Efficient iterative solutions to complex-valued nonlinear least-squares problems with mixed linear and antilinear operators

We consider a setting in which it is desired to find an optimal complex vector x ∈ C N that satisfies A ( x ) ≈ b in a least-squares sense, where b ∈ C M is a data vector (possibly noise-corrupted), and A ( · ) : C N → C M is a measurement operator. If A ( · ) were linear, this reduces to the classi...

Full description

Saved in:

Bibliographic Details
Published in:	Optimization and engineering Vol. 23; no. 2; pp. 749 - 768
Main Authors:	Kim, Tae Hyung, Haldar, Justin P.
Format:	Journal Article
Language:	English
Published:	New York Springer US 01.06.2022 Springer Nature B.V
Subjects:	Algorithms Complexity Conjugation Control Engineering Environmental Management Exact solutions Financial Engineering Inverse problems Iterative methods Iterative solution Least squares method Mathematics Mathematics and Statistics Operations Research/Decision Theory Operators Optimization Research Article Systems Theory Linear and antilinear operators Inverse problems Efficient numerical computations 65K10 47J05 65H10 47J25 47N10 Iterative least-squares algorithms 15A29
ISSN:	1389-4420, 1573-2924, 1573-2924
Online Access:	Get full text
Tags:	Add Tag No Tags, Be the first to tag this record!

Abstract	We consider a setting in which it is desired to find an optimal complex vector x ∈ C N that satisfies A ( x ) ≈ b in a least-squares sense, where b ∈ C M is a data vector (possibly noise-corrupted), and A ( · ) : C N → C M is a measurement operator. If A ( · ) were linear, this reduces to the classical linear least-squares problem, which has a well-known analytic solution as well as powerful iterative solution algorithms. However, instead of linear least-squares, this work considers the more complicated scenario where A ( · ) is nonlinear, but can be represented as the summation and/or composition of some operators that are linear and some operators that are antilinear. Some common nonlinear operations that have this structure include complex conjugation or taking the real-part or imaginary-part of a complex vector. Previous literature has shown that this kind of mixed linear/antilinear least-squares problem can be mapped into a linear least-squares problem by considering x as a vector in R 2 N instead of C N . While this approach is valid, the replacement of the original complex-valued optimization problem with a real-valued optimization problem can be complicated to implement, and can also be associated with increased computational complexity. In this work, we describe theory and computational methods that enable mixed linear/antilinear least-squares problems to be solved iteratively using standard linear least-squares tools, while retaining all of the complex-valued structure of the original inverse problem. An illustration is provided to demonstrate that this approach can simplify the implementation and reduce the computational complexity of iterative solution algorithms.
AbstractList	We consider a setting in which it is desired to find an optimal complex vector x ∈ CN that satisfies A(x) ≈ b in a least-squares sense, where b ∈ CM is a data vector (possibly noise-corrupted), and A(·) : CN → CM is a measurement operator. If A(·) were linear, this reduces to the classical linear least-squares problem, which has a well-known analytic solution as well as powerful iterative solution algorithms. However, instead of linear least-squares, this work considers the more complicated scenario where A(·) is nonlinear, but can be represented as the summation and/or composition of some operators that are linear and some operators that are antilinear. Some common nonlinear operations that have this structure include complex conjugation or taking the real-part or imaginary-part of a complex vector. Previous literature has shown that this kind of mixed linear/antilinear least-squares problem can be mapped into a linear least-squares problem by considering x as a vector in R2N instead of CN. While this approach is valid, the replacement of the original complex-valued optimization problem with a real-valued optimization problem can be complicated to implement, and can also be associated with increased computational complexity. In this work, we describe theory and computational methods that enable mixed linear/antilinear least-squares problems to be solved iteratively using standard linear least-squares tools, while retaining all of the complex-valued structure of the original inverse problem. An illustration is providedtodemonstratethatthisapproachcansimplifytheimplementationandreduce the computational complexity of iterative solution algorithms. We consider a setting in which it is desired to find an optimal complex vector x∈CN that satisfies A(x)≈b in a least-squares sense, where b∈CM is a data vector (possibly noise-corrupted), and A(·):CN→CM is a measurement operator. If A(·) were linear, this reduces to the classical linear least-squares problem, which has a well-known analytic solution as well as powerful iterative solution algorithms. However, instead of linear least-squares, this work considers the more complicated scenario where A(·) is nonlinear, but can be represented as the summation and/or composition of some operators that are linear and some operators that are antilinear. Some common nonlinear operations that have this structure include complex conjugation or taking the real-part or imaginary-part of a complex vector. Previous literature has shown that this kind of mixed linear/antilinear least-squares problem can be mapped into a linear least-squares problem by considering x as a vector in R2N instead of CN. While this approach is valid, the