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The Applied Inverse Problems Conference is held every two years and is always a highlight of the inverse problems calendar. This year the conference was held at Texas A&M University where Inverse Problems was proud to be asked to sponsor the Best Paper Prize. The award was to be presented to a participant, still in the early part of their career, who was presenting in the minisymposia or special sections. The selection of the winner was made by a group composed of the scientific committee and the plenary speakers, all of which were asked to make a note of well-presented talks on interesting topics containing novel approaches or surprising results. The prize was awarded to Dr Hamid Hezari of the Massachusetts Institute of Technology for his work on spectral inverse problems. Some 60 years ago, Marc Kac posed his famous 'Can you hear the shape of a drum' problem. This remained open for 40 years when it was shown to be false. However, the counterexamples are all for non-convex regions with non-smooth boundaries. It remains possible that one can hear the shape of a convex drum with an analytic or smooth boundary. A class of domains with this property is said to be spectrally rigid. This spectral geometry problem shares links with billiards on a domain and integrable systems and is a rich area of mathematical physics research. Birkhoff conjectured that ellipses are unique domains with integrable billiards. The main result of Dr Hezari's talk (which was a joint work with Dr Steve Zelditch) was spectral rigidity of the ellipse among domains with two symmetries.

For original paper, see ibid., vol.4, p.549 (1988). It is pointed out that the methods proposed in a recent paper in this journal are mathematically flawed.

This paper is concerned with imaging of 3D scattering objects with experimental data from the Fresnel database. The first goal of the paper is to investigate a modified version of the orthogonality sampling method (OSM) by Harris and Nguyen [2020 SIAM J. Sci. Comput. 42 B72–737] for the imaging problem. The advantage of the modified OSM over its original version lies in its applicability to more types of polarization vectors associated with the electromagnetic scattering data. We analyze the modified OSM using the factorization analysis for the far field operator and the Funk–Hecke formula. The second goal is to verify the performance of the modified OSM, the OSM, and the classical factorization method for the 3D Fresnel database. The modified OSM we propose is able to invert the sparse and limited-aperture real data in a fast, simple, and efficient way. It is also shown in the real data verification that the modified OSM performs better than its original version and the factorization method.

International audience; The paper is concerned with an inverse point source problem for the Helmholtz equation. It consists of recovering the locations and amplitudes of a finite number of radiative point sources inside a given inhomogeneous medium from the knowledge of a single boundary measurement. The main result of the paper is a new Hölder type stability estimate for the inversion under the assumption that the point sources are well separated. The proof of the stability is based on a combination of Carleman estimates and a technique for proving uniqueness of the Cauchy problem.

In this paper we discuss the unique determination of unknown spatial load $F(x)$ in the damped Euler-Bernoulli beam equation $\rho(x) u_{tt}+\mu u_{t}+(r(x)u_{xx})_{xx}=F(x)G(t)$ from final time measured output (displacement, $u_{T}(x):=u(x,T)$ or velocity, $\nu_{t,T}(x):=u_t(x,T)$). It is shown in [A. Hasanov Hasanoglu and V.G. Romanov, \emph{Introduction to Inverse Problems for Differential Equations}, Springer, New York, 2017] that the unique determination of $F(x)$ in the undamped wave equation $u_{tt}-(k(x)u_{x})_{x}=F(x)G(t)$ from final time output is not possible. This result is also valid for the undamped beam equation $\rho(x) u_{tt}+(r(x)u_{xx})_{xx}=F(x)G(t)$. We prove that in the presence of damping term $\mu u_{t}$, the spatial load can be uniquely determined by the final time output, in terms of the convergent Singular Value Expansion (SVE), as $F(x)=\sum_{n=1}^{\infty} u_{T,n}\psi_n(x)/\sigma_n$, under some acceptable conditions with respect to the final time $T>0$, the damping coefficient $\mu>0$ and the temporal load $G(t)>0$. As an alternative method we propose the Adjoint Problem Approach (APA) and derive an explicit gradient formula for the Fr\'{e}chet derivative of the Tikhonov functional $J(F)=\Vert u(\cdot,T;F)-u_T \Vert_{L^2(0,l)}^2$. Comparative analysis of numerical algorithms based on SVE and APA methods are provided for the harmonic loading $G(t)= \cos (\omega t)$, $\omega>0$, as a most common dynamic loading case. The results presented in this paper not only clearly demonstrate the key role of the damping term $\mu u_{t}$ the inverse problems arising in vibration and wave phenomena, but also allows us, firstly, to find admissible values of the final time $T>0$, at which a final time measured output can be extracted, and secondly, to reconstruct the unknown spatial load $F(x)$ in the damped Euler-Bernoulli beam equation from this measured output.

