Table of contents

We develop the theory of radar imaging from data measured by a moving antenna emitting a single-frequency waveform. We show that, under a linearized (Born) scattering model, the signal at a given Doppler shift is due to a superposition of returns from stationary scatterers on a cone whose axis is the flight velocity vector. This cone reduces to a hyperbola when the scatterers are known to lie on a planar surface. In this case, reconstruction of the scatterer locations can be accomplished by a tomographic inversion in which the scattering density function is reconstructed from its integrals over hyperbolas. We give an approximate reconstruction formula and analyse the resolution of the resulting image. We provide a numerical shortcut and show results of numerical tests in a simple case.

This paper is devoted to the regularization of a class of evolution hemivariational inequalities. The operator involved is taken to be non-coercive and the data are assumed to be known approximately. Under the assumption that the evolution hemivariational inequality is solvable, a strongly convergent approximation procedure is designed by means of the so-called Browder–Tikhonov regularization method.

We examine the problem of identifying both the location and constitutive law governing electrical current flow across a one-dimensional linear crack in a two-dimensional region when the crack only partially blocks the flow of current. We develop a constructive numerical procedure for solving the inverse problem and provide computational examples.

We report on progress in algorithms for iterative phase retrieval. The theory of convex optimization is used to develop and to gain insight into counterparts for the nonconvex problem of phase retrieval. We propose a relaxation of averaged alternating reflectors and determine the fixed-point set of the related operator in the convex case. A numerical study supports our theoretical observations and demonstrates the effectiveness of the algorithm compared to the current state of the art.

We consider the inverse problems of locating pointwise or small size conductivity defaults in a plane domain, from overdetermined boundary measurements of solutions to the Laplace equation. We express these issues in terms of best rational or meromorphic approximation problems on the boundary, with poles constrained to belong to the domain. This approach furnishes efficient and original resolution schemes.

Vector field tomography deals with the reconstruction of velocity fields from line integrals over certain components of the field, the so-called Doppler transform. Even if we consider incompressible fluids, our reconstruction contains a potential part due to reconstruction errors and the null space of the Doppler transform which consists of potential fields. In this paper we present a method of defect correction for this sort of tomography detecting the potential part of the reconstruction by solving a boundary value problem. We solve this elliptic problem with the help of boundary element methods, where differentiation of the approximate solution can be avoided. If the reconstruction algorithm is based on the method of approximate inverse, the arising Newton potentials can be expressed using the Radon transform and are computed explicitly. The paper includes numerical results for solenoidal vector fields with tangential flow on the boundary from exact and noisy data.

In this paper, we derive integral equations for estimating the density function ρ and the wave speed function c or the acoustic impedance function from boundary measurement data. A nonlinear integral equation for the special case ρ = const and c ≠ const is discussed in detail. We show that the inverse problem of estimating the acoustic impedance function from boundary data measured via a zero-offset transducer configuration can be decomposed into two successively solvable inverse problems. The first inverse problem corresponds to the estimation of the spherical mean function of the acoustic impedance function from a set of one-dimensional integral equations. The second inverse problem corresponds to the estimation of the acoustic impedance function which is closely related to the estimation of the electromagnetic absorption function in transparent media. We conclude the paper with some numerical experiments.

Singular layers modelled by a tangential diffusion process supported on an embedded closed surface (of co-dimension 1) have found applications in tomography problems. In optical tomography they may model the propagation of photons in thin clear layers, which are known to hamper the use of classical diffusion approximations. In impedance tomography they may be used to model thin regions of very high conductivity profile. In this paper we show that such surfaces can be reconstructed from boundary measurements (more precisely, from a local Neumann-to-Dirichlet operator) provided that the material properties between the measurement surface and the embedded surface are known. The method is based on the factorization technique introduced by Kirsch. Once the location of the surface is reconstructed, we show under appropriate assumptions that the full tangential diffusion process and the material properties in the region enclosed by the surface can also uniquely be determined.

We consider the problem of detecting elastic inclusions in elastic bodies by means of mechanical boundary data only, that is measurements of boundary displacement and traction. In previous work of some of the present authors, upper and lower bounds on the size (area or volume) of the inclusions were proven analytically. Following the guidelines drawn up in such previous theoretical study, an extended numerical investigation has been performed in order to prove the effectiveness of this approach. The sensitivity with respect to various relevant parameters is also analysed.

