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The modified transmission eigenvalue problem of elastic waves arises from inverse scattering theory. In this work we first analyze the well-posedness of the modified transmission problem and accomplish a rigorous error analysis of conforming finite element methods. Then we establish two types of multigrid schemes. Finally, we numerically research the effect of a flaw in the material on the modified elastic transmission eigenvalues and apply the multigrid schemes to perform parallel computing. Numerical experiments are presented to verify the theoretical convergence order and the efficiency of the proposed schemes. [ABSTRACT FROM AUTHOR]

This paper introduces a novel deep neural network architecture for solving the inverse scattering problem in frequency domain with wide-band data, by directly approximating the inverse map, thus avoiding the expensive optimization loop of classical methods. The architecture is motivated by the filtered back-projection formula in the full aperture regime and with homogeneous background, and it leverages the underlying equivariance of the problem and compressibility of the integral operator. This drastically reduces the number of training parameters, and therefore the computational and sample complexity of the method. In particular, we obtain an architecture whose number of parameters scales sub-linearly with respect to the dimension of the inputs, while its inference complexity scales super-linearly but with very small constants. We provide several numerical tests that show that the current approach results in better reconstruction than optimization-based techniques such as full-waveform inversion, but at a fraction of the cost while being competitive with state-of-the-art machine learning methods. [ABSTRACT FROM AUTHOR]

The inverse scattering problem is to reconstruct or recover some physical and/or geometric properties of an object from the measured scattered field in the process of illumination by an incident wave. The probing radiation can be an electromagnetic wave (e.g., microwave, optical wave, and X-ray), an acoustic wave, or some other waves. The inverse scattering problem is important when detailed information about the structure and composition of an object is required, but it cannot be obtained directly from the measurements. The paper is devoted to the electromagnetic inverse scattering problem for a dielectric anisotropic and magnetically isotropic media. The properties of an anisotropic medium with respect to electromagnetic waves are defined by the tensors, which give the relation between the inductions and the fields. The tensor Fourier diffraction theorem derived in the paper can be considered as a useful tool for studying tensor fields in inverse problems of electromagnetic scattering. The Fourier diffraction theorem is obtained in a coordinate-free representation by the use of tensor (dyadic) analysis techniques. The method of inversion is based on the first Born approximation. [ABSTRACT FROM AUTHOR]

Based on the work of Monk and Selgas (2022), we explore the finite element method for the modified transmission eigenvalues for inverse scattering in a fluid–solid interaction problem. Under minimal regularity on the solution, we give a complete error analysis including the a priori and the a posteriori error estimates. We also present adaptive computations. [ABSTRACT FROM AUTHOR]

A method for reconstructing the locations of point-like scatterers in an elastic half-space is developed in this article. Point-like scatterers are characterized by fluctuations of the Lamé parameters and the mass densities from the background structure of the wavefield. A representation of the wavefield is realized with the monopole and dipole Green's functions. The main contribution of the present work is to clarify the far-field properties of the Green's functions for both the monopole and dipole sources, which yields the pseudo-projection approach to the wave problem. Indicator functions that reconstruct the locations of point-like scatterers are defined by the far-field operator derived from the pseudo-projections and the near-field observation. Numerical computations were carried out to verify the accuracy of the pseudo-projection method. The sensitivities of the accuracy of the reconstruction results for the proposed method to random noise, the grid resolution at the free surface, and the analyzed frequency were also examined. [ABSTRACT FROM AUTHOR]

Abstract Many inverse problems can be described by a PDE model with unknown parameters that need to be calibrated based on measurements related to its solution. This can be seen as a constrained minimization problem where one wishes to minimize the mismatch between the observed data and the model predictions, including an extra regularization term, and use the PDE as a constraint. Often, a suitable regularization parameter is determined by solving the problem for a whole range of parameters –e.g. using the L-curve –which is computationally very expensive. In this paper we derive two methods that simultaneously solve the inverse problem and determine a suitable value for the regularization parameter. The first one is a direct generalization of the Generalized Arnoldi Tikhonov method for linear inverse problems. The second method is a novel method based on similar ideas, but with a number of advantages for nonlinear problems. [ABSTRACT FROM AUTHOR]

