## New Analytic and Geometric Methods in Inverse Problems: Lectures Given at the EMS Summer School and Conference Held in Edinburgh, Scotland 2000In inverse problems, the aim is to obtain, via a mathematical model, information on quantities that are not directly observable but rather depend on other observable quantities. Inverse problems are encountered in such diverse areas of application as medical imaging, remote sensing, material testing, geosciences and financing. It has become evident that new ideas coming from differential geometry and modern analysis are needed to tackle even some of the most classical inverse problems. This book contains a collection of presentations, written by leading specialists, aiming to give the reader up-to-date tools for understanding the current developments in the field. |

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### Contents

Metric Geometry | 3 |

Intrinsic Metrics | 7 |

Angles | 11 |

Riemannian Metrics | 12 |

Covariant Derivatives | 19 |

Curvature | 23 |

Comparison Theorems | 28 |

Alexandrov Spaces | 31 |

The Ray Transform | 211 |

Inversion of the Ray Transform | 225 |

The Modified Horizontal Derivative | 238 |

On the Local DirichlettoNeumann Map | 261 |

The Cauchy Data for the Schrödinger Equation | 269 |

Semiclassical Complex Geometrical Solutions | 275 |

EMS Conference Recent Developments in the Wave Field and Diffuse Tomographic Inverse Problems | 281 |

Remarks on the Inverse Scattering Problem for Acoustic Waves | 283 |

Meaning of Sectional Curvature Size and Pinching | 34 |

Lipschitz and GromovHausdorff Convergences | 36 |

LargeScale Geometry | 42 |

Intertwining Operators in Inverse Scattering | 51 |

Direction Dependent Fundamental Solutions of the Operator ᐃx ᐃy | 53 |

Construction of Intertwining Operators for Small Potentials | 63 |

An Expression for the Scattering Matrix and the Inverse Problem | 78 |

Carleman Type Estimates and Their Applications | 93 |

Carleman Type Estimates and PseudoConvexity | 94 |

Use of a Second Large Parameter | 102 |

Uniqueness and Stability in the Cauchy Problem | 104 |

Applications to Inverse Problems | 115 |

Sharp Uniqueness of the Continuation Results | 119 |

Gaussian Beams and Inverse Boundary Spectral Problems | 127 |

Gauge Transformations | 129 |

Boundary Spectral Data and Main Results | 131 |

Spectral Representation of Waves | 133 |

Gaussian Beams | 135 |

The HamiltonJacobi Equation and Transport Equations | 136 |

Riccati Equation | 140 |

Gaussian Beams from the Boundary | 148 |

Construction of Manifold and Boundary Distance Functions | 151 |

Analytic Methods for Inverse Scattering Theory | 165 |

Maximal Functions and Sobolev Spaces | 167 |

Mapping Properties of ᐃ + k21 | 171 |

Faddeevs Greens Function | 174 |

Uniqueness of the Inverse Scattering Problem | 176 |

Born Approximation | 181 |

Ray Transform on Riemannian Manifolds | 187 |

Twodimensional Integral Geometry Problem | 188 |

Some Questions of Tensor Analysis | 195 |

The Linear Sampling Method | 285 |

Piecewise Homogeneous Background Medium | 287 |

Asymptotic Properties of Solutions to 3particle Schrodinger Equations | 291 |

3body Schrödinger Operator | 297 |

Analysis of Ha | 301 |

Timedependent Scattering Theory and Generalized Fourier Transformation for H | 302 |

Main Theorems | 305 |

Stability and Reconstruction in Gelfand Inverse Boundary Spectral Problem | 309 |

Main Results Stability | 311 |

Main Results Approximate Reconstruction | 314 |

Construction of a Finite enet | 315 |

Concluding Remarks | 320 |

Uniqueness in Inverse Obstacle Scattering | 323 |

Global Uniqueness | 325 |

Local Uniqueness | 330 |

Geometric Methods for Anisotopic Inverse Boundary Value Problems | 337 |

Scalar Anisotropic Inverse Conductivity Problem | 339 |

Linearization | 342 |

Constrained Anisotropic Problems | 343 |

Laplacians on Forms | 344 |

Linear Elasticity | 345 |

Maxwells Equations | 346 |

Symbols and PseudoDifferential Operators | 347 |

Applications of the OscillatingDecaying Solutions to Inverse Problems | 353 |

Pointwise Determination at the Boundary from the Localized DirichlettoNeumann Map | 354 |

Identification of a Polygonal Cavity in a Conductive Medium | 360 |

Identification of Inclusion in a Conductive Medium | 363 |

TimeDependent Methods in Inverse Scattering Theory | 367 |

The Proofs | 373 |