577 Multi-Scale and High-Contrast PDE: From Modelling, to Mathematical Analysis, to Inversion Conference on Multi-Scale and High-Contrast PDE: From Modelling, to Mathematical Analysis, to Inversion June 28–July 1, 2011 University of Oxford, United Kingdom Habib Ammari Yves Capdeboscq Hyeonbae Kang Editors AmericanMathematicalSociety Multi-Scale and High-Contrast PDE: From Modelling, to Mathematical Analysis, to Inversion Conference on Multi-Scale and High-Contrast PDE: From Modelling, to Mathematical Analysis, to Inversion June 28–July 1, 2011 University of Oxford, United Kingdom Habib Ammari Yves Capdeboscq Hyeonbae Kang Editors 577 Multi-Scale and High-Contrast PDE: From Modelling, to Mathematical Analysis, to Inversion Conference on Multi-Scale and High-Contrast PDE: From Modelling, to Mathematical Analysis, to Inversion June 28–July 1, 2011 University of Oxford, United Kingdom Habib Ammari Yves Capdeboscq Hyeonbae Kang Editors AmericanMathematicalSociety Providence,RhodeIsland EDITORIAL COMMITTEE Dennis DeTurck, Managing Editor Michael Loss Kailash Misra Martin J. Strauss 2010 Mathematics Subject Classification. Primary 35B30, 35J05, 35J25, 35K35, 35R30, 65M06,76A15, 35L05, 93B05. Library of Congress Cataloging-in-Publication Data ConferenceonMulti-scaleandHigh-contrastPDE:fromModelling,toMathematicalAnalysis,to Inversion(2011Oxford,England). Multi-scale and high-contrast PDE: from modelling, to mathematical analysis, to inversion, June28–July1,2011,UniversityofOxford,UnitedKingdom/HabibAmmari,YvesCapdeboscq, HyeonbaeKang,editors. p.cm. —(Contemporarymathematics;v.577) Includesbibliographicalreferences. ISBN978-0-8218-6929-1(alk.paper) 1. Image processing–Mathematics–Congresses. 2. Multiscale modeling–Congresses. 3. Dif- ferential equations, Partial–Congresses I. Ammari, Habib. II. Capdeboscq, Yves, 1971- III.Kang,Hyeonbae. IV.Title. TA1637.C666 2010 515(cid:2).353—dc23 2012013127 Copying and reprinting. Materialinthisbookmaybereproducedbyanymeansfor edu- cationaland scientific purposes without fee or permissionwith the exception ofreproduction by servicesthatcollectfeesfordeliveryofdocumentsandprovidedthatthecustomaryacknowledg- ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercialuseofmaterialshouldbeaddressedtotheAcquisitionsDepartment,AmericanMath- ematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can [email protected]. Excludedfromtheseprovisionsismaterialinarticlesforwhichtheauthorholdscopyright. In suchcases,requestsforpermissiontouseorreprintshouldbeaddresseddirectlytotheauthor(s). (Copyrightownershipisindicatedinthenoticeinthelowerright-handcornerofthefirstpageof eacharticle.) (cid:2)c 2012bytheAmericanMathematicalSociety. Allrightsreserved. TheAmericanMathematicalSocietyretainsallrights exceptthosegrantedtotheUnitedStatesGovernment. Copyrightofindividualarticlesmayreverttothepublicdomain28years afterpublication. ContacttheAMSforcopyrightstatusofindividualarticles. PrintedintheUnitedStatesofAmerica. (cid:2)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines establishedtoensurepermanenceanddurability. VisittheAMShomepageathttp://www.ams.org/ 10987654321 171615141312 Contents Preface vii Enhancement of near-cloaking. Part III: Numerical simulations, statistical stability, and related questions H. Ammari, J. Garnier, V. Jugnon, H. Kang, H. Lee, and M. Lim 1 Looking at the world through liquid crystal glasses O. D. Lavrentovich 25 A remark on the observability of conservative linear systems E. Zuazua 47 On the scattered field generated by a ball inhomogeneity of constant index in dimension three Y. Capdeboscq, G. Leadbetter, and A. Parker 61 Pointwise bounds on the gradient and the spectrum of the Neumann-Poincar´e operator: The case of 2 discs E. Bonnetier and F. Triki 81 A high-contrast fourth-order PDE from imaging: numerical solution by ADI splitting B. Du¨ring and C.