Abstract In this paper we consider the transmission eigenvalue problem for Maxwell's equations corresponding to non-magnetic inhomogeneities with contrast in electric permittivity that has fixed sign (only) in a neighborhood of the boundary. Following the analysis made by Robbiano in the scalar case we study this problem in the framework of semiclassical analysis and relate the transmission eigenvalues to the spectrum of a Hilbert–Schmidt operator. Under the additional assumption that the contrast is constant in a neighborhood of the boundary, we prove that the set of transmission eigenvalues is discrete, infinite and without finite accumulation points. A notion of generalized eigenfunctions is introduced and a denseness result is obtained in an appropriate solution space. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we have designed a simple method for the inverse scattering problem of stratified ocean waveguide. The simple approach applies the bisection technique in parallel to estimate the boundaries of inhomogeneous media from the received partial data. In addition, the algorithm only involves the matrix–vector operations and possesses the optimal computation complexity in both two and three dimensions. In practice, it is easy to carry out, robust against noise and capable of reconstructing the penetrable obstacles of different locations and shapes. We can consider the simple method as a direct and straightforward process to supply satisfactory initial positions for the implementation of any existing more refined but computationally more demanding techniques to achieve accurate reconstructions of physical features of scatterers. [ABSTRACT FROM AUTHOR]

This study presents a deep learning based methodology for both remote sensing and design of acoustic scatterers. The ability to determine the shape of a scatterer, either in the context of material design or sensing, plays a critical role in many practical engineering problems. This class of inverse problems is extremely challenging due to their high-dimensional, nonlinear, and ill-posed nature. To overcome these technical hurdles, we introduce a geometric regularization approach for deep neural networks (DNN) based on non-uniform rational B-splines (NURBS) and capable of predicting complex 2D scatterer geometries in a parsimonious dimensional representation. Then, this geometric regularization is combined with physics-embedded learning and integrated within a robust convolutional autoencoder (CAE) architecture to accurately predict the shape of 2D scatterers in the context of identification and inverse design problems. An extensive numerical study is presented in order to showcase the remarkable ability of this approach to handle complex scatterer geometries while generating physically-consistent acoustic fields. The study also assesses and contrasts the role played by the (weakly) embedded physics in the convergence of the DNN predictions to a physically consistent inverse design. [ABSTRACT FROM AUTHOR]

The Born series provides a framework for constructing a connection between earth model parameters and a wavefield characterised by perturbation about a known reference wavefield. The inverse scattering series provides a linear and a nonlinear expression for approximating earth's model parameters. Based on our results, the linear expression provides an accurate approximation of the 1D depth varying velocity profile. However, with increasing contrast, one has to utilise the nonlinear expression for increased accuracy. In seismic inversion, a major problem relates to the bandlimited nature of recorded seismic data, limiting the inversion for absolute amplitudes. To reconstruct missing low frequencies, we employ a frequency extrapolation method, in this case, projection onto convex sets (POCS) to reconstruct missing low frequencies critical for accurate inversion and reconstruction of depth varying velocity profile. Numerical results on 1D synthetic data and succeeding results on 1D physical modelling data demonstrate the efficiency of POCS in reconstructing critical missing low frequencies and subsequent advantage of nonlinear inverse scattering in approximating the change in velocity profile. In comparison, the linear scattering expression fails to provide a reasonable estimate of reflectors beyond the first interface. • Review of Born series and inverse scattering framework for seismic inversion. • Application of frequency recovery technique to bandlimited data. • Combination of inverse scattering and frequency extrapolation to bandlimited data. • Numerical implementation of above methods on VSP data to recover model parameters. • Demonstrate viability of inverse scattering and frequency recovery on lab data. [ABSTRACT FROM AUTHOR]

This paper addresses an inverse scattering problem in a near-zone setting under a multi-monostatic multi-frequency configuration. The main goal is to design a sampling scheme for collecting the data to be convenient and efficient. Accordingly, the considered approach to achieve this goal is a non-uniform sampling strategy which functions based on approximation of the point-spread function via a migration-like inversion algorithm. This scheme benefits from a convolutional scattering operator which can capture the most significant singular values of the continuous operator. However, this scheme leads to a complicated measurement grid due to the influence of different frequencies. To overcome this problem, we exploit an iterative optimal strategy, which optimizes the sampling grid with respect to frequency samples and spatial points. This solution results in an efficient, simple and convenient grid in which the data points are distributed uniformly. We will support the theoretical findings by numerical results. [ABSTRACT FROM AUTHOR]

The focusing Zakharov–Shabat scattering problem on the infinite line with non-zero boundary conditions for the potential is studied, and sufficient conditions on the potential are identified to ensure that the problem admits only purely imaginary discrete eigenvalues. The results, which generalize previous work by Klaus and Shaw, are applicable to the study of solutions of the focusing nonlinear Schrödinger equation with non-zero background. [ABSTRACT FROM AUTHOR]

