We consider the Calderón type inverse problem of recovering an isotropic quasilinear conductivity from the Dirichlet-to-Neumann map when the conductivity depends on the solution and its gradient. We show that the conductivity can be recovered on an open subset of small gradients, hence extending a partial result of Muñoz and Uhlmann to all real analytic conductivities. We also recover non-analytic

We consider two main inverse Sturm–Liouville problems: the problem of recovery of the potential and the boundary conditions from two spectra or from a spectral density function. A simple method for practical solution of such problems is developed, based on the transmutation operator approach, new Neumann series of Bessel functions representations for solutions and the Gelfand–Levitan equation. The

In this article, for a time-fractional diffusion-wave equation ##IMG## [http://ej.iop.org/images/0266-5611/36/12/125016/ipabbc5eieqn1.gif] {${\partial }_{t}^{\alpha }u\left(x,t\right)=-Au\left(x,t\right)$} , 0 < t < T with fractional order α ∈ (1, 2), we consider the backward problem in time: determine u (⋅, t ), 0 < t < T by u (⋅, T ) and ∂ t u (⋅, T ). We prove that there exists a countably infinite

This paper considers the inverse boundary value problem for the equation ∇ ⋅ ( σ ∇ u + a |∇ u | p −2 ∇ u ) = 0. We give a procedure for the recovery of the coefficients σ and a from the corresponding Dirichlet-to-Neumann map, under suitable regularity and ellipticity assumptions.

In this paper we introduce a mathematical model of the image saturation phenomenon occurring in a charged coupled device (CCD), and we propose a novel computational method for restoring saturated images acquired by the atmospheric imaging assembly (AIA) telescope. The mathematical model takes into account both primary saturation, when the photon-induced charge reaches the CCD full well capacity, and

The x-ray transform I integrates a function f on ##IMG## [http://ej.iop.org/images/0266-5611/37/1/015007/ipabb5e0ieqn1.gif] {${\mathbb{R}}^{n}$} over lines: ##IMG## [http://ej.iop.org/images/0266-5611/37/1/015007/ipabb5e0ieqn2.gif] {$\left(If\right)\left(x,\xi \right){=\int }_{-\infty }^{\infty }f\left(x+t\xi \right)\enspace \mathrm{d}t$} . The range characterization of the x-ray transform on the Schwartz

Strain engineering is used to obtain desirable materials properties in a range of modern technologies. Direct nanoscale measurement of the three-dimensional strain tensor field within these materials has however been limited by a lack of suitable experimental techniques and data analysis tools. Scanning electron diffraction has emerged as a powerful tool for obtaining two-dimensional maps of strain

Non-Lipschitz regularization has got much attention in image restoration with additive noise removal recently, which can preserve neat edges in the restored image. In this paper, we consider a class of minimization problems with gradient compounded non-Lipschitz regularization applied to non-additive noise removal, with Poisson and multiplicative one as examples. The existence of a solution of the

We study a constrained optimization problem of stable parameter estimation given some noisy (and possibly incomplete) measurements of the state observation operator. In order to find a solution to this problem, we introduce a hybrid regularized predictor–corrector scheme that builds upon both, all-at-once formulation, recently developed by B. Kaltenbacher and her co-authors, and the so-called traditional

We derive stability estimates for the determination of time-dependent vector and scalar potentials in the relativistic Schrödinger equation from partial boundary data. For the case of space dimensions at least 3, we obtain log–log stability estimates for the determination of vector potentials (modulo gauge equivalence) and log–log–log stability estimates for the determination of scalar potentials from

Adaptive optics corrected flood imaging of the retina is a popular technique for studying the retinal structure and function in the living eye. However, the raw retinal images are usually of poor contrast and the interpretation of such images requires image deconvolution. Different from standard deconvolution problems where the point spread function (PSF) is completely known, the PSF in these retinal

Filtered back projection (FBP) methods are the most widely used reconstruction algorithms in computerized tomography (CT). The ill-posedness of this inverse problem allows only an approximate reconstruction for given noisy data. Studying the resulting reconstruction error has been a most active field of research in the 1990s and has recently been revived in terms of optimal filter design and estimating

