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The probe and singular sources methods are two well-known classical direct reconstruction methods in inverse obstacle problems governed by partial differential equations. In this paper, by considering an inverse obstacle problem governed by the Laplace equation in a bounded domain as a prototype case, an integrated theory of the probe and singular sources methods is proposed. The theory consists of three parts: (i) introducing the singular sources method combined with the notion of the probe method; (ii) finding a third indicator function whose two ways decomposition yields the indicator functions in the probe and singular sources methods; (iii) finding the completely integrated version of the probe and singular sources methods. [ABSTRACT FROM AUTHOR]

In this article we study Banach spaces of solenoidal solutions, that is, vector fields of zero divergence that satisfy the vector Helmholtz equation in R 3 . As it is known, these solutions are related to the time-harmonic Maxwell equations or Maxwell equations in a simple medium which makes this study important. We define an integral operator given by the Fourier transform of the tangential fields which are square integrable on the unit sphere S 2 that will generate pairs of vector fields satisfying the time-harmonic Maxwell equations, which we will call electromagnetic Herglotz pairs. By last, in the context of Hilbert spaces, the space of solutions is studied, seen as a reproducing kernel Hilbert space and its reproducing kernel is calculated. [ABSTRACT FROM AUTHOR]

Measurements of light typically capture amplitude information, often overlooking crucial phase details. This oversight underscores the importance of phase retrieval (PR), essential in biomedical imaging, X-ray crystallography, and microscopy, for reconstructing complex signals from intensity-only measurements. Traditional methods frequently fall short, especially in noisy conditions or when restricted to single-shot measurements. To address these challenges, we introduce a novel model that combines non-convex regularization with physical constraints. The model adopts the smoothly clipped absolute deviation (SCAD) function as a sparsity regularization term for gradients, incorporating fundamental constraints on support and absorption to establish an inverse model. Using the alternating direction method of multipliers (ADMM), we break down the problem into manageable sub-problems, implementing SCAD shrinkage in the complex domain and applying Wirtinger gradient projection methods. A thorough convergence analysis validates the stability and robustness of the algorithm. Extensive simulations confirm significant improvements in reconstruction quality compared to existing methods, with evaluations demonstrating superior performance across various noise levels and parameter settings. [ABSTRACT FROM AUTHOR]

Electrical Impedance Tomography (EIT) is a widely employed imaging technique in industrial inspection, geophysical prospecting, and medical imaging. However, the inherent nonlinearity and ill-posedness of EIT image reconstruction present challenges for classical regularization techniques, such as the critical selection of regularization terms and the lack of prior knowledge. Deep generative models (DGMs) have been shown to play a crucial role in learning implicit regularizers and prior knowledge. This study aims to investigate the potential of three DGMs – variational autoencoder networks, normalizing flow, and score-based diffusion model – to learn implicit regularizers in learning-based EIT imaging. We first introduce background information on EIT imaging and its inverse problem formulation. Next, we propose three algorithms for performing EIT inverse problems based on corresponding DGMs. Finally, we present numerical and visual experiments, which reveal that (1) no single method consistently outperforms the others across all settings, and (2) when reconstructing an object with two anomalies using a well-trained model based on a training dataset containing four anomalies, the conditional normalizing flow (CNF) model exhibits the best generalization in low-level noise, while the conditional score-based diffusion model (CSD*) demonstrates the best generalization in high-level noise settings. We hope our preliminary efforts will encourage other researchers to assess their DGMs in EIT and other nonlinear inverse problems. [ABSTRACT FROM AUTHOR]

Conductivity reconstruction in an inverse eddy current problem is considered in the present paper. With the electric field measurement on part of domain boundary, we formulate the reconstruction problem as a constrained optimization problem with total variation regularization. The existence and stability are proved for the solution to the optimization problem. The finite element method is employed to discretize the optimization problem. The gradient Lipschitz property of the objective functional is established for the discrete optimization problem. We propose a novel modification to the traditional alternating direction method of multipliers, and prove the convergence of the modified algorithm. Finally, we show some numerical experiments to illustrate the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we consider the inverse scattering problem of identifying a two-layered cavity by internal acoustic measurements under the condition that the interior interface has a mixed transmission boundary condition. We focus on the mathematical analysis of recovering the shape of the interior interface by using the linear sampling method, including reconstructing the surface conductivity by the same measurements. [ABSTRACT FROM AUTHOR]

