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\begin{document}
\vspace{-25 pt}
\begin{firstpage}
\vspace{10 pt}
\summerprogram{Geometric Methods in Inverse Problems and PDE
Control}
{July 16-27, 2001}
{Christopher B. Croke (University of Pennsylvania)\\
Irena Lasiecka (University of Virginia)\\
Gunther Uhlmann (University of Washington)\\
Michael Vogelius (Rutgers University)\\[5pt]
See also http://www.ima.umn.edu/GM/
}
%\vspace{15 pt}
%\end{firstpage}
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An important class of inverse problems involves the
determination of coefficients and parameters in a partial differential equation
from boundary measurements. These problems arise in many areas such as medical
imaging, nondestructive testing, and geophysical prospecting. Control problems
in partial differential equations arise from the need to dampen out vibration
in a structure using actuators located in accessible regions. As part of this
one needs to determine boundary conditions that will lead to a desired state
for the solution of the partial differential equation.
The core analytical questions in inverse problems are those
of uniqueness and stability, while in control problems the issues are of
controllability and stabilizability. In fact, the issues of exact
controllability and uniform stabilizability are intimately connected with
observability/reconstruction estimates. Instead, the issue of approximate
controllability and strong stabilizability are related to unique continuation
properties. Thus, the type of estimates/inequalities one seeks in both fields
are often closely related. These areas have recently seen the use of
differential geometric methods to solve some outstanding questions in those
fields. As an example, a combination of unique continuation results with the
boundary control method has lead to the solution of the inverse problem of
determining a metric of a Riemannian manifold (with boundary) from the dynamic
Dirichlet-to-Neumann map associated with the wave equation. [The
Dirichlet-to-Neumann map for the wave equation determines the boundary distance
function (travel time along geodesics connecting points on the boundary of a
Riemannian manifold). A natural question to ask if one can determine the metric
from this data alone ; this question\end{firstpage}
is at the center of the boundary rigidity
problem studied in Riemannian geometry.] An example from control is the
establishment of sufficient conditions for controllability and stabilization,
expressed in terms of geometry of the domain. There are also interesting
inverse problems for semilinear elliptic partial differential equations where
the uniqueness issue is intimately tied with the geometry of the domain. The
boundary control method used in inverse problems as well as some deep results
in exact controllability are based on the study of propagation of singularities
for partial differential equations. The propagation in turn depends on
geometric properties of the domain. While there has been some remarkable
success, many open problems still remain.
While the importance of differential geometry in the control
of PDE and inverse problems has been established, there has never been a
concerted effort to bring geometers to interact with experts in these other
areas. The workshop goals are to bring together geometers with researchers in
inverse problems and control of PDE to facilitate exchange of ideas and
encourage collaboration; to make tools of differential geometry known to those
working in inverse problems and control, and to open new areas of research in
geometry. The workshop will for instance explore the use of inverse problem-
and control methods to rigidity problems in geometry.
We are planning a 2-week workshop in which Week 1 will
emphasize the Geometric and PDE methods and Week 2 will focus on applications
of these techniques to inverse and control problems. Both weeks will begin with
overview talks designed to set the stage for the workshop. The presence of
several talks of a survey nature will make this workshop an ideal venue for
young researcher wanting to get involved in these fields. The format of the
workshop will consist in invited talks (both of a survey and of a more
specialized nature), two sessions of contributed talks and a panel discussion.
The Proceedings of the summer program will be published as an IMA Volume.
Topics that will be covered in this
program include:
\begin{enumerate}
\item Boundary Rigidity
\item Conjugacy of Geodesic Flows
\item Integral Geometry
\item Wave propagational inverse problems
(Maxwell equations, elasticity, anisotropy)
\item Propagation of singularities (Low
regularity of coefficient and domain)
\item Boundary Control methods
\item Observability/controllability
inequalities by differential geometric methods.
\item Unique continuation
\item Elliptic inverse problems with rough
coefficients
\end{enumerate}
The summer program is divided into 2 weeks:
\begin{itemize}
\item Week 1 (July 16-20): Uniqueness, Propagation of
Singularities, Carleman Estimates
\item Week 2 (July 23-27)
Applications to Inverse Problems/Control
\end{itemize}
\newpage
%=========================================================
%
%=========================================================
%
\begin{center}{\bf WEEK 1 (JULY 16-20): UNIQUENESS,
PROPAGATION OF SINGULARITIES, CARLEMAN ESTIMATES}
\end{center}
\date{Monday, July 16}
\centerline{\bf All talks are in Lecture Hall EE/CS
3-180 unless otherwise noted.}
\talk{8:30 am}
{Coffee and Registration}
{}
{Reception Room EE/CS 3-176}
{}
\vspace{-3ex}
\talk{9:10 am}
{Willard Miller, Fred Dulles,}
{\bf and Irena Lasiecka}
{Welcome and Introduction}
{}
\talk{9:30 am}
{Christopher B. Croke}
{University of Pennsylvania}
{The Boundary Rigidity and Conjugacy Rigidty Problems}
{{\em Abstract}:
This overview talk will introduce the boundary rigidity problem ``To What
extent is the Riemannian metric of a compact Riemannian manifold with boundary
determined by the distances between its boundary points?" After a brief survey
of the early history of this problem it will then cover the relationship
between the boundary rigidity problem and the conjugacy rigidity problem ``To
what extent is a Riemannian metric on a compact manifold without boundary
determined by its geodesic flow." In particular one sees that conjugacy
rigidity of closed manifolds implies the boundary rigidity of nice subdomains.
The rest of the talk is devoted to discussing what is currently known about
these problems. The hope is that this talk will lead into other talks where
more of the techniques of proof will be discussed.
}
\vspace{3ex}
\talk{10:40 am}{Break}{}{Reception Room EE/CS 3-176}{}
\vspace{-3ex}
\talk{11:10 am--\\ 12:00 pm}
{Vladimir Sharafutdinov}
{Sobolev Institute of Mathematics}
{Deformation Boundary Rigidity and Other Applications of the Ray Transform to
Inverse Problems}
{{\em Abstract}:
The ray transform of symmetric tensor fields of rank 2 first arises in the
linearization of the boundary rigidity problem. It tuns out to be also useful
in many other tomography problems. We discuss the recent progress in
deformation boundary rigidity as well as applications of the ray transform to
anisotropic inverse problems for electrodynamic and elastic waves.
}
\vspace{3ex}
\talk{2:00--\\2:50 pm}
{Bruce Kleiner}
{University of Michigan}
{Aymptotic Geometry of Negatively Curved Manifolds}
{}
\talk{4:00 pm}{IMA Tea}{}{IMA East, 400 Lind Hall}{}
\vspace{-5ex} A variety of appetizers and beverages will be served.
%\newpage
%=========================================================
%
%\centerline{\Large Updated Schedule}
\date{Tuesday, July 17}
\centerline{\bf All talks are in Lecture Hall EE/CS
3-180 unless otherwise noted.}
\talk{9:15 am}
{Coffee}{}{Reception Room EE/CS 3-176}{}
\vspace{-3ex}
\talk{9:30 am}
{Michael E. Taylor}
{University of North Carolina}
{The Wave Equation: Analytical Subject and Analytical
Tool}
{{\em Abstract}:
Since it was produced to model vibrations of various objects, from membranes to
electromagnetic fields, the wave equation has been a topic to which many tools
from analysis have been brought to bear. These tools include energy
identities, methods of harmonic analysis, geometrical optics, and symplectic
geometry, to name a few. In turn the study of wave equations has given back to
analysis, particularly the rest of PDE, many dividends. For example, many sharp
results on the Laplace operator fall out of the study of the wave equation.
This talk will survey some of this interplay, both classical and recent.
}
\vspace{3ex}
\talk{10:40 am}{Break}{}{Reception Room EE/CS 3-176}{}
\vspace{-3ex}
\talk{11:10 am--\\ 12:00 pm}
{Dmitri Burago}
{Penn State University}
{Volume/Distance Estimates, Gaussian Measures of Surfaces, and Ellipticity of
Volume Functionals}
{{\em Abstract}:
The talk is based on a joint work with S. Ivanov.
We will discuss the following topics, which happen to be closely related:
1. Optimal fillings: metrics on a manifold (with boundary) that admit no
volume-decreasing perturbations that do not decrease distances between boundary
points. In other words, we will be looking for situations when an inequality
for boundary distance functions of two metrics implies corresponding inequality
for the volumes.
2. What are the restrictions on the Gaussian measures of closed surfaces (or
surfaces with planar boundaries). For instance, under what conditions can one
construct a polyhedron (for instance, a two-dimensional polyhedral surface in
$\Bbb{R}^{4}$) with given areas and directions of faces and no boundary (or a
planar boundary).
3. What is the relationship between different types of ellipticily for surface
area functionals (an area functional is elliptic over Z (resp. R) if regions in
affine planes are area minimizers among all Lipschits chains over Z(R) with the
same boundary.
4. Asymptotic growth of volume for large balls in a periodic metric (a metric
invariant under a co-compact action of an abelian group).
}
\vspace{3ex}
\talk{2:00 pm}
{Gerard Besson}
{CNRS}
{A Margulis Lemma Without Curvature}
{{\em Abstract}:
This is an attempt, in an elementary way, to prove compactness or
precompactness theorems with as little curvature assumptions as possible. The
curvature is being replaced by an asymptotic invariant : the entropy. The talk
is intended to be elementary.