Assume that a time-harmonic elastic wave is incident onto a penetrable anisotropic elastic body embedded in a homogeneous isotropic background medium. The scattering problem is reduced to a truncated domain. Uniqueness and existence of weak solutions are proved by applying the Fredholm alternative and using properties of the Dirichlet-to-Neumann map in both two and three dimensions. The Fréchet derivative of the near-field solution operator with respect to the boundary of the scatterer is derived. As an application, a descent algorithm is designed for recovering the interface from the near-field data of one or several incident directions and frequencies. Numerical examples in 2D are presented to show the validity and accuracy of the algorithm. [ABSTRACT FROM AUTHOR]

Common particle characteristics needed for many applications may include size, eccentricity, porosity and refractive index. Determining such characteristics from scattered light is a primary goal of remote sensing. For other applications, like differentiating a hazardous particle from the natural background, information about higher fidelity particle characteristics may be required, including specific shape or chemical composition. While a complete characterization of a particle system from its scattered light through the inversion process remains unachievable, great strides have been made in providing information in the form of constraints on particle characteristics. Recent advances have been made in quantifying the characteristics of polydispersions of irregularly shaped particles by making comparisons of the light-scattering signals from model simulant particles. We show that when the refractive index is changed, the light-scattering characteristics from polydispersions of such particles behave monotonically over relatively large parameter ranges compared with those of monodisperse distributions of particles having regular shapes, like spheres, spheroids, etc. This allows for their properties to be interpolated, which results in a significant reduction of the computational load when performing inversions. [ABSTRACT FROM AUTHOR]

The multiangle dynamic light scattering (MDLS) technique can better estimate particle size distributions (PSDs) than single-angle dynamic light scattering. However, determining the inversion range, angular weighting coefficients, and scattering angle combination is difficult but fundamental to the reconstruction for both unimodal and multimodal distributions. In this paper, we propose a self-adapting regularization method called the wavelet iterative recursion nonnegative Tikhonov–Phillips–Twomey (WIRNNT-PT) algorithm. This algorithm combines a wavelet multiscale strategy with an appropriate inversion method and could self-adaptively optimize several noteworthy issues containing the choices of the weighting coefficients, the inversion range and the optimal inversion method from two regularization algorithms for estimating the PSD from MDLS measurements. In addition, the angular dependence of the MDLS for estimating the PSDs of polymeric latexes is thoroughly analyzed. The dependence of the results on the number and range of measurement angles was analyzed in depth to identify the optimal scattering angle combination. Numerical simulations and experimental results for unimodal and multimodal distributions are presented to demonstrate both the validity of the WIRNNT-PT algorithm and the angular dependence of MDLS and show that the proposed algorithm with a six-angle analysis in the 30–130° range can be satisfactorily applied to retrieve PSDs from MDLS measurements. [ABSTRACT FROM AUTHOR]

In this paper we consider the inverse scattering problem for high-contrast targets, and analyze mathematically the experimentally observed phenomenon about super-resolution in imaging the shapes of these targets. In particular, super-resolution at specific high contrast values is justified based on the novel concept of scattering coefficients and several important implications (given by two main theorems, Theorems 4.6 and 4.9 ). This is the first time that a mathematical theory of super-resolution is established in the context of imaging high-contrast inclusions. We shall also illustrate our main findings with a variety of numerical examples. These findings may help develop resonant structures for resolution enhancement. [ABSTRACT FROM AUTHOR]

In a two-layered medium, we prove that a buried inhomogeneous scatterer is uniquely determined from the wave field data measured in the upper half-space with respect to many incident point sources. Moreover, we extend the multilevel sampling method in Liu and Zou (2013) to numerically reconstruct the buried scatterer applying only few incident fields and partial scattered data. The extended recovery scheme only involves matrix–vector operations and does not need to solve any large-scale ill-posed linear systems or any optimization process. It is feasible to deal with the scatterers of different features and easy to implement, highly tolerant to noise and computationally quite cheap. We can regard it as an effective yet simple computational method to provide a reliable initial guess for the implementation in existing more accurate and refined optimization-type reconstruction algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we employ asymptotic analysis to determine information about small volume defects in a known anisotropic scattering medium from far field scattering data. The location of the defects is reconstructed via the MUSIC algorithm from the range of the multi-static response matrix derived from the asymptotic expansion of the far field pattern in the presence of small defects. Since the same data determines the transmission eigenvalues corresponding to the perturbed media, we investigate how the presence of the defects changes the transmission eigenvalues and use this information to recover the strength of the small defects. We provide convergence results on transmission eigenvalues as the size of the defects tends to zero as well as derive the first correction term in the asymptotic expansion of the simple transmission eigenvalues. Numerical examples are presented to show the viability of our imaging method. [ABSTRACT FROM AUTHOR]

