Optimal Estimation and Uncertainty Quantification for Stochastic Inverse Problems via Variational Bayesian Methods

Optimal Estimation and Uncertainty Quantification for Stochastic Inverse Problems via Variational Bayesian Methods | 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 Optimal Estimation and Uncertainty Quantification for Stochastic Inverse Problems via Variational Bayesian Methods Ruibiao Song, Liying Zhang This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6791762/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 03 Apr, 2026 Read the published version in Statistics and Computing → Version 1 posted 8 You are reading this latest preprint version Abstract The Bayesian inversion method demonstrates significant potential for solving inverse problems, enabling both point estimation and uncertainty quantification (UQ). However, Bayesian maximum a posteriori (MAP) estimation may become unstable when handling data from diverse distributions (e.g., solutions of stochastic partial differential equations (SPDEs)). Additionally, Monte Carlo sampling methods are computationally expensive. To address these challenges, we propose a novel two-stage optimization method based on optimal control theory and variational Bayesian methods. This method not only yields stable solutions for stochastic inverse problems but also efficiently quantifies the uncertainty of solutions. In the first stage, we introduce a new weighting formulation to ensure the stability of the Bayesian MAP estimation. In the second stage, we derive the necessary condition for efficiently quantifying the uncertainty of the solutions by combining the new weighting formula with variational inference. Furthermore, we establish an error estimation theorem that relates the exact solution to the optimally estimated solution under different amounts of observed data. Finally, the efficiency of the proposed method is demonstrated through numerical examples. variational Bayesian methods stochastic inverse problems uncertainty quantification two-stage optimization method Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 03 Apr, 2026 Read the published version in Statistics and Computing → Version 1 posted Editorial decision: Revision requested 07 Nov, 2025 Reviews received at journal 24 Aug, 2025 Reviewers agreed at journal 14 Jul, 2025 Reviewers agreed at journal 05 Jun, 2025 Reviewers invited by journal 04 Jun, 2025 Editor assigned by journal 03 Jun, 2025 Submission checks completed at journal 02 Jun, 2025 First submitted to journal 31 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. 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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-6791762","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":467044949,"identity":"d012c0d8-6c8f-4d63-8045-b3f20d4c1b51","order_by":0,"name":"Ruibiao Song","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA1klEQVRIie3RIQ6DMBSA4S5NqsqwJRC4AgkJ56EGxZLJCgQLBMQ2z26BG3MkJFWdR8INhpscU4gtlLmJ/rpfXl8LgEr1hyGnbZoHi7EON4c+YLGcbAmiQym4bRRp6/aCy4lNsOdpGfTcEw+NIYMrLmbmoXFJEK1I5DOaIKAXx2CZWC0nY23R60Q6WluAiHu1TEA4TRGI3so3EQi4ZCcjkW9qOaRVF/l7msMVhETT+vm0vuAhWEcwnx+ZBIJj6S5Okc5fOT5ZbOvFeZl8hH87rlKpVKqvvQCeFk6i+I14TQAAAABJRU5ErkJggg==","orcid":"","institution":"China University of Mining and Technology (Beijing)","correspondingAuthor":true,"prefix":"","firstName":"Ruibiao","middleName":"","lastName":"Song","suffix":""},{"id":467044950,"identity":"c08d7620-9a9a-4b14-8a29-7c7db0d52d27","order_by":1,"name":"Liying Zhang","email":"","orcid":"","institution":"China University of Mining and Technology (Beijing)","correspondingAuthor":false,"prefix":"","firstName":"Liying","middleName":"","lastName":"Zhang","suffix":""}],"badges":[],"createdAt":"2025-05-31 15:23:11","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-6791762/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-6791762/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s11222-026-10847-3","type":"published","date":"2026-04-03T15:59:42+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":106343578,"identity":"8f71d1b9-40b0-4327-a684-b1ae3e00ad77","added_by":"auto","created_at":"2026-04-07 16:06:31","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":2029441,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-6791762/v1_covered_d99eee1f-f727-4257-972e-e624cf0e6901.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Optimal Estimation and Uncertainty Quantification for Stochastic Inverse Problems via Variational Bayesian Methods","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":"variational Bayesian methods, stochastic inverse problems, uncertainty quantification, two-stage optimization method","lastPublishedDoi":"10.21203/rs.3.rs-6791762/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-6791762/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"The Bayesian inversion method demonstrates significant potential for solving inverse problems, enabling both point estimation and uncertainty quantification (UQ). 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