-B. Scho¨nlieb 93 Imaging with noise blending M. de Hoop, E. Fedrizzi, J. Garnier, and K. Sølna 105 Correlations of heterogeneous wave fields propagating in homogeneous media G. Bal and O. Pinaud 125 v Preface The mathematical analysis of PDE modelling materials, or tissues, presenting multiple scales have been an active area of research for more than 40 years. The study of the corresponding imaging, or reconstruction, problem is a more recent one. If the material parameters of the PDE present high contrast ratio, then the solutiontothePDEbecomesparticularlychallengingtoanalyze,orcompute. Sim- ilar difficulties occur in time dependent equations in high frequency regimes. On the other hand, high frequency regimes, or very contrasted materials, were consid- eredfirstinimaging,aswell-differentiatedareasare,atfirstsight, simplertolocate by ad-hoc methods. Over the last decade the analysis of the inversion problem at moderate frequencies, the rigorous derivation of asymptotics at high frequencies, and the regularity properties of solutions of elliptic PDE in highly heterogeneous media have received a lot of attention. Thefocusofthisvolumeisonrecentprogresstowardsacompleteunderstanding ofthedirectproblemwithhighcontrastorhighfrequencies,andunifiedapproaches totheinverseandimagingproblemsforbothsmallandlargecontrastorfrequencies. The volumeincludes contributionsontheinverse problem, bothonitsanalysis and on numerical reconstructions. It offers the reader a good overview of current researchanddirectionforfurtherpursuitonmultiscaleproblems, bothinPDEand in signal processing, and in the analysis of the equations or the computation of their solutions. Finally, a special attention is devoted to new models and problems coming from physics leading to innovative imaging methods. The tremendous success of the workshop was only possible due to the enthu- siastic participation of wonderful speakers and authors of this volume. We are thankful to all of them. We also acknowledge with gratitude the generous support from the Engineering and Physical Sciences Research Council, the Oxford Cen- tre for Nonlinear PDE, the Oxford Centre for Collaborative Applied Mathematics, the National Research Foundation of Korea, and the European Research Council Project MULTIMOD. We would also like to thank the Mathematical Institute of the University of Oxford. Habib Ammari, Yves Capdeboscq, and Hyeonbae Kang vii ContemporaryMathematics Volume577,2012 http://dx.doi.org/10.1090/conm/577/11460 Enhancement of near-cloaking. Part III: Numerical simulations, statistical stability, and related questions Habib Ammari, Josselin Garnier, Vincent Jugnon, Hyeonbae Kang, Hyundae Lee, and Mikyoung Lim Abstract. Thegoalofthispaperistoillustratetheefficiencyandthestability ofthenear-cloakingstructuresproposedin[4]and[5]. Thesenewstructures are,beforeusingtransformationoptics,layeredstructuresandaredesignedso thattheirfirstcontractedgeneralizedpolarizationtensors(inthequasi-static limit)orscatteringcoefficients(inthecaseoftheHelmholtzequation)vanish. Inside the cloaking region, any target has near-zero boundary or scattering crosssectionmeasurements. Wenumericallyshowthatthisnewconstruction significantly enhances the invisibility cloaking effect for the conductivity and the Helmholtz equations and is quite robust with respect to random fluctua- tions of the material parameters around their theoretical values. We finally extendourmulti-coatedconstructiontotheenhancedreshapingproblem. We showhowtomakeanytargetlooklike adisc withhomogeneousphysicalpa- rameters. 1. Introduction Thecloakingproblemistomakeatargetinvisiblefromfar-fieldwavemeasure- ments[22,15,10,9,16,18]. Manyschemesareunderactivecurrentinvestigation. These include exteriorcloakingin which thecloakingregionis outside thecloaking device [19, 20, 8, 7, 2, 1], active cloaking [11], and interior cloaking, which is the focus of our study. In interior cloaking, the difficulty is to construct material parameter distribu- tions of a cloaking structure such that any target placed inside the structure is undetectable to waves. One approach is to use transformation optics [22, 10, 9, 27, 12, 23]. It takes advantage of the fact that the equations governing electro- statics, electromagnetism, and acoustics have transformation laws under change of variables. Thisallowsonetodesignstructuresthatsteelwavesaroundahiddenre- gion, returning them to their original path on the far side. The change of variables 2010 MathematicsSubjectClassification. Primary35R30;Secondary35B30. Key words and phrases. cloaking, reshaping, transformation optics, conductivity problem, Helmholtzequation,Dirichlet-to-Neumannmap,boundarymeasurements,scatteringcrosssection, contractedgeneralizedpolarizationtensors,scatteringcoefficients. This work was supported by ERC Advanced Grant Project MULTIMOD–267184 and Na- tionalResearchFoundationofKoreathroughgrantsNo. 2010-0017532,2010-0004091,and2009- 0090250. (cid:3)c2012 American Mathematical Society 1