We propose novel reverse time migration (RTM) methods for the imaging of periodic obstacles using only measurements from the lower or upper side of the obstacle arrays at a fixed frequency. We analyze the resolution of the lower side and upper side RTM methods in terms of propagating modes of the Rayleigh expansion, Helmholtz–Kirchhoff equation and the distance of the measurement surface to the obstacle arrays, where the periodic structure leads to novel analysis. We give some numerical experiments to justify the competitive efficiency of our imaging functionals and the robustness against noises. Further, numerical experiments show sharp images especially for the vertical part of the periodic obstacle in the lower-RTM case, which is not shared by results for imaging bounded compactly supported obstacles. [ABSTRACT FROM AUTHOR]

Electrical Impedance Tomography (EIT) is widely applied in medical diagnosis, industrial inspection, and environmental monitoring. Combining the physical principles of the imaging system with the advantages of data-driven deep learning networks, physics-embedded deep unrolling networks have recently emerged as a promising solution in computational imaging. However, the inherent nonlinear and ill-posed properties of EIT image reconstruction still present challenges to existing methods in terms of accuracy and stability. To tackle this challenge, we propose the learned half-quadratic splitting (HQSNet) algorithm for incorporating physics into learning-based EIT imaging. We then apply Anderson acceleration (AA) to the HQSNet algorithm, denoted as AA-HQSNet, which can be interpreted as AA applied to the Gauss-Newton step and the learned proximal gradient descent step of the HQSNet, respectively. AA is a widely-used technique for accelerating the convergence of fixed-point iterative algorithms and has gained significant interest in numerical optimization and machine learning. However, the technique has received little attention in the inverse problems community thus far. Employing AA enhances the convergence rate compared to the standard HQSNet while simultaneously avoiding artifacts in the reconstructions. Lastly, we conduct rigorous numerical and visual experiments to show that the AA module strengthens the HQSNet, leading to robust, accurate, and considerably superior reconstructions compared to state-of-the-art methods. Our Anderson acceleration scheme to enhance HQSNet is generic and can be applied to improve the performance of various physics-embedded deep learning methods. [ABSTRACT FROM AUTHOR]

The modified transmission eigenvalue (MTE) problem was introduced in [S. Cogar, D. Colton, S. Meng and P. Monk, Modified transmission eigenvalues in inverse scattering theory, Inverse Problems 33 2017, 12, Article ID 125002] and used as a target signature for nondestructive testing. In this paper, we study the inverse spectral problem to reconstruct the modified transmission eigenvalues using Cauchy data. We propose a reciprocity gap functional method and show that the MTEs can be determined by solving some linear ill-posed integral equations. Numerical examples for both absorbing and non-absorbing media are presented to validate the effectiveness and robustness of the proposed method. [ABSTRACT FROM AUTHOR]

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 a useful tool for studying tensor fields in inverse problems of electromagnetic scattering. The method is based on the first Born approximation. [ABSTRACT FROM AUTHOR]

In this work, we apply the Bayesian approach for the acoustic scattering problem to reconstruct the shape of a sound-soft obstacle using the limited-aperture far-field measure data. A novel total variation prior scheme is developed for the obstacle shape parameterization. It is imposed on the Fourier coefficients of the parameterization not the parameterization itself. Using this prior, some less smooth objects can be reconstructed. We also investigate the well-posedness in the sense of the Hellinger distance, Wasserstein distance and Kullback–Leibler divergence. Extensive numerical tests are provided to illustrate the numerical performance. [ABSTRACT FROM AUTHOR]

In this paper, we study the inverse scattering problem and propose a new method to determine the shape of a cavity with Dirichlet boundary conditions from the knowledge of acoustic near-field measurements. The Fourier–Bessel method (FBM) is used to reconstruct the boundary of the cavity from scattered fields which are incited by point sources and measured on a closed curve inside the cavity. We not only analyze the stability but also present the numerical experiments to show the effectiveness of the FBM. [ABSTRACT FROM AUTHOR]

We are concerned with the inverse scattering problem of recovering the refractive indices and transmission coefficients by the corresponding acoustic far-field measurement encoded into the scattering amplitude. Our first uniqueness result is to determine a constant refractive index by the fixed incident direction scattering amplitude, the proof of which is mainly based on the discreteness of the corresponding interior transmission eigenvalues. Then motivated by the previous work (Xiang & Yan 2021), the second uniqueness result is established to recover a piecewise constant refractive index from the far-field pattern at a fixed frequency. [ABSTRACT FROM AUTHOR]

