agreed at journal 13 Jan, 2024 Reviewers invited by journal 12 Jan, 2024 Editor assigned by journal 12 Dec, 2023 First submitted to journal 08 Dec, 2023 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. 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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-3733444\",\"acceptedTermsAndConditions\":true,\"allowDirectSubmit\":false,\"archivedVersions\":[],\"articleType\":\"Research Article\",\"associatedPublications\":[],\"authors\":[{\"id\":266934448,\"identity\":\"cc2882ec-27fd-4a23-b78a-ca6097a05448\",\"order_by\":0,\"name\":\"Licong Zhang\",\"email\":\"\",\"orcid\":\"\",\"institution\":\"Northwestern Polytechnical University School of 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However, its application to nonlinear dynamical systems is limited due to a lack of efficient uncertainty propagation (UP) methods. To end this, this work develops a random-bound Chebyshev (RBC) UP method based on the framework of the interval Monte Carlo (IMC) method. First, the Chebyshev method is applied to solve the interval analysis in the IMC simulations. Here, the bounds of intervals can be regarded as random variables whose cumulative density functions (CDFs) are the CDF bounds of the P-box. Since the CDF bounds of distribution-free P-boxes are always arbitrary and non-parameterized, the data-driven polynomial chaos expansion (DD-PCE), which only requires the information of statistical moments, is introduced to solve the problem of random bounds. Then a sparse-regression strategy is utilized to deal with the \\u0026lsquo;curse of dimensionality\\u0026rsquo; of the DD-PCE for high-dimensional problems. As a result, the RBC method efficiently achieves a non-intrusive UP of nonlinear dynamics with distribution-free P-boxes. The method is also effective for hybrid UP problems with random, interval, and P-box variables. Then the RBC method is validated based on test cases, including a duffing oscillator, a vehicle ride, and an engineering application of launch-vehicle trajectory. The results verify the ability of the method to deal with complex black-box problems. 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