Table of contents

2006 has proved to be a very successful year for Inverse Problems. After an increase for the fourth successive year, we achieved our highest impact factor to date, 1.541 (Source: 2005 ISI® Journal Citation Report), and the Editorial Board is keen to build on this success by continuing to improve the service we offer to our readers and authors.

The Board has observed that Inverse Problems receives very few Letters to the Editor submissions, and that moreover those that we do receive rarely conform to the requirements for Letters to the Editor set out in the journal's editorial policy. The Board has therefore decided to merge the current Letters to the Editor section into our regular Papers section, which will now accommodate all research articles that meet the journal's high quality standards. Any submissions that would previously have been Letters to the Editor are still very welcome as Papers, and can be submitted by e-mail to ip@iop.org or online using our online submissions form at authors.iop.org/submit.

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This paper presents a new theory for nonlinear direct amplitude versus angle (AVA) inversion and data-driven depth imaging for a depth-variable acoustic medium. The method which is derived by direct inversion of the forward model of acoustic single scattering requires no information of the velocity and density potentials (subsurface properties), except for the velocity and density of the uppermost layer which is the reference medium where the source and receiver are situated at a finite distance above the scattering medium. The vertically varying velocity and density of the scattering medium are estimated in a data-driven manner solely from the angle- and depth-dependent Born potential profile. The inversion method is obtained in three main steps. In step one, the Born potential profile is computed by constant-velocity imaging (migration) of the single scattering data in the time intercept-slowness domain. Generally, interfaces in the Born potential are severely mislocated in depth compared to the true potential. In step two, 'squeezed' depth-dependent velocity and density potentials are estimated by nonlinear direct AVA inversion of the Born potential after residual-moveout correction. Step three estimates the actual depth-dependent velocity and density potentials by stretching the squeezed potentials so that their interfaces are moved towards the correct depth. In the nomenclature of seismic processing, the three steps can be described by the sequence constant-velocity (partial) migration–inversion–residual migration. In contrast to conventional, velocity-dependent depth migration, which requires an accurate estimate of the velocities of the actual medium to obtain the proper image, the depth imaging in step three requires the squeezed actual velocity potential with interfaces matching those of the zero-angle Born potential depth profile. This is exactly the velocity potential that is found in step two.

This paper examines the range of the forward map for the one-dimensional viscoelastic seismic inverse problem. It is shown that the viscoelastic problem inherits properties that characterize the range of the lossless system, in both the continuum and discrete cases. Furthermore, in the discrete case it is shown that the range of the forward map for a viscoelastic system is actually a proper subset of the range for a lossless system.

In this paper we show that the factorization Lⁿ = L_n⋆L_n−1⋆ ⋅ ⋅ ⋅ ⋆L₁ leads to a decomposition of the second Poisson bracket of the Moyal–Kadomtsev–Petviasfvili hierarchy based on Lⁿ to the direct sum of the second Poisson brackets in the initial pairs, which are associated with L_j for j = 1, ..., n.

The inexact Newton iteration REGINN for regularizing nonlinear ill-posed problems consists of two components: the (outer) Newton iteration, stopped by a discrepancy principle, and the inner iteration, which computes the Newton update by solving approximately a linearized system. The second author proved the convergence of REGINN furnished with the conjugate gradients method as inner iteration (Rieder 2005 SIAM J. Numer. Anal.43 604–22). Amongst others the following feature distinguishes REGINN from other Newton-like regularization schemes: the regularization level for the locally linearized systems can be adapted dynamically incorporating information on the local degree of ill-posedness gained during the iteration. Of course, the potential of this feature can be fully explored only by meaningful numerical experiments in a realistic setting. Therefore, we apply REGINN to the 2D-electrical impedance tomography problem with the complete electrode model. This inverse problem is known to be severely ill-posed. The achieved reconstructions are compared qualitatively and quantitatively with reconstructions from a one-step method which is closely related to the NOSER algorithm (Cheney et al 1990 Int. J. Imag. Syst. Technol.2 66–75), an often used solver in impedance tomography. Our detailed numerical comparison reveals REGINN to be a competitive solver outperforming the one-step method when noise corrupts the data and/or a moderately large number of electrodes is used.

In this paper we present a new method for solving the time-harmonic acoustic inverse scattering problem for a sound-hard crack in . Using the integral equation method to solve the inverse scattering problem, one obtains a Fredholm integral equation of the first kind. Instead of applying regularized Newton's method directly to this integral equation, we derive an equivalent system of two nonlinear integral equations for the inverse problem. In this setting, not only can the regularized Newton's method still be used to solve the inverse problem numerically, but also has the advantage of removing the need to solve a related direct problem at every iteration.