In this paper we investigate a system of partial differential equations describing the clog- ging process of non-aqueous Lithium/air batteries. The main aim is to optimize the po- sitioning of catalysts on the surface of the pore of the battery in order to maximize the efficiency of the battery. To this end we analyze the parameter-to-state map which maps the initial catalyst positions cat0(x) to the diminishing radius of the pore r(t,x) until clogging. We obtain results on the continuity and Fr echet differentiability of the parameter-to-state map and we obtain the related sensitivity system. The paper also includes numerical exper- iments, which compare a greedy approach for catalyst positioning with results obtained by an analytic optimisation procedure.

International audience; This paper showcases the theoretical and numerical performance of the Sliding Frank-Wolfe, which is a novel optimization algorithm to solve the BLASSO sparse spikes super-resolution problem. The BLASSO is a continuous (i.e. off-the-grid or grid-less) counterpart to the well-known 1 sparse regularisation method (also known as LASSO or Basis Pursuit). Our algorithm is a variation on the classical Frank-Wolfe (also known as conditional gradient) which follows a recent trend of interleaving convex optimization updates (corresponding to adding new spikes) with non-convex optimization steps (corresponding to moving the spikes). Our main theoretical result is that this algorithm terminates in a finite number of steps under a mild non-degeneracy hypothesis. We then target applications of this method to several instances of single molecule fluorescence imaging modalities, among which certain approaches rely heavily on the inversion of a Laplace transform. Our second theoretical contribution is the proof of the exact support recovery property of the BLASSO to invert the 1-D Laplace transform in the case of positive spikes. On the numerical side, we conclude this paper with an extensive study of the practical performance of the Sliding Frank-Wolfe on different instantiations of single molecule fluorescence imaging, including convolutive and non-convolutive (Laplace-like) operators. This shows the versatility and superiority of this method with respect to alternative sparse recovery technics.

This paper analyzes the Tikhonov regularization for ill-posed backward nonlinear Maxwell's equations. We propose a variational source condition (VSC), leading to power-type convergence rates for the Tikhonov regularization. By means of the complex interpolation theory, the proposed VSC is verified under a piecewise H⁸-type Sobolev regularity assumption on the exact initial data. The second part of the paper is focused on the sensitivity analysis for the nonlinear forward Maxwell system and the first-order optimality conditions for the Tikhonov regularization. As a final result, we prove a strong convergence result for the corresponding adjoint state with power-type convergence rates based on VSC.

This paper is concerned with the inverse obstacle scattering problem with phaseless far-field data at a fixed frequency. The main difficulty of this problem is the so-called translation invariance property of the modulus of the far-field pattern or the phaseless far-field pattern generated by one plane wave as the incident field, which means that the location of the obstacle cannot be recovered from such phaseless far-field data at a fixed frequency. It was recently proved in our previous work Xu et al 2018 (SIAM J. Appl. Math. 78 1737–53) that the obstacle can be uniquely determined by the phaseless far-field patterns generated by infinitely many sets of superpositions of two plane waves with different directions at a fixed frequency if the obstacle is a priori known to be a sound-soft or an impedance obstacle with real-valued impedance function. The purpose of this paper is to develop a direct imaging algorithm to reconstruct the location and shape of the obstacle from the phaseless far-field data corresponding to infinitely many sets of superpositions of two plane waves with a fixed frequency as the incident fields. Our imaging algorithm only involves the calculation of the products of the measurement data with two exponential functions at each sampling point and is thus fast and easy to implement. Further, the proposed imaging algorithm does not need to know the type of boundary conditions on the obstacle in advance and is capable to reconstruct multiple obstacles with different boundary conditions. Numerical experiments are also carried out to illustrate that our imaging method is stable, accurate and robust to noise.