}
\vspace{3ex}
\talk{2:50 pm}{Break}{}{Reception Room EE/CS 3-176}{}
\vspace{-3ex}
\talk{3:20 pm}
{Matthias Eller}
{Georgetown University}
{Unique Continuation for Solutions to Systems of Partial Differential Equations with
Non-Analytic Coefficients}
{{\em Abstract}:
The classical result of unique continuation for solutions to a PDE with non-analytic
coefficients is Hörmander's theorem (1963). This theorem guarantees unique continuation across
strongly pseudo-convex surfaces. It provides optimal uniqueness results for second order
elliptic equations but is not as useful for second order hyperbolic equations and for higher
order equations or systems of equations. Hörmander's theorem is based on Carleman estimates.
These are weighted energy estimates that carry a large parameter.
In 1995 Tataru managed to relax the strong pseudo-convexity condition proving a new uniqueness
result. This result applies to solutions to the wave equation with time independent coefficients
gives unique continuation across non-characteristic surfaces. In other words it yields the same
conclusion than Holmgren's theorem. The proof of Tataru's result relies on a new type of
Carleman estimate.
Later in the 1990s the method of Carleman estimates was applied to certain systems of
mathematical physics. Using the special structure of e.g. the isotropic Maxwell system as well
as the robustness of Carleman estimates with respect to lower order terms uniqueness results
were proved. A similar result was proved for the system of elasticity. A common feature of these
systems is the fact that they are both hyperbolic and that they can be transformed into second
order system which is coupled only through lower order terms.
A more challenging problem is the system of thermo-elasticity which combines hyperbolic and
parabolic feature. However, Isakov was able to prove a uniqueness result when he developed a new
type of Carleman estimate carrying two large parameters. Finally, through a combination of
differential geometry and Carleman estimates one can show uniqueness for solutions to an
anisotropic Maxwell system, i.e. the case where the coefficients are matrices.
}
\vspace{3ex}
%=========================================================
%
\date{Wednesday, July 18}
\centerline{\bf All talks are in Lecture Hall EE/CS
3-180 unless otherwise noted.}
\talk{9:15 am}
{Coffee}{}{Reception Room EE/CS 3-176}{}
\vspace{-3ex}
\talk{9:30 am}
{Gunther Uhlmann}
{University of Washington}
{Inside-Out: Inverse Boundary Problems}
{{\em Abstract}:
In this talk we will survey some of the significant progress obtained in the
last 20 years or so on inverse boundary problems (IBP). The type of IBP we will
discuss concern with the determination of the coefficient(s) of a partial
differential equation on a bounded domain of Euclidean space or, more
generally, a compact Riemannian manifold with boundary, by measuring the
associated Dirichlet to Neumann map.
}
\vspace{3ex}
\talk{10:40 am}{Break}{}{Reception Room EE/CS 3-176}{}
\vspace{-3ex}
\talk{11:10 am--\\ 12:00 pm}
{Werner Ballmann}
{Rheinische Friedrich-Wilhelms-\hfil\break Univ.}
{Isospectral Manifolds}
{{\em Abstract}:
I will talk about the construction of
closed Riemannian manifolds whose Laplacians
have the same spectrum. The ideas go back to
Carolyn Gordon and Dorothee Schueth, I will
present my interpretation of these ideas
and will obtain isospectral metrics on
$S^2\times S^3$.
}
\vspace{3ex}
\talk{2:00 pm}
{Stephen Zelditch}
{Johns Hopkins University}
{Inverse Spectral and Resonance Problems for Analytic Plane
Domains}
{{\em Abstract}:
We will give some inverse spectral results for analytic plane domains with one
symmetry. First, we consider bounded simply connected analytic plane domains
with one isometry that reverses a bouncing ball orbit of a fixed length L. One
may think of the domain as obtained by flipping the graph of a function with
zeros only at two endpoints around the x- axis. Among such domains, we prove
that the Dirichlet (or Neumann) spectrum determines the domain. Second, we
consider the plane minus a mirror symmetric pair of convex analytic obstacles.
We prove that the resonances of the exterior domain in a logarithmic
neighborhood of the real axis determines the obstacles. The method is based on
an exact trace formula suggested by Balian-Bloch and on an analysis of Feynman
diagrams and their amplitudes.
}
\vspace{3ex}
\talk{2:50 pm}{Break}{}{Reception Room EE/CS 3-176}{}
\vspace{-3ex}
\talk{3:20--4:10 pm}
{Allan Greenleaf}
{University of Rochester}
{Global Uniqueness in the Calderon Problem for Conormal Conductivities}
{{\em Abstract}:
In dimensions greater than or equal to three, we consider the Calderon problem
for conductivities that have conormal singularities along a hypersurface, H, of
order less than -2. This allows the conductivities to be differentiable of
order $1 + \epsilon$ but no better. Assuming the hypersurfaces satisfy a
geometric condition, we show that two such conductivities for which the
corresponding operators have the same Cauchy data must be equal. This is joint
work with M. Lassas and G. Uhlmann.
}
\vspace{3ex}
%=========================================================
%
\date{Thursday, July 19}
\centerline{\bf All talks are in Lecture Hall EE/CS
3-180 unless otherwise noted.}
\talk{9:15 am}
{Coffee}{}{Reception Room EE/CS 3-176}{}
\vspace{-3ex}
\talk{9:30 am}
{Roberto Triggiani}
{University of Virginia}
{Differential Geometric Methods in the Control of PDEs (Overview)}
{{\em Abstract}:
Over the past 4-5 years, differential (Riemann) geometric methods have emerged
as a powerful new line of research to obtain general inverse-type, a-priori
inequalities of interest in boundary control theory (continuous
observability/stabilization inequalities) for various classes of PDEs. Their
range of applicability now includes: second order hyperbolic equations;
Schrodinger-type equations; various plate-like equations; systems of
elasticity; very complicated shell models described more below, etc, all with
variable coefficients, where the 'classical' energy methods of the
early/mid-eighties proved inadequate. In all of these PDEs classes, main
features of these differential geometric methods are:
\begin{itemize}
\item 1. they apply to operators with principal part which is allowed to have
variable coefficients (in space) with low regularity, C1;
\item 2. they tolerate energy level terms which are both space- and time-dependent,
and only in L-infinity in time and space;
\item 3. they yield rather general and verifiable sufficient conditions, which may
serve for the construction of many complicated, variable coefficient examples,
as well as for counter-examples (say, in the hyperbolic case, in dimension
greater than 2), even when the control acts on the whole boudary;
\item 4. they provide a good estimate (for some classes, optimal estimate) of the
minimal time for observability in the hyperbolic case, and arbitrary short time
when there is no finite speed of propagation;
\item 5. they combine well with microlocal analysis methods needed for sharp trace
estimates and for shifting topologies, thus producing at the end very general
observability/stabilization results, with variable coefficients and with no
geometric conditions on the observed (controlled) portion of the boundary;
\item 6. ultimately, and with the same effort, they apply to these classes of PDEs
defined on Riemann manifolds.
\end{itemize}
In addition, differential geometric methods have recently provided the
intrinsic language for: (i) modelling the motion of dynamic shells far beyond
the classical approach (rooted in classical geometry), and (ii) performing
observability/stabilization energy methods on their very complicated equations,
for which the classical setting based on Christoffel symbols appears to be
unfeasible. A shell is a curved geometric object which can be modeled as a
system of two PDEs both of hyperbolic type with strong coupling depending on
the curvature: an 'elastic wave-type' equation ('curved system of elasticity')
in the in-plane displacement; and a 'curved Kirchhoff plate-like equation' for
the vertical displacement.
In this talk we shall review a recent Riemann geometric line of research for
PDEs with variable coefficients as above, or else on manifolds, yielding
general Carleman-type estimates and hence observability/stabilization
estimates. They are obtained by using energy methods (multipliers) in a
corresponding natural Riemann metric. These multipliers may be viewed as
extending the 'Carleman multipliers' in the Euclidean setting with constant
principal part (and variable energy level terms). In addition, the combination
of these Riemann methods with microlocal sharp trace estimates will be given.
Three canonical cases are chosen:
\begin{itemize}
\item (i) the control of general second order hyperbolic equations with purely
Neumann BC on both the controlled and the uncontrolled portion of the boundary;
\item (ii) the control of a general plate equation with third order derivatives on
the displacement and first order derivative on the velocity;
\item (iii) boundary stabilization of a shell by a non-linear natural feedback in the
moment and strains, with no geometric conditions on the controlled part of the
boundary.
\end{itemize}
}
\vspace{3ex}
\talk{10:40 am}{Break}{}{Reception Room EE/CS 3-176}{}
\vspace{-3ex}
\talk{11:10 am}
{John Sylvester}
{University of Washington}
{Inverse Theory and an Experiment}
{{\em Abstract}:
We will discuss a linear inverse scattering calculation that
will explain the difficulties with a conventional
electro-magnetic remote sensing experiment and suggest how to
design a better one.
The experiment (circular intensity differential scattering)
deals with determining the presence of chiral material in a
solution using a a single source (a green laser) and a single
receiver. The model is Maxwell's equations and we will show how
to use the Born approximation and a little geometry to decide
where to put the source and the receiver and how to polarize
the beam, so as to best distinguish the small chiral effect
from the much larger effects due to changes in permittivity.