Peter Monk has made numerous significant contributions to the field of inverse scattering theory. In the following I try to highlight Peter’s most significant achievements in this area with emphasis on the renaissance that took place in the mathematical and numerical treatment of inverse scattering problems that began in the mid 1980s. [ABSTRACT FROM AUTHOR]

Solitons are a class of nonlinear stable, localized waves. They arise widely in physical problems; applications include water waves, plasma physics, Bose–Einstein condensation and nonlinear optics. Such localized water waves can be traced back to research in the 1800s. In fiber optics 'bright and dark' solitons were discovered in 1973 by Hasegawa and Tappert. In the 1970s a general theory emerged which allows one to linearize and explicitly find soliton solutions to a class of nonlinear wave equations including physically significant equations such as the Korteweg–deVries, nonlinear Schrödinger (NLS) and sine-Gordon equations. Solitons are connected to eigenvalues/bound states of underlying linear scattering equations. The theory, termed the Inverse Scattering Transform (IST) by Ablowitz, Kaup, Newell, Segur, leads to linearization/solutions to broad classes of nonlinear wave equations. In 2013 the theory was extended to novel classes of nonlocal nonlinear wave equations including PT symmetric and reverse-space–time nonlinear equations. In 2022 the theory was shown to encompass fractional integrable nonlinear systems; the fractional KdV and NLS equations are paradigms. [ABSTRACT FROM AUTHOR]

A non-iterative waveform sensing approach is proposed toward (i) geometric reconstruction of penetrable fractures, and (ii) quantitative identification of their heterogeneous contact condition by elastic waves in the frequency domain. To this end, the fracture support Γ (which may be non-planar and unconnected) is first recovered without prior knowledge of the interfacial condition by way of the recently established approaches to non-iterative waveform tomography of heterogeneous fractures, e.g. the methods of generalized linear sampling and topological sensitivity. Given suitable approximation Γ ̆ of the fracture geometry, the jump in the displacement field across Γ ̆ i.e. the fracture opening displacement (FOD) profile is computed from remote sensory data via a regularized inversion of the boundary integral representation mapping the FOD to remote observations of the scattered field. Thus obtained FOD is then used as input for solving the traction boundary integral equation on Γ ̆ for the unknown (linearized) contact parameters. In this study, linear and possibly dissipative interactions between the two faces of a fracture are parametrized in terms of a symmetric, complex-valued matrix K ( ξ ) collecting the normal, shear, and mixed-mode coefficients of specific stiffness. To facilitate the high-fidelity inversion for K ( ξ ) , a 3-step regularization algorithm is devised to minimize the errors stemming from the inexact geometric reconstruction and FOD recovery. The performance of the inverse solution is illustrated by a set of numerical experiments where a cylindrical fracture, endowed with two example patterns of specific stiffness coefficients, is illuminated by plane waves and reconstructed in terms of its geometry and heterogeneous (dissipative) contact condition. [ABSTRACT FROM AUTHOR]

The scattering of stress waves in graded media is discussed in this paper using a conversion of the heterogeneous wave equation to a potential form. In this form, the scattering and inverse scattering methods of mathematical physics can be used with proper enforcement of the boundary conditions on the mechanical quantities, i.e. displacements and tractions. Two approaches for design of non-reflective media are analyzed, one for all frequencies with infinite thickness, and one with finite thickness for a target frequency range. In the latter approach, the transformed potential is set to zero, and the wave speed can have almost any desired profile, from which the profile of impedance is derived. The dispersive behavior of the finite thickness design is studied. Finally, an example is discussed in which a continuous non-reflective profile is discretized to a piece-wise constant profile and the effect of such unavoidable practical constraints on the scattering of the designed layer is shown to be minimal. [ABSTRACT FROM AUTHOR]

We study the inverse acoustic scattering problem in mathematical physics. The problem is to recover the index of refraction in an inhomogeneous medium by measuring the scattered wave fields in the far field. We transform the problem to the interior transmission problem in the study of the Helmholtz equation. We find an inverse uniqueness on the scatterer with a knowledge of a fixed interior transmission eigenvalue. By examining the solution in a series of spherical harmonics in the far field, we can determine uniquely the perturbation source for the radially symmetric perturbations. [ABSTRACT FROM AUTHOR]

