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Humanitarian clearance of minefields is most often carried out by hand, conventionally using a a metal detector and a probe. Detection is a very slow process, as every piece of detected metal must treated as if it were a landmine and carefully probed and excavated, while many of them are not. The process can be safely sped up by use of Ground-Penetrating Radar (GPR) to image the subsurface, to verify metal detection results and safely ignore any objects which could not possibly be a landmine. In this thesis, we explore the possibility of using Full Wave Inversion (FWI) to improve GPR imaging for landmine detection. Posing the imaging task as FWI means solving the large-scale, non-linear and ill-posed optimisation problem of determining the physical parameters of the subsurface (such as electrical permittivity) which would best reproduce the data. This thesis begins by giving an overview of all the mathematical and implementational aspects of FWI, so as to provide an informative text for both mathematicians (perhaps already familiar with other inverse problems) wanting to contribute to the mine detection problem, as well as a wider engineering audience (perhaps already working on GPR or mine detection) interested in the mathematical study of inverse problems and FWI.We present the first numerical 3D FWI results for GPR, and consider only surface measurements from small-scale arrays as these are suitable for our application. The FWI problem requires an accurate forward model to simulate GPR data, for which we use a hybrid finite-element boundary-integral solver utilising first order curl-conforming N\'{e}d\'{e}lec (edge) elements. We present a novel `line search' type algorithm which prioritises inversion of some target parameters in a region of interest (ROI), with the update outside of the area defined implicitly as a function of the target parameters. This is particularly applicable to the mine detection problem, in which we wish to know more about some detected metallic objects, but are not interested in the surrounding medium. We may need to resolve the surrounding area though, in order to account for the target being obscured and multiple scattering in a highly cluttered subsurface. We focus particularly on spatial sensitivity of the inverse problem, using both a singular value decomposition to analyse the Jacobian matrix, as well as an asymptotic expansion involving polarization tensors describing the perturbation of electric field due to small objects. The latter allows us to extend the current theory of sensitivity in for acoustic FWI, based on the Born approximation, to better understand how polarization plays a role in the 3D electromagnetic inverse problem. Based on this asymptotic approximation, we derive a novel approximation to the diagonals of the Hessian matrix which can be used to pre-condition the GPR FWI problem.

In the early 1980's, Kodama, Ablowitz and Satsuma, together with Santini, Ablowitz and Fokas, developed the formal inverse scattering theory of the Intermediate Long Wave (ILW) equation and explored its connections with the Benjamin-Ono (BO) and KdV equations. The ILW equation\begin{align*} u_t + \frac{1}{\delta} u_x + 2 u u_x + Tu_{xx} = 0, \end{align*} models the behavior of long internal gravitational waves in stratified fluids of depth $0< \delta < \infty$, where $T$ is a singular operator which depends on the depth $\delta$. In the limit $\delta \to 0$, the ILW reduces to the Korteweg de Vries (KdV) equation, and in the limit $\delta \to \infty$, the ILW (at least formally) reduces to the Benjamin-Ono (BO) equation. While the KdV equation is very well understood, a rigorous analysis of inverse scattering for the ILW equation remains to be accomplished. There is currently no rigorous proof that the Inverse Scattering Transform outlined by Kodama \textit{et al.} solves the ILW, even for small data. In this dissertation, we seek to help ameliorate this gap in knowledge by presenting a mathematically rigorous construction of the direct scattering map for the ILW's Inverse Scattering Transform.

We study the Cauchy problem of the derivative nonlinear Schrodinger equation in one space dimension. Using the method of inverse scattering, we prove global well-posedness of the derivative nonlinear Schrodinger equation for initial conditions in a dense and open subset of weighted Sobolev space that can support bright solitons.

The purpose of this work was to develop new methods to address inverse problems in elasticity, taking advantage of the presence of a small parameter in the considered problems by means of higher-order asymptotic expansions. The first part is dedicated to the localization and size identification of a buried inhomogeneity B true in a 3D elastic domain. In this goal, we focus on the study of functionals J(Ba ) quantifying the misfit between Btrue and a trial homogeneity Ba . Such functionals are to be minimized w.r.t. some or all the characteristics of the trial inclusion Ba (location, size, mechanical properties ...) to find the best agreement with Btrue. To this end, we produce an expansion of J with respect to the size a of Ba , providing a polynomial approximation easier to minimize. This expansion is established in a volume integral equations framework up to the sixth order, and justified by an estimate of the residual. A suited identification procedure is then given and supported by numerical illustrations for simple obstacles in full-space R3. The main purpose of this second part is to characterize a microstructured two- phases layered 1D inclusion of length L, supposing we already know its low-frequency transmission eigenvalues (TEs). Those are computed as the eigenvalues of the so-called interior transmission problem (ITP). To provide a convenient invertible model, while accounting for the microstructure effects, we rely on homogenized approximations of the exact ITP for the periodic inclusion. Focusing on the leading-order homogenized ITP, we first provide a straightforward method to recover the macroscopic parameters (L and material contrast) of such inclusion. To access the key features of the microstructure, higher-order homogenization is finally addressed, with emphasis on the need for suitable boundary conditions.