The result suggests that the prevailing experimental technique
for sensing chirality can be improved upon.
}
\vspace{3ex}
\talk{2:00 pm}
{Robert Gulliver}
{University of Minnesota}
{Boundary Control of Wave Equations via Geometry}
{{\em Abstract}:
Consider a hyperbolic PDE with smooth, time-independent coefficients
in $\Omega \times [0,T]$, where $\Omega$ is a smooth, relatively
compact domain in $\Re^n$ or in an n-dimensional manifold:
%
$$ \frac{\partial^2 u}{\partial t^2} = %
\frac{\partial}{\partial x_i}\left(g^{ij}(x)%
\frac{\partial u}{\partial x_j}\right) %
+ {\rm lower \ order \ terms}.$$
%
We have written the coefficients in a form, $g^{ij}(x)$, which
suggests that they should be used to define a Riemannian metric
$ds^2 = g_{ij}(x)\, dx^i\, dx^j$ on $\Omega$, where for each
$x \in \Omega$, $\left(g_{ij}(x)\right)$ is the inverse of the
matrix $\left(g^{ij}(x)\right)$.
Modulo first-order terms, the right-hand side of the PDE
is the Riemannian Laplace operator. We use concepts
such as convexity and geodesic curvature in their Riemannian,
and hence coordinate-independent, versions.
The boundary control problem is whether, for any initial conditions at
$t = 0$, there is a choice of (say) Dirichlet boundary values on
$\partial \Omega \times [0,T]$ so that the solution of the PDE
vanishes for $t \geq T$. Write $T_0$ for the minimum value of
$T$ for which this is possible. It has been shown (under strong
smoothness hypotheses: e.g. Bardos--Lebeau--Rauch) that $T_0$ is
the maximum length of geodesics in $\overline\Omega.$
We will outline a few recent results: (1) If there is a convex
function $v:\Omega \rightarrow [0,K]$ with
$\nabla^2 v \geq 2c\, ds^2$ then $T_0 \leq 2\sqrt{K/c}$; (2) If
$\partial\Omega$ is locally convex, and if the minimizing geodesic
joining any two points of $\partial\Omega$ is unique and nondegenerate,
then $T_0$ is the maximum of the distance between boundary points; and
(3) If $n = 2$, $\partial \Omega$ is locally convex and there are
no closed geodesics in $\Omega$, then $T_0$ is finite, with an estimate.
Result (3) uses Grayson's work on the flow of curves by curvature.
(1) is due to Lasiecka--Triggiani--Yao, with a new proof by Michael
Galbraith; (2) is joint work with Walter Littman; and (3) is joint
work with Littman and Santiago Betel\'u.
}
\vspace{3ex}
\talk{3:10 pm}{Break}{}{Reception Room EE/CS 3-176}{}
\vspace{-3ex}
\centerline{\bf Contributed Talks}
\talk{3:40 pm}
{Jianguo Cao}
{University of Notre Dame}
{The Spectrum of Non-compact Manifolds with Big Ends}
{{\em Abstract}:
In this lecture, we will discuss a new result on the bottom of the Laplace
spectrum for non-compact manifolds with sufficiently large ends. We shall show
that if such a manifold $M$ satisfies a Gromov's length-area linear isoperimetric
inequality then the bottom of the Laplace spectrum of such a manifold is
positive. Furthermore, the Dirichlet problem at infinity is solvable for such
an open space $M$. Examples of such an open space $M$ above can be construed with
prescribed ends. Therefore, the universal cover of such a space $M$ is NOT
necessarily to be diffeomorphic to the Euclidean space. For example, any
Gromov-hyperbolic cone $M$ over a $(n-1)$-dimensional space $N$ with $n > 1$ is a
Gromov-hyperbolic space with a big end.
}
\vspace{3ex}
\talk{4:00 pm}
{Chris Judge}
{Indiana University}
{Behavior of the Laplace Spectrum of Degenerating
Riemannian Manifolds}
{{\em Abstract}:
I will discuss the limiting behavior of eigenfunctions/eigenvalues of the
Laplacian of a family of Riemannian metrics that degenerates on a
hypersurface. In particular, I will describe the transition
from purely discrete to continuous spectrum and give sufficient
conditions for eigenvalue branches to have limits.
The results generalize earlier work concerning the
classical degeneration of hyperbolic surfaces.
}
\vspace{3ex}
\talk{4:20 pm}{Break}{}{Reception Room EE/CS 3-176}{}
\vspace{-3ex}
\talk{4:30 pm}
{Takashi Takiguchi}
{National Defense Academy (of Japan)}
{A Generalization of Helgason's Support Theorem}
{{\em Abstract}:
We disscuss a gengeralization of Helgason's support theorem for the Radon
transform. In this theorem, the assumption of rapid decay of functions is
essential. We restrict this rapid decay condition to an open cone and give a
generalization. We also mention that our generalization is not possible with no
global decay condition, to prove which we construct a counterexample.
}
\vspace{3ex}
\talk{4:50--\\5:10 pm}
{Nicolas Valvidia}
{Wichita State University}
{Uniqueness in Inverse Obstacle Scattering with General Conductive
Boundary Condition}
{{\em Abstract}:
We give uniqueness results for the inverse scattering problem where the
unknown scatterer D is a bounded open set and some coefficients of an
elliptic equation are unknown as well. On the boundary of D conductivity
conditions are prescribed, so we consider a penetrable medium. Our data is
the scattering amplitude A given by one frequency. The proof is based on the
method of singular solutions which is constructive and can be used for
numerical methods.
}
\vspace{3ex}
\newpage
%=========================================================
%
\date{Friday, July 20}
\centerline{\bf All talks are in Lecture Hall EE/CS
3-180 unless otherwise noted.}
\talk{9:15 am}
{Coffee}{}{Reception Room EE/CS 3-176}{}
\vspace{-3ex}
\talk{9:30 am}
{Michael Vogelius}
{Rutgers University}
{Special Purpose Electromagnetic Tomography}
{}
\talk{10:40 am}{Break}{}{Reception Room EE/CS 3-176}{}
\vspace{-3ex}
\talk{11:10 am--\\ 12:00 pm}
{Patrick Eberlein}
{University of North Carolina}
{Nilpotent Groups and the Boundary Geometry of Negatively Curved Manifolds}
{{\em Abstract}:
Let X denote a complete, simply connected Riemannian manifold with sectional
curvatures between two negative constants. U. Hamenstaedt has constructed a
pseudometric on the geodesic boundary of X that is analogous to the Gromov
pseudometric on the geodesic boundary of a simply connected space of
nonpositive curvature. If X is a symmetric space, then this pseudometric is a
metric that is isometric to a left invariant Carnot-Caratheodory metric on the
horospheres of X. In this case the horospheres are isometric a single 2-step
nilpotent, simply connected Lie group N with a left invariant metric of
``Heisenberg type."
Conversely, a simply connected nilpotent Lie group N with an appropriate left
invariant metric becomes a horosphere in a homogeneous space X defined by a
canonical procedure that mimics the symmetric space situation.
For a general X we consider relations between the geometry of its boundary and
the geometry of its horospheres. We give particular attention to the
Hamenstaedt pseudometric and the case that X admits a finite volume quotient
X/G, where G is a noncocompact lattice in the isometry group of X. In the
latter case G determines cocompact lattices G1,G2, ... Gk in horospheres N1,
N2, ... Nk that correspond to the cusps of X/G. The problem of geodesic
conjugacy rigidity for X/G suggests comparison to the problems of geodesic
conjugacy and marked length spectrum rigidity for the compact nilmanifolds
G1/N1, G2/N2, ... Gk/Nk. We discuss these two rigidity problems for 2-step
nilmanifolds G\\N, where N has a left invariant Riemannian metric.
}
\vspace{3ex}
\talk{2:00 pm}
{Gerhard Knieper}
{Ruhr-Universitaet Bochum}
{Asymptotic Geometry on Manifolds of Nonpositive Curvature}
{}
\talk{2:50 pm}{Break}{}{Reception Room EE/CS 3-176}{}
\vspace{-3ex}
\talk{3:20 pm}
{Victor Isakov}
{Wichita State University}
{Uniqueness of the Continuation, Control and
Inverse Problems for the Dynamical Lame System}
{{\em Abstract}:
We consider the classical dynamical Lame system of elasticity theory in a
(bounded) three-dimensional space domain. We show the uniquness of the
continuation across any noncharacteristic surface when the coefficients of the
system are time-independent and finitely smooth (joint work with Eller,
Nakamura and Tatary) and derive approximate controllability at the final
moment of time by boundary (displacement) data. Combining these results, the
boundary control method of Belishev and the results of Nakamura and Uhlmann on
the identification of stationary Lame systems, we obtain the uniqueness of
time-independent elastic parameters for large observation times.
By imposing additional pseudo-convexity type constraints (implying the abscence
of trapped rays) we derive Carleman type estimates and exact boundary
controllability (joint work with Cheng and Yamamoto).
A common tool is a reduction to a princpally diagonal hyperbolic system of
second order and the use of available results for hyperbolic equations.