The time domain enclosure method is one of the analytical methods for inverse obstacle problems governed by partial differential equations in the time domain. This paper considers the case when the governing equation is given by the Maxwell system and consists of two parts. The first part establishes the base of the time domain enclosure method for the Maxwell system using a single set of the solutions over a finite time interval for a general (isotropic) inhomogeneous medium in the whole space. It is a system of asymptotic inequalities for the indicator function which may enable us to apply the time domain enclosure method to the problem of finding unknown penetrable obstacles embedded in various background media. As a first step of its expected applications, the case when the background medium is homogeneous and isotropic is considered and the time domain enclosure method is realized. This is the second part. [ABSTRACT FROM AUTHOR]

We consider the inverse source problem of determining an acoustic source from multi-frequency phaseless far-field data. By supplementing some reference sources to the inverse source model, we develop a novel strategy for recovering the phase information of far-field data. This reference source technique leads to an easy-to-implement phase retrieval formula. Mathematically, the stability of the phase retrieval approach is rigorously justified. Then we employ the Fourier method to deal with the multi-frequency inverse source problem with recovered phase information. Finally, some two and three dimensional numerical results are presented to demonstrate the viability and effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Assume that an unknown integral operator living in some known subspace is observed indirectly, by evaluating its action on a discrete measure containing a few isolated Dirac masses at an unknown location. Is this information enough to recover the impulse response location and the operator with a sub-pixel accuracy? We study this question and bring to light key geometrical quantities for exact and stable recovery. We also propose an in-depth study of the presence of additive white Gaussian noise. We illustrate the well-foundedness of this theory on the challenging optical imaging problem of blind deconvolution and blind deblurring with non-stationary operators. [ABSTRACT FROM AUTHOR]

This paper is concerned with the stability of the inverse source problem for Maxwell's equations in an inhomogeneous background medium. We show that the stability estimate consists of the Lipschitz-type data discrepancy and the high frequency tail of the source function, where the latter decreases as the upper bound of the frequency increases. The analysis employs scattering theory to obtain the holomorphic domain and an upper bound for the resolvent of the elliptic operator. [ABSTRACT FROM AUTHOR]

This paper presents a detailed sensitivity analysis of the active manipulation scheme for electromagnetic (EM) fields in free space. The active EM fields control strategy is designed to construct surface current sources (electric and/or magnetic) that can manipulate the EM fields in prescribed exterior regions. The active EM field control is formulated as an inverse source problem. We follow the numerical strategies in our previous works, which employ the Debye potential representation and integral equation representation in the forward modelling. We consider two regularization approaches to the inverse problem to approximate a current source, namely the truncated singular value decomposition (TSVD) and the Tikhonov regularization with the Morozov discrepancy principle. Moreover, we discuss the sensitivity of the active scheme (concerning power budget, control accuracy, and quality factor) as a function of the frequency, the distance between the control region and the source, the mutual distance between the control regions, and the size of the control region. The numerical simulations demonstrate some challenges and limitations of the active EM field control scheme. [ABSTRACT FROM AUTHOR]

A design strategy is proposed for microwave devices built with dielectric-loaded waveguides having one- or two-dimensional discontinuity profiles. The problem is formulated as an inverse problem where the pre-defined scattering parameters are aimed to be the final response of the system. This goal is achieved by optimizing the dimensions of the filling materials. To increase the success of the optimization, the problem is reduced to the longitudinal and widthwise thickness determination of the unit cell, which constitutes the whole structure in a cascade form. To this aim, the frequency response of the unit cell is targeted and then that of the whole system, which is achieved by a two-level optimization procedure of the unit cell. The usefulness of the proposed design strategy is shown through numerical examples as well as the measurement results of two prototypes. [ABSTRACT FROM AUTHOR]

This paper, based on an extension of the Radon transform on distributions, is a mathematical contribution to the field of tomographic imaging in optics. Indeed, we tackle the reconstruction of a Lambertian convex reflector using tomography. In a bi-dimensional setup, we prove that the Lambert's cosine law can be exactly inverted by an original Radon formula. The associated reconstruction contains the geometry and the physics of the problem: it is a Radon measure supported by the reflector, and its density is the inverse of the albedo. Surprisingly, the Radon transform from X-ray transmission tomography, extended on distributions, permits here to invert reflected radiances in optics. [ABSTRACT FROM AUTHOR]

