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Liliana Borcea

Publications and source records attributed to Liliana Borcea.

51 records · Page 3Linked to original sources
arXiv2016

Polarization effects for electromagnetic wave propagation in random media

We study Maxwell's equations in random media with small fluctuations of the electric permittivity. We consider a setup where the waves propagate toward a preferred direction, called range. We decompose the electromagnetic wave field in transverse electric and transverse magnetic plane waves, called modes, with random amplitudes that model cumulative scattering effects in the medium. Their evolution in range is described by a coupled system of stochastic differential equations driven by the random fluctuations of the electric permittivity. We analyze the solution of this system with the Markov limit theorem and obtain a detailed asymptotic characterization of the electromagnetic wave field in the long range limit. In particular, we quantify the loss of coherence of the waves due to scattering by calculating the range scales (scattering mean free paths) on which the mean amplitudes of the modes decay. We also quantify the energy exchange between the modes, and consequently the loss of polarization induced by scattering, by analyzing the Wigner transform (energy density) of the electromagnetic wave field. This analysis involves the derivation of transport equations with polarization. We study in detail these equations and connect the results with the existing literature in radiative transport and paraxial wave propagation.

math.AP↗
arXiv2016

Derivation of a one-way radiative transfer equation in random media

We derive from first principles a one-way radiative transfer equation for the wave intensity resolved over directions (Wigner transform of the wave field) in random media. It is an initial value problem with excitation from a source which emits waves in a preferred, forward direction. The equation is derived in a regime with small random fluctuations of the wave speed but long distances of propagation with respect to the wavelength, so that cumulative scattering is significant. The correlation length of the medium and the scale of the support of the source are slightly larger than the wavelength, and the waves propagate in a wide cone with opening angle less than $180^o$, so that the backward and evanescent waves are negligible. The scattering regime is a bridge between that of radiative transfer, where the waves propagate in all directions and the paraxial regime, where the waves propagate in a narrow angular cone. We connect the one-way radiative transport equation with the equations satisfied by the Wigner transform of the wave field in these regimes.

math.AP↗
arXiv2015

Imaging with electromagnetic waves in terminating waveguides

We study an inverse scattering problem for Maxwell's equations in terminating waveguides, where localized reflectors are to be imaged using a remote array of sensors. The array probes the waveguide with waves and measures the scattered returns. The mathematical formulation of the inverse scattering problem is based on the electromagnetic Lippmann-Schwinger integral equation and an explicit calculation of the Green tensor. The image formation is carried with reverse time migration and with $\ell_1$ optimization.

math.NA↗
arXiv2015

Resolution analysis of imaging with $\ell_1$ optimization

We study array imaging of a sparse scene of point-like sources or scatterers in a homogeneous medium. For source imaging the sensors in the array are receivers that collect measurements of the wave field. For imaging scatterers the array probes the medium with waves and records the echoes. In either case the image formation is stated as a sparsity promoting $\ell_1$ optimization problem, and the goal of the paper is to quantify the resolution. We consider both narrow-band and broad-band imaging, and a geometric setup with a small array. We take first the case of the unknowns lying on the imaging grid, and derive resolution limits that depend on the sparsity of the scene. Then we consider the general case with the unknowns at arbitrary locations. The analysis is based on estimates of the cumulative mutual coherence and a related concept, which we call interaction coefficient. It complements recent results in compressed sensing by deriving deterministic resolution limits that account for worse case scenarios in terms of locations of the unknowns in the imaging region, and also by interpreting the results in some cases where uniqueness of the solution does not hold. We demonstrate the theoretical predictions with numerical simulations.

math.NA↗
arXiv2015

Source estimation with incoherent waves in random waveguides

We study an inverse source problem for the acoustic wave equation in a random waveguide. The goal is to estimate the source of waves from measurements of the acoustic pressure at a remote array of sensors. The waveguide effect is due to boundaries that trap the waves and guide them in a preferred (range) direction, the waveguide axis, along which the medium is unbounded. The random waveguide is a model of perturbed ideal waveguides which have flat boundaries and are filled with known media that do not change with range. The perturbation consists of fluctuations of the boundary and of the wave speed due to numerous small inhomogeneities in the medium. The fluctuations are uncertain in applications, which is why we model them with random processes, and they cause significant cumulative scattering at long ranges from the source. The scattering effect manifests mathematically as an exponential decay of the expectation of the acoustic pressure, the coherent part of the wave. The incoherent wave is modeled by the random fluctuations of the acoustic pressure, which dominate the expectation at long ranges from the source. We use the existing theory of wave propagation in random waveguides to analyze the inverse problem of estimating the source from incoherent wave recordings at remote arrays. We show how to obtain from the incoherent measurements high fidelity estimates of the time resolved energy carried by the waveguide modes, and study the invertibility of the system of transport equations that model energy propagation in order to estimate the source.