For certain initial data, we solve the Novikov-Veselov equation by the inverse scat- tering method. This is a (2+1)-dimensional completely integrable system that gen- eralizes the (1+1)-dimensional Korteweg-de-Vries equation. The method used is the inverse scattering method. To study the direct and inverse scattering maps, we prove existence and uniqueness properties of exponentially growing solutions of the two- dimensional Schrodinger equation. For conductivity-type potentials, this was done by Nachman in his work on the inverse conductivity problem. Our work expands the set of potentials for which the analysis holds, completes the study of the inverse scattering map, and show that the inverse scattering method yields global in time solutions to the Novikov-Veselov equation. This is the first proof that the inverse scattering method yields classical solutions to the Novikov-Veselov equation for the class of potentials considered here.

Biofuel produced from photosynthetic microorganisms offers a green and sustainable alternative to fossil fuels and bioethanol for next generation transportation fuels. Their simple cell structures enable them to be more efficient in photosynthesis than higher order plants. However, large scale cultivation of these microorganisms is typically hampered by the poor light utilization of photobioreactor systems. In order to increase biomass productivities, the light transfer in photobioreactors must be characterized and optimized. The radiation characteristics of photosynthetic microorganisms are essential parameters in this analysis. This study aims to (1) experimentally measure the radiation characteristics of various photosynthetic microorganisms of different morphologies and (2) to develop models that can predict these radiation characteristics.First, the spectral complex indices of refraction of unicellular spheroidal green algae Botryococcus braunii, Chlorella sp., and Chlorococcum littorale were retrieved from their experimentally measured average absorption and scattering cross-sections. Next, the temporal evolution of the scattering and absorbing cross-sections of marine eustigmatophycease Nannochloropsis oculata grown in flat-plate photobioreactor (PBR) was reported. The variations in cross-sections were found to be directly related to the up- and down-regulations of pigments and other intracellular components and vary significantly with time in response to changes in light and nutrients availability. Furthermore, this study demonstrates that the light transfer in the PBR could be predicted using constant radiation characteristics measured during the exponential growth phase with reasonable accuracy provided that the cultures were not nitrogen limited. In addition, this study presents experimental measurements of the absorption and scattering cross-sections and the spectral complex index of refraction of filamentous heterocystous cyanobacterium Anabaena cylindrica. Its filaments, consisting of long chains of polydisperse cells, were modeled as infinitely long and randomly oriented volume-equivalent cylinders. Finally, it was demonstrated that the absorption and scattering cross-sections and asymmetry factor of randomly oriented and optically soft bispheres, quadspheres, and circular rings of spheres, with either monodisperse or polydisperse monomers, can be approximated by an equivalent coated sphere with identical volume and average projected area. Based on this approximation, the spectral complex index of refraction of unicellular dumbbell-shaped cyanobacterium Synechocystis sp. was retrieved from experimental measurements of its average absorption and scattering cross-sections and size distribution.

Acoustic and electromagnetic full-wave nonlinear inverse scattering techniques are explored in both theory and experiment with the ultimate aim of noninvasively mapping the material properties of the breast. There is evidence that benign and malignant breast tissue have different acoustic and electrical properties and imaging these properties directly could provide higher quality images with better diagnostic certainty. In this dissertation, acoustic and electromagnetic inverse scattering algorithms are first developed and validated in simulation. The forward solvers and optimization cost functions are modified from traditional forms in order to handle the large or lossy imaging scenes present in ultrasonic and microwave breast imaging. An antenna model is then presented, modified, and experimentally validated for microwave S-parameter measurements. Using the antenna model, a new electromagnetic volume integral equation is derived in order to link the material properties of the inverse scattering algorithms to microwave S-parameters measurements allowing direct comparison of model predictions and measurements in the imaging algorithms. This volume integral equation is validated with several experiments and used as the basis of a free-space inverse scattering experiment, where images of the dielectric properties of plastic objects are formed without the use of calibration targets. These efforts are used as the foundation of a solution and formulation for the numerical characterization of a microwave near-field cavity-based breast imaging system. The system is constructed and imaging results of simple targets are given. Finally, the same techniques are used to explore a new self-characterization method for commercial ultrasound probes. The method is used to calibrate an ultrasound inverse scattering experiment and imaging results of simple targets are presented. This work has demonstrated the feasibility of quantitative microwave inverse scattering by way of a self-consistent characterization formalism, and has made headway in the same area for ultrasound.