In nondestructive evaluation, the maximum wavelength of the incident wave must be smaller than twice the perceptible obstacle size. Hence, in order to be able to detect a nanosize fiber of radius about a few nanometer, the frequency of the incident elastic wave should be in the order of 1 0 12 Hz, i.e., in the terahertz frequency range. The present work addresses the possibility of application of elastic waves in terahertz frequency range to detect embedded circular nanosize inhomogeneity or nanocavity for which, respectively, the interface and surface effects are remarkable. Traditionally, singular sources method (SSM) and linear sampling method (LSM) have been utilized to treat inverse problems in the context of classical elasticity theory, where the surface/interface effects are not important. In the present paper these methods are extended to operate in the mathematical framework of surface/interface elasticity theory, suitable for capturing the surface/interface effects. In this framework the boundary conditions in contrast to the common Dirichlet and Neumann types involve the second derivative of the displacement field. The validity and uniqueness of the extended LSM and SSM in this framework will be established. An exact analysis emphasizing the importance of the consideration of surface/interface effects will also be given for the direct problem of SH waves scattering. In this work, the scatterer will be reconstructed solely based on data from the incident waves and the far-field pattern. Using the proposed theory, the size and location of the embedded nano-size cavity and nano-size inhomogeneity are detected. The surface effects of a nano-size cavity within a silver plate are more significant than the copper–silver interface effects in the case of a nano-size copper inhomogeneity embedded in a silver plate. • SH-wave scattered by a nanosize inhomogeneity/cavity with interface/surface effects. • Reconstruction solely based on data from the incident waves and far-field pattern. • Application of elastic waves in terahertz frequency range. • Extend singular sources and linear sampling methods to work in surface elasticity. • Non-standard interface/boundary conditions in terms of the displacement field. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the anomalous dispersive relations, inverse scattering transform with a Riemann–Hilbert (RH) problem, and fractional multi-solitons of the integrable combined fractional higher-order mKdV (fhmKdV) hierarchy, including the fractional mKdV (fmKdV), fractional fifth-order mKdV (f5mKdV), fractional combined third–fifth-order mKdV (f35mKdV) equations, etc., which can be featured via completeness of squared scalar eigenfunctions of the ZS spectral problem. We construct a matrix RH problem to present three types of fractional N -solitons illustrating anomalous dispersions of the combined fhmKdV hierarchy for the reflectionless case. As some examples, we analyse the wave velocity of the fractional one-soliton such that we find that the fhmKdV equation predicts a power law relationship between the wave velocity and amplitude, and demonstrates the anomalous dispersion. Furthermore, we illustrate other interesting anomalous dispersive wave phenomena containing the elastic interactions of fractional bright and dark solitons, W-shaped soliton and dark soliton, as well as breather and dark soliton. These obtained fractional multi-solitons will be useful to understand the related nonlinear super-dispersive wave propagations in fractional nonlinear media. • The integrable combined fractional higher-order mKdV hierarchy is studied. • The anomalous dispersive relations and explicit forms are presented. • Three types of fractional N-solitons are found via the IST and RH problems. • The interactions of two-soliton solutions are analysed. [ABSTRACT FROM AUTHOR]

As long as the size of the embedded scatterer in comparison to the internal length scale of its surrounding elastic matrix is large, then the linear sampling method (LSM) and singular sources method (SSM) can be used in conjunction with classical theory of elasticity to reconstruct the size of the scatterer with reasonable accuracy. On the other hand, for the micro-/nano-size scatterer this treatment ceases to hold due to the shortcomings of classical theory of elasticity. Moreover, in the realm of this theory, wave propagation through a homogeneous medium is nondispersive on the macro-scale even for high frequency waves. This outcome is incompatible with the practical observations. A remedy to circumvent these shortcomings is to resort to such a higher order continuum theory as couple stress theory with micro inertia. In particular, by accounting for the micro inertia term a new length scale referred to as ”dynamic characteristic length” enters the equation of motion. Association of the media with positive or negative dispersion is a clear manifestation of the new formulation. Therefore, with attention to the aforementioned, both the direct and inverse scattering of high frequency SH-waves by a micro-/nano-size inhomogeneity embedded in an unbounded elastic isotropic medium are addressed in the mathematical framework of couple stress elasticity with micro inertia. It will be shown that for positive (negative) dispersion materials the couple stress phase velocity is greater (less) than the corresponding classical phase velocity. For the purpose of inverse scattering reconstruction of the micro-/nano-size scatterer the LSM and SSM are developed in the context of the proposed theory for the first time. It will be shown that LSM and SSM reconstructions of the size of the scatterer are unique. [ABSTRACT FROM AUTHOR]

