Inverse Problems in Analysis and Geometry
Monday 1st August 2011 to Friday 5th August 2011
09:00 to 09:55  Registration  
09:55 to 10:00  Welcome from John Toland (INI Director Designate)  INI 1  
10:00 to 10:45  Using the formalism of inverse problem in the theory of integrable models  INI 1  
10:45 to 11:15  Morning Coffee  
11:15 to 12:00 
U Leonhardt (University of St Andrews) Transformation optics: cloaking and perfect imaging
The field of transformation optics and metamaterials has been named by Science as one of the top ten research insights of the last decade (in fact, it was the only one in physics and engineering that made it into the top ten). What is it? In transformation optics manmade dielectric materials, called metamaterials, are used to implement a coordinate transformation of space (or in some cases of space and time). What can it do? For example, such transformation devices can make things invisible. They can also create perfect images with a resolution no longer limited by the wave nature of light.

INI 1  
12:30 to 13:30  Lunch at Wolfson Court  
14:00 to 14:45  Inverse gravimetry approach to attenuated tomography  INI 1  
14:45 to 15:15  Afternoon Tea  INI 1  
15:15 to 16:15  Free time  
16:15 to 17:00 
On approximate cloaking by nonsingular transformation media
We give a comprehensive study on regularized approximate electromagnetic cloaking in the spherical geometry via the transformation optics approach. The following aspects are investigated: (i) nearinvisibility cloaking of passive media as well as active/radiating sources; (ii) the existence of cloakbusting inclusions without lossy medium lining; (iii) overcoming the cloakingbusts by employing a lossy layer outside the cloaked region; (iv) the frequency dependence of the cloaking performances. We address these issues and connect the obtained asymptotic results to singular ideal cloaking. Numerical veri cations and demonstrations are provided to show the sharpness of our analytical study.

INI 1  
17:00 to 18:00  Welcome wine reception  
18:15 to 19:00  Dinner at Murray Edwards College (Residents only) 
10:00 to 10:45 
Cloaked wave amplifiers via transformation optics
The advent of transformation optics and metamaterials has made possible devices producing extreme effects on wave propagation. Here we give theoretical designs for devices, Schrodinger hats, acting as invisible concentrators of waves. These exist for any wave phenomenon modeled by either the Helmholtz or Schrodinger equations, e.g., polarized waves in electromagnetism, pressure waves in acoustics and matter waves in quantum mechanics, and occupy one part of a parameter space continuum of wavemanipulating structures which also contains standard transformation optics based cloaks, resonant cloaks and cloaked sensors. For electromagnetic and acoustic Schrodinger hats, the resulting centralized wave is a localized excitation. In quantum magnetism, the result is a new charged quasiparticle, a quasmon, which causes conditional probabilistic illusions. We discuss possible solid state implementations.

INI 1  
10:45 to 11:15  Morning Coffee  
11:15 to 12:00 
Transmission Eigenvalues and Upper Triangular Compactness
The interior transmission eigenvalue problem can be formulated as a 2x2 system of pdes, where one of the two unknown functions must satisfy too many boundary conditions, and the other too few. The system is not selfadjoint and the resolvent is not compact.
Under the hypothesis that the contrast satisfies a coercivity condition on the boundary of the domain, we show that the corresponding operator has Upper Triangular Compact Resolvent and that the analytic Fredholm theorem holds for such opertors.
As corollaries, we can show that the set of (complex) interior transmission eigenvalues is a (possibly empty) discrete set which depends contunuously on the contrast, and that eigenfunctions must be linearly independent. This is different from previous results because the contrast need not have a constant sign (or be real valued) in the interior of the domain.

INI 1  
12:30 to 13:30  Lunch at Wolfson Court  
14:00 to 14:45 
Reconstructing Conductivity from Minimal Internal Data
We consider the problem of recovering the electric conductivity of a body from knowledge of the magnitude of one curent in the interior. We show that the corresponding equipotential surfaces are area minimizing in a conformal metric determined by the given data, prove identifiability and give numerical reconstructions. We also extend the uniqueness results to the case when the object may contain perfectly conducting and/or insulating regions. (Joint work with Amir Moradifam, Alexandru Tamasan and Alexandre Timonov.)