For data obeying a Poisson statistics, the ML-EM ("Maximum Likelihood - Expectation Maximization"), also known as the Richardson-Lucy algorithm, is frequently used and its convergence properties are well known since several decades. To take into account the ubiquitous presence of background emission in several important applications, e.g. in astronomy and medical imaging, a modified algorithm is used in practice. However, despite of its popularity, the convergence of this modified algorithm with background has been established only recently by Salvo and Defrise (IEEE Trans. Med. Imaging. 38:721–729, 2019) in the usual probabilistic context of EM (Expectation Maximization) methods. We present in this paper an alternative convergence proof, which we deem simpler, in a deterministic framework and using only basic tools from convex analysis and optimization theory. [ABSTRACT FROM AUTHOR]

We are concerned with the time-harmonic elastic scattering due to an inhomogeneous elastic material inclusion located inside a uniformly homogeneous isotropic medium. We establish a sharp stability estimate of logarithmic type in determining the support of the elastic scatterer, independent of its material content, by a single far-field measurement when the support is a convex polyhedral domain in R n , n = 2 , 3 . Our argument in establishing the stability result is localized around a corner of the medium scatterer. This enables us to further establish a byproduct result by proving that if a generic medium scatterer, not necessary to be a polyhedral shape, possesses a corner, then there exists a positive lower bound of the scattered far-field patterns. The latter result indicates that if an elastic material object possesses a corner on its support, then it scatters every incident wave stably and invisibility phenomenon does not occur. [ABSTRACT FROM AUTHOR]

This paper is concerned with the two-dimensional inverse acoustic scattering by an unbounded, sound-soft rough surface. We propose a nonlinear integral equation method using multi-frequency phaseless near-field data associated with point sources to reconstruct the shape and location of the rough surface, yielding a fast imaging algorithm. Numerical examples are presented to show the effectiveness of the inverse algorithm. [ABSTRACT FROM AUTHOR]

We define a reciprocity gap functional for electromagnetic scattering in chiral media. We assume that a perfectly conducting scatterer is embedded in a piecewise homogeneous isotropic background chiral medium. The incident field is a spherical electromagnetic wave due to a point source located at a point contained in a chiral environment. We also define the chiral reciprocity gap operator and we prove that it is injective and it has dense range. Using these results and appropriate density properties of chiral Herglotz functions we solve an inverse scattering problem for the determination of the boundary of the scatterer from the knowledge of the tangential components of the electric and magnetic fields. [ABSTRACT FROM AUTHOR]

We investigate a globally convergent method for solving a one-dimensional inverse medium scattering problem using backscattering data at a finite number of frequencies. The proposed method is based on the minimization of a discrete Carleman weighted objective functional. The global convexity of this objective functional is proved. [ABSTRACT FROM AUTHOR]

In this paper, we study an inverse problem of determining the cross section of an infinitely long cylindrical-like material structure from the transverse electromagnetic scattering measurement. We establish a sharp logarithmic stability result in determining a polygonal scatterer by a single far-field measurement. The argument in establishing the stability result is localised around a corner and can be as well used to produce two highly intriguing implications for invisibility and transmission resonance in the wave scattering theory. In fact, we show that if a generic medium scatterer possesses an admissible corner on its support, then there exists a positive lower bound of the L 2 -norm of the associated far-field pattern. For the transmission resonance, we discover a quantitative connection between the regularity of the transmission eigenfunction at a corner and its analytic or Fourier extension. [ABSTRACT FROM AUTHOR]

Consider the scattering of electromagnetic waves by a penetrable homogeneous cylinder at oblique incident. The Maxwell equations are then reduced to a system of a pair of the two-dimensional Helmholtz equations for 푧-components of the electric and magnetic field through coupled oblique boundary conditions. This paper studies an inverse problem of recovering the penetrable obstacle from the far-field pattern of the electric field. The well-known linear sampling method is used to solve this problem. Compared with the usual inverse scattering problem, the coupled system and the oblique derivative boundary condition bring difficulties in theoretical analysis. Some numerical examples are presented to illustrate the validity and feasibility of the proposed method. [ABSTRACT FROM AUTHOR]

We study the global convergence of the gradient descent method of the minimization of strictly convex functionals on an open and bounded set of a Hilbert space. Such results are unknown for this type of sets, unlike the case of the entire Hilbert space. Then, we use our result to establish a general framework to numerically solve boundary value problems for quasi-linear partial differential equations with noisy Cauchy data. The procedure involves the use of Carleman weight functions to convexify a cost functional arising from the given boundary value problem and thus to ensure the convergence of the gradient descent method above. We prove the global convergence of the method as the noise tends to 0. The convergence rate is Lipschitz. Next, we apply this method to solve a highly nonlinear and severely ill-posed coefficient inverse problem, which is the so-called back scattering inverse problem. This problem has many real-world applications. Numerical examples are presented. [ABSTRACT FROM AUTHOR]