replacement of the original complex-valued optimization problem with a real-valued optimization problem can be complicated to implement, and can also be associated with increased computational complexity. In this work, we describe theory and computational methods that enable mixed linear/antilinear least-squares problems to be solved iteratively using standard linear least-squares tools, while retaining all of the complex-valued structure of the original inverse problem. An illustration is provided to demonstrate that this approach can simplify the implementation and reduce the computational complexity of iterative solution algorithms. We consider a setting in which it is desired to find an optimal complex vector x ∈ C N that satisfies A (x) ≈ b in a least-squares sense, where b ∈ C M is a data vector (possibly noise-corrupted), and A (·) : C N → C M is a measurement operator. If A (·) were linear, this reduces to the classical linear least-squares problem, which has a well-known analytic solution as well as powerful iterative solution algorithms. However, instead of linear least-squares, this work considers the more complicated scenario where A (·) is nonlinear, but can be represented as the summation and/or composition of some operators that are linear and some operators that are antilinear. Some common nonlinear operations that have this structure include complex conjugation or taking the real-part or imaginary-part of a complex vector. Previous literature has shown that this kind of mixed linear/antilinear least-squares problem can be mapped into a linear least-squares problem by considering x as a vector in R 2N instead of C N . While this approach is valid, the replacement of the original complex-valued optimization problem with a real-valued optimization problem can be complicated to implement, and can also be associated with increased computational complexity. In this work, we describe theory and computational methods that enable mixed linear/antilinear least-squares problems to be solved iteratively using standard linear least-squares tools, while retaining all of the complex-valued structure of the original inverse problem. An illustration is providedtodemonstratethatthisapproachcansimplifytheimplementationandreduce the computational complexity of iterative solution algorithms.We consider a setting in which it is desired to find an optimal complex vector x ∈ C N that satisfies A (x) ≈ b in a least-squares sense, where b ∈ C M is a data vector (possibly noise-corrupted), and A (·) : C N → C M is a measurement operator. If A (·) were linear, this reduces to the classical linear least-squares problem, which has a well-known analytic solution as well as powerful iterative solution algorithms. However, instead of linear least-squares, this work considers the more complicated scenario where A (·) is nonlinear, but can be represented as the summation and/or composition of some operators that are linear and some operators that are antilinear. Some common nonlinear operations that have this structure include complex conjugation or taking the real-part or imaginary-part of a complex vector. Previous literature has shown that this kind of mixed linear/antilinear least-squares problem can be mapped into a linear least-squares problem by considering x as a vector in R 2N instead of C N . While this approach is valid, the replacement of the original complex-valued optimization problem with a real-valued optimization problem can be complicated to implement, and can also be associated with increased computational complexity. In this work, we describe theory and computational methods that enable mixed linear/antilinear least-squares problems to be solved iteratively using standard linear least-squares tools, while retaining all of the complex-valued structure of the original inverse problem. An illustration is providedtodemonstratethatthisapproachcansimplifytheimplementationandreduce the computational complexity of iterative solution algorithms. We consider a setting in which it is desired to find an optimal complex vector $${\mathbf {x}}\in {\mathbb {C}}^N$$ x ∈ C N that satisfies $${\mathcal {A}}({\mathbf {x}}) \approx {\mathbf {b}}$$ A ( x ) ≈ b in a least-squares sense, where $${\mathbf {b}} \in {\mathbb {C}}^M$$ b ∈ C M is a data vector (possibly noise-corrupted), and $${\mathcal {A}}(\cdot ): {\mathbb {C}}^N \rightarrow {\mathbb {C}}^M$$ A ( · ) : C N → C M is a measurement operator. If $${\mathcal {A}}(\cdot )$$ A ( · ) were linear, this reduces to the classical linear least-squares problem, which has a well-known analytic solution as well as powerful iterative solution algorithms. However, instead of linear least-squares, this work considers the more complicated scenario where $${\mathcal {A}}(\cdot )$$ A ( · ) is nonlinear, but can be represented as the summation and/or composition of some operators that are linear and some operators that are antilinear. Some common nonlinear operations that have this structure include complex conjugation or taking the real-part or imaginary-part of a complex vector. Previous literature has shown that this kind of mixed linear/antilinear least-squares problem can be mapped into a linear least-squares problem by considering $${\mathbf {x}}$$ x as a vector in $${\mathbb {R}}^{2N}$$ R 2 N instead of $${\mathbb {C}}^N$$ C N . While this approach is valid, the replacement of the original complex-valued optimization problem with a real-valued optimization problem can be complicated to implement, and can also be associated with increased computational complexity. In this work, we describe theory and computational methods that enable mixed linear/antilinear least-squares problems to be solved iteratively using standard