Table of contents

This paper is concerned with the direct and inverse scattering of time-harmonic electromagnetic waves from bi-anisotropic media. For the direct problem, we establish an integro-differential equation formulation, its Fredholm property, and the uniqueness of a weak solution. Using this integro-differential formulation we study a fast spectral Galerkin method for the numerical solution to the direct problem. Numerical examples are presented and convergence of the spectral method is proved via Gårding estimates for (strongly singular) integro-differential equations. We solve the inverse problem of recovering bi-anisotropic scatterers from far-field data using orthogonality sampling methods. These methods aim to construct imaging functionals which are robust to noise, computationally cheap, and require data for only one or a few incident fields. We provide some theoretical analysis as well as numerical simulations for the proposed imaging functionals.

In this paper we present a fast numerical method for solving large-scale inverse scattering problems. The computational efficiency of the proposed method stems from the utilization of the special structure of the linear forward scattering operator, and does not require or assume any symmetries of the measurement geometry. The described approach is especially useful for inverse problems involving large data sets. As an illustration, we have performed direct numerical inversions for the problem of diffuse optical tomography in measurement geometries with up to  ∼10⁸ independent data points and  ∼ unknowns.

In this paper, we propose to solve the classical backward parabolic equation under the optimal control framework associated with the Tikhonov regularization formulation, where the initial condition is set to be the control variable while the misfit of the final condition is formulated as the Tikhonov objective functional to be minimized. The corresponding first-order necessary optimality system is discretized in a one-shot manner by a second-order finite difference scheme in space and time. The proposed optimal control setting, as discussed in the paper, provides more general framework in terms of solving this type of ill-posed inverse problem. In particular, several existing nonlocal quasi-boundary value methods appear to be the special cases of our proposed optimal control framework with appropriately chosen generalized regularization setting respectively. The relationship between our Tikhonov regularization approach and the quasi-boundary value methods is discussed in detail. The optimal control approach based on Tikhonov regularization is shown to deliver the known optimal order convergence rate. Numerical examples are provided to demonstrate both effective accuracy of approximation as well as excellent stability under the optimal control realization, comparing with three recent quasi-boundary value methods in literature.

In this paper we investigate the problem of recovering the source f  in the elliptic system from an observation z of the state u on a part of the boundary , where the functions and j  are given. For particular interest in reconstructing probably discontinuous sources, we use the standard least squares method with the total variation regularization, i.e. we consider a minimizer of the minimization problem as a reconstruction. Here denotes the unique weak solution of the above elliptic system which depends on the source term f , is the total variation of f , is the regularization parameter, the admissible set where , are given constants, and is the Banach space of all bounded total variation functions. We approximate the problem with piecewise linear and continuous finite elements, where denotes the corresponding finite element approximation of u. This leads to the minimization problem where . In theorems 3.1 and 3.5 we provide the numerical analysis for the discrete solutions f ^h of , and also propose an algorithm to stably solve this discrete minimization problem, where we are led by the algorithmic developments of Bartels (2012 SIAM J. Numer. Anal. 50 1162–80); Tian and Yuan (2016 Inverse Problems32 115011). In particular we prove that the iteration sequence generated by this algorithm converges to a minimizer of , and that a convergence measure of the kind is satisfied. Finally, a numerical experiment is presented to illustrate our theoretical findings.

One of the challenges in single particle reconstruction in cryo-electron microscopy is to find a three-dimensional model of a molecule using its two-dimensional noisy projection-images. In this paper, we propose a robust 'angular reconstitution' algorithm for molecules with n-fold cyclic symmetry, that estimates the orientation parameters of the projections-images. Our suggested method utilizes self common lines which induce identical lines within the Fourier transform of each projection-image. We show that the location of self common lines admits quite a few favorable geometrical constraints, thus allowing to detect them even in a noisy setting. In addition, for molecules with higher order rotational symmetry, our proposed method exploits the fact that there exist numerous common lines between any two Fourier transformed projection-images of such molecules, thus allowing to determine their relative orientation even under high levels of noise. The efficacy of our proposed method is demonstrated using numerical experiments conducted on simulated and experimental data.

We present a comparative study of two qualitative imaging methods in an acoustic waveguide with sound hard walls. The waveguide terminates at one end and contains unknown obstacles of compact support, to be determined from data gathered by an array of sensors that probe the obstacles with waves and measure the scattered response. The first imaging method, known as the factorization method, is based on the factorization of the far field operator. It is designed to image at single frequency and estimates the support of the obstacles by a Picard range criterion. The second imaging method, known as migration, works either with one or multiple frequencies. It forms an image by backpropagating the measured scattered wave to the search points, using the Green's function in the empty waveguide. We study the connection between these methods with analysis and numerical simulations.

