Methods for source detection in high noise environments are important for single-photon emission computed tomography medical imaging and especially crucial for homeland security applications, which is our main interest. In the latter case, one deals with passively detecting the presence of low emission nuclear sources with significant background noise (with signal to noise ratio 1% or less). In passive

Ptychography, a special case of the phase retrieval problem, is a popular method in modern imaging. Its measurements are based on the shifts of a locally supported window function. In general, direct recovery of an object from such measurements is known to be an ill-posed problem. Although for some windows the conditioning can be controlled, for a number of important cases it is not possible, for instance

Nonlinear least squares data-fitting driven by physical process simulation is a classic and widely successful technique for the solution of inverse problems in science and engineering. Known as ‘full waveform inversion (FWI)’ in application to seismology, it can extract detailed maps of earth structure from near-surface seismic observations, but also suffers from a defect not always encountered in

We consider the irregular (in the Birkhoff and even the Stone sense) transmission eigenvalue problem of the form − y ″ + q ( x ) y = ρ 2 y , y (0) = y (1) cos ρa − y ′(1) ρ −1 sin ρa = 0. The main focus is on the ‘most’ irregular case a = 1, which is important for applications. The uniqueness questions of recovering the potential q ( x ) from transmission eigenvalues were studied comprehensively. Here

We study the inverse problem of recovering a vector field in ##IMG## [http://ej.iop.org/images/0266-5611/36/10/104002/ipabaa32ieqn1.gif] {${\mathbb{R}}^{2}$} from a set of new generalized V-line transforms in three different ways. First, we introduce the longitudinal and transverse V-line transforms for vector fields in ##IMG## [http://ej.iop.org/images/0266-5611/36/10/104002/ipabaa32ieqn2.gif] {${\mathbb{R}}^{2}$}

It is proved that a connected polygonal obstacle coated by thin layers together with its surface impedance function can be determined uniquely from the far field pattern of a single incident plane wave. As a by-product, we prove that the wave field cannot be real-analytic on each corner point lying on the convex hull of the scatterer. Our arguments are based on the Schwarz reflection principle for

This paper is dedicated to design a direct sampling method of inverse electromagnetic scattering problems, which uses multi-frequency sparse backscattering far field data for reconstructing the boundaries of perfectly conducting obstacles. We show that the smallest strip containing the unknown object can be approximately determined by the multi-frequency backscattering far field data at two opposite

The aim of this paper is to develop and analyze numerical schemes for approximately solving the backward problem of subdiffusion equation involving a fractional derivative in time with order α ∈ (0, 1). After using quasi-boundary value method to regularize the ‘mildly’ ill-posed problem, we propose a fully discrete scheme by applying finite element method (FEM) in space and convolution quadrature (CQ)

We consider the inverse acoustic time domain scattering problem for absorbing scatterers modeled by impedance boundary conditions. We present and analyze a factorization method for reconstructing the obstacle boundary from far field measurements. The analysis is based on using the Laplace transform and proving the coercivity of the solution operator in suitable weighted spaces in time. This leads us

We propose a Bayesian inference framework to estimate uncertainties in inverse scattering problems. Given the observed data, the forward model and their uncertainties, we find the posterior distribution over a finite parameter field representing the objects. To construct the prior distribution we use a topological sensitivity analysis. We demonstrate the approach on the Bayesian solution of 2D inverse

The inverse scattering problem for anisotropic media with data measured inside a cavity is considered. We aim to introduce the exterior Steklov eigenvalues (ESEs) and the modified exterior Steklov eigenvalues (MESEs) to detect changes in the material properties of the medium from a knowledge of a modified near field operator each. First, a connection between a modified near field operator and the exterior

In this work, we consider the problem of reconstructing the shape of a three dimensional impenetrable sound-soft axis-symmetric obstacle from measurements of the scattered field at multiple frequencies. This problem has important applications in locating and identifying obstacles with axial symmetry in general, such as, land mines. We obtain a uniqueness result based on a single measurement and propose

This paper suggests a framework for the learning of discretizations of expensive forward models in Bayesian inverse problems. The main idea is to incorporate the parameters governing the discretization as part of the unknown to be estimated within the Bayesian machinery. We numerically show that in a variety of inverse problems arising in mechanical engineering, signal processing and the geosciences

In this paper, we study the inverse scattering problem for energy-dependent Schrödinger equations on the half-line with energy-dependent boundary conditions at the origin. Under certain positivity and very mild regularity assumptions, we transform this scattering problem to the one for non-canonical Dirac systems and show that, in turn, the latter can be placed within the known scattering theory for

