{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,27]],"date-time":"2025-06-27T13:10:08Z","timestamp":1751029808998,"version":"3.41.0"},"reference-count":23,"publisher":"Walter de Gruyter GmbH","issue":"2","license":[{"start":{"date-parts":[[2024,1,25]],"date-time":"2024-01-25T00:00:00Z","timestamp":1706140800000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["DOC78"],"award-info":[{"award-number":["DOC78"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2024,4,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>In this paper, we consider the inverse problem of vibro-acoustography, a technique for enhancing ultrasound imaging by making use of nonlinear effects.\nIt amounts to determining two spatially variable coefficients in a system of PDEs describing propagation of two directed sound beams and the wave resulting from their nonlinear interaction.\nTo justify the use of Newton\u2019s method for solving this inverse problem, on one hand, we verify well-definedness and differentiability of the forward operator corresponding to two versions of the PDE model; on the other hand, we consider an all-at-once formulation of the inverse problem and prove convergence of Newton\u2019s method for its solution.<\/jats:p>","DOI":"10.1515\/cmam-2023-0076","type":"journal-article","created":{"date-parts":[[2024,1,24]],"date-time":"2024-01-24T16:41:21Z","timestamp":1706114481000},"page":"421-438","source":"Crossref","is-referenced-by-count":2,"title":["Simultaneous Reconstruction of Speed of Sound and Nonlinearity Parameter in a Paraxial Model of Vibro-Acoustography in Frequency Domain"],"prefix":"10.1515","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3295-6977","authenticated-orcid":false,"given":"Barbara","family":"Kaltenbacher","sequence":"first","affiliation":[{"name":"Department of Mathematics , University of Klagenfurt , Klagenfurt , Austria"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3056-7086","authenticated-orcid":false,"given":"Teresa","family":"Rauscher","sequence":"additional","affiliation":[{"name":"Department of Mathematics , University of Klagenfurt , Klagenfurt , Austria"}]}],"member":"374","published-online":{"date-parts":[[2024,1,25]]},"reference":[{"key":"2025062712372271647_j_cmam-2023-0076_ref_001","doi-asserted-by":"crossref","unstructured":"S. Acosta, G. Uhlmann and J. Zhai,\nNonlinear ultrasound imaging modeled by a Westervelt equation,\nSIAM J. Appl. Math. 82 (2022), no. 2, 408\u2013426.","DOI":"10.1137\/21M1431813"},{"key":"2025062712372271647_j_cmam-2023-0076_ref_002","doi-asserted-by":"crossref","unstructured":"D. Colton and R. Kress,\nInverse Acoustic and Electromagnetic Scattering Theory, 3rd ed.,\nAppl. Math. Sci. 93,\nSpringer, New York, 2013.","DOI":"10.1007\/978-1-4614-4942-3"},{"key":"2025062712372271647_j_cmam-2023-0076_ref_003","unstructured":"L. C. Evans,\nPartial Differential Equations,\nGrad. Stud. Math. 19,\nAmerican Mathematical Society, Providence, 1998."},{"key":"2025062712372271647_j_cmam-2023-0076_ref_004","doi-asserted-by":"crossref","unstructured":"M. Fatemi and J. F. Greenleaf,\nUltrasound-stimulated vibro-acoustic spectrography,\nScience 280 (1998), 82\u201385.","DOI":"10.1126\/science.280.5360.82"},{"key":"2025062712372271647_j_cmam-2023-0076_ref_005","doi-asserted-by":"crossref","unstructured":"M. Fatemi and J. F. Greenleaf,\nVibro-acoustography: An imaging modality based on ultrasound-stimulated acoustic emission,\nProc. Nat. Acad. Sci. 96 (1999), no. 12, 6603\u20136608.","DOI":"10.1073\/pnas.96.12.6603"},{"key":"2025062712372271647_j_cmam-2023-0076_ref_006","doi-asserted-by":"crossref","unstructured":"B. Flemisch, M. Kaltenbacher and B. I. Wohlmuth,\nElasto-acoustic and acoustic-acoustic coupling on non-matching grids,\nInternat. J. Numer. Methods Engrg. 67 (2006), no. 13, 1791\u20131810.","DOI":"10.1002\/nme.1669"},{"key":"2025062712372271647_j_cmam-2023-0076_ref_007","doi-asserted-by":"crossref","unstructured":"I. G. Graham and S. A. Sauter,\nStability and finite element error analysis for the Helmholtz equation with variable coefficients,\nMath. Comp. 89 (2020), no. 321, 105\u2013138.","DOI":"10.1090\/mcom\/3457"},{"key":"2025062712372271647_j_cmam-2023-0076_ref_008","doi-asserted-by":"crossref","unstructured":"B. Kaltenbacher,\nUniqueness of some space dependent coefficients in a wave equation of nonlinear acoustics, Evol. Equ. Control Theory (2023), 10.3934\/eect.2023052.","DOI":"10.3934\/eect.2023052"},{"key":"2025062712372271647_j_cmam-2023-0076_ref_009","doi-asserted-by":"crossref","unstructured":"B. Kaltenbacher,\nPeriodic solutions and multiharmonic expansions for the Westervelt equation,\nEvol. Equ. Control Theory 10 (2021), no. 2, 229\u2013247.","DOI":"10.3934\/eect.2020063"},{"key":"2025062712372271647_j_cmam-2023-0076_ref_010","doi-asserted-by":"crossref","unstructured":"B. Kaltenbacher,\nConvergence guarantees for coefficient