}
\vspace{3ex}
\talk{4:10 pm}
{Workshop Organizers}
{}
{Discussion Session}
{}
\newpage
%\end{document}
%=========================================================
%
%=========================================================
%
\centerline{\bf WEEK 2 (JULY 23-27): APPLICATIONS TO INVERSE
PROBLEMS/CONTROL}
\date{Monday, July 23}
\centerline{\bf All talks are in Lecture Hall EE/CS
3-180 unless otherwise noted.}
\talk{9:15 am}
{Coffee}{}{Reception Room EE/CS 3-176}{}
\vspace{-3ex}
\talk{9:30 am}
{Masahiro Yamamoto}
{The University of Tokyo}
{Inverse Problems for Hyperbolic Systems
by Carleman Estimates}
{{\em Abstract}: We consider inverse problems of determining
source terms in evolutional systems in mathematical
physics (e.g. the isotropic Lame system, Maxwell's
systems, some fluid dynamics). The inverse problem is
described as follows; $u_{tt} = Lu + R(x,t)f(x)$, where
u is a state variable (e.g., displacement) and L is
a partial differential operator (e.g., the isotropic Lame
operator), R is a given matrix-valued function and
f = f(x) is unknown to be determined from boundary
measurements. We mainly discuss the global uniqueness
and the stability in this inverse problem, by means of
Carleman estimates. This kind of inverse problems
have been studied by Bukhgeim, Isakov, Klibanov and
others. Here we mainly take systems whose
principal parts are coupled and we will show a new way
for establishing the uniqueness and stability.
This work is a joint paper with Oleg Imanuvilov
(Iowa State University, Ames).
}
\vspace{3ex}
\talk{10:20 am}{Break}{}{Reception Room EE/CS 3-176}{}
\vspace{-3ex}
%NOTE: Eller talk has to be rescheduled.
\talk{11:00--\\ 11:50 am}
{Fadil Santosa}
{University of Minnesota/IMA}
{Level Set Method for Inverse Problems and Optimal Design}
{{\em Abstract}:
We consider problems in which the desired unknown is the description of a
region. Examples arising in inverse problems include determination of
scattering obstacles, aperture reconstruction, and electrical impedance
imaging. In optimal design, typical problems are ones where we seek a
geometry which optimizes a certain design objective subject to some
constraints. Both types of problems can be viewed as optimization problem
where the unknown is parameterized by a level set function. In this talk,
the speaker will give an overview of an approach based on the level set
method. It will be followed by examples from different applications.
}
\vspace{3ex}
\talk{2:00 pm}
{Walter Littman}
{University of Minnesota}
{The Scope of the "SUR" Method in Boundary Control}
{{\em Abstract}:
A decade ago a method of boundary control briefly described by "(local)Smoothing + Uniqueness +
Reversibility => controllability" (or "SUR => controllability") was devised by S. Taylor and the
speaker. It essentially consists of reducing the Control question to a Fredholm equation in the
initial manifold t = 0. The method will be illustrated by examples and its advantages and
disadvantages will be compared with those of other methods.
}
\vspace{3ex}
\talk{2:50 pm}{Break}{}{Reception Room EE/CS 3-176}{}
\vspace{-3ex}
\talk{3:30--4:20 pm}
{Matti Lassas}
{University of Helsinki}
{Inverse Boundary Spectral Problems and Gauge
Transformations}
{{\em Abstract}:
We consider the inverse boundary spectral problem for second order elliptic differential
operators on a compact manifold and the inverse problems for the corresponding hyperbolic
equations. The objective of these problems is to reconstruct the unknown manifold and the
operator on it from the boundary spectral data (the boundary, the eigenvalues and the boundary
values of the eigenfunctions), or for the wave equation, from the hyperbolic
Dirichlet-to-Neumann map or the energy flux through the boundary. It turns out that all these
boundary data determine equivalence class of the boundary spectral data in the gauge
transformations $u(x)\mapsto \kappa(x)u(x)$. To analyse these inverse problems simultaneously,
we consider the problem of reconstruction of the gauge-equivalence class of the operator from
the gauge-equivalence class of the boundary data. We also describe some methods to solve this
problem for selfadjoint [1-4], and non-selfadjoint operators [5].
[1] Belishev, M. An appropch to mult dimensional inverse
problems
for the wave equation. Dokl. Akad. Nauk SSSR 297(1987),
524--527.
[2] Belishev, M., Kurylev, Y.
A nonstationary inverse problem for the multidimenional wave
equation ``in the large." Zap. Nauk. Sem. LOMI 165(1987),
21--30.
[3] Belishev, M., Kurylev, Y. To the reconstruction of a
Riemannian
manifold via its spectral data (BC-method). Comm. Part.
Diff. Eq.
17(1992), 767--804.
[4] Katchalov A., Kurylev Y. and Lassas M.
Inverse boundary spectral problems, Chapman/CRC, in print,
290 pp.
[5] Kurylev, Y., Lassas, M.
Gelf'and inverse problem for a quadratic operator pencil.
J. Funct. Anal. 176(2000), 247--263.
Joint work with A. Katchalov and Y. Kurylev.
}
\vspace{3ex}
%=========================================================
%
\date{Tuesday, July 24}
\centerline{\bf All talks are in Lecture Hall EE/CS
3-180 unless otherwise noted.}
\talk{9:15 am}{Coffee}{}{Reception Room EE/CS 3-176}{}
\vspace{-3ex}
%Nicolas Burq cancelled
\talk{9:30 am}
{Mikhail Belishev}
{Steklov Institute of Mathematics}
{An Invariant Look at Inverse Problems: A
Possible Way to Revise the Calderon Problem}
{}
\talk{10:20 am}{Break}{}{Reception Room EE/CS 3-176}{}
\vspace{-3ex}
\talk{11:00--\\ 11:50 am}
{Shari Moscow}
{University of Florida}
{Identification of Conductivity Imperfections of Small Diameter}
{{\em Abstract}:
We show asymptotic formulae for a voltage potential in the presence of small
inhomogeneities. We then discuss several possible techniques using these
formulae to determine the location, size, and/or conductivities of these
imperfections.
Joint work with Habib Ammari and Michael Vogelius.
}
\vspace{3ex}
\talk{2:00 pm}
{Jean-Pierre Puel}
{Universite de Versailles St. Quentin}
{Some Methods for Studying Exact Controllability to
Trajectories in Parabolic Evolution Equations and Applications}
{{\em Abstract}:
We will present the general problem of exact controllability to trajectories for
parabolic dissipative problems with applications to various situations for the heat
equation and possibly to Navier Stokes equations. We will give a general method which
shows how global Carleman inequalities are involved in the resolution of this problem
and we will give some complete results following Imanuvilov's method. As an indirect
application we will comment upon the question of data assimilation for the
corresponding systems.
}
\vspace{3ex}
\talk{2:50 pm}{Break}{}{Reception Room EE/CS 3-176}{}
\vspace{-3ex}
\centerline{\bf Contributed Talks}
\talk{3:30 pm}
{Peter Gibson}
{Technische Universit\"at Darmstadt}
{Discretized Inverse Problems and ``Splitting" of Distributions}
{{\em Abstract}:
Tridiagonal matrices are the discrete analog of a
fundamental class of objects, namely second order (one-dimensional)
differential operators. As such, the spectral analysis of tridiagonal
matrices, both finite and infinite, relates naturally to a range of
physical inverse problems. In particular, the so-called interior point
problem for tridiagonal matrices is motivated by the inverse analysis of
certain physical systems. The interior point problem is ill-posed in
the sense that it's solution is not unique. In studying the geometry
of the solution set, one encounters a curious phenomenon, ```splitting" of
distributions, which will be described in this talk.
}
\vspace{3ex}
\newpage
\talk{3:50 pm}
{S\"onke Hansen}
{Universit\"at Paderborn}
{Propagation of Polarization in Elastodynamics with Residual Stress and
Travel Times}
{{\em Abstract}:
The inverse problem for the hyperbolic system of elastodynamics is to
recover the density, the Lame parameters, and the residual stress tensor from
measurements at the boundary. We show that from such measurements, encoded in the
hyperbolic DN map, one can recover, separately, the travel times of shear and
pressure
waves. This allows us to use known solutions about inverse kinematic problems in
the solution of inverse problems of elastodynamics. We obtain our results using
the methods of microlocal analysis. In particular, we study the propagation of
polarization
in elastodynamics boundary problems. The travel times and, more generally, the
canonical boundary relations associated with S and P waves are shown to be
determined by boundary measurements of high-frequency waves even when
caustics develop along rays.
This is joint work with G. Uhlmann}
\vspace{3ex}
\talk{4:10 pm}{Break}{}{Reception Room EE/CS 3-176}{}
\vspace{-3ex}
\talk{4:20 pm}
{Hyeonbae Kang}
{Seoul National University}
{Boundary Determination of Anisotropic Conductivity via the Dirichlet to Numann
Map}
{{\em Abstract}:
Suppose that $\gamma$ is a $C^m$-smooth
anisotropic conductivity. We show that the derivatives of
$\gamma$ up to order $m-1$ can be
recovered on the boundary of the given domain via the Dirichlet to Numann map
(up to isometry).
}
\vspace{3.5ex}
\talk{4:40 pm}
{V. V. Kryzhniy}
{Kuban State University of Technology}
{Regularization Algorithm of Numerical Inversion of Laplace Transform
Usage in Expansion of Expontial Signals into Partial Fractions}
{
}
%\vspace{3ex}
\talk{5:00--\\5:20 pm}
{Lizabeth V. Rachele}
{Tufts University}
{Uniqueness in Inverse Problems
for Elastic Media}
{{\em Abstract}:
In this talk we consider dynamic inverse problems for bounded,
three-dimensional isotropic and anisotropic elastic media with smoothly
varying density and elastic properties.