math.AP↗
arXiv2014

A model reduction approach to numerical inversion for a parabolic partial differential equation

We propose a novel numerical inversion algorithm for the coefficients of parabolic partial differential equations, based on model reduction. The study is motivated by the application of controlled source electromagnetic exploration, where the unknown is the subsurface electrical resistivity and the data are time resolved surface measurements of the magnetic field. The algorithm presented in this paper considers inversion in one and two dimensions. The reduced model is obtained with rational interpolation in the frequency (Laplace) domain and a rational Krylov subspace projection method. It amounts to a nonlinear mapping from the function space of the unknown resistivity to the small dimensional space of the parameters of the reduced model. We use this mapping as a nonlinear preconditioner for the Gauss-Newton iterative solution of the inverse problem. The advantage of the inversion algorithm is twofold. First, the nonlinear preconditioner resolves most of the nonlinearity of the problem. Thus the iterations are less likely to get stuck in local minima and the convergence is fast. Second, the inversion is computationally efficient because it avoids repeated accurate simulations of the time-domain response. We study the stability of the inversion algorithm for various rational Krylov subspaces, and assess its performance with numerical experiments.

math.NA↗
arXiv2013

Electromagnetic wave propagation in random waveguides

We study long range propagation of electromagnetic waves in random waveguides with rectangular cross-section and perfectly conducting boundaries. The waveguide is filled with an isotropic linear dielectric material, with randomly fluctuating electric permittivity. The fluctuations are weak, but they cause significant cumulative scattering over long distances of propagation of the waves. We decompose the wave field in propagating and evanescent transverse electric and magnetic modes with random amplitudes that encode the cumulative scattering effects. They satisfy a coupled system of stochastic differential equations driven by the random fluctuations of the electric permittivity. We analyze the solution of this system with the diffusion approximation theorem, under the assumption that the fluctuations decorrelate rapidly in the range direction. The result is a detailed characterization of the transport of energy in the waveguide, the loss of coherence of the modes and the depolarization of the waves due to cumulative scattering.

math-ph↗
arXiv2013

A quantitative study of source imaging in random waveguides

We present a quantitative study of coherent array imaging of remote sources in randomly perturbed waveguides with bounded cross-section. We study how long range cumulative scattering by perturbations of the boundary and the medium impedes the imaging process. We show that boundary scattering effects can be mitigated with filters that enhance the coherent part of the data. The filters are obtained by optimizing a measure of quality of the image. The point is that there is an optimal trade-off between the robustness and resolution of images in such waveguides, which can be found adaptively, as the data are processed to form the image. Long range scattering by perturbations of the medium is harder to mitigate than scattering by randomly perturbed boundaries. Coherent imaging methods do not work and more complex incoherent methods, based on transport models of energy, should be used instead. Such methods are nor useful, nor needed in waveguides with perturbed boundaries. We explain all these facts using rigorous asymptotic stochastic analysis of the wave field in randomly perturbed waveguides. We also analyze the adaptive coherent imaging method and obtain a quantitative agreement with the results of numerical simulations.

math.NA↗
arXiv2012

Paraxial coupling of propagating modes in three-dimensional waveguides with random boundaries

We analyze long range wave propagation in three-dimensional random waveguides. The waves are trapped by top and bottom boundaries, but the medium is unbounded in the two remaining directions. We consider scalar waves, and motivated by applications in underwater acoustics, we take a pressure release boundary condition at the top surface and a rigid bottom boundary. The wave speed in the waveguide is known and smooth, but the top boundary has small random fluctuations that cause significant cumulative scattering of the waves over long distances of propagation. To quantify the scattering effects, we study the evolution of the random amplitudes of the waveguide modes. We obtain that in the long range limit they satisfy a system of paraxial equations driven by a Brownian field. We use this system to estimate three important mode-dependent scales: the scattering mean free path, the cross-range decoherence length and the decoherence frequency. Understanding these scales is important in imaging and communication problems, because they encode the cumulative scattering effects in the wave field measured by remote sensors. As an application of the theory, we analyze time reversal and coherent interferometric imaging in strong cumulative scattering regimes.