We study the small dispersion limit of the Korteweg–de Vries (KdV) equation with periodic boundary conditions and we apply the results to the Zabusky–Kruskal experiment. In particular, we employ a WKB approximation for the solution of the scattering problem for the KdV equation [i.e., the time-independent Schrödinger equation] to obtain an asymptotic expression for the trace of the monodromy matrix and thereby of the spectrum of the problem. We then perform a detailed analysis of the structure of said spectrum (i.e., band widths, gap widths and relative band widths) as a function of the dispersion smallness parameter ϵ . We then formulate explicit approximations for the number of solitons and corresponding soliton amplitudes as a function of ϵ . Finally, by performing an appropriate rescaling, we compare our results to those in the famous Zabusky and Kruskal’s paper, showing very good agreement with the numerical results. [ABSTRACT FROM AUTHOR]

Consider the scattering of long ocean tidal waves by an island taking into account the influence of daily rotation of the Earth, which is modeled by an exterior boundary value problem for the two-dimensional Helmholtz equation with generalized oblique derivative boundary condition. In this paper, we are concerned with a corresponding inverse scattering problem which is to reconstruct the unknown obstacle (island) from the far-field data. After proving the unique solvability of the direct scattering problem in a suitable function space required for our inverse scattering problem, we establish the linear sampling method (LSM) for reconstructing the boundary of the obstacle from the far-field data. To clarify the validity of such a sampling-type method which essentially depends on the solvability of an interior boundary value problem, we show that, except a discrete set of wave numbers, such an interior problem has a unique solution. Finally, some numerical examples are presented to demonstrate the efficiency of the reconstruction scheme. [ABSTRACT FROM AUTHOR]

Using the inverse diffractive grating problem as an example, we demonstrate how a super-resolution can be achieved stably by using far-field data. The idea is to place a slab of a homogeneous medium with a large index of refraction above the grating surface, and more propagating wave modes can be utilized from the far-field data which contributes to the reconstruction resolution. [ABSTRACT FROM AUTHOR]

The local and non-local vector Non-linear Schrödinger Equation (NLSE) with a general cubic non-linearity are considered in presence of a linear term characterized, in general, by a non-hermitian matrix which under certain condition incorporates balanced loss and gain and a linear coupling between the complex fields of the governing non-linear equations. It is shown that the systems posses a Lax pair and an infinite number of conserved quantities and hence integrable. Apart from the particular form of the local and non-local reductions, the systems are integrable when the matrix representing the linear term is pseudo hermitian with respect to the hermitian matrix comprising the generic cubic non-linearity. The inverse scattering transformation method is employed to find exact soliton solutions for both the local and non-local cases. Further, it is shown that the presence of the linear term restricts the possible form of the norming constants and hence the polarization vector. • Local and non-local vector Non-linear Schrödinger Equation with balanced loss and gain is shown to be integrable. • The Lax pair is constructed and an infinite number of conserved quantities are found. • The inverse scattering transformation method is employed to find exact soliton solutions. • The presence of the balanced loss and gain terms determine the possible form of the polarization vector. [ABSTRACT FROM AUTHOR]

Conventional regularization methods have been widely used for estimating particle size distribution (PSD) in single-angle dynamic light scattering, but they could not be used directly in multiangle dynamic light scattering (MDLS) measurements for lack of accurate angular weighting coefficients, which greatly affects the PSD determination and none of the regularization methods perform well for both unimodal and multimodal distributions. In this paper, we propose a recursive regularization method—Recursion Nonnegative Tikhonov–Phillips–Twomey (RNNT-PT) algorithm for estimating the weighting coefficients and PSD from MDLS data. This is a self-adaptive algorithm which distinguishes characteristics of PSDs and chooses the optimal inversion method from Nonnegative Tikhonov (NNT) and Nonnegative Phillips–Twomey (NNPT) regularization algorithm efficiently and automatically. In simulations, the proposed algorithm was able to estimate the PSDs more accurately than the classical regularization methods and performed stably against random noise and adaptable to both unimodal and multimodal distributions. Furthermore, we found that the six-angle analysis in the 30–130° range is an optimal angle set for both unimodal and multimodal PSDs. [ABSTRACT FROM AUTHOR]