Surface data for the inverse problem is modeled by the hyperbolic
\DN \ on a finite time interval.
In the case of isotropic elastodynamics
we have shown that the compressional and shear wave speeds are determined
uniquely by the Dirichlet-to-Neumann boundary data.
A new result is the uniqueness of the third parameter for isotropic
elastodynamics, the density.
Using information about the propagation of ``lower-order polarization,''
we show that the Dirichlet-to-Neumann map determines a certain ray transform.
We then apply results of Pestov and Sharafutdinov, who have
shown that the ray transform for symmetric tensor fields
may be inverted, up to their potential parts, in certain cases.
We next consider elastodynamics for anisotropic and more general elastic
media (for example, isotropic elastic media with residual stress).
We observe that the transformation of the elastic
medium by a change of coordinates, via any diffeomorphism that fixes
the boundary, does not preserve the
symmetry properties of the elasticity tensor $c_{ijkl}$ and so does not
preserve the form of the operator for anisotropic elastodynamics,
\begin{equation}\label{eq:defPani}
(Pu)_i \ = \ a(x) \frac{\partial^2 u_i}{\partial t^2} \ - \
\mathop{\displaystyle\Sigma}\limits_{j,k,l=1}^{3} \partial_{x_{j}}
\Bigl( c_{ijkl}(x)\frac{\partial u_k}{\partial x_l} \Bigr),
\end{equation}
for example.
It follows that change of coordinates is not an obstacle to
uniqueness in the case of anisotropic elastodynamics.
We then consider hyperbolic systems of operators of the form (\ref{eq:defPani})
with the property that the determinant of the principal symbol
can be factored as
$
\deter \sigma_{pr}(P) \ = \
- a(x)^3 \, \mathop{\Pi}_{k} \ \bigl(\tau^2 - \xi^t A_k(x) \xi
\bigr)^{r_k}
\hspace{-.1in}.
\, \ $
We associate the metrics $g_k = (\Sym A_k)^{-1}$ with these operators;
wave propagation occurs along the geodesics of the $g_k. \ $
We show that, under certain natural conditions, the metrics $g_k$
are determined by the \DN \ in the interior,
up to pullback by diffeomorphisms $\psi_k$ that fix the boundary.
That is, the geometry of the wave paths is determined, up to isometry,
by the Dirichlet-to-Neumann map.
We then apply this result to inverse problems for media modeled by certain
hyperbolic systems, and, in particular, to the inverse problem for
isotropic elastodynamics with residual stress,
modelled here by
$$ %\begin{equation}
%\hspace{-.9in}
P u \ = \ \rho\partial_t^2 u - \nabla_x \cdot
\Biggl[ \lambda \, \tr (\nabla_x \otimes u) + \mu (\nabla_x \otimes u) +
(\nabla_x \otimes u)^t \bigl( \mu I + \frac{1}{\rho}\mathcal{T} \bigr) \Biggr],
$$ %\end{equation}
where $\mathcal{T}(x)$ is a symmetric coefficient matrix.
We also show that the nine parameters
for elastodynamics with residual stress
%%the density $\rho, \ $ the Lam\'e parameters
%%$\lambda$ and $\mu, \ $ and the residual stress tensor $\mathcal{T}, \ $
are determined at the boundary by the \DN.
}
\vspace{3ex}
%=========================================================
%
\date{Wednesday, July 25}
\centerline{\bf All talks are in Lecture Hall EE/CS
3-180 unless otherwise noted.}
\talk{9:15 am}
{Coffee}{}{Reception Room EE/CS 3-176}{}
\vspace{-3ex}
\talk{9:30 am}
{Giovanni Alessandrini}
{University of Trieste}
{Size Estimates of Inclusions in an Elastic Body by Boundary
Measurements}
{{\em Abstract}:
We consider the problem of determining an inclusion D in an elastic isotropic body
made from boundary measurements of traction and displacement. We assume that the
inclusion D is made of a different elastic material, either harder or softer, than the
unperturbed specimen. We prove that the volume of D can be estimated, from above and
below, by an easily expressed quantity related to work, only depending on the boundary
traction and displacement.
Joint work with Antonino Morassi and Edi Rosset.
}
\vspace{3ex}
\talk{10:20 am}{Break}{}{Reception Room EE/CS 3-176}{}
\vspace{-3ex}
\talk{11:00--\\ 11:50 am}
{Habib Ammari}
{CNRS}
{Reconstruction of Small Electromagnetic Inhomogeneities}
{{\em Abstract}:
We consider solutions to the (full) time-harmonic Maxwell's equations. Our first goal
is to provide a rigorous derivation of the leading order boundary perturbations
resulting from the presence of a finite number of interior inhomogeneities. Our second
goal is to apply these asymptotic formulae for the purpose of identifying the location
and certain properties of the shapes (polarization tensors) of the inhomogeneities
from boundary measurements. In contrast to a boundary least squares fit to the
measured data, we present a method based on appropriate averaging, using particular
background solutions as weights. We also discuss the reconstruction of the small
inhomogeneities in the time-dependent case and show how to generalize our approach to
treat the case where we are only in possession of boundary measurements on part of the
boundary. Our main idea for solving these inverse problems is to reduce them to
calculations of inverse Fourier transforms.
}
\vspace{3ex}
\talk{2:00 pm}
{James G. Berryman}
{Lawrence Livermore National Laboratory}
{Time-Reversal Acoustics and Maximum-Entropy Imaging}
{{\em Abstract}:
A common problem in acoustics and radar imaging is target
location either from passive or active data collection and inversion.
The passive case is source localization. The active case is reflection
imaging. Time-reversal acoustics has the important characteristic that it
provides a means of determining the eigenfunctions and eigenvalues of the
scattering operator for either of these problems. Each eigenfunction
may be approximately associated with an individual scatterer. The
resulting decoupling of the scattered field from the various targets
is a very important aid to locating the targets, and suggests a number
of imaging and localization algorithms. One of these is
maximum-entropy imaging.
}
\vspace{3ex}
\talk{2:50 pm}{Break}{}{Reception Room EE/CS 3-176}{}
\vspace{-3ex}
%\newpage
\talk{3:20 pm}
{Jin Keun Seo}
{Yonsei University}
{A Real Time Algorithm for the Location
Search of Discontinuous Conductivities with One Measurement}
{{\em Abstract}:
We consider an inverse problem for finding the anomaly of discontinuous electrical conductivity by one
current-voltage observation. We develop a real time algorithm for determining the location of the anomaly.
This new idea is based on the observation of the pattern of a simple weighted combination of the input
current and the output voltage. Combined with the size estimation result, this algorithm gives a good initial
guess for Newton-type schemes. We give the rigorous proof for the location search algorithm. Both the
mathematical analysis and its numerical implementation indicate our location search algorithm is very fast,
stable and efficient.
This is joint work with Ohin Kwon and Jeong-Rock Yoon.
}
\vspace{3ex}
\vspace{3ex}
\talk{4:10 pm}{Break}{}{Reception Room EE/CS 3-176}{}
\vspace{-3ex}
\newpage
\centerline{\bf Contributed Talks}
\talk{4:20 pm}
{Paolo Albano}
{University of Bologna}
{Observability Estimates for a Parabolic-hyperbolic Coupled System}
{{\em Abstract}:
We describe some (sufficient) conditions ensuring that a solution to a
coupled parabolic-hyperbolic system can be observed from the boundary of a
given cilinder. The main technical tool needed for such an analysis are
Carleman estimates with singular (in time) weights. Moreover, we discuss
the special case of operators with constant coefficients.
Joint work with D. Tataru (Northwestern University).
}
\vspace{3ex}
\talk{4:40 pm}
{Seongjai Kim}
{University of Kentucky}
{Smooth Detectors of Linear Phase}
{{\em Abstract}:
Functions with linear phase depend on linear combinations of the independent
variables. For example plane waves are functions of a single linear combination
of the variables. Detection of phase is an important problem in seismic
velocity analysis and ocean acoustic signal processing, amongst other
applications. Robust estimation of phase by local optimization requires the
construction of smooth objective functionals. For this purpose it is useful to
characterize those quadratic functionals of functions with linear phase which
are smooth in the phase: all such smooth phase detectors are
pseudodifferential. Some of these pseudodifferential quadratic forms are
globally convex in the phase, hence permit phase estimation using local smooth
optimization methods.
}
\vspace{3ex}
\talk{5:00--\\5:20 pm}
{Rakesh}
{University of Delaware}
{Inverse Problems for Hyperbolic PDE}
{{\em Abstract}:
We discuss results for one space dimensional problems motivated by formally
determined multidimentsional problems. We prove injectivity of the forward map
(the ``uniqueness"), characterize its range, and construct its inverse, for some
problems. These are partly based on work done with Paul Sacks.