We derive a new 3D model for magnetic particle imaging (MPI) that is able to incorporate realistic magnetic fields in the reconstruction process. In real MPI scanners, the generated magnetic fields have distortions that lead to deformed magnetic low-field volumes with the shapes of ellipsoids or bananas instead of ideal field-free points (FFP) or lines (FFL), respectively. Most of the common model-based

Choosing an appropriate regularization term is necessary to obtain a meaningful solution to an ill-posed linear inverse problem contaminated with measurement errors or noise. The ℓ p norm covers a wide range of choices for the regularization term since its behavior critically depends on the choice of p and since it can easily be combined with a suitable regularization matrix. We develop an efficient

We prove a Hölder-logarithmic stability estimate for the problem of finding a sufficiently regular compactly supported function v on ##IMG## [http://ej.iop.org/images/0266-5611/36/12/125003/ipabb5dfieqn1.gif] {${\mathbb{R}}^{d}$} from its Fourier transform ##IMG## [http://ej.iop.org/images/0266-5611/36/12/125003/ipabb5dfieqn2.gif] {$\mathcal{F}v$} given on [− r , r ] d . This estimate relies on a Hölder

We present a new inner–outer iterative algorithm for edge enhancement in imaging problems. At each outer iteration, we formulate a Tikhonov-regularized problem where the penalization is expressed in the two-norm and involves a regularization operator designed to improve edge resolution as the outer iterations progress, through an adaptive process. An efficient hybrid regularization method is used to

We consider the inverse scattering problem of reconstructing a perfect conductor from the far field pattern of a scattered time harmonic electromagnetic wave generated by one incident plane wave. In order to apply iterative regularization schemes for the severely ill-posed problem the first and the second domain derivative of the far field pattern with respect to variations of the domain are established

Let f ( x ), ##IMG## [http://ej.iop.org/images/0266-5611/36/12/124008/ipabb2fbieqn1.gif] {$x\in {\mathbb{R}}^{2}$} , be a piecewise smooth function with a jump discontinuity across a smooth surface ##IMG## [http://ej.iop.org/images/0266-5611/36/12/124008/ipabb2fbieqn2.gif] {$\mathcal{S}$} . Let f Λ ϵ denote the Lambda tomography (LT) reconstruction of f from its discrete Radon data ##IMG## [http://ej

Over the last decades, the total variation (TV) has evolved to be one of the most broadly-used regularisation functionals for inverse problems, in particular for imaging applications. When first introduced as a regulariser, higher-order generalisations of TV were soon proposed and studied with increasing interest, which led to a variety of different approaches being available today. We review several

The classic regularization theory for solving inverse problems is built on the assumption that the forward operator perfectly represents the underlying physical model of the data acquisition. However, in many applications, for instance in microscopy or magnetic particle imaging, this is not the case. Another important example represent dynamic inverse problems, where changes of the searched-for quantity

Calderón’s method is a direct linearized reconstruction method for the inverse conductivity problem with the attribute that it can provide absolute images of both conductivity and permittivity with no need for forward modeling. In this work, an explicit relationship between Calderón’s method and the D-bar method is provided, facilitating a ‘higher-order’ Calderón’s method in which a correction term

In this paper, we consider the problem of estimating the internal displacement field of an object which is being subjected to a deformation, from optical coherence tomography images before and after compression. For the estimation of the internal displacement field we propose a novel algorithm, which utilizes particular speckle information to enhance the quality of the motion estimation. We present

The sparsity-based approaches have demonstrated promising performance in image processing. In this paper, for better preservation of the salient edge structures of images, we propose an ℓ 0 + ℓ 2 -norm based analysis model， which requires solving a challenging non-separable ℓ 0 -norm related minimization problem, and we also propose an inexact augmented Lagrangian method with proven convergence to

Time-harmonic far-field source array imaging in a two-dimensional waveguide is analyzed. A low-frequency situation is considered in which the diameter of the waveguide is slightly larger than the wavelength, so that the waveguide supports a limited number of guided modes, and the diameter of the antenna array is smaller than the wavelength, so that the standard resolution formulas in open media predict

In this paper, a weak adversarial network approach is developed to numerically solve a class of inverse problems, including electrical impedance tomography and dynamic electrical impedance tomography problems. The weak formulation of the partial differential equation for the given inverse problem is leveraged, where the solution and the test function are parameterized as deep neural networks. Then