We study the issue of numerically solving inverse singular value problems (ISVPs). Motivated by the Newton-type method introduced in [3] for solving ISVPs with distinct and positive singular values, we propose an extended Newton-type method working for ISVPs with multiple and/or zero singular values. Because of the absence of some important and crucial properties, the approach/technique used in the

In this article, we propose a strategy for the active manipulation of scalar Helmholtz fields in bounded near-field regions of an active source while maintaining desired radiation patterns in prescribed far-field directions. This control problem is considered in two environments: free space and homogeneous ocean of constant depth, respectively. In both media, we proved the existence of and characterized

The averaged Kaczmarz iteration is a hybrid of the Landweber method and Kaczmarz method with easy implementation and increased stability for solving problems with multi nonlinear equations. In this paper, we propose an accelerated averaged Kaczmarz type iterative method by introducing the search direction of homotopy perturbation Kaczmarz and a projective strategy. The new iterate is updated by using

In this paper we investigate the non-linear and ill-posed inverse problem of simultaneously identifying the conductivity and the reaction in diffuse optical tomography with noisy measurement data available on an accessible part of the boundary. We propose an energy functional method and the total variational regularization combining with the quadratic stabilizing term to formulate the identification

We develop and analyze a new ‘relax-and-split’ (RS) approach for inverse problems modeled using nonsmooth nonconvex optimization formulations. RS uses a relaxation technique together with partial minimization, and brings classic techniques including direct factorization, matrix decompositions, and fast iterative methods to bear on nonsmooth nonconvex problems. We also extend the approach to robustify

In this work we consider a scattering problem governed by the two-dimensional Helmholtz equation, where some objects of different nature (sound-hard, sound-soft and penetrable) are present in the background medium. First we propose and analyze a system of boundary integral equations to solve the direct problem. After that, we propose a numerical method based on the computation of a multifrequency topological

In seismic imaging the aim is to obtain an image of the subsurface using reflection data. The reflection data are generated using sound waves and the sources and receivers are placed at the surface. The target zone, for example an oil or gas reservoir, lies relatively deep in the subsurface below several layers. The area above the target zone is called the overburden. This overburden will have an imprint

Stochastic gradient descent (SGD) is a promising numerical method for solving large-scale inverse problems. However, its theoretical properties remain largely underexplored in the lens of classical regularization theory. In this note, we study the classical discrepancy principle, one of the most popular a posteriori choice rules, as the stopping criterion for SGD, and prove the finite-iteration termination

Recovering a signal from its Fourier magnitude is referred to as phase retrieval, which occurs in different fields of engineering and applied physics. This paper gives a new characterization of the phase retrieval problem. Particularly useful is the analysis revealing that the common gradient-based regularization does not restrict the set of solutions to a smaller set. Specifically focusing on binary

One of the most powerful approaches to imaging at the nanometer length scale is coherent diffraction imaging using x-ray sources. For amorphous (non-crystalline) samples, raw data collected in the far-field can be interpreted as the modulus of the two-dimensional continuous Fourier transform of the unknown object. The goal is then to recover the phase through computational means by exploiting prior

We consider the problem of atmospheric tomography, as it appears for example in adaptive optics systems for extremely large telescopes. We derive a frame decomposition, i.e., a decomposition in terms of a frame, of the underlying atmospheric tomography operator, extending the singular-value-type decomposition results of Neubauer and Ramlau (2017 SIAM J. Appl. Math. 77 838–853) by allowing a mixture

The forward problem here is the Cauchy problem for a 1D hyperbolic PDE with a variable coefficient in the principal part of the operator. That coefficient is the spatially distributed dielectric constant. The inverse problem consists of the recovery of that dielectric constant from backscattering boundary measurements. The data depend on one variable, which is time. To address this problem, a new version

In this paper we propose a motion-aware variational approach to reconstruct moving objects from sparse dynamic data. The motivation of this work stems from x-ray imaging of plants perfused with a liquid contrast agent, aimed at increasing the contrast of the images and studying the phloem transport in plants over time. The key idea of our approach is to deploy 3D shearlets as a space-temporal prior

In this work we consider the stable numerical solution of large-scale ill-posed nonlinear least squares problems with nonzero residual. We propose a non-stationary Tikhonov method with inexact step computation, specially designed for large-scale problems. At each iteration the method requires the solution of an elliptical trust-region subproblem to compute the step. This task is carried out employing

In this paper we describe an investigation into the application of deep learning methods for low-dose and sparse angle computed tomography using small training datasets. To motivate our work we review some of the existing approaches and obtain quantitative results after training them with different amounts of data. We find that the learned primal-dual method has an outstanding performance in terms

Electrical impedance tomography (EIT) is an imaging modality where a patient or object is probed using harmless electric currents. The currents are fed through electrodes placed on the surface of the target, and the data consists of voltages measured at the electrodes resulting from a linearly independent set of current injection patterns. EIT aims to recover the internal distribution of electrical

We give asymptotic formulas for finding the scattering amplitude at fixed frequency and angles (scattered far field) from the scattering wave function given at n points in dimension d ⩾ 2. These formulas are explicit and their precision is proportional to n . To our knowledge these formulas are new for n ⩾ 2; whereas they reduce to a classical result for n = 1.