linear least-squares tools, while retaining all of the complex-valued structure of the original inverse problem. An illustration is provided to demonstrate that this approach can simplify the implementation and reduce the computational complexity of iterative solution algorithms. We consider a setting in which it is desired to find an optimal complex vector x ∈ C N that satisfies A ( x ) ≈ b in a least-squares sense, where b ∈ C M is a data vector (possibly noise-corrupted), and A ( · ) : C N → C M is a measurement operator. If A ( · ) were linear, this reduces to the classical linear least-squares problem, which has a well-known analytic solution as well as powerful iterative solution algorithms. However, instead of linear least-squares, this work considers the more complicated scenario where A ( · ) is nonlinear, but can be represented as the summation and/or composition of some operators that are linear and some operators that are antilinear. Some common nonlinear operations that have this structure include complex conjugation or taking the real-part or imaginary-part of a complex vector. Previous literature has shown that this kind of mixed linear/antilinear least-squares problem can be mapped into a linear least-squares problem by considering x as a vector in R 2 N instead of C N . While this approach is valid, the replacement of the original complex-valued optimization problem with a real-valued optimization problem can be complicated to implement, and can also be associated with increased computational complexity. In this work, we describe theory and computational methods that enable mixed linear/antilinear least-squares problems to be solved iteratively using standard linear least-squares tools, while retaining all of the complex-valued structure of the original inverse problem. An illustration is provided to demonstrate that this approach can simplify the implementation and reduce the computational complexity of iterative solution algorithms. We consider a setting in which it is desired to find an optimal complex vector ∈ that satisfies ( ) ≈ in a least-squares sense, where ∈ is a data vector (possibly noise-corrupted), and (·) : → is a measurement operator. If (·) were linear, this reduces to the classical linear least-squares problem, which has a well-known analytic solution as well as powerful iterative solution algorithms. However, instead of linear least-squares, this work considers the more complicated scenario where (·) is nonlinear, but can be represented as the summation and/or composition of some operators that are linear and some operators that are antilinear. Some common nonlinear operations that have this structure include complex conjugation or taking the real-part or imaginary-part of a complex vector. Previous literature has shown that this kind of mixed linear/antilinear least-squares problem can be mapped into a linear least-squares problem by considering as a vector in instead of . While this approach is valid, the replacement of the original complex-valued optimization problem with a real-valued optimization problem can be complicated to implement, and can also be associated with increased computational complexity. In this work, we describe theory and computational methods that enable mixed linear/antilinear least-squares problems to be solved iteratively using standard linear least-squares tools, while retaining all of the complex-valued structure of the original inverse problem. An illustration is providedtodemonstratethatthisapproachcansimplifytheimplementationandreduce the computational complexity of iterative solution algorithms.
Author	Haldar, Justin P. Kim, Tae Hyung
AuthorAffiliation	1 Department of Electrical and Computer Engineering, University of Southern California, Los Angeles, CA 90089, USA
AuthorAffiliation_xml	– name: 1 Department of Electrical and Computer Engineering, University of Southern California, Los Angeles, CA 90089, USA
Author_xml	– sequence: 1 givenname: Tae Hyung surname: Kim fullname: Kim, Tae Hyung email: taehyung@usc.edu organization: Department of Electrical and Computer Engineering, University of Southern California – sequence: 2 givenname: Justin P. surname: Haldar fullname: Haldar, Justin P. organization: Department of Electrical and Computer Engineering, University of Southern California
BackLink	https://www.ncbi.nlm.nih.gov/pubmed/35656362$$D View this record in MEDLINE/PubMed
BookMark	eNp9kktvFDEMxyNURB_wBTigkbhwGcg7kwsSqkqLVIkLnKNsxtOmyiTbJLMt4suTZbc8eujBipP8bP8t-xgdxBQBodcEvycYqw-FEDyQHtNmWmLe82foiAjFeqopP2g-G3TPOcWH6LiUG4yJFHR4gQ6ZkEIySY_Qz7Np8s5DrJ2vkG31G-hKCkv1KZaups6leR3gvt_YsMDYNQ3BR7C5C2BL7cvtYjOUbp3TKsBcujtfr7vZ3zd2D9o4Nqt-f03rbZ2Uy0v0fLKhwKv9eYK-fz77dnrRX349_3L66bJ3XPHaW-GAMd0aA25Vcwc28oG49qymYeKjpmQgExGjHbHQo6KKOFitiOKa6omzE_Rxl3e9rGYYXWs222DW2c82_zDJevP_T_TX5iptTKup5YBbgnf7BDndLlCqmX1xEIKNkJZiqFSMCaEFaejbR-hNWnJs7TVKUkUlZ7JRb_5V9EfKw1waMOwAl1MpGSbjfLXbmTSBPhiCzXYFzG4FTFsB83sFzLZb-ij0IfuTQWwXVBocryD_lf1E1C91rsae
CitedBy_id	crossref_primary_10_1109_TSP_2023_3257989 crossref_primary_10_1002_mrm_30156 crossref_primary_10_1109_JRFID_2023_3284670
Cites_doi	10.1145/355984.355989 10.1002/mrm.24229 10.1109/TMI.2015.2427157 10.1002/mrm.25685 10.1002/mrm.20492 10.1109/MSP.2019.2949570 10.1137/110826497 10.1109/TMI.2013.2293974 10.2307/2372313 10.1007/s00020-010-1825-4 10.1002/mrm.21284 10.1109/42.802758 10.1016/j.jmr.2005.06.004 10.6028/jres.049.044 10.1016/j.laa.2010.07.011 10.1109/ISBI.2007.357026 10.1109/ISBI.2015.7164018
ContentType	Journal Article
Copyright	The Author(s) 2021. corrected publication 2021 The Author(s) 2021. corrected publication 2021. This work is published under http://creativecommons.org/licenses/by/4.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.