We propose a new direct reconstruction method based on series inversion for electrical impedance tomography (EIT) and the inverse scattering problem for diffuse waves. The standard Born series for the forward problem has the limitation that the series requires that the contrast lies within a certain radius for convergence. Here, we instead propose a modified Born series which converges for the forward

We study the inversion of the conical Radon transform which integrates a function on the surface of a cone. The conical Radon transform recently got significant attention due to its relevance in various imaging applications such as Compton camera imaging and single scattering optical tomography. The unrestricted conical Radon transform is over-determined because the manifold of all cones depends on

In this article, for a two dimensional fractional diffusion equation, we study an inverse problem for simultaneous restoration of the fractional order and the source term from the sparse boundary measurements. By using a sequence of harmonic functions, we construct useful quantitative relation between the unknowns and measurements. From Laplace transform and the knowledge in complex analysis, the uniqueness

Electrical impedance tomography (EIT) is an emerging non-invasive medical imaging modality. It is based on feeding electrical currents into the patient, measuring the resulting voltages at the skin, and recovering the internal conductivity distribution. The mathematical task of EIT image reconstruction is a nonlinear and ill-posed inverse problem. Therefore any EIT image reconstruction method needs

In this article we deal with a class of geometric inverse problem for bottom detection by one single measurement on the free surface in water-waves. We found upper and lower bounds for the size of the region enclosed between two different bottoms, in terms of Neumann and/or Dirichlet data on the free surface. Starting from the general water-waves system in bounded domains with side walls, we manage

In compressed sensing a sparse vector is approximately retrieved from an under-determined equation system Ax = b . Exact retrieval would mean solving a large combinatorial problem which is well known to be NP-hard. For b of the form Ax 0 + ϵ , where x 0 is the ground truth and ϵ is noise, the ‘oracle solution’ is the one you get if you a priori know the support of x 0 , and is the best solution one

In this paper, we consider the inverse eigenvalue problem for the positive doubly stochastic matrices, which aims to construct a positive doubly stochastic matrix from the prescribed realizable spectral data. By using the real Schur decomposition, the inverse problem is written as a nonlinear matrix equation on a matrix product manifold. We propose monotone and nonmonotone Riemannian inexact Newton-CG

Chemical imaging provides information about the distribution of chemicals within a target. When combined with structural information about the target, in situ chemical imaging opens the door to applications ranging from tissue classification to industrial process monitoring. The combination of infrared spectroscopy and optical microscopy is a powerful tool for chemical imaging of thin targets. Unfortunately

The second-order derivative information plays an important role for large-scale full waveform inversion problems. However, exploiting this information requires massive computations and memory requirements. In this study, we develop two inexact Newton methods based on the Lanczos tridiagonalization process to consider the second-order derivative information. Several techniques are developed to improve

This paper proposes a solution method for identification problems in the context of contact mechanics when overabundant data are available on a part Γ m of the domain boundary while data are missing from another part of this boundary. The first step is then to find a solution to a Cauchy problem. The method used by the authors for solving Cauchy problems consists of expanding the displacement field

The inverse coefficient problem of recovering the potential q ( x ) in the damped wave equation ##IMG## [http://ej.iop.org/images/0266-5611/36/11/115011/ipabb8e8ieqn1.gif] {$m\left(x\right){u}_{tt}+\mu \left(x\right){u}_{t}={\left(r\left(x\right){u}_{x}\right)}_{x}+q\left(x\right)u$} , ( x , t ) ∈ Ω T ≔ (0, ℓ ) × (0, T ) subject to the boundary conditions r (0) u x (0, t ) = f ( t ), u ( ℓ , t ) =

We introduce a general framework for the reconstruction of periodic multivariate functions from finitely many and possibly noisy linear measurements. The reconstruction task is formulated as a penalized convex optimization problem, taking the form of a sum between a convex data fidelity functional and a sparsity-promoting total variation based penalty involving a suitable spline-admissible regularizing

In this article we study the stability in an inverse problem of recovering the magnetic field and the electric potential in a bounded smooth domain from boundary observation of the corresponding wave equation. We prove that the knowledge of the partial Dirichlet-to-Neumann map measured on arbitrary subset of the boundary determines the electric potential and the magnetic field. Next, we apply this

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