Minisymposia Abstracts
Speaker:  Liliana Borcea Rice University 
Title:  Wave propagation in time dependent, randomly layered media 
Abstract: 
I will describe the cumulative scattering effects on wave front propagation in
time dependent randomly layered media. It is well known that the wave front has
a deterministic characterization in time independent media, aside from a small
random shift in the travel time. That is, the pulse shape is stable, but faded
and smeared as described mathematically by a convolution kernel determined by
the second order statistics of the random fluctuations of the wave speed. I
will describe the extension of the pulse stabilization results to time
dependent randomly layered media. 
Speaker:  John Schotland University of Michigan 
Title:  Imaging with Quantum Fluctuations 
Abstract: 
We consider the inverse scattering problem that arises in twophoton quantum
imaging with interferometric measurements. We show that the twopoint
correlation function of the field contains information about the scattering
medium zat a spatial frequency of twice the Rayleigh bandwidth. The
linearized inverse problem, however, yields reconstructions with a
resolution of λ/2, where λ/ is the wavelength of light.

Speaker:  Richard Tsai University of Texas, Austin 
Title:  Discovery of point sources in the Helmholtz equation posed in domains with unknown obstacles 
Abstract:  We consider an inverse source problem for the Helmholtz equation in domains
with possibly unknown obstacles, which are much larger than the wavelength. The
inverse problem is to determine the number and locations of point sources in
the domain based on sparse measurements of the wave field. Our proposed
strategy relies on solving a local inverse scattering problem to obtain the
incoming directions of waves passing through an observation location. We
formulate the problem as an L1 constrained minimization problem and solve it
using the Bregman iterative procedure. The wave direction rays are then traced
back and sources are uniquely determined at the intersection of the rays from
several observing locations. We present examples in 2D, however all of the
formulas and methods used have direct analogues in 3D.

Speaker:  Ricardo Alonso Rice University 
Title:  Statistical properties for the response matrix singular values in randomly layered media. 
Abstract: 
Echoes from small reflectors buried in heavy clutter are weak and difficult to
distinguish from the medium backscatter. An effective approach to resolve such
reflectors has been proposed recently by considering the singular values of the
medium array response matrix. In this talk we explain how the theory of G.
Szegö on the asymptotic behavior of eigenvalues of Hermitian forms and the
diffusive theory approximation are used to obtain statistical properties for
the singular values that give a quantitative justification of the proposed approach.

Speaker:  Fioralba Cakoni University of Delaware 
Title:  The imaging of anisotropic media using electromagnetic waves 
Abstract: 
We discuss two inverse problems related to anisotropic media for Maxwell's
equations. The first one is the inverse scattering problem of determining the
anisotropic surface impedance of a bounded obstacle from a knowledge of
electromagnetic scattered field due to incident plane waves. Such an
anisotropic boundary condition can arise from surfaces covered with patterns of
conducting and insulating patches. We show that the anisotropic impedance is
uniquely determined if sufficient data is available, and characterize the
nonuniqueness present if a single incoming wave is used. We derive an
integral equation for the surface impedance in terms of solutions of a certain
interior impedance boundary value problem. These solutions can be
reconstructed from far field data using the Herglotz theory underlying the
Linear Sampling Method. The second problem is to obtain information about
matrix index of refraction of an anisotropic media again from a knowledge of
electromagnetic scattered field due to incident plane waves. This problem
plays a special role in inverse scattering theory due to the fact that the
(matrix) index of refraction is not uniquely determined from the scattered
fields even if multifrequency data is available. Our imaging tool is a new
class of eigenvalues associated with the scattering by inhomogeneous media,
known as transmission eigenvalues. In this presentation we describe how
transmission eigenvalues can be determined from scattering data and be used to
obtain upper and lower bounds on the norm of the index of refraction.
Preliminary numerical results will be shown for both problems.

Speaker:  Lexing Ying University of Texas, Austin 
Title:  Sweeping Preconditioners for the Helmholtz Equation 
Abstract: 
Numerical solution of the variable coefficient Helmholtz equation in the high
frequency regime is a challenging computational problem due to the
indefiniteness of the operator and the large size of the discrete system. In
this talk, we introduce the sweeping preconditioners for the rapid solution of
the variable coefficient Helmholtz equation. The novelties of this new class of
preconditioners are a specific order of eliminating the unknowns and efficient
representations of the Schur complement matrices. For a problem with N
unknowns, these preconditioners take essentially O(N) steps to apply, give
iteration numbers that are independent of the frequency, and hence provide a
linearcomplexity method for solving the variable coefficient Helmholtz
equation. This is a joint work with Bjorn Engquist.

Speaker:  Thomas Callaghan Rice University 
Title:  Joint Motion Estimation and Autofocus in Synthetic Aperture Radar Imaging 
Abstract: 
In synthetic aperture radar (SAR) imaging, two important applications are
formation of high resolution images and motion estimation of moving targets on
the ground. The formation of high resolution images requires a process known as
autofocus that compensates for unknown SAR platform perturbations that cause
images to appear blurred. In previous work, we described and analyzed a phase
space approach using Wigner transforms and ambiguity functions for motion
estimation and autofocus using small, successive sub apertures. However, in
simple imaging scenes, we cannot simultaneously estimate both speed of a moving
target and the platform trajectory perturbations that we need for autofocus. Recently, we have been developing novel data preprocessing strategies that complement and extend the applicability range of our phase space velocity estimation and autofocus methods to more complex imaging scenes with both stationary and moving targets. The goal of the data preprocessing is to decompose it into parts that correspond to the stationary targets and the moving ones. Once that is accomplished, autofocus can be applied to the data from the stationary targets. The velocity estimation and imaging of the moving targets can then be carried out separately. Our approach is to use information from preliminary images as a guide to applying the ideas of robust principal component analysis and constructing travel time transformations to filter out the stationary topics. Numerical simulations will also be presented. 
Speaker:  Rami Nammour Rice University 
Title:  Approximate MultiParameter Inverse Scattering Using Pseudodifferential Scaling 
Abstract: 
I will present a computationally efficient method to approximate the inverse of the
normal operator arising in the multiparameter linearized inverse problem of
reflection seismology in two and three spatial dimensions. The application of the normal operator requires solving expensive large scale PDE problems. Furthermore, it is not computationally feasible to use direct matrix methods like Gaussian elimination to invert the normal operator. However, under certain conditions, the normal operator is a matrix of pseudodifferential operators. This presentation shows how to generalize Cramer's rule to approximately invert matrices of pseudodifferential operators in two steps: first, apply the normal operator several times to specific permutations of the normal equations' right hand side; second, estimate and apply a scalar pseudodifferential correction (``scaling'') to the result of the first step. The cost of the method is a few applications of the normal operator. The method is validated on variable density acoustics models (two parameter models). 
Please address administrative questions to aipc@math.tamu.edu. Scientific questions should be addressed to the chair of the Scientific Program Committee: rundell AT math.tamu.edu