This paper is concerned with the inverse scattering and the transmission eigenvalues for anisotropic periodic layers. For the inverse scattering problem, we study the Factorization method for shape reconstruction of the periodic layers from near field scattering data. This method provides a fast numerical algorithm as well as a unique determination for the shape reconstruction of the scatterer. We present a rigorous justification and numerical examples for the factorization method. The transmission eigenvalue problem in scattering have recently attracted a lot of attentions. Transmission eigenvalues can be determined from scattering data and they can provide information about the material parameters of the scatterers. In this paper, we formulate the interior transmission eigenvalue problem and prove the existence of infinitely many transmission eigenvalues for the scattering from anisotropic periodic layers. [ABSTRACT FROM AUTHOR]

We present a general framework to study uniqueness, stability and reconstruction for infinite-dimensional inverse problems when only a finite-dimensional approximation of the measurements is available. For a large class of inverse problems satisfying Lipschitz stability we show that the same estimate holds even with a finite number of measurements. We also derive a globally convergent reconstruction algorithm based on the Landweber iteration. This theory applies to nonlinear ill-posed problems such as electrical impedance tomography (EIT), inverse scattering and quantitative photoacoustic tomography (QPAT), under the assumption that the unknown belongs to a finite-dimensional subspace. In particular, we derive Lipschitz stability estimates for EIT with a matrix approximation of the Neumann-to-Dirichlet map; for the inverse scattering problem with measurements of the scattering amplitude at a finite number of directions on S 2 × S 2 ; and for QPAT with a low-pass filter of the internal energy. [ABSTRACT FROM AUTHOR]

We consider the inverse scattering problem of determining the shape of an impenetrable two-dimensional obstacle with a generalized impedance boundary condition from near-field measurements. We propose the reverse time migration method where the imaging functional is defined as the imaginary part of the cross-correlation. Our analysis shows that, on the one hand, the imaging functional has contrast at the boundary of the scatterer and decays outside the scatterer; On the other hand, the imaging functional is always positive thus may have better stability properties. These imply that the approach has powerful imaging quality which is confirmed in our numerical experiments. [ABSTRACT FROM AUTHOR]

This paper is concerned with the inverse scattering problem which aims to determine the spatially distributed dielectric constant coefficient of the 2D Helmholtz equation from multifrequency backscatter data associated with a single direction of the incident plane wave. We propose a globally convergent convexification numerical algorithm to solve this nonlinear and ill-posed inverse problem. The key advantage of our method over conventional optimization approaches is that it does not require a good first guess about the solution. First, we eliminate the coefficient from the Helmholtz equation using a change of variables. Next, using a truncated expansion with respect to a special Fourier basis, we approximately reformulate the inverse problem as a system of quasilinear elliptic PDEs, which can be numerically solved by a weighted quasi-reversibility approach. The cost functional for the weighted quasi-reversibility method is constructed as a Tikhonov-like functional that involves a Carleman Weight Function. Our numerical study shows that, using a version of the gradient descent method, one can find the minimizer of this Tikhonov-like functional without any advanced a priori knowledge about it. [ABSTRACT FROM AUTHOR]

In this work, we prove the possibility of actively controlling the acoustic field in an ocean consisting of two homogeneous layers of constant depth using a surface source embedded in one of the layers. For a class of prescribed fields on some bounded control regions in either layer, we show the existence of a boundary input on the source, either acoustic pressure or normal velocity so that the propagated field approximates the prescribed fields. Also, we provide a constructive scheme to find the required boundary input using a Morozov discrepancy principle-based Tikhonov regularization scheme. Several numerical simulations are also provided to support our analyses. [ABSTRACT FROM AUTHOR]

In this paper, we study an inverse transmission scattering problem of a time-harmonic acoustic wave from the viewpoint of Bayesian statistics. In Bayesian inversion, the solution of the inverse problem is the posterior distribution of the unknown parameters conditioned on the observational data. The shape of the scatterer will be reconstructed from full-aperture and limited-aperture far-field measurement data. We first prove a well-posedness result for the posterior distribution in the sense of the Hellinger metric. Then, we employ the Markov chain Monte Carlo method based on the preconditioned Crank-Nicolson algorithm to extract the posterior distribution information. Numerical results are given to demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

We consider convergence of alternating projections between non-convex sets and obtain applications to convergence of the Gerchberg-Saxton error reduction method, of the Gaussian expectation-maximization algorithm, and of Cadzow's algorithm. [ABSTRACT FROM AUTHOR]