{"id":"https://openalex.org/W2963053625","doi":"https://doi.org/10.1137/18m1196820","title":"Uniqueness in Inverse Scattering Problems with Phaseless Far-Field Data at a Fixed Frequency. II","display_name":"Uniqueness in Inverse Scattering Problems with Phaseless Far-Field Data at a Fixed Frequency. II","publication_year":2018,"publication_date":"2018-01-01","ids":{"openalex":"https://openalex.org/W2963053625","doi":"https://doi.org/10.1137/18m1196820","mag":"2963053625"},"language":"en","primary_location":{"id":"doi:10.1137/18m1196820","is_oa":false,"landing_page_url":"https://doi.org/10.1137/18m1196820","pdf_url":null,"source":{"id":"https://openalex.org/S77200987","display_name":"SIAM Journal on Applied Mathematics","issn_l":"0036-1399","issn":["0036-1399","1095-712X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Applied Mathematics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5067196344","display_name":"Xiaoxu Xu","orcid":null},"institutions":[{"id":"https://openalex.org/I19820366","display_name":"Chinese Academy of Sciences","ror":"https://ror.org/034t30j35","country_code":"CN","type":"government","lineage":["https://openalex.org/I19820366"]},{"id":"https://openalex.org/I4210120485","display_name":"Academy of Mathematics and Systems Science","ror":"https://ror.org/02jkmyk67","country_code":"CN","type":"facility","lineage":["https://openalex.org/I19820366","https://openalex.org/I4210120485"]},{"id":"https://openalex.org/I4210165038","display_name":"University of Chinese Academy of Sciences","ror":"https://ror.org/05qbk4x57","country_code":"CN","type":"education","lineage":["https://openalex.org/I19820366","https://openalex.org/I4210165038"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Xiaoxu Xu","raw_affiliation_strings":["Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China and School of Mathematical Sciences,","University of Chinese Academy of Sciences, Beijing 100049, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China and School of Mathematical Sciences,","institution_ids":["https://openalex.org/I4210120485","https://openalex.org/I19820366"]},{"raw_affiliation_string":"University of Chinese Academy of Sciences, Beijing 100049, China","institution_ids":["https://openalex.org/I4210165038"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5100335192","display_name":"Bo Zhang","orcid":"https://orcid.org/0000-0001-7355-0941"},"institutions":[{"id":"https://openalex.org/I19820366","display_name":"Chinese Academy of Sciences","ror":"https://ror.org/034t30j35","country_code":"CN","type":"government","lineage":["https://openalex.org/I19820366"]},{"id":"https://openalex.org/I4210120485","display_name":"Academy of Mathematics and Systems Science","ror":"https://ror.org/02jkmyk67","country_code":"CN","type":"facility","lineage":["https://openalex.org/I19820366","https://openalex.org/I4210120485"]},{"id":"https://openalex.org/I4210165038","display_name":"University of Chinese Academy of Sciences","ror":"https://ror.org/05qbk4x57","country_code":"CN","type":"education","lineage":["https://openalex.org/I19820366","https://openalex.org/I4210165038"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Bo Zhang","raw_affiliation_strings":["University of Chinese Academy of Sciences, Beijing 100049, China","LSEC and Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China and School of Mathematical Sciences,"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Chinese Academy of Sciences, Beijing 100049, China","institution_ids":["https://openalex.org/I4210165038"]},{"raw_affiliation_string":"LSEC and Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China and School of Mathematical Sciences,","institution_ids":["https://openalex.org/I4210120485","https://openalex.org/I19820366"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5047634561","display_name":"Haiwen Zhang","orcid":"https://orcid.org/0000-0003-4035-4754"},"institutions":[{"id":"https://openalex.org/I19820366","display_name":"Chinese Academy of Sciences","ror":"https://ror.org/034t30j35","country_code":"CN","type":"government","lineage":["https://openalex.org/I19820366"]},{"id":"https://openalex.org/I4210120485","display_name":"Academy of Mathematics and Systems Science","ror":"https://ror.org/02jkmyk67","country_code":"CN","type":"facility","lineage":["https://openalex.org/I19820366","https://openalex.org/I4210120485"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Haiwen Zhang","raw_affiliation_strings":["Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China"],"raw_orcid":"https://orcid.org/0000-0003-4035-4754","affiliations":[{"raw_affiliation_string":"Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China","institution_ids":["https://openalex.org/I4210120485","https://openalex.org/I19820366"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":9.8453,"has_fulltext":false,"cited_by_count":36,"citation_normalized_percentile":{"value":0.99198415,"is_in_top_1_percent":true,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"78","issue":"6","first_page":"3024","last_page":"3039"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11205","display_name":"Numerical