[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83695-en":3,"doc-seo-83695-105":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},83695,4810365810221,"Aurora","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","Direct Sampling Methods for Inverse Interface Problems","This work investigates two inverse interface problems in settings where only limited Cauchy data on the outer boundary is available. Both problems stem from a Laplace equation with a Robin-type flux jump across an internal interface. The first reconstructs crack locations along a known interface, while the second determines the position of an unknown interface. An efficient Direct Sampling Method (DSM) is developed, with refined boundary conditions and a noise-robust DSM refinement, validated by Fourier analysis, numerical experiments, and performance metrics like MLE and CNR.","arXiv :2607 .02913v1 [math .NA] 3 Jul 2026  \nDirect Sampling Methods for Inverse Interface Problems  \nKazufumi Ito∗ Tong Wu† Jun Zou‡  \nAbstract  \nThis work investigates two types of inverse interface problems in scenarios where only very limited Cauchy data is available. These problems are associated with the Laplace equation featuring a Robin-type flux jump across an internal interface. The first problem focuses on reconstructing the location of cracks along a known interface using Cauchy data measured on the outer boundary. The second problem involves determining the location of an unknown interface based on Cauchy data from the outer boundary. To address these challenges, we adopt an efficient Direct Sampling Method (DSM) and introduce innovative enhancements to the boundary conditions in the reference system, thereby maximizing the utility of the available Cauchy data. Additionally, we propose a novel refinement to further improve the robustness of the DSM against noise.  \nWe provide a detailed exposition of the general principles underlying the DSM and systematically present its computational implementation steps. Through detailed Fourier analysis and computations, we illustrate the theoretical background of the DSM as well as the effectiveness of our refinement approach. A series of numerical experiments demonstrates that our method yields highly satisfactory results, even when processing incomplete and noisy Cauchy data on the outer boundary. We introduce quantitative metrics, such as Mean Localization Error (MLE) and Contrast-to-Noise Ratio (CNR), to rigorously evaluate the performance of our method. These findings underscore the exceptional effectiveness and broad applicability of the proposed approach.  \n1 Introduction  \n1.1 Background of inverse interface problems and direct sampling methods  \nInterface problems are ubiquitous in various scientific and engineering disciplines. For instance, when a physical domain is composed of two different materials or media, an interface naturally emerges between them, leading to an interface problem. In the context of Partial Differential Equations (PDEs), numerical solutions to interface problems often require partitioning the computational domain into distinct regions or subdomains. These regions may exhibit different material properties or physical conditions and are typically separated by interfaces or boundaries. For example, in electrical conduction problems, resistivity varies across different materials, resulting in irregularities such as discontinuous parameters, jump conditions, and singular source terms in the governing differential equations atthe interfaces or boundaries. Such complexities pose significant challenges in solving interface problems.  \nA flux jump condition describes a discontinuous change in the flux (or flow) of a quantity across an interface. This phenomenon typically arises when there is a sharp transition or inhomogeneity in the physical properties atthe interface, and is particularly relevant in problems involving composite materials, phase boundaries, or fractured media.  \nIn this paper, we focus on inverse interface problems in elliptic equations. While addressing two physically distinct scenarios, we unify them mathematically as the problem of recovering the spatial support of a singular perturbation. The first part of our work addresses the identification of the flux jump location at a known interior interface with a flux boundary condition, where the interface represents a surface with cracks, defects, or damages. In the second part, we identify the location of an entirely unknown interface. Both problems are formulated and solved using limited Cauchy measurements on the outer boundary. These problems find applications in diverse fields, including electrical conductivity problems [25], multi-material compressible flow simulations [17], medical diffusion MRI imaging [22], and geological surface reconstruction [11] .  \n∗ Department of Mathema","cbCaiem8AnVwQcrp","https://ap.wps.com/l/cbCaiem8AnVwQcrp","pdf",3142236,2,1,35,"English","en",105,"# Introduction\n## Background of inverse interface problems and direct sampling methods\n## Direct Sampling Method (DSM) approach","[{\"question\":\"What inverse interface scenarios are studied in this work?\",\"answer\":\"The paper studies two scenarios: locating cracks along a known interior interface and determining the location of an unknown interface. Both are formulated for a Laplace equation with a Robin-type flux jump and use limited Cauchy data from the outer boundary.\"},{\"question\":\"How is the boundary data used to recover interface information?\",\"answer\":\"Reconstruction is performed using limited Cauchy measurements on the outer boundary. The method extracts probing information tied to the interface and uses it to infer the spatial support of the singular perturbation.\"},{\"question\":\"What improvements make the DSM more effective in the presence of noise?\",\"answer\":\"The work introduces innovative enhancements to boundary conditions in the reference system and proposes a new DSM refinement to improve robustness under noisy and incomplete Cauchy data. Performance is assessed using metrics such as Mean Localization Error (MLE) and Contrast-to-Noise Ratio (CNR).\"}]",1784189780,88,{"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},"direct-sampling-methods-for-inverse-interface-problems","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/direct-sampling-methods-for-inverse-interface-problems/83695/",4,{"url":51,"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","2026-07-23","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 interface scenarios are studied in this work?","Question",{"text":75,"@type":76},"The paper studies two scenarios: locating cracks along a known interior interface and determining the location of an unknown interface. Both are formulated for a Laplace equation with a Robin-type flux jump and use limited Cauchy data from the outer boundary.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is the boundary data used to recover interface information?",{"text":80,"@type":76},"Reconstruction is performed using limited Cauchy measurements on the outer boundary. The method extracts probing information tied to the interface and uses it to infer the spatial support of the singular perturbation.",{"name":82,"@type":73,"acceptedAnswer":83},"What improvements make the DSM more effective in the presence of noise?",{"text":84,"@type":76},"The work introduces innovative enhancements to boundary conditions in the reference system and proposes a new DSM refinement to improve robustness under noisy and incomplete Cauchy data. Performance is assessed using metrics such as Mean Localization Error (MLE) and Contrast-to-Noise Ratio (CNR).","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"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":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]