Copyright_xml	– notice: The Author(s) 2021. corrected publication 2021 – notice: The Author(s) 2021. corrected publication 2021. This work is published under http://creativecommons.org/licenses/by/4.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.
DBID	C6C AAYXX CITATION NPM 7TB 8FD FR3 KR7 7X8 5PM
DOI	10.1007/s11081-021-09604-4
DatabaseName	Springer Nature OA Free Journals CrossRef PubMed Mechanical & Transportation Engineering Abstracts Technology Research Database Engineering Research Database Civil Engineering Abstracts MEDLINE - Academic PubMed Central (Full Participant titles)
DatabaseTitle	CrossRef PubMed Civil Engineering Abstracts Engineering Research Database Technology Research Database Mechanical & Transportation Engineering Abstracts MEDLINE - Academic
DatabaseTitleList	Civil Engineering Abstracts MEDLINE - Academic CrossRef PubMed
Database_xml	– sequence: 1 dbid: NPM name: PubMed url: http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=PubMed sourceTypes: Index Database – sequence: 2 dbid: 7X8 name: MEDLINE - Academic url: https://search.proquest.com/medline sourceTypes: Aggregation Database
DeliveryMethod	fulltext_linktorsrc
Discipline	Engineering Mathematics
EISSN	1573-2924
EndPage	768
ExternalDocumentID	PMC9159680 35656362 10_1007_s11081_021_09604_4
Genre	Journal Article
GroupedDBID	-5D -5G -BR -EM -Y2 -~C .86 .DC .VR 06D 0R~ 0VY 123 1N0 1SB 203 29N 2J2 2JN 2JY 2KG 2LR 2P1 2VQ 2~H 30V 4.4 406 408 409 40D 40E 5VS 67Z 6NX 8TC 8UJ 95- 95. 95~ 96X AAAVM AABHQ AACDK AAHNG AAIAL AAJBT AAJKR AANZL AARHV AARTL AASML AATNV AATVU AAUYE AAWCG AAYIU AAYQN AAYTO AAYZH ABAKF ABBBX ABBXA ABDZT ABECU ABFSI ABFTD ABFTV ABHLI ABHQN ABJNI ABJOX ABKCH ABKTR ABMNI ABMQK ABNWP ABQBU ABQSL ABSXP ABTEG ABTHY ABTKH ABTMW ABULA ABWNU ABXPI ACAOD ACBXY ACDTI ACGFO ACGFS ACHSB ACHXU ACIWK ACKNC ACMDZ ACMLO ACOKC ACOMO ACPIV ACREN ACSNA ACZOJ ADHHG ADHIR ADINQ ADKNI ADKPE ADRFC ADTPH ADURQ ADYFF ADYOE ADZKW AEBTG AEFQL AEGAL AEGNC AEJHL AEJRE AEKMD AEMSY AENEX AEOHA AEPYU AESKC AETLH AEVLU AEXYK AFBBN AFGCZ AFLOW AFQWF AFWTZ AFYQB AFZKB AGAYW AGDGC AGJBK AGMZJ AGQEE AGQMX AGRTI AGWIL AGWZB AGYKE AHAVH AHBYD AHKAY AHSBF AHYZX AIAKS AIGIU AIIXL AILAN AITGF AJBLW AJRNO AJZVZ ALMA_UNASSIGNED_HOLDINGS ALWAN AMKLP AMTXH AMXSW AMYLF AMYQR AOCGG ARMRJ ASPBG AVWKF AXYYD AYJHY AZFZN B-. BA0 BAPOH BDATZ BGNMA BSONS C6C CAG COF CS3 CSCUP DDRTE DL5 DNIVK DPUIP DU5 EBLON EBS EIOEI EJD ESBYG FEDTE FERAY FFXSO FIGPU FINBP FNLPD FRRFC FSGXE FWDCC GGCAI GGRSB GJIRD GNWQR GQ6 GQ7 GQ8 GXS H13 HF~ HG5 HG6 HMJXF HQYDN HRMNR HVGLF HZ~ I09 IHE IJ- IKXTQ IWAJR IXC IXD IXE IZIGR IZQ I~X I~Z J-C J0Z J9A JBSCW JCJTX JZLTJ KDC KOV LAK LLZTM M4Y MA- N2Q N9A NB0 NPVJJ NQJWS NU0 O9- O93 O9J OAM OVD P2P P9R PF0 PT4 Q2X QOS R89 R9I RNI RNS ROL RPX RSV RZC RZE S16 S1Z S27 S3B SAP SDH SHX SISQX SJYHP SMT SNE SNPRN SNX SOHCF SOJ SPISZ SRMVM SSLCW STPWE SZN T13 TEORI TSG TSK TSV TUC U2A UG4 UOJIU UTJUX UZXMN VC2 VFIZW W23 W48 WK8 YLTOR Z45 Z7R Z7S Z7X Z7Y Z7Z Z83 Z88 ZMTXR ~A9 8AO AAPKM AAYXX ABBRH ABDBE ABFSG ABJCF ABRTQ ACSTC ADHKG AEZWR AFDZB AFFHD AFHIU AFKRA AFOHR AGQPQ AHPBZ AHWEU AIXLP ATHPR AYFIA BENPR BGLVJ CCPQU CITATION HCIFZ M7S PHGZM PHGZT PQGLB PTHSS S0W NPM 7TB 8FD FR3 KR7 7X8 5PM
ID	FETCH-LOGICAL-c474t-a5ce339157e4a7e3383d481c5ce7f8f4d92181f15dad059d7271cebb174929f43
IEDL.DBID	RSV
ISICitedReferencesCount	6
ISICitedReferencesURI	http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=000628098800001&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D
ISSN	1389-4420 1573-2924