methods in inverse problems","score":1.0,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11205","display_name":"Numerical methods in inverse problems","score":1.0,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11739","display_name":"Microwave Imaging and Scattering Analysis","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2204","display_name":"Biomedical Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10662","display_name":"Ultrasonics and Acoustic Wave Propagation","score":0.9987999796867371,"subfield":{"id":"https://openalex.org/subfields/2211","display_name":"Mechanics of Materials"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/uniqueness","display_name":"Uniqueness","score":0.7448184490203857},{"id":"https://openalex.org/keywords/inverse-scattering-problem","display_name":"Inverse scattering problem","score":0.6263303756713867},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6091391444206238},{"id":"https://openalex.org/keywords/scattering","display_name":"Scattering","score":0.5898139476776123},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.5880041122436523},{"id":"https://openalex.org/keywords/a-priori-and-a-posteriori","display_name":"A priori and a posteriori","score":0.5332934856414795},{"id":"https://openalex.org/keywords/obstacle","display_name":"Obstacle","score":0.4967091679573059},{"id":"https://openalex.org/keywords/plane-wave","display_name":"Plane wave","score":0.4853455126285553},{"id":"https://openalex.org/keywords/refraction","display_name":"Refraction","score":0.4624080955982208},{"id":"https://openalex.org/keywords/field","display_name":"Field (mathematics)","score":0.454324871301651},{"id":"https://openalex.org/keywords/plane","display_name":"Plane (geometry)","score":0.4383096992969513},{"id":"https://openalex.org/keywords/inverse-problem","display_name":"Inverse problem","score":0.39135098457336426},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.3157328963279724},{"id":"https://openalex.org/keywords/optics","display_name":"Optics","score":0.2659122943878174},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.2069287896156311},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.1461879312992096}],"concepts":[{"id":"https://openalex.org/C2777021972","wikidata":"https://www.wikidata.org/wiki/Q22976830","display_name":"Uniqueness","level":2,"score":0.7448184490203857},{"id":"https://openalex.org/C174558057","wikidata":"https://www.wikidata.org/wiki/Q11639409","display_name":"Inverse scattering problem","level":3,"score":0.6263303756713867},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6091391444206238},{"id":"https://openalex.org/C191486275","wikidata":"https://www.wikidata.org/wiki/Q210028","display_name":"Scattering","level":2,"score":0.5898139476776123},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5880041122436523},{"id":"https://openalex.org/C75553542","wikidata":"https://www.wikidata.org/wiki/Q178161","display_name":"A priori and a posteriori","level":2,"score":0.5332934856414795},{"id":"https://openalex.org/C2776650193","wikidata":"https://www.wikidata.org/wiki/Q264661","display_name":"Obstacle","level":2,"score":0.4967091679573059},{"id":"https://openalex.org/C40308292","wikidata":"https://www.wikidata.org/wiki/Q122518","display_name":"Plane wave","level":2,"score":0.4853455126285553},{"id":"https://openalex.org/C205318122","wikidata":"https://www.wikidata.org/wiki/Q72277","display_name":"Refraction","level":2,"score":0.4624080955982208},{"id":"https://openalex.org/C9652623","wikidata":"https://www.wikidata.org/wiki/Q190109","display_name":"Field (mathematics)","level":2,"score":0.454324871301651},{"id":"https://openalex.org/C17825722","wikidata":"https://www.wikidata.org/wiki/Q17285","display_name":"Plane (geometry)","level":2,"score":0.4383096992969513},{"id":"https://openalex.org/C135252773","wikidata":"https://www.wikidata.org/wiki/Q1567213","display_name":"Inverse problem","level":2,"score":0.39135098457336426},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.3157328963279724},{"id":"https://openalex.org/C120665830","wikidata":"https://www.wikidata.org/wiki/Q14620","display_name":"Optics","level":1,"score":0.2659122943878174},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.2069287896156311},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.1461879312992096},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"score":0.0},{"id":"https://openalex.org/C111472728","wikidata":"https://www.wikidata.org/wiki/Q9471","display_name":"Epistemology","level":1,"score":0.0},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/18m1196820","is_oa":false,"landing_page_url":"https://doi.org/10.1137/18m1196820","pdf_url":null,"source":{"id":"https://openalex.org/S77200987","display_name":"SIAM Journal on Applied