}
\vspace{3ex}
%=========================================================
%
\vspace{3ex}
\date{Thursday, July 26}
\centerline{\bf All talks are in Lecture Hall EE/CS
3-180 unless otherwise noted.}
\talk{9:15 am}
{Coffee}{}{Reception Room EE/CS 3-176}{}
\vspace{-3ex}
\talk{9:30 am}
{Yaroslav Kurylev}
{Loughborough University}
{Geometric Convergence for Manifolds with Boundary and
Reconstruction of a Riemannian Manifold}
{{\em Abstract}:
We consider the problem of a Riemannian manifold reconstruction. The inverse data used is the boundary
spectral data of the Neumann Laplacian or its finite approximation. We describe (pre)compacts in
Gromov-Hausdorff toplogy which provide stable identification of a Riemannian manifold. Under additional
conditions on geometry, we describe a stable algorithm to find a metric approximation to the unknown
manifold.
Bibliography;
[1] Belishev, M. An approach to multidimensional inverse
problems
for the wave equation. {\it Dokl. Akad. Nauk SSSR} {\bf 297}
(1987), 524--527.
[2] Belishev, M., Kurylev, Y. To the reconstruction of a
Riemannian manifold via its
spectral data (BC-method). {\it Comm. Part. Diff. Eq.} {\bf
17} (1992),
no. 5-6, 767--804.
[3] Katsuda A., Kurylev Y. and Lassas M. Stability and
reconstruction
in the inverse
boundary spectral problem. {\it to appear}.
[4] Katchalov A., Kurylev Y. and Lassas M.
Inverse boundary spectral problems, Chapman/CRC (2001),
xx+290 pp.
[5] Kurylev, Y. Multidimensional Gel'fand inverse problem
and
boundary distance map.
{\it in: Inv. Probl. related to Geom. (ed. H.Soga)}, (1997),
1--15.
Joint work with A. Katsuda and M. Lassas.
}
\vspace{3ex}
\talk{10:20 am}{Break}{}{Reception Room EE/CS 3-176}{}
\vspace{-3ex}
\talk{11:00--\\ 11:50 am}
{Leonid N. Pestov}
{Sobolev Institute of Mathematics}
{On Uniqueness and Solvability of Integral Geometry Problems
on a Riemannian Manifold}
{}
\talk{2:00 pm}
{Gen Nakamura}
{Hokkaido University}
{Discretization of Dirichlet to Neumann Map
and Inverse Boundary Value Problem}
{{\em Abstract}: The Dirichlet to Neumann map has been used
in extensive
studies for inverse boundary value problems.
These studies have shown the importance of Dirichlet to
Neumann map for the theoretical studies of the
inverse boundary value problems. However, the Dirichlet to
Neumann map is the observation data by
infinitely many measurements and it is really impractical
observation data. If there is a systematic
way to discretize the Dirichlet to Neumann map and
possibility to use the discretized Dirichlet to
Neumann map for identifying the unknown approximately, all
of these theoretical studies which looked
quite impractical become more meaningful. In my talk, a
general principle of discretizing the Dirichlet
to Neumann will be presented and this principle will be
proved for the inverse boundary value problem for
identifying the potential of the Schrodinger equation. Also,
the rate of the convergence and the accuracy
of the Tikhonov regularized solution to the true solution
will be discussed when we take a discretized
Diriclet to Neumann map with some noise as observation data.
This is a joint work with Jin Cheng.
}
\vspace{3ex}
\talk{2:50 pm}{Break}{}{Reception Room EE/CS 3-176}{}
\vspace{-3ex}
\centerline{\bf Contributed Talks}
\talk{3:30 pm}
{George Avalos}
{University of Nebraska - Lincoln}
{Exact Controllabilty of Structural Acoustic Interactions}
{{\em Abstract}:
In this paper, we work to discern exact controllability properties of two
coupled wave equations, one of which holds on the interior of a bounded open
domain $\Omega $, and the other on a segment $\Gamma _{0}$ of the boundary
$\partial \Omega $. Moreover, the coupling is accomplished through terms on the
boundary. \ Because of the particular physical application involved--the
attenuation of acoustic waves within a chamber by means of active controllers
on the chamber walls--control is to be implemented on the boundary only. We
give here concise results of exact controllability for this system of
interactions, with the control functions being applied through $\partial \Omega
$. In particular, it is seen that for special geometries, control may be
exerted on the boundary segment $\Gamma _{0}$ only. We make use here of
microlocal estimates derived for the Neumann-control of wave equations, as well
as a special vector field which is now known to exist under certain geometrical
situations.
}
\vspace{3ex}
\talk{3:50 pm}
{Michael Galbraith}
{University of Minnesota}
{A Geometric-Optics Proof of a Theorem on Boundary Control Given
a Convex Function}
{{\em Abstract}:
We consider a hyperbolic equation of the form
$$u_{tt}- \sum_{i,j=1}^n \dfrac{\partial}{\partial x_{i}} \left(a_{ij}(x)
\dfrac{\partial u}{\partial x_{j}} \right) + lower\ order\ terms = 0$$ in a
cylindrical region $\Omega \times (0,T)$ in space-time. In the important
paper ``Inverse/Observability Estimates for Second-Order
Hyperbolic Equations with Variable Coefficients'' by Lasiecka, Triggiani
and Yao (see below), those authors use Carleman estimates to
show that the equation can be controlled from a subset of the boundary
$\partial\Omega\times(0,T)$ if there is a positive function $v$
on $\Omega$ which is strictly convex with respect to the metric defined by
the coefficients of the equation, if that convex function has non-positive
outward normal derivative on the uncontrolled part of the boundary. The
time needed for control is a function of the maximum value of $v$ on $\Omega$
and a lower bound on its convexity. In the present paper we will show that
control in the same time is established by a simpler geometric optics
argument---in fact it comes down to a short calculus computation on the
value of $v$ along a bicharacteristic of the equation.
Lasiecka, Irene, Roberto Triggiani and Peng-Fei Yao:
Inverse/observability estimates for second-order hyperbolic
equations with variable coefficients. {\it J. Math. Anal.
Applications} {\bf 235} (1999), 13-57.
}
\vspace{3ex}
\talk{4:10 pm}{Break}{}{Reception Room EE/CS 3-176}{}
\vspace{-3ex}
\talk{4:20 pm}
{Judith Vancostenoble}
{Universit\'e Paul Sabatier}
{Optimality of Energy Estimates for
the Wave Equation with Nonlinear Boundary Velocity Feedbacks}
{{\em Abstract}:
We consider the wave equation damped by a nonlinear
boundary velocity feedback $q(u_t)$.
We consider the case where $q$ has a linear growth at infinity.
We prove that the usual decay rate estimates
proved by M. Nakao, A. Haraux, F. Conrad et al., E. Zuazua, V. Komornik
when $q$ has a polynomial behavior at zero and
by the second author in the general case are in fact optimal
in one space dimension.
More generally, we prove that the energy decays exactly
like the solution of an explicit and simple ordinary differential
equation.
This work is joint Patrick Martinez.