IngestDate	Tue Nov 04 02:00:34 EST 2025 Sun Nov 09 14:37:51 EST 2025 Thu Sep 25 01:05:18 EDT 2025 Thu Apr 03 07:08:58 EDT 2025 Tue Nov 18 22:22:57 EST 2025 Sat Nov 29 01:39:09 EST 2025 Fri Feb 21 02:47:35 EST 2025
IsDoiOpenAccess	true
IsOpenAccess	true
IsPeerReviewed	true
IsScholarly	true
Issue	2
Keywords	Linear and antilinear operators Inverse problems Efficient numerical computations 65K10 47J05 65H10 47J25 47N10 Iterative least-squares algorithms 15A29
Language	English
License	OpenAccess This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
LinkModel	DirectLink
MergedId	FETCHMERGED-LOGICAL-c474t-a5ce339157e4a7e3383d481c5ce7f8f4d92181f15dad059d7271cebb174929f43
Notes	ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 content type line 23
OpenAccessLink	https://link.springer.com/10.1007/s11081-021-09604-4
PMID	35656362
PQID	2662726436
PQPubID	326265
PageCount	20
ParticipantIDs	pubmedcentral_primary_oai_pubmedcentral_nih_gov_9159680 proquest_miscellaneous_2673355951 proquest_journals_2662726436 pubmed_primary_35656362 crossref_citationtrail_10_1007_s11081_021_09604_4 crossref_primary_10_1007_s11081_021_09604_4 springer_journals_10_1007_s11081_021_09604_4
PublicationCentury	2000
PublicationDate	2022-06-01
PublicationDateYYYYMMDD	2022-06-01
PublicationDate_xml	– month: 06 year: 2022 text: 2022-06-01 day: 01
PublicationDecade	2020
PublicationPlace	New York
PublicationPlace_xml	– name: New York – name: United States – name: Dordrecht
PublicationSubtitle	International Multidisciplinary Journal to Promote Optimization Theory & Applications in Engineering Sciences
PublicationTitle	Optimization and engineering
PublicationTitleAbbrev	Optim Eng
PublicationTitleAlternate	Optim Eng
PublicationYear	2022
Publisher	Springer US Springer Nature B.V
Publisher_xml	– name: Springer US – name: Springer Nature B.V
References	Markovsky (CR18) 2011; 32 Luenberger (CR17) 1969 Bydder (CR2) 2010; 433 Landweber (CR15) 1951; 73 Willig-Onwuachi, Yeh, Grant, Ohliger, McKenzie, Sodickson (CR22) 2005; 176 Blaimer, Heim, Neumann, Jakob, Kannengiesser, Breuer (CR1) 2016; 75 CR5 CR7 Kay (CR13) 1993 Rudin (CR20) 1991 Haldar (CR6) 2014; 33 CR14 Haldar, Setsompop (CR8) 2020; 37 CR11 Huhtanen, Ruotsalainen (CR12) 2011; 69 Lew, Pineda, Clayton, Spielman, Chan, Bammer (CR16) 2007; 58 Hestenes, Stiefel (CR10) 1952; 49 Paige, Saunders (CR19) 1982; 8 Varadarajan, Haldar (CR21) 2015; 34 Bydder, Robson (CR3) 2005; 53 Haldar, Wedeen, Nezamzadeh, Dai, Weiner, Schuff, Liang (CR9) 2013; 69 Erdogan, Fessler (CR4) 1999; 18 M Bydder (9604_CR3) 2005; 53 D Varadarajan (9604_CR21) 2015; 34 JP Haldar (9604_CR6) 2014; 33 JP Haldar (9604_CR8) 2020; 37 H Erdogan (9604_CR4) 1999; 18 JP Haldar (9604_CR9) 2013; 69 L Landweber (9604_CR15) 1951; 73 C Lew (9604_CR16) 2007; 58 9604_CR5 9604_CR14 M Blaimer (9604_CR1) 2016; 75 9604_CR7 9604_CR11 I Markovsky (9604_CR18) 2011; 32 CC Paige (9604_CR19) 1982; 8 W Rudin (9604_CR20) 1991 JD Willig-Onwuachi (9604_CR22) 2005; 176 SM Kay (9604_CR13) 1993 M Huhtanen (9604_CR12) 2011; 69 DG Luenberger (9604_CR17) 1969 MR Hestenes (9604_CR10) 1952; 49 M Bydder (9604_CR2) 2010; 433
