Shari Moskow

Convergence and stability of the inverse scattering series for diffuse waves

We analyze the inverse scattering series for diffuse waves in random media. In previous work the inverse series was used to develop fast, direct image reconstruction algorithms in optical tomography. Here we characterize the convergence, stability and approximation error of the series.

Numerical studies of the inverse Born series for diffuse waves

We consider the inverse scattering problem for diffuse waves. We analyze the convergence of the inverse Born series and study its use in numerical simulations for the case of a spherically symmetric absorbing medium in two and three dimensions.

Local inversions in ultrasound-modulated optical tomography

Ultrasound-modulated optical tomography is a hybrid imaging modality that aims to combine the high contrast of optical waves with the high resolution of ultrasound. We follow the model of the influence of ultrasound modulation on the light intensity measurements developed in Bal and Schotland (2010 Phys. Rev. Lett. 104 043902). We present sufficient conditions ensuring that the absorption and diffusion coefficients modeling light propagation can locally be uniquely and stably reconstructed from the corresponding available information. We present an iterative procedure to solve such a problem based on the analysis of linear elliptic systems of redundant partial differential equations.

AN APPROXIMATE METHOD FOR SCATTERING BY THIN STRUCTURES

Scattering of waves by a thin structure is considered in this work. The Helmholtz equation with variable coefficient models the wave phenomena. The scatterer is assumed to have a high index of refraction while at the same time it is very small in one of the dimensions. We show that if the index scales as O(1/h), where h is the thickness of the scatterer, then an approximate solution, based on perturbation analysis, can be obtained. The approximate solution consists of a leading order term plus a corrector, each of which solves an integral equation in two dimensions for a three-dimensional problem. We provide error analysis on the approximation. The approximate method can be viewed as an efficient computational approach since it can potentially greatly simplify scattering calculations. Numerical results provide an assessment of the accuracy of the approximate solution.

Asymptotic and Numerical Techniques for Resonances of Thin Photonic Structures

We consider the problem of calculating resonance frequencies and radiative losses of an optical resonator. The optical resonator is in the form of a thin membrane with variable dielectric properties. This work provides two very different approaches for doing such calculations. The first is an asymptotic method which exploits the small thickness and high index of the membrane. We derive a limiting resonance problem as the thickness goes to zero, and for the case of a simple resonance, find a first order correction. The limiting problem and the correction are in one less space dimension, which can make the approach very efficient. Convergence estimates are proved for the asymptotics. The second approach, based on the finite element method with a truncated perfectly matched layer, is not restricted to thin structures. We demonstrate the use of these methods in numerical calculations which further illustrate their differences. The asymptotic method finds resonance by solving a dense, but small, nonlinear eige...

Inverse Born series for the Calderon problem

We propose a direct reconstruction method for the Calderon problem based on inversion of the Born series. We characterize the convergence, stability and approximation error of the method and illustrate its use in numerical reconstructions. (Some figures may appear in colour only in the online journal)

Nonlinear eigenvalue approximation for compact operators

In the work of Osborn [Math. Comput. 29, 712–725 (1975)], a general spectral approximation theory was developed for compact operators on a Banach space which does not require that the operators be self-adjoint and also provides a first order correction term. Here, we extend some of the results of that paper to nonlinear eigenvalue problems. We present examples of its application that arise in electromagnetics and numerical analysis.

Asymptotic expansions for transmission eigenvalues for media with small inhomogeneities

We consider the transmission eigenvalue problem for an inhomogeneous medium containing a finite number of diametrically small inhomogeneities of different refractive index. We prove a convergence result for the transmission eigenvalues and eigenvectors corresponding to media with small homogeneities as the diameter of small inhomogeneities goes to zero. In addition we derive rigorously a formula for the perturbations in the real transmission eigenvalues caused by the presence of these small inhomogeneities.

On the Homogenization of a Scalar Scattering Problem for Highly Oscillating Anisotropic Media

We study the homogenization of a transmission problem arising in the scattering theory for bounded inhomogeneities with periodic coefficients modeled by the anisotropic Helmholtz equation. The coefficients are assumed to be periodic functions of the fast variable, specified over the unit cell with characteristic size

Asymptotic analysis of resonances of small volume high contrast linear and nonlinear scatterers

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