}
\vspace{3ex}
\talk{6:00 pm}{Workshop Dinner}{}{Bangkok Thai Restaurant}{}
%=========================================================
%
\date{Friday, July 27}
\centerline{\bf All talks are in Lecture Hall EE/CS
3-180 unless otherwise noted.}
\talk{10:30am}
{Coffee}{}{Lind Hall 400}{}
\vspace{-3ex}
\centerline{Free day, no workshop events scheduled.}
% \talk{9:30 am}
% {TBA}
% {}
% {}
% {}
%
%
%
% \vspace{3ex}
% \talk{10:20 am}{Break}{}{Reception Room EE/CS 3-176}{}
% \vspace{-3ex}
%
% \talk{11:10--\\ 11:50 am}
% {TBA}
% {}
% {}
% {}
\nlbar
%=========================================================
%
%=========================================================
%
%\newpage
\centerline{WORKSHOP PARTICIPANTS (as of 14 July 2001):}
%\small
%\hrule
%Use ~/gulliver/bin/greformular
%
% \begin{tabbing}
%
% Breuer, H.P.aaaaaaaaaaaaaaasssssssss \= bbbbbbbbbbbbbbAlbert-Ludwigs-University\=MAR
% 20 - MAR 25\kill
% AHN, CHI YOUNG \> Yonsei University \> JUL 9 - 27 \\
% AKTOSUN, TUNCAY \> North Dakota State University \> JUL 15 - 28 \\
% ALBANO, PAOLO \> Universitya di Roma "Tor Vergata" \> JUL 15 - 27 \\
% ALESSANDRINI, GIOVANN \> University of Trieste \> I JUL - 01 JUL \\
% AMMARI, HABIB \> CNRS \> JUL 15 - 28 \\
% AMMARI, KAIS \> University of Nancy-I \> JUL 15 - 27 \\
% ANDERSON, TONY \> University of Minnesota \> JUN 1 - AUG 31 \\
% AVALOS, GEORGE \> Texas Tech University \> JUL 15 - 27 \\
% BALLMANN, WERNER \> Rheinische Friedrich-Wilhelms-Univers \> JUL 14 - 20 \\
% BARCELO, BARTOLOME \> Universidad Autunoma de Madrid \> JUL 15 - 27 \\
% BELISHEV, MIKHAIL \> Steklov Institute of Mathematics \> JUL 14 - 27 \\
% BERRYMAN, JAMES G. \> Lawrence Livermore National Laborator \> JUL 15 - 27 \\
% BESSON, GERARD \> CNRS \> JUL 15 - 27 \\
% BROWN, RUSSELL \> University of Kentucky \> JUL 17 - 27 \\
% BURAGO, DMITRI \> Penn State University \> JUL 15 - 27 \\
% CAGNOL, JOHN \> University Leonard de Vinci \> JUL 15 - 27 \\
% CAO, JIANGUO \> University of Notre Dame \> JUL 15 - 22 \\
% CAPDEBOSCQ, YVES \> Rutgers University \> JUL 15 - 28 \\
% CHAPPA, EDUARDO \> University of Washington \> JUL 15 - 28 \\
% COCKBURN, BERNARDO \> University of Minnesota \> SEP 1 - AUG 31 \\
% COX, ALLEN \> Honeywell, Inc. \> JUL 31 - 31 \\
% CROKE, CHRISTOPHER \> University of Pennsylvania \> JUL 14 - 20 \\
% DOBRANSKI, MICHAEL \> University of Kentucky \> JUL 15 - 28 \\
% DULLES, FRED \> IMA \> SEP 1 - AUG 31 \\
% EBERLEIN, PATRICK \> University of North Carolina \> JUL 15 - 27 \\
% ELLER, MATTHIAS \> Georgetown University \> JUL 15 - 21 \\
% FADDA, GIUSEPPE \> University of Padua \> MAR 1 - AUG 31 \\
% GARRETT, PAUL \> University of Minnesota \> SEP 1 - AUG 31 \\
% GIBSON, PETER C. \> Technische Universitaet Darmstadt \> JUL 15 - 27 \\
% GILG, ALBERT \> Siemens AG \> JUL 29 - 31 \\
% GOODMAN, GERALD \> University of Florence \> APR 1 - AUG 5 \\
% GREENLEAF, ALLAN \> University of Rochester \> JUL 15 - 22 \\
% GULLIVER, ROBERT \> University of Minnesota \> SEP 1 - AUG 31 \\
% HANSEN, SCOTT \> Iowa State University \> JUL 15 - 27 \\
% HANSEN, SOENKE \> University of Paderborn \> JUL 15 - 27 \\
% HORN, MARY ANN \> Vanderbilt University \> JUL 15 - 27 \\
% HOU, L. STEVEN \> Iowa State University \> JUL 15 - 27 \\
% HU, BEI \> University of Notre Dame \> JUL 12 - 27 \\
% HUANG, KAI \> Michigan State University \> JUL 15 - 27 \\
% ISAKOV, VICTOR \> Wichita State University \> JUL 15 - 27 \\
% JUDGE, CHRIS \> Indiana University \> JUL 14 - 21 \\
% KANG, HYEONBAE \> Seoul National University \> JUL 15 - AUG 14 \\
% KETTENRING, JON \> Telcordia Technologies (Bellcore) \> JUL 30 - 31 \\
% KIM, CHANG-WAN \> Korea Advanced Institute of Science \& \> JUL 14 - 27 \\
% KIM, HYEJOO \> Seoul National University \> MAR 17 - AUG 31 \\
% KIM, JEONG-HOON \> Yonsei University \> JUL 9 - 14 \\
% KIM, JINSEOG \> Seoul National University \> MAR 17 - AUG 31 \\
% KIM, JUMI \> Brain 21 Korea/Yonsei University \> JUL 9 - 27 \\
% KIM, SEONGJAI \> University of Kentucky \> JUL 15 - 27 \\
% KLEINER, BRUCE \> University of Michigan \> JUL 15 - 27 \\
% KNIEPER, GERHARD \> Ruhr-Universitaet Bochum \> JUL 15 - 27 \\
% KNUDSEN, KIM \> Aalborg University \> JUL 15 - 28 \\
% KURYLEV, YAROSLAV \> Loughborough University \> JUL 15 - 27 \\
% LASIECKA, IRENA \> University of Virginia \> JUL 15 - 27 \\
% LASSAS, MATTI \> University of Helsinki \> JUL 15 - 27 \\
% LEBIEDZIK, CATHERINE \> University of Virginia \> JUL 15 - 27 \\
% LI, SHUWANG \> University of Minnesota \> JUN 1 - AUG 31 \\
% LIM, HYEONA \> Michigan State University \> JUL 15 - 27 \\
% LIM, MI KYOUNG \> Seoul National University \> JUL 15 - 28 \\
% LITTMAN, WALTER \> University of Minnesota \> JUL 16 - 27 \\
% LU, GUOZHEN \> Wayne State University \> JUL 15 - 29 \\
% LYUBEZNIK, GENNADY \> University of Minnesota \> SEP 1 - AUG 31 \\
% MACKLIN, PAUL \> University of Minnesota \> JUN 1 - AUG 31 \\
% MARTINIZ, PATRICK \> Universite Paul Sabatier \> JUL 15 - 27 \\
% MCDOWALL, STEPHEN \> University of Rochester \> JUL 15 - 28 \\
% MILLER, WILLARD \> IMA \> SEP 1 - AUG 31 \\
% MOELLER, STEEN \> Aalborg University \> JUL 14 - 27 \\
% MOSKOW, SHARI \> University of Florida \> JUL 15 - 27 \\
% MOUBACHIR, MARWAN \> Laboratoire Central des Ponts et Chau \> JUL 15 - 27 \\
% NAKAMURA, GEN \> Hokkaido University \> JUL 15 - 27 \\
% NIGAM, NILIMA \> MCGill University \> JUL 16 - 27 \\
% OLVER, PETER \> University of Minnesota \> SEP 1 - AUG 31 \\
% PESTOV, LEONID N. \> Sobolev Institute of Mathematics \> JUL 15 - 27 \\
% PUEL, JEAN-PIERRE \> Universite de Versailles St. Quentin \> JUL 15 - 25 \\
% RACHELE, LIZABETH \> Tufts University \> JUL 14 - 29 \\
% RAKESH \> University of Delaware \> JUL 15 - 27 \\
% RASMUSSEN, JAN MARTHE \> Technical University of Denmark \> DAL JUL - 01 JUL \\
% SACKS, PAUL \> Iowa State University \> JUL 15 - 27 \\
% SANTOSA, FADIL \> IMA and MCIM \> SEP 1 - AUG 31 \\
% SEO, JIN KEUN \> Yonsei University \> JUL 16 - 27 \\
% SHARAFUTDINOV, VLADIM \> Sobolev Institute of Mathematics \> IR JUL - 01 JUL \\
% SHIN, JAEMIN \>Yonsei University \> JUL 9 - 27 \\
% STEFANOV, PLAMEN D. \> Purdue University \> JUL 16 - 28 \\
% SVERAK, VLADIMIR \> University of Minnesota \> SEP 1 - AUG 31 \\
% SYLVESTER, JOHN \> University of Washington \> JUL 15 - 27 \\
% TAKIGUCHI, TAKASHI \> National Defense Academy \> JUL 15 - 22 \\
% TAMASAN, ALEXANDRU \> University of Washington \> JUL 15 - 27 \\
% TAYLOR, MICHAEL \> University of North Carolina \> JUL 15 - 27 \\
% TRIGGIANI, ROBERTO \> University of Virginia \> JUL 15 - 27 \\
% UHLMANN, GUNTHER \> University of Washington \> JUL 15 - 27 \\
% VALVIDIA, NICOLAS \> Wichita State University \> JUL 18 - 28 \\
% VANCOSTENOBLE, JUDITH \> Universite Paul Sabatier \> JUL 15 - 27 \\
% VOGELIUS, MICHAEL \> Rutgers University \> JUL 15 - 27 \\
% WANG, JENN-NAN \> National Cheng Kung University \> JUL 3 - 31 \\
% WANG, XIAOMING \> Iowa State University \> JUL 15 - 27 \\
% WIENHARD, ANNA \> Rheinische Friedrich-Wilhelms Univers \> JUL 15 - 28 \\