References_xml	– volume: 8 start-page: 43 year: 1982 end-page: 71 ident: CR19 article-title: LSQR: An algorithm for sparse linear equations and sparse least squares publication-title: ACM Trans Math Softw doi: 10.1145/355984.355989 – volume: 69 start-page: 277 year: 2013 end-page: 289 ident: CR9 article-title: Improved diffusion imaging through SNR-enhancing joint reconstruction publication-title: Magn Reson Med doi: 10.1002/mrm.24229 – volume: 34 start-page: 2191 year: 2015 end-page: 2202 ident: CR21 article-title: A majorize-minimize framework for Rician and non-central chi MR images publication-title: IEEE Trans Med Image doi: 10.1109/TMI.2015.2427157 – year: 1991 ident: CR20 publication-title: Functional analysis – volume: 75 start-page: 1086 year: 2016 end-page: 1099 ident: CR1 article-title: Comparison of phase-constrained parallel MRI approaches: analogies and differences publication-title: Magn Reson Med doi: 10.1002/mrm.25685 – year: 1969 ident: CR17 publication-title: Optimization by vector space methods – volume: 53 start-page: 1393 year: 2005 end-page: 1401 ident: CR3 article-title: Partial Fourier partially parallel imaging publication-title: Magn Reson Med doi: 10.1002/mrm.20492 – volume: 37 start-page: 69 year: 2020 end-page: 82 ident: CR8 article-title: Linear predictability in MRI reconstruction: Leveraging shift-invariant Fourier structure for faster and better imaging publication-title: IEEE Signal Process Mag doi: 10.1109/MSP.2019.2949570 – ident: CR14 – volume: 32 start-page: 987 year: 2011 end-page: 992 ident: CR18 article-title: On the complex least squares problem with constrained phase publication-title: SIAM J Matrix Anal Appl doi: 10.1137/110826497 – volume: 33 start-page: 668 year: 2014 end-page: 681 ident: CR6 article-title: Low-rank modeling of local k-space neighborhoods (LORAKS) for constrained MRI publication-title: IEEE Trans Med Imag doi: 10.1109/TMI.2013.2293974 – volume: 73 start-page: 615 year: 1951 end-page: 624 ident: CR15 article-title: An iteration formula for Fredholm integral equations of the first kind publication-title: Am J Math doi: 10.2307/2372313 – ident: CR11 – year: 1993 ident: CR13 publication-title: Fundamentals of statistical signal processing, Volume I: estimation theory – volume: 69 start-page: 113 year: 2011 end-page: 132 ident: CR12 article-title: Real linear operator theory and its applications publication-title: Int Equ Oper Theory doi: 10.1007/s00020-010-1825-4 – volume: 58 start-page: 910 year: 2007 end-page: 921 ident: CR16 article-title: SENSE phase-constrained magnitude reconstruction with iterative phase refinement publication-title: Magn Reson Med doi: 10.1002/mrm.21284 – ident: CR5 – ident: CR7 – volume: 18 start-page: 801 year: 1999 end-page: 814 ident: CR4 article-title: Monotonic algorithms for transmission tomography publication-title: IEEE Trans Med Imag doi: 10.1109/42.802758 – volume: 176 start-page: 187 year: 2005 end-page: 198 ident: CR22 article-title: Phase-constrained parallel MR image reconstruction publication-title: J Magn Reson doi: 10.1016/j.jmr.2005.06.004 – volume: 49 start-page: 409 year: 1952 end-page: 436 ident: CR10 article-title: Methods of conjugate gradients for solving linear systems publication-title: J Res Natl Bur Stand doi: 10.6028/jres.049.044 – volume: 433 start-page: 1719 year: 2010 end-page: 1721 ident: CR2 article-title: Solution of a complex least squares problem with constrained phase publication-title: Linear Algebra Appl doi: 10.1016/j.laa.2010.07.011 – volume-title: Fundamentals of statistical signal processing, Volume I: estimation theory year: 1993 ident: 9604_CR13 – ident: 9604_CR14 – volume: 69 start-page: 277 year: 2013 ident: 9604_CR9 publication-title: Magn Reson Med doi: 10.1002/mrm.24229 – volume: 34 start-page: 2191 year: 2015 ident: 9604_CR21 publication-title: IEEE Trans Med Image doi: 10.1109/TMI.2015.2427157 – ident: 9604_CR5 – volume: 33 start-page: 668 year: 2014 ident: 9604_CR6 publication-title: IEEE Trans Med Imag doi: 10.1109/TMI.2013.2293974 – volume: 32 