Working within the class of piecewise constant conductivities, the inverse problem of electrical impedance tomography can be recast as a shape optimization problem where the discontinuity interface is the unknown. Using Gröger’s ##IMG## [http://ej.iop.org/images/0266-5611/36/9/095006/ipab9f87ieqn1.gif] {${W}_{p}^{1}$} -estimates for mixed boundary value problems, the averaged adjoint method is extended

For dynamical systems of the form ẋ = Ax + c , henceforth called affine systems, we study the problem of determining system parameters A , c from data derived from a single trajectory. We establish necessary and sufficient conditions for identifiability of parameters from either a full trajectory or a set of discrete data points sampled from a single trajectory, expressed as geometrical conditions

We are concerned with the Calderón problem of determining the unknown conductivity of a body from the associated boundary measurement. We establish a logarithmic type stability estimate in terms of the Hausdorff distance in determining the support of a convex polygonal inclusion by a single partial boundary measurement. We also derive the uniqueness result in a more general scenario where the conductivities

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The authors consider a randomized solution to ill-posed operator equations in Hilbert spaces. In contrast to statistical inverse problems, where randomness appears in the noise, here randomness arises in the low-rank matrix approximation of the forward operator, which results in using a Monte Carlo method to solve the inverse problems. In particular, this approach follows the paradigm of the study

Acousto-optic imaging (AOI) is a hybrid imaging process. By perturbing the to-be-reconstructed tissues with acoustic waves, one introduces the interaction between the acoustic and optical waves, leading to a more stable reconstruction of the optical properties. The mathematical model was described in [27], with the radiative transfer equation serving as the forward model for the optical transport.

Inverse scattering problems of the reconstructions of physical properties of a medium from boundary measurements are substantially challenging ones. This work aims to verify the performance on experimentally collected data of a newly developed convexification method for a 3D coefficient inverse problem for the case of unknown objects buried in a sandbox. The measured backscatter data are generated

In this work, we study the inverse spectral stability for the transmission eigenvalue problem from finitely many data with errors. It is shown that when a ⩽ 1, all potentials, whose transmission eigenvalues are ɛ -close in some disc centered at the origin, are also close.

In this paper, we are interested in designing and analyzing a finite element data assimilation method for laminar steady flow described by the linearized incompressible Navier–Stokes equation. We propose a weakly consistent stabilized finite element method which reconstructs the whole fluid flow from noisy velocity measurements in a subset of the computational domain. Using the stability of the continuous

For the large-scale linear discrete ill-posed problem min‖ Ax − b ‖ or Ax = b with b contaminated by white noise, the Golub–Kahan bidiagonalization based LSQR method and its mathematically equivalent CGLS, the conjugate gradient (CG) method applied to A T Ax = A T b , are most commonly used. They have intrinsic regularizing effects, where the iteration number k plays the role of regularization parameter

We introduce a framework for the reconstruction and representation of functions in a setting where these objects cannot be directly observed, but only indirect and noisy measurements are available, namely an inverse problem setting. The proposed methodology can be applied either to the analysis of indirectly observed functional images or to the associated covariance operators, representing second-order

Data-consistent inversion is a recently developed measure-theoretic framework for solving a stochastic inverse problem involving models of physical systems. The goal is to construct a probability measure on model inputs (i.e., parameters of interest) whose associated push-forward measure matches (i.e., is consistent with) a probability measure on the observable outputs of the model (i.e., quantities

We consider an inverse obstacle scattering problem for the Helmholtz equation with obstacles that carry mixed Dirichlet and Neumann boundary conditions. We discuss far field operators that map superpositions of plane wave incident fields to far field patterns of scattered waves, and we derive monotonicity relations for the eigenvalues of suitable modifications of these operators. These monotonicity

For ##IMG## [http://ej.iop.org/images/0266-5611/36/8/085001/ipab7d2aieqn1.gif] {$\mathcal{O}$} a bounded domain in ##IMG## [http://ej.iop.org/images/0266-5611/36/8/085001/ipab7d2aieqn2.gif] {${\mathbb{R}}^{d}$} and a given smooth function ##IMG## [http://ej.iop.org/images/0266-5611/36/8/085001/ipab7d2aieqn3.gif] {$g:\mathcal{O}\to \mathbb{R}$} , we consider the statistical nonlinear inverse problem