This paper is devoted to the uniqueness in inverse scattering problems for the Helmholtz equation with phaseless far-field data. Some novel techniques are developed to overcome the difficulty of translation invariance induced by a single incident plane wave. In this paper, based on adding a reference ball as an extra artificial impenetrable obstacle (resp. penetrable homogeneous medium) to the inverse obstacle (resp. medium) scattering system and then using superpositions of a plane wave and a fixed point source as the incident waves, we rigorously prove that the location and shape of the obstacle as well as its boundary condition or the refractive index can be uniquely determined by the modulus of far-field patterns. The reference ball technique in conjunction with the superposition of incident waves brings in several salient benefits. First, the framework of our theoretical analysis can be applied to both the inverse obstacle and medium scattering problems. Second, for inverse obstacle scattering, the underlying boundary condition could be of a general type and be uniquely determined. Third, only a single frequency is needed. Finally, it provides a very simple proof of the uniqueness.

In this paper we investigate convergence and stability properties of an adaptation to Banach spaces of the algorithm REGINN (Rieder 1999 Inverse Problems 15 309–27). This inexact Newton method solves nonlinear inverse problems by means of linearizing the equation around the current iterate and subsequently applying a regularization technique in the so-called inner iteration in order to obtain a stable approximation of the resulting linearized system. The current iterate is then updated by adding this approximate solution. Using a generic gradient-like method as inner iteration, we provide a whole converge analysis of this Newton-type method, proving, under reasonable assumptions, strong convergence of the generated sequence to a solution of the inverse problem in the noiseless situation and the regularization property in the noisy-data case. Some numerical experiments are performed at the end of the paper for providing the necessary support to the theoretical results

In this paper, we study the gradient descent algorithm generated by a robust loss function over a reproducing kernel Hilbert space (RKHS). The loss function is defined by a windowing function G and a scale parameter σ, which can include a wide range of commonly used robust losses for regression. There is still a gap between theoretical analysis and optimization process of empirical risk minimization based on loss: the estimator needs to be global optimal in the theoretical analysis while the optimization method can not ensure the global optimality of its solutions. In this paper, we aim to fill this gap by developing a novel theoretical analysis on the performance of estimators generated by the gradient descent algorithm. We demonstrate that with an appropriately chosen scale parameter σ, the gradient update with early stopping rules can approximate the regression function. Our elegant error analysis can lead to convergence in the standard L 2 norm and the strong RKHS norm, both of which are optimal in the mini-max sense. We show that the scale parameter σ plays an important role in providing robustness as well as fast convergence. The numerical experiments implemented on synthetic examples and real data set also support our theoretical results.

The inversion of geophysical potential field data can be formulated as an optimization problem with a constraint in the form of a partial differential equation (PDE). It is common practice, if possible, to provide an analytical solution for the forward problem and to reduce the problem to a finite dimensional optimization problem. In an alternative approach the optimization is applied to the problem and the resulting continuous problem which is defined by a set of coupled PDEs is subsequently solved using a standard PDE discretization method, such as the finite element method (FEM). In this paper, we show that under very mild conditions on the data misfit functional and the forward problem in the three-dimensional space, the continuous optimization problem and its FEM discretization are well-posed including the existence and uniqueness of respective solutions. We provide error estimates for the FEM solution. A main result of the paper is that the FEM spaces used for the forward problem and the Lagrange multiplier need to be identical but can be chosen independently from the FEM space used to represent the unknown physical property. We will demonstrate the convergence of the solution approximations in a numerical example. The second numerical example which investigates the selection of FEM spaces, shows that from the perspective of computational efficiency one should use 2 to 4 times finer mesh for the forward problem in comparison to the mesh of the physical property.