We provide a general scheme, in the combined frameworks of mathematical scattering theory and factorization method, for inverse scattering for the couple of self-adjoint operators , where is the free Laplacian in and is one of its singular perturbations, i.e. such that the set is dense. Typically corresponds to a self-adjoint realization of the Laplace operator with some kind of boundary conditions imposed on a null subset; in particular our results apply to standard, either separating or semi-transparent, boundary conditions at , where is a bounded Lipschitz domain. Similar results hold in the case the boundary conditions are assigned only on , a relatively open subset with a Lipschitz boundary. We show that either the obstacle or the screen are determined by the knowledge of the scattering matrix, equivalently of the far field operator, at a single frequency.

We present a convex-nonconvex variational approach for the additive decomposition of noisy scalar fields defined over triangulated surfaces into piecewise constant and smooth components. The energy functional to be minimized is defined by the weighted sum of three terms, namely an fidelity term for the noise component, a Tikhonov regularization term for the smooth component and a total variation (TV)-like non-convex term for the piecewise constant component. The last term is parametrized such that the free scalar parameter allows to tune its degree of non-convexity and, hence, to separate the piecewise constant component more effectively than by using a classical convex TV regularizer without renouncing to convexity of the total energy functional. A method is also presented for selecting the two regularization parameters. The unique solution of the proposed variational model is determined by means of an efficient ADMM-based minimization algorithm. Numerical experiments show a nearly perfect separation of the different components.

We consider the Kelvin–Voigt model for the viscoelasticity, and prove a Carleman estimate for functions without compact supports. Then we apply this Carleman estimate to prove the Lipschitz stability in determining a spatial varying function in an external source term of Kelvin–Voigt model by a single measurement.

Variable-order time-fractional diffusion equations provide very competitive modeling capabilities of challenging phenomena including anomalously subdiffusive transport of solutes in heterogeneous porous media and memory effect as constant-order time-fractional diffusion equations do, while eliminating the nonphysical singularity of the solutions near the initial time of the latter. Moreover, variable-order time-fractional diffusion equations themselves occur in many applications.

We study the initial-boundary value problem of variable-order time-fractional diffusion equations and prove the uniqueness of determining the variable order in the initial-boundary value problem, from the observations of its solution on a sufficiently small open spatial interval over a sufficiently small time interval.

Imaging point sources in heterogeneous environments from boundary or far-field measurements has been extensively studied in the past. In most existing results, the environment, represented by the refractive index function in the model equation, is assumed known in the imaging process. In this work, we investigate the impact of environment uncertainty on the reconstruction of point sources inside it. Following the techniques developed by El Badia and El Hajj (2012 C. R. Acad. Sci. Paris I 350 1031–5), we derive stability of reconstructing point sources in heterogeneous media with respect to measurement error as well as smooth changes in the environment, that is, the refractive index. Numerical simulations with synthetic data are presented to further explore the derived stability properties.

Estimation of the discrete-time Fourier transform (DTFT) at points of a finite domain arises in many imaging applications. A new approach to this task, the golden angle linogram Fourier domain (GALFD), is presented, together with a computationally fast and accurate tool, named golden angle linogram evaluation (GALE), for approximating the DTFT at points of a GALFD. A GALFD resembles a linogram Fourier domain (LFD), which is efficient and accurate. A limitation of linograms is that embedding an LFD into a larger one requires many extra points, at least doubling the domain's cardinality. The GALFD, on the other hand, allows for incremental inclusion of relatively few data points. Approximation error bounds and floating point operations counts are presented to show that GALE computes accurately and efficiently the DTFT at the points of a GALFD. The ability to extend the data collection in small increments is beneficial in applications such as magnetic resonance imaging. Experiments for simulated and for real-world data are presented to substantiate the theoretical claims. The mathematical analysis, algorithms, and software developed in the paper are equally suitable to other angular distributions of rays and therefore we bring the benefits of linograms to arbitrary radial patterns.

Homotopy perturbation iteration is an effective and fast method for solving nonlinear ill-posed problems. It only needs approximately half the computation time of Landweber iteration to reach the similar recovery precision. In this paper, a Nesterov-type accelerated sequential subspace optimization method based on homotopy perturbation iteration is proposed for solving nonlinear inverse problems. The convergence analysis is provided under the general assumptions for iterative regularization methods. The numerical experiments on inverse potential problem indicate that the proposed method dramatically reduces the total number of iterations and time consumption to obtain satisfying approximations, especially for the problems with costly solution of forward calculation.

Optimal order stability estimates of Hölder type for the backward Caputo time-fractional abstract parabolic equations are obtained. This ill-posed problem is regularized by a non-local boundary value problem method with a priori and a posteriori parameter choice rules which guarantee error estimates of Hölder type. Numerical implementations are presented to show the validity of the proposed scheme.

