
THEMATIC PROGRAMS 

July 20, 2024  
Thematic Program in Partial Differential EquationsSymposium on Inverse Problems

Wednesday, October 1, 2003  
11:30  12:30 
Adrian Nachman 
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 
Friday, October 3, 2003  
10:10  11:30 
Gunther Uhlmann 
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 
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 socalled
pseudoconvexity 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 pseudoconvexity 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 socalled 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 BukhgeimKlibanov
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 operatorvalued 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 DirichlettoNeumann
map for the LaplaceBeltrami 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 DirichlettoNeumann (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.