In this paper we develop theoretical analysis and numerical reconstruction techniques for the solution of an inverse boundary value problem dealing with the nonlinear, time-dependent monodomain equation, which models the evolution of the electric potential in the myocardial tissue. The goal is the detection of an inhomogeneity omega(e) (where the coefficients of the equation are altered) located inside a domain Omega starting from observations of the potential on the boundary partial derivative Omega. Such a problem is related to the detection of myocardial ischemic regions, characterized by severely reduced blood perfusion and consequent lack of electric conductivity. In the first part of the paper we provide an asymptotic formula for electric potential perturbations caused by internal conductivity inhomogeneities of low volume fraction in the case of three-dimensional, parabolic problems. In the second part we implement a reconstruction procedure based on the topological gradient of a suitable cost functional. Numerical results obtained on an idealized three- dimensional left ventricle geometry for different measurement settings assess the feasibility and robustness of the algorithm.

In this paper, we present a generalization to nonlinear models of the four-dimensional variational dual method, the 4D-PSAS algorithm. The idea of 4D-PSAS (physical space analysis system) is to perform the minimization in the space of the observations, rather than in the model space as in the primal 4D-VAR scheme. Despite the formal equivalence between 4D-VAR and 4D-PSAS in a linear situation (both for model equations and observation operators), the dual method has several important advantages: in oceanographic cases, the observation space is smaller than the model space, which should improve the minimization process; for no additional cost, it provides an estimation of the model error; and finally, it does not have any singularities when the covariance error matrices tend to zero. The idea of this paper is to extend this algorithm to a fully nonlinear situation, as has been done in previous years with other classical data assimilation schemes: the 4D-VAR and the Kalman filter. For this purpose, we consider a nonlinear multi-layer quasi-geostrophic ocean model, which mimics quite well the mid-latitude circulation. We recall the standard primal 4D-VAR scheme applied to this model, and then introduce an extended 4D-PSAS algorithm in the particular case of this nonlinear QG model. We report then the results of extensive numerical experiments that have been carried out to compare this extended algorithm to the classical variational formulation, and to study its sensitivity to many parameters such as the nonlinearities, the number of available observations, the presence of an unknown term in the assimilation model and to study the detection of the model error. As a matter of fact, it is found that this extended algorithm has kept the same advantages as in the linear case (model error detection, smaller sensitivity to various perturbations, more efficient minimization process). All these experiments suggest that it is an efficient assimilation scheme for oceanographic problems.

In this paper we consider level-set-based methods for anomaly reconstruction for applications involving low-sensitivity data concentrating specifically on the problem of electrical resistance tomography (ERT). Typical descent-based inversion methods suffer from poor reconstruction precision in low-sensitivity regions and typically display quite slow convergence. In this paper, we develop a method for constructing the level-set speed function that is capable of overcoming these problems. In the case of the gradient-descent-based level-set evolution, the speed function is defined in terms of the dot products of residual and sensitivity functions. For Gauss–Newton-type methods, however, the speed function is the solution of the linearized inverse problem at each iteration. Here we propose a projection-based approach where the displacement of the level zero, i.e. estimated anomaly contour, at each point depends on the correlation coefficients between the sensitivity of the data to conductivity perturbations and the residual error. In other words, the proposed velocity field is invariant to the absolute amplitudes of both residual and sensitivity, but rather is a reflection of the angle between these two quantities. We demonstrate that our method is a descent-based reconstruction and we relate the mathematical formulation of the projection-based speed function to that of the gradient-descent method and the Gauss–Newton-type approach. The comparison suggests that the proposed technique can be seen as a diagonal approximation of the Gauss–Newton formulation. Using a quasi-linear source-type formulation of the forward problem, we describe an efficient implementation of the projection-based approach for finite-domain imaging problems. The proposed algorithm is simulated with numerical 2D and 3D data and its performance and efficiency are compared to those of the gradient-descent method.