We consider an integrable scalar partial differential equation (PDE) that is second order in time. By rewriting it as a system and applying the Wahlquist–Estabrook prolongation algebra method, we obtain the zero curvature representation of the equation, which leads to a Lax representation in terms of an energy-dependent Schrödinger spectral problem of the type studied by Antonowicz and Fordy. The solutions of this PDE system, and of its associated hierarchy of commuting flows, display weak Painlevé behaviour, i.e. they have algebraic branching. By considering the travelling wave solutions of the next flow in the hierarchy, we find an integrable perturbation of the case (ii) Hénon–Heiles system which has the weak Painlevé property. We perform separation of variables for this generalized Hénon–Heiles system and describe the corresponding solutions of the PDE.

A variable Krasnosel'skii–Mann algorithm generates a sequence {x_n} via the formula x_n+1 = (1 − α_n)x_n + α_nT_nx_n, where {α_n} is a sequence in [0, 1] and {T_n} is a sequence of nonexpansive mappings. We will show, in a fairly general Banach space, that the sequence {x_n} generated converges weakly. This result is used to solve the split feasibility problem which is to find a point x with the property that x ∊ C and Ax ∊ Q, where C and Q are closed convex subsets of Hilbert spaces H₁ and H₂, respectively, and A is a bounded linear operator from H₁ to H₂. The multiple-set split feasibility problem recently introduced by Censor et al is stated as finding a point x ∊ ∩^N_i=1C_i such that Ax ∊ ∩^M_j=1Q_j, where N and M are positive integers, {C₁, ..., C_N} and {Q₁, ..., Q_M} are closed convex subsets of H₁ and H₂, respectively, and A is again a linear bounded operator from H₁ to H₂. One of the purposes of this paper is to introduce more iterative algorithms that solve this problem in the framework of infinite-dimensional Hilbert spaces.

Anti-reflective boundary conditions (BCs) have been introduced recently in connection with fast deblurring algorithms. In the noise free case, it has been shown that they substantially reduce artefacts called ringing effects with respect to other classical choices (zero Dirichlet, periodic, reflective BCs) and lead to O(n²log(n)) arithmetic operations, where n² is the size of the image. In the one-dimensional case, for noisy data, we proposed a successful approach called re-blurring: more specifically, when the PSF is symmetric, the normal equations product A^TA is replaced by A², where A is the blurring operator (see Donatelli et al2005 Inverse Problems21 169–82). Our present goal is to extend the re-blurring idea to nonsymmetric point spread functions (PSFs) in two dimensions. In this more general framework, suitable for real applications, the new proposal is to replace A^T by A' in the normal equations, where A' is the blurring matrix related to the current BCs with PSF rotated by 180°. We notice that, although with zero Dirichlet and periodic BCs the re-blurring approach is equivalent to the normal equations scheme, since there A' = A^T, the novelty concerns both reflective BCs and anti-reflective BCs, where in general A' ≠ A^T. We show that the re-blurring with anti-reflective BCs is computationally convenient and leads to a large reduction of the ringing effects arising in classical deblurring schemes. A wide set of numerical experiments concerning 2D images and nonsymmetric PSFs confirms the effectiveness of our proposal.

In the present paper the inverse scattering problem for the Schrödinger-type equation with a polynomial energy-dependent potential is investigated on the entire real line (−∞; + ∞). The scattering data of the scattering problem are defined and the properties of the data are studied. The system of the main integral equations of the scattering problem is derived and the potential functions are recovered.

This paper introduces and examines some new finite difference methods for computing the (generally nonsymmetric) potential of a Sturm–Liouville operator from its first m Dirichlet eigenvalues and its first m or m + 1 Dirichlet–Neumann eigenvalues. The methods use an asymptotic correction technique of Paine, de Hoog and Anderssen, and its extension to Numerov's method by Andrew and Paine. Numerical results suggest that Numerov's method has even greater advantages over related second-order methods for this problem than those recently reported for problems with symmetric potential.

New formulae on the inverse problem for the continuous skew-self-adjoint Dirac-type system are obtained. For the discrete skew-self-adjoint Dirac-type system the solution of a general-type inverse spectral problem is also derived in terms of the Weyl functions. The description of the Weyl functions on the interval is given. Borg–Marchenko-type uniqueness theorems are also derived for both discrete and continuous non-self-adjoint systems.