An ultrawide band (UWB) time domain (TD) radar has been developed for Time Reversal (TR) synthetic inverse scattering. This work describes the fabrication and test of a time domain TR “mirror” (TRM), which is the system that transmits, receives, and processes TR signals. Time reversal electromagnetics is a family of signal processing techniques that exploit the time reversal invariance of Maxwell equations, and allow focusing of waves in space and time with “super resolution”, i.e. beyond the classical resolution limit. In a typical TR experiment, first a pulsed signal is emitted and scattered by a target, which then acts as a weak emitter “source” toward the TRM. Then the TRM senses and records the weak scattered signal. Next, the recorded signals are time-reversed for their subsequent normalization and reradiation into the medium. The result is a signal that focuses in time and space where the target originally was, permitting localization and identification applications.A 12 mW low power short range experimental verification of the above process in the microwave domain is demonstrated in this work. The developed system exhibits a 40 GHz useful bandwidth such as to be able to follow 12 ps transition duration signals without distortion. An antenna system was designed and fabricated to satisfy the above bandwidth specification. The backward propagation in the TR process is executed synthetically on account of the current state of the required technology to recreate ultra-fast signals of arbitrary shape.Owing to its intrinsic TD nature, the finite difference time-domain (FDTD) method was applied as the numerical engine to simulate the electromagnetic fields in the TR process, facilitating the link between acquired data and its interpretation. The system designed showed capable of recreating in TD, with this extended bandwidth, the standard TR experiment as predicted by the theory. Hence, from real measurements the system was able to recreate synthetically space and time focusing. This kind of system could have application in subsurface target detection, trough-wall imaging and novel wireless communications. Additionally, in this work a numerical study introducing a novel technique to estimate the (mean) effective permittivity of a random media applying TR techniques is presented, and for which high performance computing was required to process the data. The results showed that TR can be applied to determine effective permittivity of random media difficult to otherwise model deterministically.

Qualitative inverse scattering methods have been a subject of much interest in the inverse scattering community for the past two decades. In contrast to image reconstruction methods, such as the celebrated back-projection methods,qualitative methods do not rely on weak scattering approximations and are thus appropriate for multiple scattering imaging. In particular the most well established of these methods known as the Linear Sampling Method is based on solving a certain linear and ill-posed integral equation. Thus for sufficiently dense and large enough aperture viewing angles the method promotes promotes simplicity,speed, and good qualityreconstructions. The main drawback of this method is the assumption of complete redundant multi-static data. In practice however, such data might not be possible or perhaps too costly to obtain. An example of such a case would be in the field of radar imaging, where increasing the number of sensors also increases the likelihood of enemy detection.Our work is based on LSM imaging with radar imaging as the main target application. Specifically we deal with the case of sparse and/or aperture obstacle scattering. We attempt to improve the reconstruction quality for thesesituations by focusing on introducing physically meaningful constraints into the regularization of the so called far-field equation. The first portion of our study focused on constraining our desired solutions to those that exhibit sparse spatial gradients. The second portion of our proposed study is focused on multi-frequency LSM. The LSM method is primarily a single frequency approach, to our knowledge very little work exists on exploiting multi-frequencydata within the LSM framework. We show that solutions to the far-field equation exhibit a Lipschitz like continuity with respect to frequency. We develop a frequency based regularization method that exploits this a priori knowledge.

We consider the analysis of the scattering problem. Assume that an incoming time harmonic wave is scattered by a surface of an impenetrable obstacle. The reflected wave is determined by the surface impedance of the obstacle. In this paper we will investigate the problem of choosing the surface impedance so that a desired scattering amplitude is achieved. We formulate this control problem within the framework of the minimization of a Tikhonov functional. In particular, questions of the existence of an optimal solution and the derivation of the optimality conditions will be addressed.