In a celebrated paper 'Can one hear the shape of a drum?' (Kac 1966 Am. Math. Mon. 73 1–23) Kac asked his famous question on the consequences of isospectrality, which was eventually answered negatively in Gordon et al (1992 Invent. Math. 110 1–22) by the construction of non-congruent planar isospectral pairs. In this paper, we pose a similar question for Lie-type geometries, give partial answers and show that the question can be reduced to an important geometrical problem.

This paper is concerned with nonlinear inverse problems where data and solution are vector valued and, moreover, where the solution is assumed to have a sparse expansion with respect to a preassigned frame. We especially focus on such problems where the different channels of the solution exhibit a common or so-called joint sparsity pattern encoding special characteristics of the function under consideration (e.g. a coupling of non-vanishing channel components). Quite recently, an iterative strategy for linear inverse problems with such joint sparsity constraints was presented. Here, we develop an iterative approach for nonlinear inverse problems for which we show norm convergence and regularization properties. The focus throughout the paper is in the context of color image inpainting/recolorization.

This paper considers inverse problems of shape recovery from noisy boundary data, where the forward problem involves the inversion of elliptic PDEs. The piecewise constant solution, a scaling and translation of a characteristic function, is described in terms of a smoother level set function. A fast and simple dynamic regularization method has been recently proposed that has a robust stopping criterion and typically terminates after very few iterations. Direct linear algebra methods have been used for the linear systems arising in both forward and inverse problems, which is suitable for problems of moderate size in 2D. For larger problems, especially in 3D, iterative methods are required. In this paper we extend our previous results to large-scale problems by proposing and investigating iterative linear system solvers in the present context. Perhaps contrary to one's initial intuition, the iterative methods are particularly useful for the inverse rather than the forward linear systems. Moreover, only very few preconditioned conjugate gradient iterations are applied towards the solution of the linear system for the inverse problem, allowing the regularizing effects of such iterations to take centre stage. The efficacy of the obtained method is demonstrated.

In this paper, the topic of optimal experiment design in electrical impedance tomography (EIT) is studied. More specifically, we consider determination of optimal current patterns in EIT in cases of time-varying targets. The reconstruction problem associated with EIT imaging is known to be an ill-posed inverse problem. Statistical inversion methods have been shown to be advantageous in many cases in EIT. In Kaipio et al (2004 Inverse Problems 20 919–36), we considered the problem of optimal experiment design in statistical framework and we proposed an approach for determining optimal current patterns in cases of imaging of time-invariant targets. The approach was based on the statistical interpretation of the reconstruction problem and optimal current patterns were obtained by minimizing the trace of an approximate posterior covariance matrix. In this paper, we utilize a similar approach to determining optimal current patterns in cases of time-varying targets. The image reconstruction problem of EIT is formulated as a state estimation problem. As in the time-invariant case, the optimality criterion is based on the posterior covariances but instead of considering one specific time instant we minimize the time-averaged mean posterior variance. It is shown in a numerical study that the uncertainties of the estimates obtained with optimized current patterns are smaller than those obtained with conventional current patterns. In addition, the results indicate that in time-varying problems a single optimized current pattern may be sufficient to achieve good accuracy, i.e., multiple optimized current patterns do not provide substantial further information on the target. We also demonstrate that in some cases the increase in the number of current patterns can even decrease the reliability of the estimates. This is one of the reasons for the topic of optimal current patterns being quite important in the case of imaging time-varying targets.

In this paper, we investigate the convergence behaviour of the Krasnoselski–Mann (KM) iteration and its generalizations. The KM iteration may be written as follows: where N is a nonexpansive operator on a Hilbert space . This scheme aims to find fixed points of the operator N. Many problems from various fields, including the inverse problems area, can be expressed as a fixed point problem of a certain operator N. In earlier articles, the convergence of the KM iteration and its generations have been investigated in the case when the operator N is nonexpansive and has the fixed points. This paper further studies the convergence behaviour of the algorithms discussed. We first extend the convergence result for the KM iteration to the case when the operator N is firmly nonexpansive, in which case the relaxation parameters are allowed to be in the interval [0, 2], instead of [0, 1]. Then, we show that this result remains valid for the generalized KM iterations. Furthermore, we prove that the sequences generated from the KM iteration or the generalized KM iterations are unbounded if the operators related to the iterations have no fixed points.