We consider the reconstruction of vertical concentration profiles of atmospheric gases from a spectral distribution of radiation measured from a space-borne infrared spectrometer. Under some separability assumptions of the gases' spectral absorption coefficients, we obtain uniqueness results on the reconstruction of concentration profiles from (multiple-wavenumber) radiance measurements and provide an explicit reconstruction procedure. We show that the reconstruction is a severely ill-posed problem. To address the reconstruction of localized layers, such as ozone or dust layers, we model the reconstruction of strong localized variations in the concentration profiles by using asymptotic expansions in the layer thickness. Assuming the background is known, we obtain that the location as well as the product of the concentration variability within the layer multiplied and the thickness of the layer may be reconstructed from moderately noisy data. The reconstructions of both the concentration and the thickness of the layer require more accurate data.

Anti-reflective boundary conditions have been introduced recently in connection with fast de-blurring algorithms: in the noise-free case, it has been shown that they reduce substantially artefacts called ringing effects with respect to other classical choices (zero Dirichlet, periodic, Neumann) and lead to algorithms costing O(n^dlog(n)) arithmetic operations where n^d is the size of the signal if d = 1 or of the image if d = 2. Here we limit our analysis to the case of signals i.e. d = 1. More precisely, our study considers the role of the noise and how to connect the choice of appropriate boundary conditions with classical regularization schemes. It turns out that a successful approach is close to the Tikhonov technique: we call it re-blurring where the normal equations product A^TA is replaced by A² with A being the blurring operator. A wide numerical experimentation confirms the effectiveness of the proposed idea.

Stochastic inverse problems in heat conduction with consideration of uncertainties in the measured temperature data, temperature sensor locations and thermophysical properties are addressed using a Bayesian statistical inference method. Both parameter estimation and thermal history reconstruction problems, including boundary heat flux and heat source reconstruction, are studied. Probabilistic specification of the unknown variables is deduced from temperature measurements. Hierarchical Bayesian models are adopted to relax the prior assumptions on the unknowns. The use of a hierarchical Bayesian method for automatic selection of the regularization parameter in the function estimation inverse problem is discussed. In addition, the method explores the length scales in the estimation of thermal variables varying in space and time. Markov chain Monte Carlo (MCMC) simulation is conducted to explore the high dimensional posterior state space. The methodologies presented are general and applicable to a number of data-driven engineering inverse problems.

Physical optics is an asymptotic scattering regime where the wavelength of the incident field is much smaller than the linear dimensions of the scatterer. In such conditions, the inverse scattering problem of determining the object profile from measurements of the far-field pattern can be formulated in terms of a linear integral equation of the first kind. We apply an iterative algorithm with convex projections to scattering data corresponding to different conducting objects and discuss the complementarity between this physical optics approach and the reconstruction algorithm based on the linear sampling method. Finally, the two approaches are combined to increase the reconstruction accuracy.

This paper examines and extends a method, recently proposed by the author, for recovering from eigenvalues a symmetric potential of a Sturm–Liouville operator with Dirichlet boundary conditions. It uses Numerov's method and an extension by Andrew and Paine of an asymptotic correction technique of Paine, de Hoog and Anderssen. The method is extended to deal with natural boundary conditions and its convergence properties are investigated. Numerical results show that the method can extract more information from a given set of data than a related earlier method which uses a second-order discretization of the differential equation. Non-symmetric problems are also considered.

We consider the inverse problem of determining the shape of some inaccessible portion of the boundary of a region in n dimensions from Cauchy data for the heat equation on an accessible portion of the boundary. The inverse problem is quite ill-posed and nonlinear. We develop a Newton-like algorithm for solving the problem, with a simple and efficient means for computing the required derivatives, develop methods for regularizing the process, and provide computational examples.

For the Schrödinger equation at fixed energy with a potential supported in a bounded domain we give formulae and equations for finding scattering data from the Dirichlet-to-Neumann map with nonzero background potential. For the case of zero background potential these results were obtained in (Novikov 1988 Multidimensional inverse spectral problem for the equation −Δψ + (v(x) − Eu(x))ψ = 0 Funkt. Anal. Ego Pril.22 (4) 11–22).

Consider the semilinear parabolic equation with the initial condition Dirichlet boundary conditions and a sufficiently regular source term q(⋅), which is assumed to be known a priori on the range of u₀(x). We investigate the inverse problem of determining the function q(⋅) outside this range from measurements of the Neumann boundary data

Via the method of Carleman estimates, we derive global uniqueness of a solution (u, q) to this inverse problem and Hölder stability of the functions u and q with respect to errors in the Neumann data ψ₀, ψ₁, the initial condition u₀ and the a priori knowledge of the function q (on the range of u₀). The results are illustrated by numerical tests. The results of this paper can be extended to more general nonlinear parabolic equations.