We discuss in this paper the possibility of detecting slow slip events (such as silent earthquakes, or earthquake nucleation phases) in the vicinity of geological faults, and the possible localization of those faults from GPS observations. An eigenvalue problem (of Steklov type), modelling the slow evolution of the slip, is stated as a direct problem. The recovery of an active fault from surface observations is formulated as the related inverse problem.

We use an asymptotic formula for the observed surface displacement to infer two inversion techniques for the recovery of faults from surface observations. The first one involves a least-square minimization method; the second one uses the momentum method. The recovered information contains only the depth of the fault, its horizontal position and the 'normalized seismic moment', which is related to the fault shape. We test the two inversion methods for line segment faults in numerical simulations. We are led to conclude that the momentum method gives a very good initial guess for the least-square minimization method, which turns out to be sharp, robust and computationally inexpensive. Unexpectedly, the latter method is also very efficient for faults that are close to the observation surface, despite the fact that our asymptotic approximation may not be valid in that region. Finally we assess how our method for detecting active faults is affected by the sensitivity of the observation apparatus and the stepsize for the grid of surface observation points. The maximum permissible stepsize for such a grid is computed for different values of fault depth and orientation.

We develop an exact filtered backprojection reconstruction algorithm for dynamic-pitch helical trajectories when the convexity condition is violated. The algorithm is a modification of the original inversion formula, which was proposed earlier by one of the authors for a constant-pitch helix. Two filtering lines are added to the algorithm to guarantee exactness. We also study the region where the algorithm applies. It does not extend all the way to the helix, but its size depends on the severity of the violation. If the violation is not very strong, then the region is sufficiently large. Results of numerical experiments are presented.

Decomposition of the time-reversal operator for an array, or equivalently the singular value decomposition of the multistatic response matrix, has been used to improve imaging and localization of targets in complicated media. Typically, each singular value is associated with one scatterer even though it has been shown in several cases that a single scatterer can generate several singular values. In earlier papers Chambers and Berryman (2004 Phys. Rev. Lett.92 0239021–4; 2004 IEEE Trans. Ant. Prop.52 1729–38) showed that a small spherical scatterer can generate up to six singular values depending on the array geometry and sphere composition. It was shown that the existence and characteristics of multiple singular values for each scatterer can, in principle, be used to determine certain properties of the scatterers, e.g. conducting or non-conducting material. In this paper, we extend this analysis to non-spherical targets and show how orientation information about the target may be obtained from the spectrum of singular values. The general properties of the decomposition for small non-spherical dielectric (and possibly conductive) targets in an electromagnetic field are derived and detailed results are obtained for the specific cases of non-magnetic and perfectly conducting needles and discs. It is shown that scatterer orientation can be estimated by tracking the singular values of a linear array as it is rotated around its midpoint.

Certain explicit solutions to the Korteweg–de Vries equation in the first quadrant of the xt-plane are presented. Such solutions involve algebraic combinations of truly elementary functions, and their initial values correspond to rational reflection coefficients in the associated Schrödinger equation. In the reflectionless case such solutions reduce to pure N-soliton solutions. An illustrative example is provided.

In this study we consider the reconstruction of the smooth subdomain boundaries of piecewise constant coefficients of the radiative transfer equation (RTE) from optical tomography data. The assumption is that the values of the absorption and scattering coefficients (μ_a, μ_s) of the different subdomains are known a priori but the smooth subdomain boundaries where μ_a and μ_s are discontinuous are unknown. For the reconstruction of (μ_a, μ_s) it is then sufficient to find the subdomain boundaries separating different values of the coefficients. This results in a nonlinear ill-posed inverse problem. In this study we propose a numerical algorithm for this inverse problem. The approach is based on the finite element discretization of the RTE. We formulate the forward problem as a mapping from a set of shape coefficients representing the shapes of the subdomain boundaries to optical tomography data, and derive the Jacobian of this forward mapping. Then an iterative Newton-type algorithm which seeks a boundary configuration minimizing the residual norm between measured and predicted data is implemented. The performance of the method is tested with simulated frequency domain optical tomography data from diffusive domains containing low-scattering (void) subdomains.

An inverse scattering method is developed for the Camassa–Holm equation. As an illustration of our approach the solutions corresponding to the reflectionless potentials are constructed in terms of the scattering data. The main difference with respect to the standard inverse scattering transform lies in the fact that we have a weighted spectral problem. We therefore have to develop different asymptotic expansions.