start-page: 987 year: 2011 ident: 9604_CR18 publication-title: SIAM J Matrix Anal Appl doi: 10.1137/110826497 – volume: 433 start-page: 1719 year: 2010 ident: 9604_CR2 publication-title: Linear Algebra Appl doi: 10.1016/j.laa.2010.07.011 – volume: 176 start-page: 187 year: 2005 ident: 9604_CR22 publication-title: J Magn Reson doi: 10.1016/j.jmr.2005.06.004 – volume: 49 start-page: 409 year: 1952 ident: 9604_CR10 publication-title: J Res Natl Bur Stand doi: 10.6028/jres.049.044 – volume-title: Functional analysis year: 1991 ident: 9604_CR20 – volume-title: Optimization by vector space methods year: 1969 ident: 9604_CR17 – volume: 58 start-page: 910 year: 2007 ident: 9604_CR16 publication-title: Magn Reson Med doi: 10.1002/mrm.21284 – volume: 75 start-page: 1086 year: 2016 ident: 9604_CR1 publication-title: Magn Reson Med doi: 10.1002/mrm.25685 – volume: 37 start-page: 69 year: 2020 ident: 9604_CR8 publication-title: IEEE Signal Process Mag doi: 10.1109/MSP.2019.2949570 – volume: 73 start-page: 615 year: 1951 ident: 9604_CR15 publication-title: Am J Math doi: 10.2307/2372313 – volume: 8 start-page: 43 year: 1982 ident: 9604_CR19 publication-title: ACM Trans Math Softw doi: 10.1145/355984.355989 – volume: 53 start-page: 1393 year: 2005 ident: 9604_CR3 publication-title: Magn Reson Med doi: 10.1002/mrm.20492 – ident: 9604_CR11 doi: 10.1109/ISBI.2007.357026 – volume: 18 start-page: 801 year: 1999 ident: 9604_CR4 publication-title: IEEE Trans Med Imag doi: 10.1109/42.802758 – ident: 9604_CR7 doi: 10.1109/ISBI.2015.7164018 – volume: 69 start-page: 113 year: 2011 ident: 9604_CR12 publication-title: Int Equ Oper Theory doi: 10.1007/s00020-010-1825-4
SSID	ssj0016528
Score	2.2926114
Snippet	We consider a setting in which it is desired to find an optimal complex vector x ∈ C N that satisfies A ( x ) ≈ b in a least-squares sense, where b ∈ C M is a... We consider a setting in which it is desired to find an optimal complex vector $${\mathbf {x}}\in {\mathbb {C}}^N$$ x ∈ C N that satisfies $${\mathcal... We consider a setting in which it is desired to find an optimal complex vector ∈ that satisfies ( ) ≈ in a least-squares sense, where ∈ is a data vector... We consider a setting in which it is desired to find an optimal complex vector x∈CN that satisfies A(x)≈b in a least-squares sense, where b∈CM is a data vector... We consider a setting in which it is desired to find an optimal complex vector x ∈ C N that satisfies A (x) ≈ b in a least-squares sense, where b ∈ C M is a... We consider a setting in which it is desired to find an optimal complex vector x ∈ CN that satisfies A(x) ≈ b in a least-squares sense, where b ∈ CM is a data...
SourceID	pubmedcentral proquest pubmed crossref springer
SourceType	Open Access Repository Aggregation Database Index Database Enrichment Source Publisher
StartPage	749
SubjectTerms	Algorithms Complexity Conjugation Control Engineering Environmental Management Exact solutions Financial Engineering Inverse problems Iterative methods Iterative solution Least squares method Mathematics Mathematics and Statistics Operations Research/Decision Theory Operators Optimization Research Article Systems Theory
Title	Efficient iterative solutions to complex-valued nonlinear least-squares problems with mixed linear and antilinear operators
URI	https://link.springer.com/article/10.1007/s11081-021-09604-4 https://www.ncbi.nlm.nih.gov/pubmed/35656362 https://www.proquest.com/docview/2662726436 https://www.proquest.com/docview/2673355951 https://pubmed.ncbi.nlm.nih.gov/PMC9159680
Volume	23