INI 1  
14:45 to 15:30 
M Salo (University of Helsinki) An inverse problem for the pLaplacian
We study an inverse problem for strongly nonlinear elliptic equations modelled after the pLaplacian. It is proved that the boundary values of a conductivity coefficient are uniquely determined by a nonlinear DirichlettoNeumann map. The proofs work with the nonlinear equation directly instead of being based on linearization, and involve complex geometrical optics type solutions based on pharmonic exponentials and certain pharmonic functions introduced by Wolff. This is joint work with Xiao Zhong (University of Jyväskylä).

INI 1  
15:30 to 16:15  Afternoon Tea  
16:15 to 17:00 
C Guillarmou (École Normale Supérieure) Inverse problem for systems in 2 dimension
Using the new method of Bukhgeim, we study the Calderon problem in 2 dimensions for certain elliptic systems on Riemann surfaces with boundary.

INI 1  
18:15 to 19:00  Dinner at Murray Edwards College (Residents only) 
10:00 to 10:45 
The inverse crack problem
I shall outline some of the open issues related to the three dimensional inverse problem of cracks and present some new stability results obtained in joint work with Eva Sincich.

INI 1  
10:45 to 11:15  Morning Coffee  
11:15 to 12:00 
An Inverse Spectral Theorem of Ambarzumyan
We prove a substantial extension of an inverse spectral theorem of Ambarzumyan, and show that it can be applied to arbitrary compact Riemannian manifolds, compact quantum graphs and finite combinatorial graphs, subject to the imposition of Neumann (or Kirchhoff) boundary conditions.

INI 1  
12:00 to 12:45 
A Vasy (Stanford University) New nonelliptic methods in the analysis of the Laplacian on conformally compact (asymptotically hyperbolic) spaces
I will explain how to analyze the resolvent of the Laplacian on conformally compact spaces by transforming the spectral family to a family of operators on a compact manifold without boundary. One easy consequence of this approach is high energy estimates for the resolvent, uniform in strips, which are crucial, for instance, for understanding the decay of waves. The methods involved are also applicable in many other settings.

INI 1  
12:45 to 13:30  Lunch at Wolfson Court  
14:00 to 14:45 
Transmission Eigenvalues for a Spherically Stratified Medium
We consider the transmission eigenvalue problem for a spherically stratified medium and note that this eigenvalue problem is not selfadjoint. Nevertheless, considerable information on the the spectral theory of transmission eigenvalues can be obtained using methods from the theory of entire functions. In particular we will show that if the index of refraction is constant then complex eigenvalues exist and show that "most"
of these complex eigenvalues lie near the real axis. We also consider the inverse spectral problem associated with this eigenvalue problem and show that that the solution of this problem depends heavily on whether or not the index of refraction is less than one or greater than one.

INI 1  
14:45 to 15:30 
The Factorization Method for an Electromagnetic Inverse Scattering Problem
First, we recall some facts from the (direct) scattering of timeharmonic electromagnetic waves by an inhomogeneous medium. We study the question of existence of weak solutions by an integrodifferential equation, introduce the far field pattern and derive important properties of the far field operator. Then we turn to the inverse problem to determine the contrast of the refractive index from the knowledge of the far field patterns for all incident plane waves. We apply the Factorization Method which provides an explicit characterization of the shape of the contrast by the given far field data.

INI 1  
15:30 to 16:00  Afternoon Tea  
16:00 to 16:45 
Transmission Eigenvalues in Inverse Scattering Theory
The transmission eigenvalue problem is a new class of eigenvalue problems that has recently appeared in inverse scattering theory for inhomogeneous media. Such eigenvalues provide information about material properties of the scattering object and can be determined from scattering data, hence can play an important role in a variety of problems in target identification. The transmission eigenvalue problem is nonselfadjoint and nonlinear which make its mathematical investigation very interesting.
In this lecture we will describe how the transmission eigenvalue problem arises in scattering theory, how transmission eigenvalues can be computed from scattering data and what is known mathematically about these eigenvalues. The investigation of transmission eigenvalue problem for anisotropic media will be discussed and FaberKrahn type inequalities for the first real transmission eigenvalue will be presented. We conclude our presentation with some recent preliminary results on transmission eigenvalues for absorbing and dispersive media, i.e. with complex valued index of refraction, as well as for anisotropic media with contrast that changes sign.
Our presentation contains a collection of results obtained with several collaborators, in particular with David Colton, Drossos Gintides, Houssem Haddar and Andreas Kirsch.