This study presents experimental measurements of the radiation characteristics of unicellular freshwater cyanobacterium Synechocystis sp. during their exponential growth in F medium. Their scattering phase function at 633 nm average spectral absorption and scattering cross-sections between 400 and 750 nm were measured. In addition, an inverse method was used for retrieving the spectral effective complex index of refraction of overlapping or touching bispheres and quadspheres from their absorption and scattering cross-sections. The inverse method combines a genetic algorithm and a forward model based on Lorenz–Mie theory, treating bispheres and quadspheres as projected area and volume-equivalent coated spheres. The inverse method was successfully validated with numerically predicted average absorption and scattering cross-sections of suspensions consisting of bispheres and quadspheres, with realistic size distributions, using the T-matrix method. It was able to retrieve the monomers׳ complex index of refraction with size parameter up to 11, relative refraction index less than 1.3, and absorption index less than 0.1. Then, the inverse method was applied to retrieve the effective spectral complex index of refraction of Synechocystis sp. approximated as randomly oriented aggregates consisting of two overlapping homogeneous spheres. Both the measured absorption cross-section and the retrieved absorption index featured peaks at 435 and 676 nm corresponding to chlorophyll a , a peak at 625 nm corresponding to phycocyanin, and a shoulder around 485 nm corresponding to carotenoids. These results can be used to optimize and control light transfer in photobioreactors. The inverse method and the equivalent coated sphere model could be applied to other optically soft particles of similar morphologies. [ABSTRACT FROM AUTHOR]

In order to compare the effect of polarization on linear sampling method, a comprehensive comparison is carried out. Several recoveries based on TE and TM polarization are done and different aspects of linear sampling including minimum number for needed Tx and Rx, optimum value for decision making threshold and resonance phenomena are considered. For each comparison aspect, differences and similarities are revealed. In some aspects two polarizations acts quite similar, while in other aspects they are different. Finally, based on comparison the best choice is introduced. [ABSTRACT FROM AUTHOR]

We consider a two-stage numerical procedure for imaging of objects buried in dry sand using time-dependent backscattering experimental radar measurements. These measurements are generated by a single point source of electric pulses and are collected using a microwave scattering facility which was built at the University of North Carolina at Charlotte. Our imaging problem is formulated as the inverse problem of the reconstruction of the spatially distributed dielectric constant ε r ( x ) , x ∈ R 3 , which is an unknown coefficient in Maxwell’s equations. On the first stage the globally convergent method of Beilina and Klibanov (2012) is applied to get a good first approximation for the exact solution. Results of this stage were presented in Thành et al. (2014). On the second stage the locally convergent adaptive finite element method of Beilina (2011) is applied to refine the solution obtained on the first stage. The two-stage numerical procedure results in accurate imaging of all three components of interest of targets: shapes, locations and refractive indices. In this paper we briefly describe methods and present new reconstruction results for both stages. [ABSTRACT FROM AUTHOR]

In this paper we investigate an iterative method to reconstruct the unknown scatter in R 3 , with only the far-field pattern for just one incident wave. We introduce a general boundary condition and prove its Fréchet derivative. Then this iterative method is able to implement without the boundary condition of the scatterer provided. In practical, one can obtain a better reconstruction without a forward solver at each step. The numerical experiments show the feasibility of this method. [ABSTRACT FROM AUTHOR]

The concept of topological sensitivity (TS) is extended to enable simultaneous 3D reconstruction of fractures with unknown boundary condition and characterization of their interface by way of elastic waves. Interactions between the two surfaces of a fracture, due to e.g. presence of asperities, fluid, or proppant, are described via the Schoenberg’s linear slip model. The proposed TS sensing platform is formulated in the frequency domain, and entails point-wise interrogation of the subsurface volume by infinitesimal fissures endowed with interfacial stiffness. For completeness, the featured elastic polarization tensor – central to the TS formula – is mathematically described in terms of the shear and normal specific stiffness ( κ s , κ n ) of a vanishing fracture. Simulations demonstrate that, irrespective of the contact condition between the faces of a hidden fracture, the TS (used as a waveform imaging tool) is capable of reconstructing its geometry and identifying the normal vector to the fracture surface without iterations. On the basis of such geometrical information, it is further shown via asymptotic analysis – assuming “low frequency” elastic-wave illumination, that by certain choices of ( κ s , κ n ) characterizing the trial (infinitesimal) fracture, the ratio between the shear and normal specific stiffness along the surface of a nearly-planar (finite) fracture can be qualitatively identified. This, in turn, provides a valuable insight into the interfacial condition of a fracture at virtually no surcharge – beyond the computational effort required for its imaging. The proposed developments are integrated into a computational platform based on a regularized boundary integral equation (BIE) method for 3D elastodynamics, and illustrated via a set of canonical numerical experiments. [ABSTRACT FROM AUTHOR]

