[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84537-en":3,"doc-seo-84537-105":29,"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":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},84537,34359740700684,"Finn","https://ap-avatar.wpscdn.com/avatar/1f400023980c374ae676?_k=1777273430885731487",8,"Research & Report","Identifying defective units in infinite periodic arrays of point sources","This paper addresses identification of defective units in unbounded periodic arrays of point sources using boundary measurements. Defects break the array’s periodicity, complicating inversion compared with classical free-space inverse source problems. The study reformulates the problem via the Floquet–Bloch transform into a quasi-periodic inverse source setting. Uniqueness theorems are proved for both formulations, and a new numerical method is proposed to infer the number, locations, and intensities of defective sources from single-frequency data, validated by numerical experiments.","arXiv :2607 .00241v1 [math .NA] 30 Jun 2026  \nIdentifying defective units in infinite periodic arrays of  \npoint sources  \nDinh-Liem Nguyen∗ Nhung H. Nguyen† Thi-Phong Nguyen‡  \nAbstract  \nThis paper focuses on identifying defective units in unbounded periodic arrays of point sources using boundary data. The study is motivated by the noninvasive evaluation of large-scale periodic source systems. Unlike classical inverse source problems in free space, the key challenge here lies in the disruption of periodicity caused by defective sources in the infinite array. To address this, we employ the Floquet–Bloch transform to reformulate the original inverse source problem as aquasi-periodic inverse source problem. We first establish uniqueness theorems for both the original and the quasi-periodic formulations. Then, we develop a new numerical method for identifying defective sources. This method combines a sampling indicator function with an algebraic technique to determine not only the number of defective sources, but also their locations and intensities. Numerical experiments are presented to validate the effectiveness of the proposed method.  \nKeywords: inverse source problems, defective sources identification, unique determination, periodic arrays, Floquet-Bloch transform, imaging function  \n1 Introduction  \nWe consider a one-dimensional array of point sources that is unbounded, periodic along the x 1-direction with period L > 0, and bounded in the x2-direction between −h and h, for some h > 0. Each unit cell of length L contains one point source. We denote by  \nΩ := R × (−h, h), Ω0 := 􀀒 − L2 , L2􀀓 × (−h, h),  \nand let x0 be the location of the source in Ω0 . The locations of all other sources in the array are then given by xj := x0 + (jL,0)⊤ for j ∈ Z. These sources are represented by the delta distribution δ(x;x0 + (jL,0)⊤ ) for x ∈ Ω and have corresponding intensities γj .  \n∗ Department of Mathematics, Kansas State University, Manhattan, KS 66506 ([dlnguyen@ksu.edu](dlnguyen@ksu.edu))  \n†Department of Mathematics, Kansas State University, Manhattan, KS 66506 ([nhungnh@ksu.edu](nhungnh@ksu.edu))  \n‡Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102 ([thiphong.nguyen@njit.edu](thiphong.nguyen@njit.edu))  \nIn this work, we assume that some sources are not functioning properly, in the sense that their intensities differ from 1 . In particular, there are N ∈ N∗ defective sources located at xm ℓ, each having intensity γm ℓ  1, for ℓ = 1 ,..., N. We denote by I the set of indices of all defective sources, i.e. , I := {m : γm  1} and assume that I is a finite set.  \nThe wave generated by all point sources in a homogeneous background with a wave number k > 0 satisfies the Helmholtz equation  \n∆u + k2 u = − mI δ (·, x0 + (mL,0)⊤ ) +  γmδ(·, x0 + (mL,0)⊤ ) ,  \nwhich is equivalent to  \n∆u + k2 u = − δper(·, x0 ) +  σm δ (·, xm)! , (1)  \nwhere  \nσm := γm − 1, δper(·, x0 ) :=X δ(·, x0 + (mL,0)⊤ ), in Ω .  \nm∈Z  \nWe denote by Gper(·, x0 ) the outgoing periodic Green’s function with period L associated with  \nthe point source at x0 ; that is, Gper(·, x0 ) admits the spectral representation Gper(x, x0 ) =  2iLθn ei ~~2π~~L~~n~~ 􀀐x1−x(1)0􀀑 +iθn 􀀌x2−x(2)0 􀀌 , x, x0 ∈ Ω0 , x  x0 ,  \nwhere θn = qk 2 − (2πn/L)2 , θn  0, with Im(θn) ≥ 0. Then we impose the outgoing behavior of u by requiring that u − Gper(·, x0 ) satisfies the Sommerfeld radiation condition. Then the outgoing wave solution v to (1) is given by  \nu (x) = Gper(x, x0 ) +X σmi4H(1)0(k|x − xm|), (2)  \nm∈I  \nwhere H(1)0 is a Hankel function of the first kind and zero order.  \nThe objective of this work is to determine the set of indices of all defective sources I and the corresponding intensities γm for all m ∈ I using measurements of v on horizontal lines Γ±t := R × {±t} for some t > h. We consider the following inverse problem below.  \nInverse problem: Determine the indices m ∈ I corresponding to the defective sources located at xm = x0 +(mL,0)⊤ an","cbCaifbvVFx6UdVc","https://ap.wps.com/l/cbCaifbvVFx6UdVc","pdf",823007,1,18,"English","en",105,"# Abstract\n# Introduction\n## Problem setting and inverse formulation\n## Motivation and related work\n# Main method and theory","[{\"question\":\"What makes defective-source identification hard in an infinite periodic array?\",\"answer\":\"Defective units disrupt the perfect periodicity of the infinite array, so the inverse problem differs from classical inverse source problems in free space or bounded domains.\"},{\"question\":\"How is the original inverse problem reformulated to handle the infinite periodic geometry?\",\"answer\":\"The paper applies the Floquet–Bloch transform to convert the inverse problem on an unbounded periodic array into a quasi-periodic inverse source problem defined on a single unit cell.\"},{\"question\":\"What does the proposed numerical method determine about the defects?\",\"answer\":\"It identifies how many defective sources exist and estimates their locations and intensities using a sampling indicator function combined with an algebraic technique.\"}]",1784196483,45,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"identifying-defective-units-in-infinite-periodic-arrays-of-point-sources","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/identifying-defective-units-in-infinite-periodic-arrays-of-point-sources/84537/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","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 makes defective-source identification hard in an infinite periodic array?","Question",{"text":75,"@type":76},"Defective units disrupt the perfect periodicity of the infinite array, so the inverse problem differs from classical inverse source problems in free space or bounded domains.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is the original inverse problem reformulated to handle the infinite periodic geometry?",{"text":80,"@type":76},"The paper applies the Floquet–Bloch transform to convert the inverse problem on an unbounded periodic array into a quasi-periodic inverse source problem defined on a single unit cell.",{"name":82,"@type":73,"acceptedAnswer":83},"What does the proposed numerical method determine about the defects?",{"text":84,"@type":76},"It identifies how many defective sources exist and estimates their locations and intensities using a sampling indicator function combined with an algebraic 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