Reconstruction of Small Inhomogeneities from Boundary MeasurementsSpringer, 30 Ağu 2004 - 242 sayfa This is the first book to provide a systematic exposition of promising techniques for the reconstruction of small inhomogeneities from boundary measurements. In particular, theoretical results and numerical procedures for the inverse problems for the conductivity equation, the Lamé system, as well as the Helmholtz equation are discussed in a readable and informative manner. The general approach developed in this book is based on layer potential techniques and modern asymptotic analysis of partial differential equations. The book is particularly suitable for graduate students in mathematics. |
İçindekiler
21 Some Notations and Preliminaries | 11 |
22 Layer Potentials for the Laplacian | 14 |
23 Neumann and Dirichlet Functions | 30 |
24 Representation Formula | 34 |
25 Energy Identities | 39 |
31 Definition | 42 |
32 Uniqueness Result | 45 |
33 Symmetry and Positivity of GPTs | 47 |
71 Asymptotic Expansion in Free Space | 128 |
72 Properties of EMTs | 132 |
73 EMTs Under Linear Transforms | 140 |
74 EMTs for Ellipses | 143 |
75 EMTs for Elliptic Holes and Hard Ellipses | 147 |
81 Full Asymptotic Expansions | 151 |
91 Detection of EMTs | 158 |
92 Representation of the EMTs by Ellipses | 163 |
34 Bounds for the Polarization Tensor of PólyaSzegö | 49 |
35 Estimates of the Weighted Volume and the Center of Mass | 52 |
41 Energy Estimates | 67 |
42 Asymptotic Expansion | 71 |
43 Derivation of the Asymptotic Formula for Closely Spaced Small Inclusions | 76 |
51 Constant Current Projection Algorithm Reconstruction of Single Inclusion | 80 |
52 Quadratic Algorithm Detection of Closely Spaced Inclusions | 85 |
53 LeastSquares Algorithm | 88 |
54 Variational Algorithm | 89 |
55 Linear Sampling Method | 91 |
61 Layer Potentials for the Lamé System | 109 |
62 Kelvin Matrix Under Unitary Transforms | 113 |
63 Transmission Problem | 115 |
64 Complex Representation of Displacement Vectors | 123 |
93 Detection of the Location | 164 |
94 Numerical Results | 167 |
101 Existence and Uniqueness of a Solution | 179 |
111 Preliminary Results | 185 |
112 Representation Formulae | 188 |
121 Asymptotic Expansion | 196 |
131 Asymptotic Expansion of a Weighted Combination of VoltagetoCurrent Pairs | 207 |
A1 Theorem of Coifman McIntosh and Meyer | 215 |
A2 Continuity Method | 216 |
A3 Collectively Compact Operators | 217 |
A42 Uniqueness of Disks With One Measurement | 220 |
Blank Page | 239 |
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Sık kullanılan terimler ve kelime öbekleri
algorithm apply associated assume asymptotic expansion asymptotic formula background boundary bounded bounded Lipschitz domain chapter compact complete compute conclude condition conductivity consider constant contains corresponding defined definition denote depends derive Detection determine disk domain elastic ellipse equation equivalent estimate exists fact formula Fourier function given GPT’s harmonic hence holds identity inclusions independent inequality integral invertible Kºp Lamé layer potentials Lemma linear Lipschitz matrix measurements method Neumann Note Observe obtain operator Öſ pair polarization tensor positive problem Proof properties prove reconstruction relation representation respectively satisfies solution solving space Step Suppose symmetric takes term Theorem transform values vector yields zero