math.AP↗
arXiv2012

Motion Estimation and Imaging of Complex Scenes with Synthetic Aperture Radar

We study synthetic aperture radar (SAR) imaging and motion estimation of complex scenes consisting of stationary and moving targets. We use the classic SAR setup with a single antenna emitting signals and receiving the echoes from the scene. The known motion estimation methods for such setups work only in simple cases, with one or a few targets in the same motion. We propose to extend the applicability of these methods to complex scenes, by complementing them with a data pre-processing step intended to separate the echoes from the stationary targets and the moving ones. We present two approaches. The first is an iteration designed to subtract the echoes from the stationary targets one by one. It estimates the location of each stationary target from a preliminary image, and then uses it to define a filter that removes its echo from the data. The second approach is based on the robust principle component analysis (PCA) method. The key observation is that with appropriate pre-processing and windowing, the discrete samples of the stationary target echoes form a low rank matrix, whereas the samples of a few moving target echoes form a high rank sparse matrix. The robust PCA method is designed to separate the low rank from the sparse part, and thus can be used for the SAR data separation. We present a brief analysis of the two methods and explain how they can be combined to improve the data separation for extended and complex imaging scenes. We also assess the performance of the methods with extensive numerical simulations.

math.NA↗
arXiv2012

Synthetic Aperture Radar Imaging and Motion Estimation via Robust Principle Component Analysis

We consider the problem of synthetic aperture radar (SAR) imaging and motion estimation of complex scenes. By complex we mean scenes with multiple targets, stationary and in motion. We use the usual setup with one moving antenna emitting and receiving signals. We address two challenges: (1) the detection of moving targets in the complex scene and (2) the separation of the echoes from the stationary targets and those from the moving targets. Such separation allows high resolution imaging of the stationary scene and motion estimation with the echoes from the moving targets alone. We show that the robust principal component analysis (PCA) method which decomposes a matrix in two parts, one low rank and one sparse, can be used for motion detection and data separation. The matrix that is decomposed is the pulse and range compressed SAR data indexed by two discrete time variables: the slow time, which parametrizes the location of the antenna, and the fast time, which parametrizes the echoes received between successive emissions from the antenna. We present an analysis of the rank of the data matrix to motivate the use of the robust PCA method. We also show with numerical simulations that successful data separation with robust PCA requires proper data windowing. Results of motion estimation and imaging with the separated data are presented, as well.

cs.IT↗
arXiv2011

Study of noise effects in electrical impedance tomography with resistor networks

We present a study of the numerical solution of the two dimensional electrical impedance tomography problem, with noisy measurements of the Dirichlet to Neumann map. The inversion uses parametrizations of the conductivity on optimal grids. The grids are optimal in the sense that finite volume discretizations on them give spectrally accurate approximations of the Dirichlet to Neumann map. The approximations are Dirichlet to Neumann maps of special resistor networks, that are uniquely recoverable from the measurements. Inversion on optimal grids has been proposed and analyzed recently, but the study of noise effects on the inversion has not been carried out. In this paper we present a numerical study of both the linearized and the nonlinear inverse problem. We take three different parametrizations of the unknown conductivity, with the same number of degrees of freedom. We obtain that the parametrization induced by the inversion on optimal grids is the most efficient of the three, because it gives the smallest standard deviation of the maximum a posteriori estimates of the conductivity, uniformly in the domain. For the nonlinear problem we compute the mean and variance of the maximum a posteriori estimates of the conductivity, on optimal grids. For small noise, we obtain that the estimates are unbiased and their variance is very close to the optimal one, given by the Cramer-Rao bound. For larger noise we use regularization and quantify the trade-off between reducing the variance and introducing bias in the solution. Both the full and partial measurement setups are considered.

math-ph↗
arXiv2011

Wave propagation in waveguides with random boundaries

We give a detailed analysis of long range cumulative scattering effects from rough boundaries in waveguides. We assume small random fluctuations of the boundaries and obtain a quantitative statistical description of the wave field. The method of solution is based on coordinate changes that straighten the boundaries. The resulting problem is similar from the mathematical point of view to that of wave propagation in random waveguides with interior inhomogeneities. We quantify the net effect of scattering at the random boundaries and show how it differs from that of scattering by internal inhomogeneities.

math.AP↗
arXiv2011

Resistor network approaches to electrical impedance tomography

We review a resistor network approach to the numerical solution of the inverse problem of electrical impedance tomography (EIT). The networks arise in the context of finite volume discretizations of the elliptic equation for the electric potential, on sparse and adaptively refined grids that we call optimal. The name refers to the fact that the grids give spectrally accurate approximations of the Dirichlet to Neumann map, the data in EIT. The fundamental feature of the optimal grids in inversion is that they connect the discrete inverse problem for resistor networks to the continuum EIT problem.

math-ph↗