INI 1  
19:30 to 22:00  Conference Dinner at Emmanuel College 
09:15 to 10:00  Carleman estimate for stratified media: the case of a diffusive interface  INI 1  
10:00 to 10:45 
Some issues on the inverse conductivity problem with a complex coefficient
In this talk I will present some results concerning the possibility of recovering some features of a complex valued coefficient form boundary data of its solutions.

INI 1  
10:45 to 11:15  Morning Coffee  
11:15 to 12:00 
The Identification problem in SPECT: uniqueness, nonuniqueness and stability
We study the problem of recovery both the attenuation $a$ and the source $f$ in the attenuated Xray transform in the plane. We study the linearization as well. It turns out that there is a natural Hamiltonian flow that determines which singularities we can recover. If the perturbation $\delta a$ is supported in a compact set that is nontrapping for that flow, then the problem is well posed. Otherwise, it may not be, and at least in the case of radial $a$, $f$, it is not. We present uniqueness and nonuniqueness results both for the linearized and the nonlinear problem; as well as a H\"older stability estimate.

INI 1  
12:00 to 12:45 
A Lechleiter (École Polytechnique) Sampling methods for time domain inverse scattering problems
We consider inverse scattering problems for the wave equation in the time domain: find the shape of a Dirichlet scattering object from time domain measurements of scattered waves. For this time domain inverse problem, we introduce sampling methods, a wellknown family of techniques for corresponding frequency domain inverse scattering problems.
The problem setting and the time domain algorithm incorporate two basic features: The data consists of measurement of causal waves, and the inversion algorithm directly works on the time domain data without using a Fourier transformation.
Timedomain sampling methods naturally incorporate a continuum of frequencies in the inversion algorithm. Consequently, they induce the potential to improve the reconstruction quality when compared to methods working at one single frequency.

INI 1  
12:45 to 13:30  Lunch at Wolfson Court  
18:15 to 19:00  Dinner at Murray Edwards College (Residents only) 
09:15 to 10:00  Inverse scattering from cusp on generalized arithmetic surfaces  INI 1  
10:00 to 10:45 
L Robbiano (Université de Versailles SaintQuentinenYvelines) Carleman estimate for Zaremba boundary condition
The Zaremba boundary condition is a mixed boundary condition of the following type, on a part of the boundary we impose Dirichlet boundary condition and on the other part we impose the Neumann boundary condition. For such a problem we prove an logarithme type estimate for the decrease of energy of a solution for a damped wave equation. In the talk we shall explain the plan of the proof. The main part is to prove a Carleman estimate in a neighborhood of the boundary where the type of boundary conditions changes.

INI 1  
10:45 to 11:15  Morning Coffee  
11:15 to 12:00  Enhancement of nearcloaking using GPT vanishing structures  INI 1  
12:00 to 12:45 
Stability of Calderón Inverse Problem
In this talk we study the different kinds of stability in the inverse Calderon problem in EIT. We study in particular the stability in the case of partial data.

INI 1  
12:45 to 13:30  Lunch at Wolfson Court  
14:00 to 14:45 
Adaptive timefrequency detection and filtering for imaging in strongly heterogeneous background media
We consider the problem of detecting and imaging the location of compactly supported reflectors embedded in strongly heterogeneous background media. Imaging in such regimes is quite challenging as the incoherent wave field that is produced from reflections by the background medium overwhelms the scattered field from the object that wish to image. To detect the presence of a reflector in such regimes we introduce an adaptive timefrequency representation of the array response matrix followed by a Singular Value Decomposition (SVD). The detection is adaptive because the time windows that contain the primary echoes from the reflector are not determined in advance. Their location and width is determined by searching through the timefrequency binary tree of the LCT. After detecting the presence of the reflector we filter the array response matrix to retain information only in the time windows that have been selected. We also project the filtered array response matrix to the subspace associated with the top singular value and then image using travel time migration. We show with extensive numerical simulations that this approach to detection and imaging works well in heavy clutter that is calibrated using random matrix theory so as to simulate regimes close to experiments. While the detection and filtering algorithm that we present works well in general clutter it has been analyzed theoretically only for the case of randomly layered media.

INI 1  
18:15 to 19:00  Dinner at Murray Edwards College (Residents only) 