reconstruction in PDEs from boundary measurements by variational and Newton-type methods via range invariance,\nIMA J. Numer. Anal. (2023), 10.1093\/imanum\/drad044.","DOI":"10.1093\/imanum\/drad044"},{"key":"2025062712372271647_j_cmam-2023-0076_ref_011","doi-asserted-by":"crossref","unstructured":"B. Kaltenbacher,\nOn the inverse problem of vibro-acoustography,\nMeccanica 58 (2023), no. 6, 1061\u20131072.","DOI":"10.1007\/s11012-022-01485-w"},{"key":"2025062712372271647_j_cmam-2023-0076_ref_012","doi-asserted-by":"crossref","unstructured":"B. Kaltenbacher and W. Rundell,\nOn an inverse problem of nonlinear imaging with fractional damping,\nMath. Comp. 91 (2021), no. 333, 245\u2013276.","DOI":"10.1090\/mcom\/3683"},{"key":"2025062712372271647_j_cmam-2023-0076_ref_013","doi-asserted-by":"crossref","unstructured":"B. Kaltenbacher and W. Rundell,\nOn the identification of the nonlinearity parameter in the westervelt equation from boundary measurements,\nInverse Probl. Imaging 15 (2021), 865\u2013891.","DOI":"10.3934\/ipi.2021020"},{"key":"2025062712372271647_j_cmam-2023-0076_ref_014","doi-asserted-by":"crossref","unstructured":"B. Kaltenbacher and W. Rundell,\nNonlinearity parameter imaging in the frequency domain,\nInverse Probl. Imaging (2023), 10.3934\/ipi.2023037","DOI":"10.3934\/ipi.2023037"},{"key":"2025062712372271647_j_cmam-2023-0076_ref_015","doi-asserted-by":"crossref","unstructured":"B. Kaltenbacher and W. Rundell,\nOn the simultaneous reconstruction of the nonlinearity coefficient and the sound speed in the Westervelt equation,\nInverse Problems 39 (2023), no. 10, Paper No. 105001.","DOI":"10.1088\/1361-6420\/aceef2"},{"key":"2025062712372271647_j_cmam-2023-0076_ref_016","doi-asserted-by":"crossref","unstructured":"C. E. Kenig,\nLecture notes: Global well-posedness, scattering and blow up for the energy-critical, focusing, non-linear Schr\u00f6dinger and wave equations,\nJourn. \u00c9qu. D\u00e9riv. Partielles (2007), 10.5802\/jedp.40.","DOI":"10.1007\/s11511-008-0031-6"},{"key":"2025062712372271647_j_cmam-2023-0076_ref_017","doi-asserted-by":"crossref","unstructured":"A. E. Malcolm, F. Reitich, J. Yang, J. F. Greenleaf and M. Fatemi,\nA combined parabolic-integral equation approach to the acoustic simulation of vibro-acoustic imaging,\nUltrasonics 48 (2008), 553\u2013558.","DOI":"10.1016\/j.ultras.2008.04.006"},{"key":"2025062712372271647_j_cmam-2023-0076_ref_018","doi-asserted-by":"crossref","unstructured":"A. E. Malcolm, F. Reitich, J. Yang, J. F. Greenleaf and M. Fatemi,\nNumerical modeling for assessment and design of ultrasound vibro-acoustography systems,\nBiomedical Applications of Vibration and Acoustics for Imaging and Characterizations,\nASME Press, New York (2007), https:\/\/doi.org\/10.1115\/1.802731.ch2.","DOI":"10.1115\/1.802731.ch2"},{"key":"2025062712372271647_j_cmam-2023-0076_ref_019","unstructured":"J. M. Melenk,\nOn Generalized Finite-Element Methods,\nProQuest LLC, Ann Arbor, 1995;\nThesis (Ph.D.), University of Maryland, College Park."},{"key":"2025062712372271647_j_cmam-2023-0076_ref_020","unstructured":"T. Rauscher,\nA paraxial approach for the inverse problem of vibroacoustic imaging in frequency domain,\npreprint (2023), https:\/\/arxiv.org\/abs\/2310.03367."},{"key":"2025062712372271647_j_cmam-2023-0076_ref_021","doi-asserted-by":"crossref","unstructured":"A. Rozanova,\nThe Khokhlov\u2013Zabolotskaya\u2013Kuznetsov equation,\nC. R. Math. Acad. Sci. Paris 344 (2007), no. 5, 337\u2013342.","DOI":"10.1016\/j.crma.2007.01.010"},{"key":"2025062712372271647_j_cmam-2023-0076_ref_022","doi-asserted-by":"crossref","unstructured":"M. Yamamoto and B. Kaltenbacher,\nAn inverse source problem related to acoustic nonlinearity parameter imaging,\nTime-Dependent Problems in Imaging and Parameter Identification,\nSpringer, New York (2021), 413\u2013456.","DOI":"10.1007\/978-3-030-57784-1_14"},{"key":"2025062712372271647_j_cmam-2023-0076_ref_023","unstructured":"E. A. Zabolotskaya and R. V. Khokhlov,\nQuasi-plane waves in the non-linear acoustics of confined beams,\nSov. Phys.-Acoust. 15 (1969), 35\u201340."}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2023-0076\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2023-0076\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,27]],"date-time":"2025-06-27T12:37:41Z","timestamp":1751027861000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2023-0076\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,1,25]]},"references-count":23,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2024,1,2]]},"published-print":{"date-parts":[[2024,4,1]]}},"alternative-id":["10.1515\/cmam-2023-0076"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2023-0076","relation":{},"ISSN":["1609-4840","1609-9389"],"issn-type":[{"type":"print","value":"1609-4840"},{"type":"electronic","value":"1609-9389"}],"subject":[],"published":{"date-parts":[[2024,1,25]]}}}