The inverse scattering problem on the half-line has been studied in the literature in detail. V. Marchenko presented the solution to this problem. In this paper, the invertibility of the steps of the inversion procedure is discussed and a new set of necessary and sufficient conditions on the scattering data is given for the scattering data to be generated by a potential q ε L 1,1 . Our proof is new and in contrast with Marchenko's proof does not use equations on the negative half-line. [ABSTRACT FROM AUTHOR]

Existing damage imaging algorithms for detecting and quantifying structural defects, particularly those based on diffraction tomography, assume far-field conditions for the scattered field data. This paper presents a major extension of diffraction tomography that can overcome this limitation and utilises a near-field multi-static data matrix as the input data. This new algorithm, which employs numerical solutions of the dynamic Green’s functions, makes it possible to quantitatively image laminar damage even in complex structures for which the dynamic Green’s functions are not available analytically. To validate this new method, the numerical Green’s functions and the multi-static data matrix for laminar damage in flat and stiffened isotropic plates are first determined using finite element models. Next, these results are time-gated to remove boundary reflections, followed by discrete Fourier transform to obtain the amplitude and phase information for both the baseline (damage-free) and the scattered wave fields. Using these computationally generated results and experimental verification, it is shown that the new imaging algorithm is capable of accurately determining the damage geometry, size and severity for a variety of damage sizes and shapes, including multi-site damage. Some aspects of minimal sensors requirement pertinent to image quality and practical implementation are also briefly discussed. [ABSTRACT FROM AUTHOR]

The topological sensitivity analysis consists in providing an asymptotic expansion of a shape functional with respect to emerging of small holes in the interior of the domain occupied by the body. In this paper, such an expansion is obtained for the modified Helmholtz equation with an impedance condition prescribed on the boundary of a hole. The topological derivative is then used for numerical simulations for an inverse problem. [ABSTRACT FROM AUTHOR]

We present a Tikhonov parameter choice approach for three-dimensional reconstructions based on a maximum product criterion (MPC) which provides a regularization parameter located in the concave part of the L-curve in log-log scale. Our method, baptized Improved Maximum Product Criterion (IMPC), is an extension of the MPC method developed by Bazán et al. for two-dimensional reconstructions. In the 3D framework, IMPC computes the regularization parameter via a fast iterative algorithm and requires no a priori knowledge of the noise level in the data. It is applied on the linear sampling method for solving the electromagnetic inverse medium problem in the 3D framework. The effectiveness of IMPC is illustrated with numerical examples involving more than one scatterer. [ABSTRACT FROM AUTHOR]

X-ray phase-contrast imaging (XPCI) holds great promise for medical X-ray imaging with high soft-tissue contrast. Obviating optical elements in the imaging chain, propagation-based XPCI (PB-XPCI) has definite advantages over other XPCI techniques in terms of cost, alignment and scalability. However, it imposes strict requirements on the spatial coherence of the source and the resolution of the detector. In this study, we demonstrate that using a coded X-ray source and sparsity-based reconstruction, we can significantly relax these requirements. Using numerical simulation, we assess the feasibility of our approach and study the effect of system parameters on the reconstructed image. The results are demonstrated with images obtained using a bench-top micro-focus XPCI system. [ABSTRACT FROM AUTHOR]

Abstract: Discerning the geometry of spheroidal scatterers of micron order is an important topic in identifying marine microbes. Optical diffraction tomography theory indicates that under the first-order Born approximation for weak scattering, scattering amplitude in the far zone and scattering potential of the scatterer have a Fourier relationship. In this paper, we describe a method based on diffraction tomography theory and determine the size and the shape of spheroidal scatterers by reconstructing the distribution of scattering potential from angular resolved scattered field. As a demonstration of this method, the scattering from spheroidal particles with equal-volume-sphere radii of 0.5429, 1.00, and 2.00μm and an aspect ratio that varies from 0.4 to 1.5 was modeled by using T-matrix theory and used as test data. Simulation results show that in the case of low contrast, size and shape determination can be achieved with sub-wavelength precision. [Copyright &y& Elsevier]