Mathematics","issn_l":"0036-1399","issn":["0036-1399","1095-712X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Applied Mathematics","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G2772607984","display_name":"\u969c\u788d\u4e0e\u4ecb\u8d28\u6df7\u5408\u53cd\u6563\u5c04\u95ee\u9898\u7684\u6570\u5b66\u7406\u8bba\u548c\u6570\u503c\u7b97\u6cd5","funder_award_id":"11571355","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G4969315547","display_name":null,"funder_award_id":"91630309","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G503881559","display_name":null,"funder_award_id":"11501558","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320336876","display_name":"National Center for Mathematics and Interdisciplinary Sciences, Chinese Academy of Sciences","ror":"https://ror.org/00s97k668"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":39,"referenced_works":["https://openalex.org/W1964510609","https://openalex.org/W1971231406","https://openalex.org/W1983223664","https://openalex.org/W2003659905","https://openalex.org/W2009766219","https://openalex.org/W2021457366","https://openalex.org/W2028817727","https://openalex.org/W2030764277","https://openalex.org/W2031445862","https://openalex.org/W2039771789","https://openalex.org/W2059427263","https://openalex.org/W2082531160","https://openalex.org/W2089332382","https://openalex.org/W2089337236","https://openalex.org/W2132468176","https://openalex.org/W2332783578","https://openalex.org/W2416067360","https://openalex.org/W2500401920","https://openalex.org/W2509331849","https://openalex.org/W2512673524","https://openalex.org/W2515007495","https://openalex.org/W2535155561","https://openalex.org/W2538725373","https://openalex.org/W2594185299","https://openalex.org/W2597150118","https://openalex.org/W2607200862","https://openalex.org/W2741129808","https://openalex.org/W2758271156","https://openalex.org/W2803261465","https://openalex.org/W2963366171","https://openalex.org/W2963829305","https://openalex.org/W2964056273","https://openalex.org/W2964185390","https://openalex.org/W3098670860","https://openalex.org/W3098877181","https://openalex.org/W3098920388","https://openalex.org/W3101123351","https://openalex.org/W3104701575","https://openalex.org/W3105593508"],"related_works":["https://openalex.org/W2035579502","https://openalex.org/W4213260229","https://openalex.org/W2520812953","https://openalex.org/W2128608246","https://openalex.org/W3101167575","https://openalex.org/W2545426741","https://openalex.org/W1567523647","https://openalex.org/W2144075718","https://openalex.org/W4321641047","https://openalex.org/W2755915683"],"abstract_inverted_index":{"This":[0],"paper":[1],"is":[2,91,111,182],"concerned":[3],"with":[4,10,76,162,210],"uniqueness":[5,187],"in":[6,121,188],"inverse":[7],"acoustic":[8],"scattering":[9,159,176],"phaseless":[11,62,196],"far-field":[12,36,63,197],"data":[13],"at":[14,79,217],"a":[15,80,85,95,137,154,163,190,218],"fixed":[16,81,219],"frequency.":[17,220],"In":[18,131],"our":[19,179],"previous":[20],"work":[21],"[SIAM":[22],"J.":[23],"Appl.":[24],"Math.,":[25],"78":[26],"(2018),":[27],"pp.":[28],"1737--1753],":[29],"by":[30,60,66,152],"utilizing":[31],"spectral":[32],"properties":[33],"of":[34,50,52,70,72,104,107,124,146,148,166,204,206],"the":[35,41,45,48,61,84,89,102,108,122,136,141,144,149,158,175,195,214],"operator":[37],"we":[38,134],"proved":[39],"for":[40,126,174],"first":[42],"time":[43],"that":[44,88,115],"obstacle":[46,90,100,142],"and":[47,101,113,143,170],"index":[49,103,145],"refraction":[51,105,147],"an":[53],"inhomogeneous":[54,109,150],"medium":[55,110,151],"can":[56],"be":[57,94],"uniquely":[58],"determined":[59],"patterns":[64,198],"generated":[65],"infinitely":[67,201],"many":[68,202],"sets":[69,203],"superpositions":[71,205],"two":[73,207],"plane":[74,208],"waves":[75,209],"different":[77,211],"directions":[78,212],"frequency":[82],"under":[83],"priori":[86,138],"assumption":[87,139],"known":[92],"to":[93,157,185,200],"sound-soft":[96],"or":[97,119],"nonabsorbing":[98],"impedance":[99],"$n$":[106],"real-valued":[112],"satisfies":[114],"either":[116],"$n-1\\ge":[117],"c_1$":[118],"$n-1\\le-c_1$":[120],"support":[123],"$n-1$":[125],"some":[127],"positive":[128],"constant":[129],"$c_1$.":[130],"this":[132],"paper,":[133],"remove":[135],"on":[140],"adding":[153],"reference":[155],"ball":[156],"system":[160],"together":[161],"simpler":[164],"method":[165,181],"using":[167],"Rellich's":[168],"lemma":[169],"Green's":[171],"representation":[172],"formula":[173],"solutions.":[177],"Further,":[178],"new":[180],"also":[183],"used":[184],"prove":[186],"determining":[189],"locally":[191],"rough":[192],"surface":[193],"from":[194],"corresponding":[199],"as":[213],"incident":[215],"fields":[216]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":4},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":5},{"year":2020,"cited_by_count":8},{"year":2019,"cited_by_count":13},{"year":2018,"cited_by_count":1}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}