Uncertainty Quantification for Linear Inverse Problems with Besov Prior: A Randomize-Then-Optimize Method

Uncertainty Quantification for Linear Inverse Problems with Besov Prior: A Randomize-Then-Optimize Method | 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 Uncertainty Quantification for Linear Inverse Problems with Besov Prior: A Randomize-Then-Optimize Method Andreas Horst, Babak Maboudi Afkham, Yiqiu Dong, Jakob Lemvig This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4528903/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 23 May, 2025 Read the published version in Statistics and Computing → Version 1 posted 9 You are reading this latest preprint version Abstract In this work, we investigate the use of Besov priors in the context of Bayesian inverse problems. The solution to Bayesian inverse problems is the posterior distribution which naturally enables us to interpret the uncertainties. Besov priors are discretization invariant and can promote sparsity in terms of wavelet coefficients. We propose the randomize-then-optimize method to draw samples from the posterior distribution with Besov priors under a general parameter setting and estimate the modes of the posterior distribution. The performance of the proposed method is studied through numerical experiments of a 1D inpainting problem, a 1D deconvolution problem, and a 2D computed tomography problem. Further, we discuss the influence of the choice of the Besov parameters and the wavelet basis in detail, and we compare the proposed method with the state-of-the-art methods. The numerical results suggest that the proposed method is an effective tool for sampling the posterior distribution equipped with general Besov priors. Bayesian inverse problems Besov priors sampling methods randomize-then-optimize method Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 23 May, 2025 Read the published version in Statistics and Computing → Version 1 posted Editorial decision: Revision requested 22 Apr, 2025 Reviews received at journal 27 Jan, 2025 Reviewers agreed at journal 26 Jan, 2025 Reviewers agreed at journal 23 Jan, 2025 Reviewers agreed at journal 21 Jan, 2025 Reviewers invited by journal 20 Jan, 2025 Editor assigned by journal 05 Jun, 2024 Submission checks completed at journal 05 Jun, 2024 First submitted to journal 04 Jun, 2024 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-4528903","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":314236789,"identity":"6ea90d77-4959-486f-95c7-07144ba8a93c","order_by":0,"name":"Andreas Horst","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAj0lEQVRIiWNgGAWjYDACCSD+wAZipZGghXEGmwGJWph5SNLCP7v52Gebsj+JDexpCURacudY8uyccwaJDTzPDhBpzY0cY+bcNqAWifQG4nTIg7RYkqTFAKSFEawljUiHGd5IS2bsOWds3MbzLIE4LXI3kg8z/CiTk+1nTzMgTgscsJGofhSMglEwCkYBPgAAYLEoQ5lDveUAAAAASUVORK5CYII=","orcid":"","institution":"Technical university of Denmark","correspondingAuthor":true,"prefix":"","firstName":"Andreas","middleName":"","lastName":"Horst","suffix":""},{"id":314236790,"identity":"3cbea2c6-1b4a-4c4a-b09a-3c17ae2dae4e","order_by":1,"name":"Babak Maboudi Afkham","email":"","orcid":"","institution":"Technical university of Denmark","correspondingAuthor":false,"prefix":"","firstName":"Babak","middleName":"Maboudi","lastName":"Afkham","suffix":""},{"id":314236791,"identity":"43e36ef2-b1d5-4fed-93a2-96bf0ef34c40","order_by":2,"name":"Yiqiu Dong","email":"","orcid":"","institution":"Technical university of Denmark","correspondingAuthor":false,"prefix":"","firstName":"Yiqiu","middleName":"","lastName":"Dong","suffix":""},{"id":314236792,"identity":"2c43d9ad-bdf9-4014-bba1-24f5bafd2c41","order_by":3,"name":"Jakob Lemvig","email":"","orcid":"","institution":"Technical university of Denmark","correspondingAuthor":false,"prefix":"","firstName":"Jakob","middleName":"","lastName":"Lemvig","suffix":""}],"badges":[],"createdAt":"2024-06-04 14:48:27","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-4528903/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-4528903/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s11222-025-10638-2","type":"published","date":"2025-05-23T15:57:39+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":83460170,"identity":"9da7b677-6b0d-43c5-8614-b8d5d2ffc39c","added_by":"auto","created_at":"2025-05-26 16:11:30","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1315513,"visible":true,"origin":"","legend":"","description":"","filename":"paper.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4528903/v1_covered_63c9c860-61d5-4d7c-9151-2eeacca37d4b.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Uncertainty Quantification for Linear Inverse Problems with Besov Prior: A Randomize-Then-Optimize Method","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":"Bayesian inverse problems, Besov priors, sampling methods, randomize-then-optimize method","lastPublishedDoi":"10.21203/rs.3.rs-4528903/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-4528903/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"In this work, we investigate the use of Besov priors in the context of Bayesian inverse problems. The solution to Bayesian inverse problems is the posterior distribution which naturally enables us to interpret the uncertainties. Besov priors are discretization invariant and can promote sparsity in terms of wavelet coefficients. We propose the randomize-then-optimize method to draw samples from the posterior distribution with Besov priors under a general parameter setting and estimate the modes of the posterior distribution. The performance of the proposed method is studied through numerical experiments of a 1D inpainting problem, a 1D deconvolution problem, and a 2D computed tomography problem. Further, we discuss the influence of the choice of the Besov parameters and the wavelet basis in detail, and we compare the proposed method with the state-of-the-art methods. The numerical results suggest that the proposed method is an effective tool for sampling the posterior distribution equipped with general Besov priors.\n","manuscriptTitle":"Uncertainty Quantification for Linear Inverse Problems with Besov Prior: A Randomize-Then-Optimize Method","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-06-18 18:04:10","doi":"10.21203/rs.3.rs-4528903/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2025-04-22T04:54:24+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-01-27T20:17:04+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"136882219333892753642347464575276548460","date":"2025-01-26T08:32:32+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"225846329884055794264132356200306378996","date":"2025-01-23T21:10:56+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"314751059477438438006038389194512308808","date":"2025-01-21T06:55:45+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2025-01-21T01:54:47+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2024-06-06T01:02:15+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2024-06-05T07:56:51+00:00","index":"","fulltext":""},{"type":"submitted","content":"Statistics and Computing","date":"2024-06-04T14:47:05+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":"3931ac42-f1e4-4eb7-a0ae-7039b5386167","owner":[],"postedDate":"June 18th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[],"tags":[],"updatedAt":"2025-05-26T16:05:57+00:00","versionOfRecord":{"articleIdentity":"rs-4528903","link":"https://doi.org/10.1007/s11222-025-10638-2","journal":{"identity":"statistics-and-computing","isVorOnly":false,"title":"Statistics and Computing"},"publishedOn":"2025-05-23 15:57:39","publishedOnDateReadable":"May 23rd, 2025"},"versionCreatedAt":"2024-06-18 18:04:10","video":"","vorDoi":"10.1007/s11222-025-10638-2","vorDoiUrl":"https://doi.org/10.1007/s11222-025-10638-2","workflowStages":[]},"version":"v1","identity":"rs-4528903","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-4528903","identity":"rs-4528903","version":["v1"]},"buildId":"qtupq5eGEP_6zYnWcrvyt","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}