THEMATIC PROGRAMS

Thematic Program in Partial Differential Equations

Symposium on Inverse Problems

October 1 - 4, 2003

Organizer: A. Nachman (Toronto)

Speakers: C. Croke (Pennsylvania), V. Isakov (Wichita State), H. Isozaki (Tokyo Metropolitan), S. Kurylev (Loughborough), G. Uhlmann (Washington)

The symposium takes place as part of the 2003-04 Theme Year in Partial Differential Equations hosted by the Fields Institute for Mathematical Sciences in Toronto, Canada.
The lectures are aimed at a general audience, and aim to present some of the beautiful questions in the mathematical theory of Inverse Problems, the various approaches brought to bear on these questions (microlocal analysis, differential geometry, control theory, Carlemann estimates, scattering theory) and the recent resulting breakthroughs in the field.

Audio and Slides from the symposium are available here.

Schedule:

Wednesday, October 1, 2003
11:30 - 12:30	Adrian Nachman Introduction to Inverse Problems
12:30 - 2:10	Lunch Break
2:10 - 3:30	Hiroshi Isozaki Inverse Problems and Hyperbolic Manifolds: I
3:30 - 4:00	Afternoon Tea
Thursday, October 2, 2003
9:30 - 10:50	Christopher Croke The Boundary Rigidity Problem: I
10:55 - 11:10	Coffee Break
11:10 - 12:00	Gunther Uhlmann The Dirichlet to Neumann Map and the Boundary Distance Function: I
12:00 - 2:10	Lunch Break
2:10 - 3:30	Victor Isakov Carleman Estimates Uniqueness and Stability in the Cauchy Problem: I
3:30 - 4:00	Afternoon Tea
4:10 - 6:00	Slava Kurylev Gel'fand Inverse Boundary Problem in Multidimensions: I and II
Friday, October 3, 2003
10:10 - 11:30	Gunther Uhlmann The Dirichlet to Neumann Map and the Boundary Distance Function: II
11:35 - 12:25	Slava Kurylev Gel'fand Inverse Boundary Problem in Multidimensions: III
12:25 - 2:15	Lunch Break
2:15 - 3:35	Christopher Croke The Boundary Rigidity Problem: II
3:35 - 4:00	Tea Break
Saturday, October 4, 2003
9:30 - 10:50	Hiroshi Isozaki Inverse Problems and Hyperbolic Manifolds: II
10:50 - 11:10	Coffee Break
11:10 - 12:30	Victor Isakov Uniqueness and Stability in the Cauchy Problem: II Applications to Inverse Problems and Optimal Control

Abstracts:

To view all abstracts in pdf format, click here.

Christopher Croke (Pennsylvania)
The Boundary Rigidity Problem
This series of talks is an introduction to the boundary rigidity problem. A long term goal would be to determine a Riemannian metric on a manifold with boundary from the distances between its boundary points. This would have applications in areas from medical imaging to
seismology. Unfortunately, it is not always possible to do this. The boundary rigidity problem is to determining when it is possible. We
consider Riemannian manifolds (M,B,g) with boundary B and metric g. We let d, the "boundary distance function", be the real valued function on BxB giving the distance in M (i.e. the "chordal distance") between boundary points. The question is whether there is a unique g for a given d (up to an isometry which leaves the boundary fixed). We will talk about the various conjectures, theorems and counter examples that have been developed over the years.

Victor Isakov (Wichita State)
Carleman Estimates
We will discuss weighted $L^2$-estimates of solutions of general partial differential equations of second order. We introduce the so-called pseudo-convexity condition for the weight function and give examples of such functions for elliptic and hyperbolic operators. Then we formulate Carleman estimates with boundary terms, and give an elementary proof for a particular case of the Helmholtz operator. This proof illustrates the general case and gives new estimates with constants not depending on the wave number.

Uniqueness and stability in the Cauchy problem
Here, following the classical Carleman idea, we apply Carleman estimates to derive uniqueness results and stability estimates of the continuation of solutions to partial differential equations. We give the counterexample of Fritz John which shows importance of pseudo-convexity and outline recent progress in increased stability for the Helmholtz equation.

Applications to inverse problems and optimal control
By studying an "adjoint" problem we show that uniqueness of the continuation implies the so-called approximate controllability by solutions of PDE. For hyperbolic equations we will derive from Carleman estimates a stronger property called an exact controllability and its dual which is a Lipschitz stability estimate of the initial data by the lateral boundary data. Finally we outline the method of Bukhgeim-Klibanov which under certain conditions transform Carleman estimates into uniqueness results for unknown source terms and coeffieints of hyperbolic PDE. In conclusion we discuss open problems and further possibilities of Carleman estimates.

Hiroshi Isozaki (Tokyo Metropolitan)
Inverse Problems and Hyperbolic Manifolds

Slava Kurylev (Loughborough)
Gel'fand Inverse Boundary Problem in Multidimensions
Gel'fand inverse boundary problem consists of determination of an unknown elliptic operator on a bounded domain/manifold from the restriction to the boundary of its resolvent kernel. This kernel is assumed to be known, as a meromorphic operator-valued function, for all values of the spectral parameter. In our lectures we concentrate on the case of a Laplace operator on an unknown Riemannian manifold. Using the geometric version of the Boundary Control method we show that the Gel'fand inverse boundary problem is uniquely solvable and provide a procedure to recover the manifold and the metric. Using the theory of geometric convergence, we also study geometric conditions on an unknown manifold to guarantee stability of this inverse problem.

Adrian Nachman (Toronto)
Introduction to Inverse Problems
This talk will give a graduate level introduction to the inverse boundary value problem of Calderon, its applications to medical and geophysical imaging, and its analysis using exponentially growing solutions of an elliptic equation. Several open problems in the field will also be presented. In the anisotropic case, the problem becomes one of recovering a metric in a Riemannian manifold with boundary from the corresponding Dirichlet-to-Neumann map for the Laplace-Beltrami operator. This leads to beautiful connections to differential geometry which will be further brought out in several of the lectures in the Symposium.

Gunther Uhlmann (Washington)
The Dirichlet to Neumann Map and the Boundary Distance Function
We will consider in these introductory lectures the inverse boundary problem of Electrical Impedance Tomography (EIT). This inverse method consists in determining the electrical conductivity inside a body by making voltage and current measurements at the boundary. The boundary information is encoded in the Dirichlet-to-Neumann (DN) map and the inverse problem is to determine the coefficients of the conductivity equation (an elliptic partial differential equation) knowing the DN map. We will also consider the anisotropic case which can be formulated, in dimension three or larger, as the question of determining a Riemannian metric from the associated DN map. We will discuss a connection of this latter problem with the boundary rigidity problem which will be the topic of C. Croke's lectures. In this case the information is encoded in the boundary distance function which measures the lengths of geodesics joining points in the boundary of a compact Riemannian manifold with boundary.