Imaging, Multi-Scale and High Contrast Partial Differential Equations: Seoul Icm 2014 Satellite Conference Imaging, Multi-scale and High Contrast Pdes ... Daejeon, Korea

160 Pages·2016·2.193 MB·English

by  Habib Ammari, Yves Capdeboscq, Hyeonbae Kang, Imbo Sim

660
Imaging, Multi-scale and High
Contrast Partial Differential
Equations
Seoul ICM 2014 Satellite Conference
Imaging, Multi-scale and High Contrast PDEs
August 7–9, 2014
Daejeon, Korea
Habib Ammari
Yves Capdeboscq
Hyeonbae Kang
Imbo Sim
Editors
AmericanMathematicalSociety
Imaging, Multi-scale and High
Contrast Partial Differential
Equations
Seoul ICM 2014 Satellite Conference
Imaging, Multi-scale and High Contrast PDEs
August 7–9, 2014
Daejeon, Korea
Habib Ammari
Yves Capdeboscq
Hyeonbae Kang
Imbo Sim
Editors

660
Imaging, Multi-scale and High
Contrast Partial Differential
Equations
Seoul ICM 2014 Satellite Conference
Imaging, Multi-scale and High Contrast PDEs
August 7–9, 2014
Daejeon, Korea
Habib Ammari
Yves Capdeboscq
Hyeonbae Kang
Imbo Sim
Editors
AmericanMathematicalSociety
Providence,RhodeIsland
EDITORIAL COMMITTEE
Dennis DeTurck, Managing Editor
Michael Loss Kailash Misra Catherine Yan
2010 Mathematics Subject Classification. Primary 35B10, 35B30, 35J05, 35J25, 35R30,
35R60, 65N15, 65T60, 86A15.
Library of Congress Cataloging-in-Publication Data
Names: Ammari,Habib,editor.
Title: Imaging,multi-scaleandhigh-contrastpartialdifferentialequations: SeoulICM2014Satel-
liteConference,August7–9,2014,Daejeon,Korea/HabibAmmari[andthreeothers],editors.
Description: Providence,RhodeIsland: AmericanMathematicalSociety,(cid:2)c 2016. |Series: Con-
temporarymathematics;volume660|Includesbibliographicalreferences.
Identifiers: LCCN2015037186|ISBN9781470419233(alk. paper)
Subjects: LCSH: Differential equations, Partial–Congresses. | Signal processing–Congresses. |
AMS:Partialdifferentialequations–Qualitativepropertiesofsolutions–Periodicsolutions. msc
|Partialdifferentialequations–Qualitativepropertiesofsolutions–Dependenceofsolutionson
initial and boundary data, parameters. msc | Partial differential equations – Elliptic equations
andsystems–Laplacianoperator,reducedwaveequation(Helmholtzequation),Poissonequation.
msc | Partial differential equations – Elliptic equations and systems – Boundary value problems
for second-order elliptic equations. msc | Partial differential equations – Miscellaneous topics –
Inverseproblems. msc|Partialdifferentialequations–Miscellaneoustopics–Partialdifferential
equations with randomness, stochastic partial differential equations. msc | Numerical analysis –
Partialdifferentialequations,boundaryvalueproblems–Errorbounds. msc|Numericalanalysis
–NumericalmethodsinFourieranalysis–Wavelets. msc|Geophysics–Geophysics–Seismology.
msc
Classification: LCCQA377.I4842016|DDC006.401/515353–dc23
LCrecordavailableathttp://lccn.loc.gov/2015037186
ContemporaryMathematicsISSN:0271-4132(print);ISSN:1098-3627(online)
DOI:http://dx.doi.org/10.1090/conm/660
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Contents
Preface vii
Wavelet methods for shape perception in electro-sensing
Habib Ammari, St´ephane Mallat, Ir`ene Waldspurger,
and Han Wang 1
Time-domain multiscale shape identification in electro-sensing
Habib Ammari and Han Wang 23
Estimation of stress in the presence of closely located elastic inclusions: A
numerical study
Hyeonbae Kang and Eunjoo Kim 45
Array dependence of effective parameters of dilute periodic elastic composite
Hyundae Lee and Jaeyoung Lee 59
A review on the enhancement of near-cloaking using the multilayer structure
Mikyoung Lim 73
On local non-zero constraints in PDE with analytic coefficients
Giovanni S. Alberti and Yves Capdeboscq 89
Daylight imaging for virtual reflection seismology
Josselin Garnier 99
Mode-matching solution of a scattering problem in flexible waveguide with
abrupt geometric changes
Muhammad Afzal, Muhammad Ayub, Rab Nawaz,
and Abdul Wahab 113
Direct scattering by a sound hard small body
Durga Prasad Challa 131
Helmholtz equation in unbounded domains: Some convergence results for a
constrained optimization problem
Giulio Ciraolo 139
v