Inverse problems are widely studied today, and appear in a large range of applications: tomography and imaging, material constitutive property identification, non destructive control... The present subject comes within the scope of this last prospect. It concerns the research of new methods, fast numerically, allowing qualitative object identification (inclusions, cavities, cracks...) embedded in linear elastic solid medium, knowing (at least partially) the surface response to dynamical loadings. Most of the classical methods implemented to solve this kind of problems are dealing with an iterative minimization process, requiring high number of direct simulations. In the present context (three-dimensional elastic waves propagation), these are very expensive. The recent emergence of non-iterative probing methods allows to consider the study of this type of problems in a new light. Earlier works have shown in particular, within the framework of the hypothesis adopted in this subject, the interest of methods such as the Topological Sensitivity or the Linear Sampling for an approximate but fast detection. The present study comes within the scope of the development of the two methods mentioned with application within the framework of the mechanics of deformable solids, i.e. inverse scattering problems in acoustic and elastic media. This work has been done within the framework of a joint Ph.D. program between the Department of Civil Engineering at the University of Minnesota and the Laboratoire de Mécanique des Solides at the Ecole Polytechnique (France).

We consider optimization problems in the presence of uncertainty for three different scientific examples. The first optimization problem addressed is in the area of parameter estimation. We review the asymptotic theory for standard errors in classical ordinary least squares (OLS) inverse or parameter estimation problems involving general nonlinear dynamical systems where sensitivity matrices can be used to compute the asymptotic covariance matrices. We discuss possible pitfalls in computing standard errors in regions of low parameter sensitivity and/or near a steady state solution of the underlying dynamical system. Next we consider electromagnetic evasion-interrogation games where the evader can use ferroelectric material coatings to avoid detection while the interrogator can manipulate the interrogating frequencies and angles of incidence to enhance detection. Each player in this two-player game wishes to change the amount of reflected signal created in a way that will benefit them the most. Thus both players will attempt to optimize their chances of either remaining undetected or detecting their opponent. With the introduction of uncertainty, the resulting game is carried out over spaces of probability measures. Finally, a one player dynamical game is formulated. The premise here is for the evader to manipulate the ferroelectric material coatings to avoid detection based upon updated information about the frequencies being sent out. The uncertainty is found in the frequencies that the interrogator will employ. We incorporate both drift and diffusion into this optimization problem.

Three related electromagnetic scattering problems, namely, high frequency electromagnetic (EM) ray tracing, scattering feature extraction, and inverse scattering are studied in this dissertation. New approaches are presented to advance the state of the art in each of the areas. The presented study in electromagnetic ray tracing leads to an alternative ray tracing algorithm which can outperform the traditional algorithms for complex targets. The performance of the proposed techniques demonstrates their potential application to the study of high-frequency EM scattering prediction. Second, a genetic algorithm (GA)-based algorithm with an adaptive-feeding technique is developed to simultaneously extract both scattering centers and resonances. Scattering feature extraction algorithms are then developed with the consideration of the visibility of scattering centers. Inverse scattering problems with strong multiple scattering effects are also studied. A GA-based method is presented to invert the shapes with multiple scattering effects. An approach combining hybrid GA with the tabu list idea are then developed to further improve the performance of the GA-based inversion algorithms.

The Krein integral equation of one-dimensional (1-D) inverse scattering and the Wiener-Hopf integral equation of linear least-squares estimation are Fredholm equations with a symmetric Toeplitz kernel. They are transformed using a wavelet-based Galerkin method, into a symmetric block slanted Toeplitz (BST) system of equations. This solution method is compared with direct discretization using numerical examples. A Levinson-like fast algorithm is developed for solving the symmetric BST system of equations. This algorithm can be used even if the kernel of the integral equation is not a Calderon-Zygmund operator (in which case the wavelet transform may not sparsify it). Our algorithms exploit the symmetric block slanted Toeplitz structure of the system matrix, and do not rely on sparsity. We also present a wavelet-based Schur-like layer stripping fast algorithm which yields a multiresolution solution to the lattice equations of the 1-D inverse scattering problem without discretization and without having to solve any systems of equations. Furthermore, the Schur-like algorithm can be run in parallel with the Levinson-like algorithm to compute the reflection coefficients of the Levinson-like algorithm, thereby speeding up the operation of the Levinson-like algorithm. The Levinson-like and Schur-like algorithms are shown to yield fast factorizations of the symmetric BST system matrix and its inverse. The Levinson-like algorithm is also used to develop a Gohberg-Semencul-like formula and tests for positive definiteness of the symmetric BST matrix. Symmetric forms of the Levinson-like and Schur-like algorithms are also given. From these, we develop the split Levinson-like and split Schur-like fast algorithms. Finally, we present two algorithms for incorporating a priori information about the reflectivity function into the 1-D inverse scattering solution procedure. The first of these is based on the classical Levinson and Schur algorithms and yields a discrete solution while the second is based on the Levinson-like and Schur-like algorithms and hence yields a wavelet-based solution.