% YAMAMOTO, MASAHIRO \> The University of Tokyo \> JUL 15 - 27 \\
% ZAJIC, TIM \> Lockheed Martin \> SEP 1 - AUG 31 \\
% ZELDITCH, STEPHEN \> Johns Hopkins University \> JUL 15 - 18 \\
%
%
%\end{tabbing}
\begin{center}
\begin{supertabular}{lll}
AHN, CHI YOUNG & Yonsei Univ. (Mathematics) & JUL 9 -- 27 \\
AKTOSUN, TUNCAY & North Dakota State Univ. (Mathematics) & JUL 15 -- 20 \\
ALBANO, PAOLO & Univ. of Bologna (Mathematics) & JUL 15 -- 27 \\
ALESSANDRINI, GIOVANNI & Univ. of Trieste & JUL 21 -- 28 \\
AMMARI, HABIB & CNRS (Center Applied Mathematics) & JUL 16 -- 28 \\
AMMARI, KAIS & Univ. of Nancy-I (Mathematics) & JUL 15 -- 27 \\
AVALOS, GEORGE & Texas Tech Univ. (Mathematics) & JUL 16 -- 27 \\
BALLMANN, WERNER & Rheinische Friedrich-Wilhelms-U ( & JUL 14 -- 20 \\
BARCELO, BARTOLOME & Univ. Autunoma de Madrid & JUL 15 -- 27 \\
BELISHEV, MIKHAIL & Steklov Mathematics & JUL 14 -- 27 \\
BERRYMAN, JAMES G. & Lawrence Livermore National Lab & JUL 15 -- 27 \\
BESSON, GERARD & CNRS & JUL 15 -- 27 \\
BROWN, RUSSELL & Univ. of Kentucky (Mathematics) & JUL 17 -- 27 \\
BURAGO, DMITRI & Penn State Univ. (Mathematics) & JUL 16 -- 18 \\
CAGNOL, JOHN & Paris, France (University Leonardo da Vinci) & JUL 15 -- 27 \\
CAO, JIANGUO & Univ. of Notre Dame (Mathematics) & JUL 15 -- 22 \\
CAPDEBOSCQ, YVES & Rutgers Univ. (Mathematics) & JUL 15 -- 28 \\
CHAPPA, EDUARDO & Univ. of Washington (Mathematics) & JUL 15 -- 28 \\
CROKE, CHRISTOPHER & Univ. of Pennsylvania & JUL 14 -- 20 \\
DOBRANSKI, MICHAEL & Univ. of Kentucky (Mathematics) & JUL 15 -- 28 \\
DULLES, FRED & IMA & SEP 1 -- AUG 31 \\
EBERLEIN, PATRICK & Univ. of North Carolina (Mathematics) & JUL 15 -- 25 \\
ELLER, MATTHIAS & Georgetown Univ. (Mathematics) & JUL 15 -- 21 \\
GALBRAITH, MICHAEL & Univ. of Minnesota (Mathematics) & JUL 16 -- 27 \\
GIBSON, PETER C. & Technische Univ. Darmstadt & JUL 15 -- 27 \\
GREENLEAF, ALLAN & Univ. of Rochester (Mathematics) & JUL 15 -- 22 \\
GULLIVER, ROBERT & Univ. of Minnesota (School Mathematics) & SEP 1 -- AUG 31 \\
HANSEN, SCOTT & Iowa State Univ. (Mathematics) & JUL 15 -- 27 \\
HANSEN, SOENKE & Univ. of Paderborn & JUL 15 -- 27 \\
HORN, MARY ANN & Vanderbilt Univ. (Mathematics) & JUL 15 -- 27 \\
HOU, L. STEVEN & Iowa State Univ. (Mathematics) & JUL 15 -- 27 \\
HUANG, KAI & Michigan State Univ. (Mathematics) & JUL 15 -- 27 \\
ISAKOV, VICTOR & Wichita State Univ. (Mathematics and S) & JUL 15 -- 27 \\
JUDGE, CHRIS & Rawles Hall (Indiana University) & JUL 14 -- 21 \\
KANG, HYEONBAE & Seoul National Univ. (Mathematics) & JUL 15 -- AUG 14 \\
KIM, CHANG-WAN & Korea Advanced Institute of Sci (Mathematics) & JUL 14 -- 27 \\
KIM, HYEJOO & Seoul National Univ. (Statistics) & MAR 17 -- AUG 31 \\
KIM, JEONG-HOON & Brain21 Korea/Yonsei Univ. (Mathematics) & JUL 9 -- 14 \\
KIM, JINSEOG & Seoul National Univ. (Statistics) & MAR 17 -- AUG 31 \\
KIM, JUMI & Yonsei Univ. (Mathematics) & JUL 9 -- 27 \\
KIM, SEONGJAI & Univ. of Kentucky (Mathematics) & JUL 15 -- 27 \\
KLEINER, BRUCE & Univ. of Michigan (Mathematics) & JUL 15 -- 27 \\
KNIEPER, GERHARD & Ruhr-Univ. Bochum & JUL 15 -- 27 \\
KNUDSEN, KIM & Aalborg Univ. & JUL 15 -- 28 \\
KURYLEV, YAROSLAV & Loughborough Univ. (Mathematical Scie) & JUL 15 -- 27 \\
LASIECKA, IRENA & Univ. of Virginia (Mathematics) & JUL 15 -- 27 \\
LASSAS, MATTI & Univ. of Helsinki & JUL 15 -- 27 \\
LEBIEDZIK, CATHERINE & Univ. of Virginia (Mathematics) & JUL 15 -- 27 \\
LIM, HYEONA & Michigan State Univ. (Mathematics) & JUL 15 -- 27 \\
LIM, MI KYOUNG & Seoul National Univ. (Mathematics) & JUL 15 -- 28 \\
LITTMAN, WALTER & Univ. of Minnesota (School Mathematics) & JUL 16 -- 27 \\
MARTINIZ, PATRICK & Univ. Paul Sabatier (Mathematiques Informatique) & JUL 15 -- 27 \\
MCDOWALL, STEPHEN & Univ. of Rochester (Mathematics) & JUL 15 -- 28 \\
MILLER, WILLARD & IMA & SEP 1 -- AUG 31 \\
MOELLER, STEEN & Aalborg Univ. & JUL 14 -- 27 \\
MOSKOW, SHARI & Univ. of Florida (Mathematics) & JUL 15 -- 27 \\
MOUBACHIR, MARWAN & Laboratoire Central des Ponts & JUL 15 -- 27 \\
NAKAMURA, GEN & Hokkaido Univ. & JUL 15 -- 27 \\
NIGAM, NILIMA & Univ. de Montreal & JUL 14 -- 27 \\
PUEL, JEAN-PIERRE & Univ. de Versailles St. Qu & JUL 15 -- 25 \\
RACHELE, LIZABETH & Tufts Univ. (Mathematics) & JUL 14 -- 29 \\
RAKESH & Univ. of Delaware (Mathematical Scie) & JUL 15 -- 27 \\
RASMUSSEN, JAN MARTHEDAL & Technical Univ. of Denmark & JUL 15 -- 27 \\
SACKS, PAUL & Iowa State Univ. (Mathematics) & JUL 15 -- 27 \\
SEO, JIN KEUN & Yonsei Univ. (Mathematics) & JUL 16 -- 27 \\
SHARAFUTDINOV, VLADIMIR & Sobolev Institute (Mathematic) & JUL 15 -- 27 \\
SHIN, JAEMIN & Brain 21 Korea /Yonsei Univ. (Mathematics) & JUL 9 -- 27 \\
STEFANOV, PLAMEN D. & Purdue Univ. (Mathematics) & JUL 16 -- 28 \\
SYLVESTER, JOHN & Univ. of Washington (Mathematics) & JUL 15 -- 27 \\
TAKIGUCHI, TAKASHI & National Defense Academy (Mathematics) & JUL 15 -- 22 \\
TAMASAN, ALEXANDRU & Univ. of Washington (Mathematics) & JUL 15 -- 27 \\
TAYLOR, MICHAEL & Univ. of North Carolina (Mathematics) & JUL 15 -- 27 \\
TRIGGIANI, ROBERTO & Univ. of Virginia (Mathematics) & JUL 15 -- 27 \\
UHLMANN, GUNTHER & Univ. of Washington (Mathematics) & JUL 15 -- 27 \\
VALDIVIA, NICOLAS & Wichita State Univ. (Mathematics) & JUL 15 -- 22 \\
VANCOSTENOBLE, JUDITH & Univ. Paul Sabatier (Laboratoire Mathematiques ) & JUL 15 -- 27 \\
VOGELIUS, MICHAEL & Rutgers Univ. (Mathematics) & JUL 15 -- 27 \\
WANG, JENN-NAN & National Cheng Kung Univ. (Mathematics) & JUL 12 -- 31 \\
WANG, XIAMING & Iowa State Univ. (Mathematics) & JUL 19 -- 21 \\
WIENHARD, ANNA & Rheinische Friedrich-Wilhelms U (Mathematisches) & JUL 15 -- 28 \\
YAMAMOTO, MASAHIRO & The Univ. of Tokyo (Mathematical Scie) & JUL 20 -- 26 \\
ZELDITCH, STEPHEN & Johns Hopkins Univ. (Mathematics) & JUL 15 -- 18 \\
\end{supertabular}
\end{center}
%\newpage
\vspace{4ex}
\begin{center}
POSTDOCTORAL MEMBERS FOR 2000-2001 PROGRAM YEAR
\begin{tabular}{ll}
NAME & PREVIOUS INSTITUTION \\
\hline
ARMENDARIZ, JAVIER & Northwestern University \\
BETELU, SANTIAGO & Univ. Nacional del Centro de la
Prov. de Buenos Aires \\
CARTER, JAMYLLE & UCLA \\
%CHENG, LI-TIEN & UCLA \\
EFENDIEV, YALCHIN & California Institute of
Technology \\
ESEDOGLU, SELIM & Courant Institute of Mathematical
Sciences\\
HAN, BIN & Princeton University \\
HAWA, TAKUMI & Rensselaer Polytechnic Institute
\\
KIM, YONG & University of Wisconsin \\
NOVIKOV, ALEXEI & Stanford University \\
QIAN, JIANLIANG & Rice University\\
\end{tabular}
%\end{center}
\vspace{4ex}
%\begin{center}
%\newpage
POSTDOCTORAL MEMBERSHIPS IN INDUSTRIAL MATHEMATICS
\begin{tabular}{lll}
NAME & PREVIOUS INSTITUTION & INDUSTRIAL AFFILIATION
\\
\hline
CHENG, CHRISTINE & Johns Hopkins University & Telcordia Technologies \\
GOPALAKRISHNAN, JAY & Texas A\& M University & Medtronic \\
KIRILL, DIMITRI & Northwestern University & Motorola
\\
%NIGAM, NILIMA & University of Delaware & Seagate
\\
%VARGHESE, ANTHONY & University of Oxford &
%Endocardial Solutions \\
%ZATEZALO, ALEKSANDAR & Univ. of Minnesota &
%Lockheed Martin
\end{tabular}
\end{center}
\end{document}