Preface
The mathematical analysis of partial differential equation modelling materials,
or tissues, presenting multiple scales has been a very active area of research. The
study of the corresponding imaging, or reconstruction, problem is a more recent
one. If the material parameters of the partial differential equation present high
contrast ratio, then the solution to the partial differential equation becomes par-
ticularlychallenging toanalyze, or compute. Onthe other hand, imaging in highly
heterogeneous media poses significant challenges to the mathematical community.
Thefocusofthisbookisonrecentprogressestowardsacompleteunderstanding
ofthedirectproblemwithhighcontrastorhighfrequencies,andunifiedapproaches
to the inverse and imaging problems for both small and large contrast or frequen-
cies. Of particular importance in imaging are shape representation techniques and
regularizationapproaches. Specialattentionisdevotedtonewmodelsandproblems
coming from physics leading to innovative imaging and signal processing methods.
This book is the proceedings of the workshop, “Seoul ICM 2014 Satellite Con-
ference on Imaging, Multi-scale and High Contrast PDEs”, held at the National
Institute of Mathematical Sciences, Daejeon, Korea, August 7-9, 2014. The work-
shop is the fourth one of the series of workshops. The first one was held in Seoul
(2005), the second in Paris (2008), and the third in Oxford (2011).
The tremendous success of this fourth workshop was only possible due to the
enthusiastic participation of wonderful speakers and authors of this volume. We
are thankful to all of them. We also acknowledge with gratitude the generous sup-
port from the National Institute of Institute of Mathematical Sciences in Daejeon,
the National Research Foundation of Korea, and the European Research Council
Project MULTIMOD.
Habib Ammari, Yves Capdeboscq, Hyeonbae Kang, and Imbo Sim
vii

ContemporaryMathematics
Volume660,2016
http://dx.doi.org/10.1090/conm/660/13255
Wavelet methods for shape perception in electro-sensing
Habib Ammari, St´ephane Mallat, Ir`ene Waldspurger, and Han Wang
Abstract. This paper aims at presenting a new approach to the electro-
sensing problem using wavelets. It provides an efficient algorithm for recog-
nizing the shape of a target from micro-electrical impedance measurements.
Stability and resolution capabilities of the proposed algorithm are quantified
innumericalsimulations.
1. Introduction
The aim of electro-sensing is to learn geometric parameters and material com-
positions of a target via electrical measurements. In this paper, we suppose that
the target is composed of a homogeneous material with a known electrical prop-
erty and focus uniquely on the problem of geometry. Geometric identification of
a target may mean to recognize it from a collection of known shapes (up to rigid
transformations and scaling), or to reconstruct its boundary.
In the recent work [2], an approach based on polynomial basis has been pro-
posed for the far-field measurement system. Using Taylor expansion of the Green
functions, on one hand, the geometric information of the target can be coded in
somefeatures,whicharetheactionofaboundaryintegraloperatoronhomogeneous
polynomials of different orders, and on the other hand the measurement system is
separated into a linear operator relating the features to the data. The features are
then extracted by solving a linear inverse problem and can be used to identify the
target in a database. Unlike other methods (e.g. in electrical impedance tomog-
raphy [9]) which attempt to reconstruct directly the target, this approach is more
effective and computationally efficient in the applications of shape recognition.
From a more general point of view, the problem is to know, given the physical
configurationofthemeasurementsystem,howtochoosethebasisforrepresentation
of features and how to extract them from data for identification. The ill-posedness
inelectro-sensingisinherenttothediffusioncharacterofthecurrentsandcannotbe
removed by a change of basis. Nonetheless, the particularity of a basis can modify
totally the way in which information is organized in the feature and the manner in
which it should be reconstructed.
2010 MathematicsSubjectClassification. Primary65T60,35R30,35B30.
Key words and phrases. Electro-sensing, classification, recognition, shape descriptors,
wavelets.
ThisworkwassupportedbyERCAdvancedGrantProjectMULTIMOD–267184.
(cid:2)c2016 American Mathematical Society
1