[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84205-en":3,"doc-seo-84205-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},84205,962075114765,"Quinn","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","The Stability of the Backward Problem for Photoacoustic Imaging in Attenuating Media via Carleman Estimates","This paper studies the time-backward inverse problem for photoacoustic tomography (PAT) in attenuating media. Photoacoustic imaging under attenuation is modeled by a spatial fractional-order damping operator, and the associated inverse problem is ill-posed in the Hadamard sense. The work develops a new family of Carleman estimates that is independent of spatial variables, yielding conditional stability estimates. It further proposes a Tikhonov-type regularization functional, derives its adjoint system, and proves convergence rates. Numerical experiments validate the theory.","arXiv :2607 .07244v1 [math .AP] 8 Jul 2026  \nThe Stability of the Backward Problem for Photoacoustic Imaging in Attenuating Media via Carleman Estimates  \nQihang Chen 1 , Zhiyuan Li∗2, and Song Xu3  \n1,2,3 School of Mathematics and Statistics, Ningbo University, Ningbo 315211, China  \nAbstract  \nThis paper investigates the backward problem in time for photoacoustic tomography (PAT) in attenuating media. It is well-established that photoacoustic imaging in attenuating media can be accurately modeled by spatial fractional-order damping. This inverse problem is ill-posed in the sense of Hadamard. In this work, we construct a novel class of Carleman estimates independent of spatial variables, and by virtue of these estimates, we establish conditional stability estimates for this problem for the first time. Building upon this, we propose a Tikhonov-type regularization functional and derive its associated adjoint system. Furthermore, leveraging the established conditional stability results, we derive the convergence rate of the proposed regularization approach. Finally, we validate the effectiveness of our theoretical findings through extensive numerical experiments.  \nKeywords—Photoacoustic tomography, wave equation, nonlocal damping, Carleman estimate, Tikhonov regularization  \nMSC2020: 35R30, 35L05  \n1 Introduction and main results  \nThe classical wave equation serves as the fundamental governing equation for photoacoustic tomography (PAT) in ideal non-attenuating media [1, 27] . In practical PAT applications, however, acoustic attenuation is ubiquitous and cannot be neglected in high-precision image reconstruction, which necessitates the replacement of the classical wave equation with damped wave models. Damped wave equations have been widely adopted to characterize attenuating wave propagation across diverse physics and engineering disciplines [12, 20 , 24] . Traditional PAT models typically assume a velocity-proportional damping mechanism, which introduces a conventional ∂t ∆u damping term. Nevertheless, practical acoustic attenuation exhibits prominent frequency-dependent characteristics that generally follow a power-law distribution, as extensively verified in existing PAT and wave propagation studies [3] .  \nTo accurately capture such power-law frequency-dependent damping behavior, various modified wave models have been developed in the literature. Among these modeling strategies, the fractional Laplacian-based wave equation, which employs fractional powers of the Laplacian or general second-order uniformly elliptic operators, has emerged as the most effective and widely used framework [32, 29] . Against this backdrop, this paper investigates the inverse problem for PAT under frequency-dependent attenuation, covering both classical integer-order and generalized fractional-order damped wave imaging models. The core of this PAT inverse problem is to recover the unknown initial pressure distribution of the imaging medium from finite measured wave field data. Specifically, when wave field measurements are acquired over the entire computational domain Ω at the fixed terminal time T [9], the forward wave propagation problem is converted to a time-reversed backward inverse problem, which constitutes the key research object of this work.  \n∗ Corresponding author 1: [lizhiyuan@nbu.edu.cn](lizhiyuan@nbu.edu.cn), supported by the National Natural Science Foundation of China (no. 12271277), Ningbo Youth Leading Talent Project (no. 2024QL045) .  \nBased on the above physical mechanism and modeling framework, we establish a generalized spatial damped wave equation to describe photoacoustic wave propagation in attenuating media. Let Ω ⊂ Rn denote a bounded open domain in the n-dimensional Euclidean space, and let T > 0 be a fixed terminal observation time. We define the space-time cylinder for the imaging system as ΩT := Ω × (0, T) . In this domain, we study a generalized multi-term damped wave equation with mixed integer-and fractional-or","cbCain01JRqAc6vP","https://ap.wps.com/l/cbCain01JRqAc6vP","pdf",1178373,4,1,24,"English","en",105,"# Abstract\n# Introduction and Main Results\n## Photoacoustic modeling in attenuating media\n## Fractional Laplacian damped wave equation formulation","[{\"question\":\"What inverse problem does the paper focus on in photoacoustic tomography?\",\"answer\":\"The paper focuses on the time-backward inverse problem: recovering the unknown initial pressure distribution from finite measured wave field data in an attenuating medium.\"},{\"question\":\"How is attenuation incorporated into the PAT model?\",\"answer\":\"Attenuation is modeled through spatial fractional-order damping, leading to a damped wave equation with mixed integer-and fractional-order terms that capture frequency-dependent (power-law) behavior.\"},{\"question\":\"What role do Carleman estimates and Tikhonov regularization play?\",\"answer\":\"Carleman estimates (independent of spatial variables) provide conditional stability for the ill-posed inverse problem. Based on these results, a Tikhonov-type regularization functional is constructed, its adjoint system is derived, and convergence rates for the regularization are established.\"}]",1784193943,60,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"the-stability-of-the-backward-problem-for-photoacoustic-imaging-in-attenuating-media-via-carleman-estimates","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":20},"https://docshare.wps.com/document/the-stability-of-the-backward-problem-for-photoacoustic-imaging-in-attenuating-media-via-carleman-estimates/84205/",{"url":52,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-28","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What inverse problem does the paper focus on in photoacoustic tomography?","Question",{"text":75,"@type":76},"The paper focuses on the time-backward inverse problem: recovering the unknown initial pressure distribution from finite measured wave field data in an attenuating medium.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is attenuation incorporated into the PAT model?",{"text":80,"@type":76},"Attenuation is modeled through spatial fractional-order damping, leading to a damped wave equation with mixed integer-and fractional-order terms that capture frequency-dependent (power-law) behavior.",{"name":82,"@type":73,"acceptedAnswer":83},"What role do Carleman estimates and Tikhonov regularization play?",{"text":84,"@type":76},"Carleman estimates (independent of spatial variables) provide conditional stability for the ill-posed inverse problem. Based on these results, a Tikhonov-type regularization functional is constructed, its adjoint system is derived, and convergence rates for the regularization are established.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,109,114,119,122,127,130,134],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":29,"slug":108},5,"Comic","comic",{"id":110,"doc_module":4,"doc_module_name":46,"category_name":111,"show_sort_weight":112,"slug":113},6,"Technology",50,"technology",{"id":115,"doc_module":4,"doc_module_name":46,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":46,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":46,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":46,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":46,"category_name":136,"show_sort_weight":106,"slug":137},19,"General","general"]