Geometric Inverse Problems: With Emphasis on Two Dimensions

✍ Scribed by Gabriel P. Paternain, Mikko Salo, Gunther Uhlmann

• Publisher: Cambridge University Press
• Year: 2023
• Tongue: English
• Leaves: 369
• Series: Cambridge Studies in Advanced Mathematics 204
• Category: Library

✦ Synopsis

This up-to-date treatment of recent developments in geometric inverse problems introduces graduate students and researchers to an exciting area of research. With an emphasis on the two-dimensional case, topics covered include geodesic X-ray transforms, boundary rigidity, tensor tomography, attenuated X-ray transforms and the Calderón problem. The presentation is self-contained and begins with the Radon transform and radial sound speeds as motivating examples. The required geometric background is developed in detail in the context of simple manifolds with boundary. An in-depth analysis of various geodesic X-ray transforms is carried out together with related uniqueness, stability, reconstruction and range characterization results. Highlights include a proof of boundary rigidity for simple surfaces as well as scattering rigidity for connections. The concluding chapter discusses current open problems and related topics. The numerous exercises and examples make this book an excellent self-study resource or text for a one-semester course or seminar.

✦ Table of Contents

Contents
Foreword -- András Vasy
Preface
Acknowledgements
1. The Radon Transform in the Plane
1.1 Uniqueness and Stability
1.2 Range and Support Theorems
1.3 The Normal Operator and Singularities
1.4 The Funk Transform
2. Radial Sound Speeds
2.1 Geodesics of a Radial Sound Speed
2.2 Travel Time Tomography
2.3 Geodesics of a Rotationally Symmetric Metric
2.4 Geodesic X-ray Transform
2.5 Examples and Counterexamples
3. Geometric Preliminaries
3.1 Non-trapping and Strict Convexity
3.2 Regularity of the Exit Time
3.3 The Geodesic Flow and the Scattering Relation
3.4 Complex Structure
3.5 The Unit Circle Bundle of a Surface
3.6 The Unit Sphere Bundle in Higher Dimensions
3.7 Conjugate Points and Morse Theory
3.8 Simple Manifolds
4. The Geodesic X-ray Transform
4.1 The Geodesic X-ray Transform
4.2 Transport Equations
4.3 Pestov Identity
4.4 Injectivity of the Geodesic X-ray Transform
4.5 Stability Estimate in Non-positive Curvature
4.6 Stability Estimate in the Simple Case
4.7 The Higher Dimensional Case
5. Regularity Results for the Transport Equation
5.1 Smooth First Integrals
5.2 Folds and the Scattering Relation
5.3 A General Regularity Result
5.4 The Adjoint I_A
6. Vertical Fourier Analysis
6.1 Vertical Fourier Expansions
6.2 The Fibrewise Hilbert Transform
6.3 Symmetric Tensors as Functions on SM
6.4 The X-ray Transform on Tensors
6.5 Guillemin–Kazhdan Identity
6.6 The Higher Dimensional Case
7. The X-ray Transform in Non-positive Curvature
7.1 Tensor Tomography
7.2 Stability for Functions
7.3 Stability for Tensors
7.4 Carleman Estimates
7.5 The Higher Dimensional Case
8. Microlocal Aspects, Surjectivity of I~0
8.1 The Normal Operator
8.2 Surjectivity of I_0
8.3 Stability Estimates Based on the Normal Operator
8.4 The Normal Operator with a Matrix Weight
9. Inversion Formulas and Range
9.1 Motivation
9.2 Properties of Solutions of the Jacobi Equation
9.3 The Smoothing Operator W
9.4 Fredholm Inversion Formulas
9.5 Revisiting the Euclidean Case
9.6 Range
9.7 Numerical Implementation
10. Tensor Tomography
10.1 Holomorphic Integrating Factors
10.2 Tensor Tomography
10.3 Range for Tensors
11. Boundary Rigidity
11.1 The Boundary Rigidity Problem
11.2 Boundary Determination
11.3 Determining the Lens Data and Volume
11.4 Rigidity in a Given Conformal Class
11.5 Determining the Dirichlet-to-Neumann Map
11.6 Calderón Problem
11.7 Boundary Rigidity for Simple Surfaces
12. The Attenuated Geodesic X-ray Transform
12.1 The Attenuated X-ray Transform in the Plane
12.2 Injectivity Results for Scalar Attenuations
12.3 Surjectivity of I
12.4 Discussion on General Weights
13. Non-Abelian X-ray Transforms
13.1 Scattering Data
13.2 Pseudo-linearization Identity
13.3 Elementary Background on Connections
13.4 Structure Equations Including a Connection
13.5 Scattering Rigidity and Injectivity for Connections
13.6 An Alternative Proof of Tensor Tomography
13.7 General Skew-Hermitian Attenuations
13.8 Injectivity for Connections and Higgs Fields
13.9 Scattering Rigidity for Connections and Higgs Fields
13.10 Matrix Holomorphic Integrating Factors
13.11 Stability Estimate
14. Non-Abelian X-ray Transforms II
14.1 Scattering Rigidity and Injectivity Results for gl(n,C)
14.2 A Factorization Theorem from Loop Groups
14.3 Proof of Theorems 14.1.1 and 14.1.2
14.4 General Lie Groups
14.5 Range of I{A,0} and I_{A,⊥}
14.6 Surjectivity of I_{A,0} and I_{A,⊥}
14.7 Adding a Matrix Field
15. Open Problems and Related Topics
15.1 Open Problems
15.2 Related Topics
References
A B
C D
E F G
H
I J K
L M
N
O P
Q R S
T U V W Z
Index