WOSCitedRecordID	wos000628098800001&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D
hasFullText	1
inHoldings	1
isFullTextHit
isPrint
journalDatabaseRights	– providerCode: PRVAVX databaseName: SpringerLINK Contemporary 1997-Present customDbUrl: eissn: 1573-2924 dateEnd: 99991231 omitProxy: false ssIdentifier: ssj0016528 issn: 1389-4420 databaseCode: RSV dateStart: 20000601 isFulltext: true titleUrlDefault: https://link.springer.com/search?facet-content-type=%22Journal%22 providerName: Springer Nature
link	http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnV1LT9wwEB7x6KEcKJS2hAIyEjewtNk4cXJECNRDQYiX9hYltiNW2mbpJouQ-PPMOE6WLQWpPeRgeZI447H9OeP5BmAfpzgVqtxwo3qCi8IUPFF5hB2CNTLrqTC3JK4_5fl5PBgkFy4orGpPu7cuSTtTz4LdfFy-OB0pINiND16EZVzuYhqOl1e3ne8gCm1GVfLAcSH6PRcq8_dnzC9HrzDm66OSf_hL7TJ0-un_PmANVh3sZEeNnazDgik_w8oLMkIsnXUMrtUGPJ1YbglsD2uIl3FWZJ2dsnrM7GF088iJLtxoVjacG9mEjSgdEK9-Tym0ibmMNRWjP77s1_ARZZ1gVmq86qErju-N9flXX-Dm9OT6-Ad3iRq4ElLUPAuVCYhpXhqRSUO7Xi1iH43AyCIuhE4ISBR-qDONcE4jZvKVyXPcDSE6K0TwFZawlWYTmM5CPxcS9ZXkiOR8nKyjWOTaBNKy3Xngt_2VKsdiTsk0RumMf5nUnKKaU6vmVHhw0N1z33B4vCu93ZpB6sZzlfaJJx-xYxB5sNdV40gk90pWmvGUZGSA6A0hqwffGqvpXhcQbkas4IGcs6dOgFi-52vK4Z1l-0a1JlHc8-CwtapZs97-iq1_E_8OH_sU12F_L23DUj2Zmh34oB7qYTXZhUU5iHftKHsGUogiJg
linkProvider	Springer Nature
linkToHtml	http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnV1NT9wwEB0VigQ9tHy1TQvUSNzA0oY4cXKsKhAVy6pqAXGLEtsRK22zsMlWSP3znXGcbLcUJDjkYHmSOOOx_ZwZvwHYwylOhSo33Kie4KIwBU9UHmGHYI3MeirMLYlrXw4G8dVV8s0dCqvaaPfWJWln6tlhNx-XL04hBQS78cEL8FLgikWBfN9_XHa-gyi0GVXJA8eFOOy5ozL_f8b8cnQPY94PlfzHX2qXoeM3z_uAVXjtYCf73NjJGrww5Tq8-ouMEEtnHYNrtQG_jyy3BLaHNcTLOCuyzk5ZPWY2GN3ccaILN5qVDedGNmEjSgfEq9spHW1iLmNNxeiPL_s5vENZJ5iVGq966IrjG2N9_tUmXBwfnX854S5RA1dCippnoTIBMc1LIzJpaNerReyjERhZxIXQCQGJwg91phHOacRMvjJ5jrshRGeFCN7CIrbSvAems9DPhUR9JTkiOR8n6ygWuTaBtGx3Hvhtf6XKsZhTMo1ROuNfJjWnqObUqjkVHux399w0HB6PSm-1ZpC68Vylh8STj9gxiDzY7apxJJJ7JSvNeEoyMkD0hpDVg3eN1XSvCwg3I1bwQM7ZUydALN_zNeXw2rJ9o1qTKO55cNBa1axZD3_Fh6eJf4Llk_Ozftr_Ojj9CCv0R6yJh9uCxXoyNduwpH7Vw2qyY8faHx0gJCc
linkToPdf	http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwpV3db9MwED-NDqHxwAZskFGGkXgDa0njxMkjYqtAlGoSH9pblNiOVmlLS5NOlfbP78756LoxJMRDHiJfEudytn_O3f0O4B1OcSpQmeFGuYKL3OQ8VlmIHwRbZOqqILMkriM5Hkenp_HJjSx-G-3euiTrnAZiaSqqw5nOD1eJbx4uZZzCCwiC40MewKagokG0X__-q_MjhIGtrkreOC7EwG3SZv58j_Wl6Q7evBs2ect3apek4fb_v8wOPGngKPtY289T2DDFM3h8g6QQz751zK7lc7g6tpwTeHdWEzLjbMk6-2XVlNkgdbPkRCNuNCtqLo50zs6pTBAvfy8o5Yk1lWxKRn-C2cVkibKNYFpoPKpJczqdGRsLUO7Cz-Hxj0-feVPAgSshRcXTQBmfGOilEak0tBvWIvLQOIzMo1zomABG7gU61QjzNGIpT5ksw10SorZc-HvQw16al8B0GniZkKivOEOE5-EkHkYi08aXlgXPAa_9dolq2M2pyMZ5suJlJjUnqObEqjkRDrzvrpnV3B5_le63JpE047xMBsSfj5jSDx142zXjCCW3S1qY6YJkpI-oDqGsAy9qC-oe5xOeRgzhgFyzrU6A2L_XW4rJmWUBR7XGYeQ68KG1sFW37n-L_X8TfwOPTo6GyejL-Osr2BpQ6of9A9WHXjVfmNfwUF1Wk3J-YIfdNVZGLQY
openUrl	ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Efficient+iterative+solutions+to+complex-valued+nonlinear+least-squares+problems+with+mixed+linear+and+antilinear+operators&rft.jtitle=Optimization+and+engineering&rft.au=Kim%2C+Tae+Hyung&rft.au=Haldar%2C+Justin+P&rft.date=2022-06-01&rft.pub=Springer+Nature+B.V&rft.issn=1389-4420&rft.eissn=1573-2924&rft.volume=23&rft.issue=2&rft.spage=749&rft.epage=768&rft_id=info:doi/10.1007%2Fs11081-021-09604-4&rft.externalDBID=NO_FULL_TEXT
thumbnail_l	http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=1389-4420&client=summon
thumbnail_m	http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=1389-4420&client=summon
thumbnail_s	http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=1389-4420&client=summon