Chirp-pulse microwave computerized tomography (CP-MCT) represents an innovative medical imaging technique. One of the advantages of CP-MCT is that data reduction is based on the standard algorithm of x-ray tomography, namely filtered back-projection and therefore is very fast, even if not very accurate. For this reason we recently proposed a modification of this algorithm based on a linear model, hence on a computationally efficient approach. The method has been validated by means of real data obtained with the Niigata prototype of the CP-MCT scanner. However, the applicability of the underlying model has not yet been completely analysed. In this paper, we first investigate the assumptions leading to the linear model and then we show, by means of numerical simulations, that it can provide quantitative maps of the attenuation constant of the body in the low-contrast case.

An integral operator is considered which can be represented as the sum of the convolution operator and a rank-one operator. The reconstruction of the convolution component, provided that the other term is known a priori, from a part of the characteristic numbers is investigated. This inverse problem is reduced to the so-called main nonlinear integral equation with a singularity, which we solve globally. The uniqueness of the solution of the inverse problem in an appropriate class is proved and conditions are obtained that are necessary and sufficient for its solvability. A constructive procedure for solving the inverse problem is given.

We present a new implementation of the linear sampling method in which the set of discretized far-field equations for all sampling points is replaced by a single-functional equation formulated in a Hilbert space defined as a direct sum of L² spaces. The squared norm of the regularized solution of such an equation is used as an indicator function and is analytically determined together with its Fourier transform. This provides some theoretical hints about the spatial resolution achievable by the method.

One of the main problems of astrophysics is to determine the mean field potential of galaxies. The astronomical observations, as well as the numerical simulations, lead to the determination of families of star orbits in a galaxy. On this basis, using the tools of the inverse problem of dynamics, it is possible to find the gravitational potential which gives rise to such motions. The problem can be formulated in various ways; we consider here that the particle trajectories are given by a spatial two-parameter family of curves. From these trajectories we obtain certain functions known as orbital functions, and look for potentials of a special form. If the orbital functions satisfy some differential conditions, a step-by-step procedure offers the expression of the potential. Similar problems arise in thermodynamics and nuclear physics, where axially symmetric potentials are used as models for deformed nuclei.

For solving linear ill-posed problems with noisy data, regularization methods are required. In the present paper, regularized approximations in Hilbert scales are obtained by a general regularization scheme. The analysis of such schemes is based on new results for interpolation in Hilbert scales. Error bounds are obtained under general smoothness conditions.

This work is motivated by the inverse conductivity problem of identifying an embedded object in an infinite slab. The novelty of our approach is that we use complex spherical waves rather than classical Calderón-type functions. For Calderón-type functions, they grow exponentially on one side of a hyperplane and decay exponentially on the other side. Without extra modifications, they are inadequate for treating inverse problems in unbounded domains such as the infinite slab. The obvious reason for this is that Calderón-type functions are not integrable on hyperplanes. So they cannot be used as measurements on infinite boundaries. For complex spherical waves used here, they blow up faster than any given positive polynomial order on the inner side of the unit sphere and decay to zero faster than any given negative polynomial order on the outer side of the unit sphere. We shall construct these special solutions for the conductivity equation in the unbounded domain by a Carleman estimate. Using complex spherical waves, we can treat the inverse problem of determining the object in the infinite slab like the problem in the bounded domain. Most importantly, we can easily localize the boundary measurement, which is of great value in practice. On the other hand, since the probing fronts are spheres, it is possible to detect some concave parts of the object.

We discuss a natural linearization approach to the identification of a diffusion coefficient in a quasilinear parabolic equation from observations of its solution during a short period of time. In this context, a natural linearization means a reduction of the original nonlinear inverse problem to two linear ill-posed problems, when the solution of one of them is used as data for the next one. For both these problems no information concerning solution smoothness is usually given. Therefore, an a posteriori regularization parameter choice strategy should be implemented for their regularization. Even if such a strategy allows an order-optimal regularization parameter choice for the first linear problem, it does not provide us with a reliable regularization error estimate. It means that the data perturbation level for the second linear problem is not given exactly. The main accomplishment of the present paper is to propose a new strategy of the regularization parameter choice which allows an order-optimal regularization of the second linear problem under these circumstances. We also estimate the best possible order of accuracy that in principle can be obtained within a natural linearization approach, and discuss its correlation with known results. Numerical experiments on model problems support a theoretical analysis.

We consider an inverse scattering problem for a perfect conductor that is partially coated by a dielectric. We investigate a method for determining the shape of the object from the Cauchy data of the total field measured on the boundary of a domain containing the object in its interior. We then give a variational characterization of the supremum of the surface impedance and validate the method with some numerical examples. Applications are given to the target identification of a buried partially coated perfect conductor from a knowledge of the electric and magnetic field on the surface of the earth.

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