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 two-photon quantum imaging with interferometric measurements. We show that the two-point 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 non-uniqueness 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 multi-frequency 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 linear-complexity 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 pre-processing 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 pre-processing 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 Multi-Parameter Inverse Scattering Using Pseudodifferential Scaling
Abstract: I will present a computationally efficient method to approximate the inverse of the normal operator arising in the multi-parameter 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).

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