A persistent-homology-based Bayesian prior for potential coefficient reconstruction in an elliptic partial differential equation

A persistent-homology-based Bayesian prior for potential coefficient reconstruction in an elliptic partial differential equation | 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 persistent-homology-based Bayesian prior for potential coefficient reconstruction in an elliptic partial differential equation Zhiliang Deng, Haiyang Liu, Xiaofei Guan, Zhiyuan Wang, Xiaomei Yang This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7578095/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 9 You are reading this latest preprint version Abstract We address the reconstruction of a potential coefficient in an elliptic partial differential equation from distributed observations within the Bayesian framework. The choice of prior distribution is crucial in such inverse problems, particularly when the target function exhibits sharp discontinuities that conventional Gaussian priors fail to capture effectively. To overcome this limitation, we introduce a novel prior based on persistent homology (PH), which quantifies and encodes the topological features of candidate functions through their persistent pairs. To ensure a well-defined distribution in infinite-dimensional spaces, the prior is constructed with respect to a Gaussian reference measure. A significant advantage over classical approaches is that the PH prior only requires the unknown functions to belong to a suitable topological space, which substantially enhances its applicability. Numerical results demonstrate that the proposed PH prior outperforms the Gaussian prior and achieves a modest yet consistent improvement over the classical total variation (TV) prior. Inverse potential problems topological prior Bayesian approach persistent homology TV prior Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Editorial decision: Revision requested 28 Nov, 2025 Reviews received at journal 28 Nov, 2025 Reviews received at journal 19 Nov, 2025 Reviewers agreed at journal 26 Sep, 2025 Reviewers agreed at journal 18 Sep, 2025 Reviewers invited by journal 18 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. 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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-7578095","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":521215389,"identity":"d909e3ac-4302-4969-a463-340806cecb20","order_by":0,"name":"Zhiliang 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