We study the following version of the inverse problem in dynamics: given a two-parametric family f(u, v, b) = c of regular curves on a regular surface S: , we determine the generalized force field for which these curves are trajectories. We produce a new linear partial differential relation of first order in the unknown covariant components Q₁(u, v), Q₂(u, v) of the generalized force field corresponding to the given family of orbits. Making use of this equation we find that the problem may have a solution or not depending on the given surface S and the corresponding family of regular curves lying on it. Based on given criteria, we describe a large variety of cases which may appear. Several examples are presented.

We develop quasi-Newton (QN) methods for distributed parameter estimation problems which evolve from electromagnetics, where the forward problem is governed by some form of Maxwell's equations. A Tikhonov-style regularization approach yields an optimization problem with a special structure, where the gradients are calculated using the adjoint method. In many cases, standard QN methods (such as L-BFGS) are not very effective and tend to converge slowly. Taking advantage of the special structure of the problem and the quantities that are calculated in typical gradient descent methods, we develop a class of highly effective methods for the solution of the problem. We demonstrate the merits and effectiveness of our algorithm on two realistic model problems.

We consider the Schrödinger operator −f'' + qf in , with a real compactly supported potential q. We give the solution of two inverse problems (including characterization): q → {zeros of the reflection coefficient} and q → {bound states and resonances}. We describe the set of 'isoresonance potentials', i.e., we obtain all potentials with the same resonances and bound states.

We study three-dimensional homogeneous potentials of degree m from the viewpoint of their compatibility with preassigned two-parametric families of spatial regular orbits given in the solved form f(x, y, z) = c₁, g(x, y, z) = c₂ where each of the functions f and g is also homogeneous in x, y, z (of any degree n_f and n_g, respectively) and where c₁, c₂ are real constants. The orbital elements identifying each family may be represented uniquely by a pair of functions α(x, y, z) and β(x, y, z), both homogeneous of zero degree. Then, depending on the case at hand, we find certain differential conditions (including m and partial derivatives of the given orbital elements) which, if satisfied, ensure the existence of a homogeneous potential V(x, y, z) generating the preassigned orbits. Finally, we offer certain examples which cover the general case as well as special cases.

Dynamic inverse problems, which occur in medical imaging and other fields, are inverse problems in which the quantities to be reconstructed vary in time, although they are related to the measurements through spatial operators only. Traditional methods solve these problems by frame-by-frame reconstruction, then extract temporal behaviour of the objects or regions of interest through curve fitting and other image-based processing. These approaches solve the inverse problem while exploiting only the spatial relationship between the object and the measurement data at each time instant, without using any temporal dynamics of the underlying process, and thus are not optimal unless the solution is temporally uncorrelated. If the spatial operators are linear, and if one, by contrast, solves the whole spatio-temporal process jointly, it falls into the category of general linear least-squares problems. Such approaches are generally difficult, both due to the challenge of modelling the temporal dynamics appropriately as well as to the high dimensionality of the associated large linear system. Several recent reports have approached this problem in different ways, making different prior assumptions on the spatial and temporal behaviour. In this paper we discuss three such approaches, which have been introduced from different points of view, in a common statistical regularization framework, and illuminate their relationships. The three methods are a state-space model, the separability condition and a multiple constraints model. The key result is that there is a clear relationship among the three methods; specifically, the inverse of the spatio-temporal autocovariance matrix has a block tri-diagonal form, a Kronecker product form or a Kronecker sum form, respectively. Some simple simulation examples are presented to illustrate the theoretical analysis.

A method is given for determining the shape of an object from the Cauchy data of the total field measured on the boundary of a domain containing the object in its interior. An application is given to the problem of determining the shape of an object buried in the earth from measurements on the surface of the earth. The advantages of the method are that (1) no a priori knowledge is needed of the physical properties of the scattering object and (2) there is no need to compute Green's function for the background medium.

In the convergence theory of regularization methods for ill-posed problems, so far deterministic error concepts have dominated, which leads to worst-case error estimates. Since this is sometimes not desirable, we aim at providing a framework for proving convergence rates in the Prokhorov metric for the regularization of ill-posed problems with stochastic noise. This allows us to assess uncertainty in the sense of a confidence region for the probability that the deviation between the exact and regularized solutions stays below a given bound with given probability. We exemplify this method for the special case of Tikhonov regularization for linear ill-posed problems and apply the result to the problem of deblurring an image contaminated by random blurring.

The probe method gives a general idea to obtain a reconstruction formula of unknown objects embedded in a known background medium from a mathematical counterpart (the Dirichlet-to-Neumann map) of the measured data of some physical quantity on the boundary of the medium. It is based on the sequence of special solutions of the governing equation for the background medium related to a singular solution of the equation. In this paper the blowup property of the sequence is clarified. Moreover a new formulation of the probe method based on the property is given in some typical inverse boundary value problems.