{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T18:13:25Z","timestamp":1776795205518,"version":"3.51.2"},"reference-count":23,"publisher":"American Mathematical Society (AMS)","issue":"278","license":[{"start":{"date-parts":[[2012,9,20]],"date-time":"2012-09-20T00:00:00Z","timestamp":1348099200000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>This paper deals with the problem of reconstructing the electromagnetic parameters and the shape of a target from multi-static response matrix measurements at a single frequency. The target is of characteristic size less than the operating wavelength. Using small-volume asymptotic expansions of the measurements of high-order, we show how the electromagnetic parameters and the equivalent ellipse of the target can be reconstructed. The asymptotic expansions of the measurements are written in terms of the new concept of frequency dependent polarization tensors. Moreover, we extend the optimization approach proposed in Part I to image geometric details of an electromagnetic target that are finer than the equivalent ellipse. The equivalent ellipse still provides a good initial guess for the optimization procedure. However, compared to the conductivity case in Part I, the cost functional measures the discrepancy between the computed and measured high-order frequency dependent polarization tensors rather than between the generalized polarization tensors. The main reason for such a modification of the cost functional is the fact that the (measured) frequency dependent polarization tensors can be easily obtained from multistatic measurements by solving a linear system while the derivation of the generalized polarization tensors from measurements requires more delicate treatment. The proposed methods are numerically implemented to demonstrate their validity and efficiency.<\/p>","DOI":"10.1090\/s0025-5718-2011-02534-2","type":"journal-article","created":{"date-parts":[[2011,9,20]],"date-time":"2011-09-20T12:01:36Z","timestamp":1316520096000},"page":"839-860","source":"Crossref","is-referenced-by-count":20,"title":["The generalized polarization tensors for resolved imaging Part II: Shape and electromagnetic parameters reconstruction of an electromagnetic inclusion from multistatic measurements"],"prefix":"10.1090","volume":"81","author":[{"given":"Habib","family":"Ammari","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hyeonbae","family":"Kang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Eunjoo","family":"Kim","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"June-Yub","family":"Lee","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2011,9,20]]},"reference":[{"key":"1","unstructured":"H. Ammari, J. Garnier, H. Kang, M. Lim, and K. Solna, Multistatic imaging of extended targets, SIAM J. Imag. Sci., to appear."},{"key":"2","doi-asserted-by":"crossref","unstructured":"H. Ammari, J. Garnier, H. Kang, W.-K. Park, and K. Solna, Imaging schemes for perfectly conducting cracks, SIAM J. Appl. Math., 71 (2011), 68\u201391.","DOI":"10.1137\/100800130"},{"issue":"1","key":"3","doi-asserted-by":"publisher","first-page":"130","DOI":"10.1137\/040612518","article-title":"Two numerical methods for recovering small inclusions from the scattering amplitude at a fixed frequency","volume":"27","author":"Ammari, Habib","year":"2005","journal-title":"SIAM J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/1064-8275","issn-type":"print"},{"issue":"3","key":"4","doi-asserted-by":"publisher","first-page":"597","DOI":"10.1137\/040610854","article-title":"A MUSIC algorithm for locating small inclusions buried in a half-space from the scattering amplitude at a fixed frequency","volume":"3","author":"Ammari, Habib","year":"2005","journal-title":"Multiscale Model. Simul.","ISSN":"https:\/\/id.crossref.org\/issn\/1540-3459","issn-type":"print"},{"issue":"2","key":"5","doi-asserted-by":"publisher","first-page":"674","DOI":"10.1137\/050640655","article-title":"MUSIC-type electromagnetic imaging of a collection of small three-dimensional inclusions","volume":"29","author":"Ammari, Habib","year":"2007","journal-title":"SIAM J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/1064-8275","issn-type":"print"},{"key":"6","series-title":"Lecture Notes in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/b98245","volume-title":"Reconstruction of small inhomogeneities from boundary measurements","volume":"1846","author":"Ammari, Habib","year":"2004","ISBN":"https:\/\/id.crossref.org\/isbn\/3540224831"},{"issue":"1","key":"7","doi-asserted-by":"publisher","first-page":"190","DOI":"10.1016\/j.jmaa.2004.04.003","article-title":"Boundary layer techniques for solving the Helmholtz equation in the presence of small inhomogeneities","volume":"296","author":"Ammari, Habib","year":"2004","journal-title":"J. Math. Anal. