Mathematical Methods in Elasticity Imaging

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Contents
Introduction
Chapter 1: Layer Potential Techniques
	1.1 Sobolev Spaces
	1.2 Elasticity Equations
	1.3 Radiation Condition
	1.4 Integral Representation of Solutions to the Lamé System
	1.5 Helmholtz-Kirchhoff Identities
	1.6 Eigenvalue Characterizations and Neumann and Dirichlet Functions
	1.7 A Regularity Result
Chapter 2: Elasticity Equations with High Contrast Parameters
	2.1 Problem Setting
	2.2 Incompressible Limit
	2.3 Limiting Cases of Holes and Hard Inclusions
	2.4 Energy Estimates
	2.5 Convergence of Potentials and Solutions
	2.6 Boundary Value Problems
Chapter 3: Small-Volume Expansions of the Displacement Fields
	3.1 Elastic Moment Tensor
	3.2 Small-Volume Expansions
Chapter 4: Boundary Perturbations due to the Presence of Small Cracks
	4.1 A Representation Formula
	4.2 Derivation of an Explicit Integral Equation
	4.3 Asymptotic Expansion
	4.4 Topological Derivative of the Potential Energy
	4.5 Derivation of the Representation Formula
	4.6 Time-Harmonic Regime
Chapter 5: Backpropagation and Multiple Signal Classification Imaging of Small Inclusions
	5.1 A Newton-Type Search Method
	5.2 A MUSIC-Type Method in the Static Regime
	5.3 A MUSIC-Type Method in the Time-Harmonic Regime
	5.4 Reverse-Time Migration and Kirchhoff Imaging in the Time-Harmonic Regime
	5.5 Numerical Illustrations
Chapter 6: Topological Derivative Based Imaging of Small Inclusions in the Time-Harmonic Regime
	6.1 Topological Derivative Based Imaging
	6.2 Modified Imaging Framework
Chapter 7: Stability of Topological Derivative Based Imaging Functionals
	7.1 Statistical Stability with Measurement Noise
	7.2 Statistical Stability with Medium Noise
Chapter 8: Time-Reversal Imaging of Extended Source Terms
	8.1 Analysis of the Time-Reversal Imaging Functionals
	8.2 Time-Reversal Algorithm for Viscoelastic Media
	8.3 Numerical Illustrations
Chapter 9: Optimal Control Imaging of Extended Inclusions
	9.1 Imaging of Shape Perturbations
	9.2 Imaging of an Extended Inclusion
Chapter 10: Imaging from Internal Data
	10.1 Inclusion Model Problem
	10.2 Binary Level Set Algorithm
	10.3 Imaging Shear Modulus Distributions
	10.4 Numerical Illustrations
Chapter 11: Vibration Testing
	11.1 Small-Volume Expansions of the Perturbations in the Eigenvalues
	11.2 Eigenvalue Perturbations due to Shape Deformations
	11.3 Splitting of Multiple Eigenvalues
	11.4 Reconstruction of Inclusions
	11.5 Numerical Illustrations
Appendix A: Introduction to Random Processes
	A.1 Random Variables
	A.2 Random Vectors
	A.3 Gaussian Random Vectors
	A.4 Conditioning
	A.5 Random Processes
	A.6 Gaussian Processes
	A.7 Stationary Gaussian Random Processes
	A.8 Multi-valued Gaussian Processes
Appendix B: Asymptotics of the Attenuation Operator
	B.1 Stationary Phase Theorem
	B.2 Derivation of the Asymptotics
Appendix C: The Generalized Argument Principle and Rouché’s Theorem
	C.1 Notation and Definitions
	C.2 Generalized Argument Principle
	C.3 Generalization of Rouché’s Theorem
References
Index