A convexification-based numerical method for a coefficient inverse problem for a parabolic PDE is presented. The key element of this method is the presence of the so-called Carleman weight function in the numerical scheme. Convergence analysis ensures the global convergence of this method, as opposed to the local convergence of the conventional least squares minimization techniques. Numerical results

The need to solve discrete ill-posed problems arises in many areas of science and engineering. Solutions of these problems, if they exist, are very sensitive to perturbations in the available data. Regularization replaces the original problem by a nearby regularized problem, whose solution is less sensitive to the error in the data. The regularized problem contains a fidelity term and a regularization

The potential to perform attenuation and scatter compensation (ASC) in single-photon emission computed tomography (SPECT) without a separate transmission scan is highly significant. In this context, attenuation in SPECT is primarily due to Compton scattering, where the probability of Compton scatter is proportional to the attenuation coefficient of the tissue and the energy of the scattered photon

We are concerned with the inverse scattering problem of recovering an inhomogeneous medium by the associated acoustic wave measurement. We prove that under certain assumptions, a single far-field pattern determines the values of a perturbation to the refractive index on the corners of its support. These assumptions are satisfied, for example, in the low acoustic frequency regime. As a consequence if

We consider the inverse source problem of thermo- and photoacoustic tomography, with data registered on an open surface partially surrounding the source of acoustic waves. Under the assumption of constant speed of sound we develop an explicit non-iterative reconstruction procedure that recovers the Radon transform of the sought source, up to an infinitely smooth additive error term. The source then

We analyze the inverse spectral problem on the half line associated with elastic surface waves. Here, we extend the treatment of Love waves [5] to Rayleigh waves. Under certain conditions, and assuming that the Poisson ratio is constant, we establish uniqueness and present a reconstruction scheme for the S-wave speed with multiple wells from the semiclassical spectrum of these waves.

This work investigates the elasticity imaging inverse problem of tumor identification in a fully incompressible medium through a family of inverse problems in a nearly incompressible medium. We develop an inversion framework for saddle point problems that goes far beyond the elasticity imaging inverse problem and applies to a wide variety of inverse problems. We introduce a family of convex optimization

We present new joint reconstruction and regularization techniques inspired by ideas in microlocal analysis and lambda tomography, for the simultaneous reconstruction of the attenuation coefficient and electron density from x-ray transmission (i.e., x-ray CT) and backscattered data (assumed to be primarily Compton scattered). To demonstrate our theory and reconstruction methods, we consider the ‘parallel

The iteratively regularized Gauss–Newton method (IRGNM) is a prominent method for solving nonlinear inverse problems. Based on a modified discrepancy principle, in this paper we propose for the IRGNM in Banach spaces a heuristic rule which is purely data driven and requires no information on the noise level. Under the tangential cone condition on the forward operator and the variational source conditions

In seismic waveform inversion, the reconstruction of the subsurface properties is usually carried out using approximative wave propagation models to ensure computational efficiency. The viscoelastic nature of the subsurface is often unaccounted for, and two popular approximations—the acoustic and linearized Born inversion—are widely used. This leads to reconstruction errors since the approximations

This paper is concerned with the mathematical analysis of experimental methods for the estimation of the power of an uncorrelated, extended aeroacoustic source from measurements of correlations of pressure fluctuations. We formulate a continuous, infinite dimensional model describing these experimental techniques based on the convected Helmholtz equation in ##IMG## [http://ej.iop.org/images/0266-5

We shall study in this paper the convergence rates of the Tikhonov regularized solutions for the recovery of the radiativities in elliptic and parabolic systems in general dimensional spaces. The conditional stability estimates are first derived. Due to the difficulty of the verification of the existing source conditions or nonlinearity conditions of the inverse radiativity problems in high dimensional

In this paper, we consider the minimization of a Tikhonov functional with an ℓ 1 penalty for solving linear inverse problems with sparsity constraints. One of the many approaches used to solve this problem uses the Nemskii operator to transform the Tikhonov functional into one with an ℓ 2 penalty term but a nonlinear operator. The transformed problem can then be analyzed and minimized using standard

The ensemble Kalman filter method can be used as an iterative particle numerical scheme for state dynamics estimation and control-to-observable identification problems. In applications it may be required to enforce the solution to satisfy equality constraints on the control space. In this work we deal with this problem from a constrained optimization point of view, deriving corresponding optimality

We consider the variational regularization for inverse problems in a general form. Based on the discrepancy principle, we propose a heuristic parameter choice rule for choosing the regularization parameter which does not require the information on the noise level and is therefore purely data driven. Under variational source conditions, we obtain a posteriori error estimates. According to the Bakushinskii