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0022-247X","issn-type":"print"},{"key":"8","series-title":"Applied Mathematical Sciences","isbn-type":"print","volume-title":"Polarization and moment tensors","volume":"162","author":"Ammari, Habib","year":"2007","ISBN":"https:\/\/id.crossref.org\/isbn\/9780387715650"},{"key":"9","doi-asserted-by":"crossref","unstructured":"H. Ammari, H. Kang, M. Lim, and H. Zribi, The generalized polarization tensors for resolved imaging. Part I: Shape reconstruction of a conductivity inclusion, Math. Comp., 81 (2012), 367\u2013386.","DOI":"10.1090\/S0025-5718-2011-02533-0"},{"issue":"4","key":"10","doi-asserted-by":"publisher","first-page":"501","DOI":"10.1007\/s00211-007-0130-x","article-title":"A MUSIC-type algorithm for detecting internal corrosion from electrostatic boundary measurements","volume":"108","author":"Ammari, Habib","year":"2008","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"11","unstructured":"H. Ammari, H. Kang, E. Kim, M. Lim, and K. Louati, Ultrasound imaging of internal corrosion, SIAM J. Numer. Anal., to appear."},{"issue":"2","key":"12","doi-asserted-by":"publisher","first-page":"894","DOI":"10.1137\/090749013","article-title":"Asymptotic imaging of perfectly conducting cracks","volume":"32","author":"Ammari, Habib","year":"2010","journal-title":"SIAM J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/1064-8275","issn-type":"print"},{"issue":"7","key":"13","doi-asserted-by":"publisher","first-page":"749","DOI":"10.1016\/S0021-7824(03)00033-3","article-title":"Electromagnetic scattering by small dielectric inhomogeneities","volume":"82","author":"Ammari, Habib","year":"2003","journal-title":"J. Math. Pures Appl. (9)","ISSN":"https:\/\/id.crossref.org\/issn\/0021-7824","issn-type":"print"},{"issue":"12","key":"14","doi-asserted-by":"publisher","first-page":"4638","DOI":"10.1016\/j.jcp.2009.03.031","article-title":"Numerical solution of an inverse medium scattering problem for Maxwell\u2019s equations at fixed frequency","volume":"228","author":"Bao, Gang","year":"2009","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"issue":"1","key":"15","doi-asserted-by":"publisher","first-page":"75","DOI":"10.1137\/07069290X","article-title":"Edge illumination and imaging of extended reflectors","volume":"1","author":"Borcea, Liliana","year":"2008","journal-title":"SIAM J. Imaging Sci."},{"issue":"4","key":"16","doi-asserted-by":"publisher","first-page":"635","DOI":"10.1007\/s002110200409","article-title":"A direct impedance tomography algorithm for locating small inhomogeneities","volume":"93","author":"Br\u00fchl, Martin","year":"2003","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"17","isbn-type":"print","doi-asserted-by":"publisher","first-page":"35","DOI":"10.1007\/978-3-540-78547-7_3","article-title":"Adjoint fields and sensitivities for 3D electromagnetic imaging in isotropic and anisotropic media","author":"Dorn, Oliver","year":"2008","ISBN":"https:\/\/id.crossref.org\/isbn\/9783540785453"},{"key":"18","isbn-type":"print","doi-asserted-by":"publisher","first-page":"61","DOI":"10.1090\/conm\/494\/09643","article-title":"An asymptotic factorization method for inverse electromagnetic scattering in layered media. II. A numerical study","author":"Griesmaier, Roland","year":"2009","ISBN":"https:\/\/id.crossref.org\/isbn\/9780821847459"},{"issue":"4","key":"19","doi-asserted-by":"publisher","first-page":"1151","DOI":"10.1088\/0266-5611\/22\/4\/003","article-title":"A direct imaging algorithm for extended targets","volume":"22","author":"Hou, Songming","year":"2006","journal-title":"Inverse Problems","ISSN":"https:\/\/id.crossref.org\/issn\/0266-5611","issn-type":"print"},{"issue":"1","key":"20","doi-asserted-by":"publisher","first-page":"317","DOI":"10.1016\/j.jcp.2004.02.010","article-title":"Imaging of location and geometry for extended targets using the response matrix","volume":"199","author":"Hou, Songming","year":"2004","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"issue":"4","key":"21","doi-asserted-by":"publisher","first-page":"723","DOI":"10.1051\/m2an:2000101","article-title":"Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of inhomogeneities of small diameter","volume":"34","author":"Vogelius, Michael S.","year":"2000","journal-title":"M2AN Math. Model. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0764-583X","issn-type":"print"},{"issue":"3","key":"22","doi-asserted-by":"publisher","first-page":"189","DOI":"10.1016\/S0165-2125(03)00047-7","article-title":"Numerical methods for locating small dielectric inhomogeneities","volume":"38","author":"Volkov, Darko","year":"2003","journal-title":"Wave Motion","ISSN":"https:\/\/id.crossref.org\/issn\/0165-2125","issn-type":"print"},{"issue":"3","key":"23","doi-asserted-by":"publisher","first-page":"725","DOI":"10.1137\/S0036139902415282","article-title":"Analysis of the response matrix for an extended target","volume":"64","author":"Zhao, Hongkai","year":"2004","journal-title":"SIAM J. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1399","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2012-81-278\/S0025-5718-2011-02534-2\/S0025-5718-2011-02534-2.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2012-81-278\/S0025-5718-2011-02534-2\/S0025-5718-2011-02534-2.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T17:08:59Z","timestamp":1776791339000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2012-81-278\/S0025-5718-2011-02534-2\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,9,20]]},"references-count":23,"journal-issue":{"issue":"278","published-print":{"date-parts":[[2012,4]]}},"alternative-id":["S0025-5718-2011-02534-2"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-2011-02534-2","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2011,9,20]]}}}