logo New Analytic and Geometric Methods in Inverse Problems
EuroSummerSchool
24 July to 2 August 2000
Heriot-Watt University, Edinburgh
Recent Developments in the Wave Field and Diffuse Tomographic Inverse Problems
EuroConference
3-5 August 2000
Heriot-Watt University, Edinburgh
Satellite conference for ICMP,2000
Home Page | Domestic and Financial Details | Timetable | Scientific Programme
MELIN
Suggesting reading list on distribution theory and partial differential equations

The material can be taken from the first 7 chapters in

[H] Hörmander, L., The analysis of linear partial differential operators I, Springer Verlag 1983

or from

[F] Friedlander, G. & Joshi, M., The introduction to the theory of distributions (2nd edn), Cambridge University Press 1998.

The material recommended to be taken from one or both of these books includes the following:
  1. The definition of distribution and basic operations on distributions.
  2. The Fourier transform and its inverse on tempered distributions.
  3. Homogeneous distributions (Sec. 4.2 in [F] and the more extensive Sec. 3.2 in [H]).
  4. Distributions with compact support.
  5. The fundamental solution of a partial differential operator with constant coefficients (Secs. 3.3, 4.4, 6.2 in [H] and Sec. 5.4 of [F]).
  6. The kernel theorem of Schwartz (Theorem 5.2.1 in [H] and Theorem 6.1.1 in [F]).
  7. Sobolev spaces and Plancherel's theorem (Chap 9 in [F] and Sec. 7.1 in [H]).
Back to Timetable

UHLMANN

Abstract
In these lectures we will consider the problem of determining anisotropic conductivities of a body by making current and voltage measurements on the boundary of the body. Anisotropic means here that the conductivity depends on direction as well as position. Muscle tissue in the human body is a prime example of an anisotropic conductor.
The rough outline of the course is as follows:
1) An introduction to Electrical Impedance Tomography.
2) Review of the isotropic case.
3) Determining the boundary values of the conductivity in the anisotropic case.
4) The linearized anisotropic problem.
5) The two dimensional anisotropic problem.
6) The three dimensional case.
Suggested reading list
The first reference is the most crucial. The others can be treated as supplementary
  1. G. Uhlmann, Developments in inverse problems since Calderon's foundational paper, chapter 19 of the book Harmonic analysis and partial differential equations, edited by M. Christ, C. Kenig and C. Sadosky, University of Chicago Press.
  2. G. Alessandrini and R. Gaburro, Determining conductivity with special anisotropy by boundary measurements.
  3. R. Brown and G. Uhlmann, Uniqueness in the inverse conductivity problem for nonsmooth conductivities in two dimensions, Comm. PDE 22 (1997), 1009-1027.
  4. R. Kohn and M. Vogelius, Identification of an unknown conductivity by means of measurements a the boundary, in Inverse Problems, edited by D. McLaughlin, SIAM-AMS Proc. No 14, Providence (1984), 113-123.
  5. J. Lee and G. Uhlmann, Determining anisotropic real-analytic conductivities by boundary measurements, Comm. Pure and Appl. Math., 42 (1989), 1097-1112.
  6. M. Lassas and G. Uhlmann, On determining a Riemannian manifold from the Dirichlet-to-Neumann map, preprint.
  7. A. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem. Ann. of Math. 143 (1996), 143-171.
  8. Z. Sun and G. Uhlmann, Inverse problems in quasilinear anisotropic media, Amer. J. Math. 119 (1997), 771-797.
  9. J. Sylvester, An anisotropic inverse boundary value problem, Comm. Pure and Appl. Math. 43 (1990), 201-232.
  10. J. Sylvester and G. Uhlmann, Inverse problems in anisotropic media, Contemp. Math 122 91991), 105-117.
  11. J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math. 125 (1987), 153-169.
Back to Timetable

ISAKOV

Introductory literature
  1. Hormander, L., Linear Partial Differential Operators, Springer-Verlag, 1963. Sections 1.7, 2.4, 8.1-8.4.
  2. Isakov, V., Inverse problems for PDE, Springer-Verlag, 1998. Sections 3.1-3.4, 8.2.
Back to Timetable

LASSAS & KATCHALOV

Boundary control method

Abstract
Inverse problems of recovering parameters of media by boundary measurement is one of the most important and challenged problem in applications. In this connection we want to remember that most of Geophysical investigation of searching minerals, oil and gas, nondestructive evaluation of materials and many other applied problems are just of the kind of problems. The boundary control method is one of the most advanced method to solve the problem. In the lecture course we give a solution of well known Gelfand problem of recovering of an elliptic selfadjoint second order differential operator on a differential manifold via its boundary spectral data. We also show how to use the solution and method in general to different applied problems. The method of solution based on investigation the wave equation related to the operator. In the process of reconstruction the manifold and the operator we widely use ideas of differential (Riemannian) geometry.

The main concepts which lie at the background of the method are
  1. We can calculate the inner products of waves or solutions of the wave equation via boundary spectral data.
  2. The concept of slice generated by a boundary set.
  3. The concept of boundary distance functions.
  4. A special class of smooth solutions of the wave equation (Gaussian beams or quasiphotons) which which remind a particlesin their properties. The type of solutions give an answer on one of the Einstein question.
To understand the lecture course it is necessary to have some basic background in Riemannian geometry and partial differential equations. Most of necessary results will be given in other lecture courses of the Summer School. To understand the course the student should know the basic ideas of Riemannian geometry. The ideas will be presented in other courses on the Summer School. It is also recommended to know the basic ideas of initial--boundary value problems for the wave equation. The main ideas can be found in text books

L.C.Evans Partial Differential Equations Grad. Studies in Math., v.19, AMS, 1998 (Ch.7, Sect. 2.1-2.4)
O.A.Ladyzhenskaya The Boundary Value Problems of Math. Physics Springer, 1985. (ch.4, Sect. 1-4 and 7)

Back to Timetable

BURAGO

Riemannian geometry

Prerequisites
To get the most out of this course, participants should have some familiarity with the following. (Items marked by * are desirable, but not essential)
  1. Metric topology: Metric spaces; completeness; compactness; Arzelà-Ascoli lemma; Hausdorff distance*.
  2. Smooth topology: manifolds and tangent vectors; inverse and implicit function theorems. Riemannian manifolds - just the definition. Lie brackets*.
  3. ODE: Existence, uniqueness, and smooth dependence on initial conditions and parameters for solutions of ODE.
  4. Hyperbolic geometry: Poincare model (basics)*.
  5. Differential geometry: curvature (for curves in R3; shape operator*, principal curvatures*, mean and Gaussian curvatures* for surfaces in R3
Topics covered
There follows a preliminary list. Some topics will be dealt with in greater depth than others.
  1. Length structures and intrinsic metrics. Induced metrics. Shortest paths. Isometries, gluing, polyhedral spaces, quotients, products, cones. Angles and directions.
  2. Curvature bounds. Examples. Globalization. Geometric meaning of curvature bounds.
  3. Riemannian length structures. Normal coordinates. Conjugate points and cut locus. Riemannian connection and curvature. Jacobi Equation. Cartan-Alexandrov-Toponogov Theorem.
  4. Manifolds of (non)positive and (non)negative curvature. Equidistant variations; Busemann (distance-like) functions.
  5. Gromov-Hausdorf distance. Compactness theorems and collapse. Large-scale geometry.
If time permits, maybe also: Ricci curvature bounds, or curvature free volume estimates.

Back to Timetable

This page last updated 29 June 2000
Back to ICMS Website