In this study, we derive two new inversion formulae for the n  +  1-dimensional conical Radon transform that integrates a given n  +  1-dimensional function on the upper half space over all conical surfaces with vertices on a hyperplane and a central axis orthogonal to this hyperplane: both formulae are based on the orthonormal bases of certain Hilbert spaces. One formula is derived from a singular value decomposition, a valuable tool in the study of ill-posed problems, of the conical Radon transform.

In this paper, we consider the problem of reconstructing functions in local multiply generated shift invariant spaces from convolution random samples. The sampling set is randomly chosen with one kind of probability distribution over a bounded cube and the available data are the sampled convolution of the original function on the sampling set. We obtain an explicit reconstruction formula. This reconstruction formula succeeds with overwhelming probability when the sampling size is sufficiently large.

This paper presents a novel regularization with a non-convex, non-smooth term of the form with parameters to solve ill-posed linear problems with sparse solutions. We investigate the existence, stability and convergence of the regularized solution. It is shown that this type of regularization is well-posed and yields sparse solutions. Under an appropriate source condition, we get the convergence rate in the -norm for a priori and a posteriori parameter choice rules, respectively. A numerical algorithm is proposed and analyzed based on an iterative threshold strategy with the generalized conditional gradient method. We prove the convergence even though the regularization term is non-smooth and non-convex. The algorithm can easily be implemented because of its simple structure. Some numerical experiments are performed to test the efficiency of the proposed approach. The experiments show that regularization with performs better in comparison with the classical sparsity regularization and can be used as an alternative to the regularizer.

This paper is concerned with the inverse electromagnetic scattering by a two-layered medium in the non-magnetic case. We prove that both the electric permittivity and the conductivity can be uniquely recovered in an unknown annular domain by the internal wave-field measurements at one fixed frequency, if and are both assumed to be constant. Furthermore, we prove that the unknown annular domain can be also uniquely recovered using the same measurements. The method proposed in this paper essentially depends on the uniform a priori estimate for the scattered magnetic field in the sense when the incident fields are induced by a sequence of deformed electric dipoles. Another important ingredient for our method is to reconstruct well-defined PDE systems in a sufficiently small domain by using the solutions to the electromagnetic scattering problem in the two-layered cavity.

In this paper, we consider the subspace optimization method for solving nonlinear inverse problems in Banach spaces, which is based on the sequential Bregman projections with uniformly convex penalty term. The penalty term is allowed to be non-smooth, including L¹ and total variation like penalty functionals, to reconstruct the special features of solutions such as sparsity and discontinuities. Instead of just utilizing the current gradient like the Landweber iteration, the method uses multiple search directions in each iteration to accelerate the convergence. Moreover, their step lengths are calculated by the projection onto the subspace that contains the solution set of the unperturbed problem. Under certain assumptions, we present the detailed convergence analysis when the data is given exactly. For the data containing noise, we use the discrepancy principle as stopping rule and then obtain the regularization result of the method. Finally, some numerical simulations for parameter identification problems are provided to illustrate the effectiveness of capturing the property of exact solutions and the acceleration effect of the method.

We develop a novel wave imaging scheme for reconstructing the shape of an inhomogeneous scatterer and we consider the inverse acoustic obstacle scattering problem as a prototype model for our study. There exists a wealth of reconstruction methods for the inverse obstacle scattering problem and many of them intentionally avoid the interior resonant modes. Indeed, the occurrence of the interior resonance may cause the failure of the corresponding reconstruction. However, based on the observation that the interior resonant modes actually carry the geometrical information of the underlying obstacle, we propose an inverse scattering scheme of using those resonant modes for the reconstruction. To that end, we first develop a numerical procedure in determining the interior eigenvalues associated with an unknown obstacle from its far-field data based on the validity of the factorization method. Then we propose two efficient optimization methods in further determining the corresponding eigenfunctions. Using the afore-determined interior resonant modes, we show that the shape of the underlying obstacle can be effectively recovered. Moreover, the reconstruction yields enhanced imaging resolution, especially for the concave part of the obstacle. We provide rigorous theoretical justifications for the proposed method. Numerical examples in 2D and 3D verify the theoretically predicted effectiveness and efficiency of the method.

In many practical situations, the recovery of information about some phenomenon of interest f  reduces to performing deconvolution on indirect measurements , corresponding to the convolution of f  with a known kernel (point spread function) p . However, in practice, only discrete measurements of g will be available. Consequently, the construction of discrete approximations to f  reduces to deriving discrete versions of . How this is done depends crucially on what is assumed about the properties of the kernel p . Here, it is assumed that p  is symmetric and integrable, and that the Fourier transform of p  is strictly positive.

Different discrete versions are obtained depending on how the discretisation of is performed. After establishing convergence of the truncated schemes, we discuss the underlying difficulties reflecting their ill-posedness and detail how approximations have to be treated with due care. Some of the technical issues underlying the arguments are treated in the appendix.