The paper discusses inverse problems in which the unknown is a function that is assumed to be piecewise smooth with discontinuities of unknown location and size. If the locations of the discontinuities were known a priori, this prior information could be encoded into a structural smoothness prior. Since the locations are unknown, the smoothness prior itself can be considered as part of the unknown to be determined. A natural framework for this type of problem is provided by hierarchical Bayesian models. In this paper, we propose a hierarchical model suitable for this type of inverse problems, and we show that a single estimate based on this model can be effectively calculated. Being a fully Bayesian solution, it also allows sampling-based studies of the stability and reliability. We also demonstrate that the approach is closely related to bootstrap priors previously investigated by the authors in connection with image inpainting and deblurring. Computed examples with a deconvolution problem are presented.

This paper deals with the inverse problem of detecting the level of corrosion at the interface of an inclusion given thermal measurements at the accessible boundary of the sample. This leads to a transmission problem for the heat equation with an unknown coefficient in a transmission condition. We consider both time-harmonic and delta-pulse excitations. In both cases we prove uniqueness results for the inverse problem. To reconstruct the unknown level of corrosion numerically, we study a non-iterative method and a regularized Newton method and compare their performances in a number of numerical experiments. Photothermal techniques are suitable means of inspecting composite materials with nondestructive tests. We are interested in a technique that consists in heating the accessible side of the material by a defocused laser beam. The goal is to reconstruct internal properties of the material (to detect structural defects, reconstruct the size, depth, orientation of the inclusions and/or physical properties of them) from measurements of the temperature at the side that has been thermically excited. Some recent papers on physical experiments with this kind of techniques are [12, 27, 33, 34]. In this work we study the detection of the level of corrosion at the interface separating the internal inhomogeneities from the matrix material. We explore two different kinds of thermal excitement, produced by time-harmonic incident heating fields and by delta-pulse excitations. In the first one we deal with a steady-state problem. In the corresponding forward problem one looks for time-harmonic solutions of an elliptic conductive-transmission problem. This kind of solution is commonly referred to as thermal waves. For a detailed study of these waves and their uses we refer to [3, 22–24] and references therein. The numerical solution of the forward

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.

We present a modification of the BC-method in the inverse hyperbolic problems. The main novelty is the study of the restrictions of the solutions to the characteristic surfaces instead of the fixed time hyperplanes. The main result is that the time-dependent Dirichlet-to-Neumann operator prescribed on a part of the boundary uniquely determines the coefficients of the self-adjoint hyperbolic operator up to a diffeomorphism and a gauge transformation. In this paper we prove the crucial local step. The global step of the proof will be presented in the forthcoming paper., We corrected the proof of the main Lemma 2.1 by assuming that potentials A(x),V(x) are real valued

In this paper, we consider a one-dimensional inverse problem of estimating an input signal that a priori is known to be a smooth function with a few jump discontinuities. A popular method of regularizing the inverse problem with such prior information is to use a total variation (TV) penalty as a regularizing functional. In this paper, we adopt a different approach, considering the problem from the Bayesian statistics viewpoint and using a hierarchical Gaussian smoothness prior. We demonstrate that the approach allows us to construct a local regularization scheme that is computationally effective and reproduces well the jump discontinuities. The approach avoids the non-differentiability problems encountered by TV methods. Once discretized, the algorithm is completely data driven, the parameter selections requiring no user intervention. The method is applied to an inverse problem in metabolic modelling, where the objective is to estimate the mitochondrial oxygen consumption during muscle activities based on noisy observations of the time decay of the oxygen concentration on the muscle surface.

This paper extends the results of a previous paper designated I hereafter in which the one- and two-soliton solutions of the Degasperis–Procesi (DP) equation were obtained and their peakon limit was considered. Here, we present the general N-soliton solution of the DP equation and investigate its property. We show that it has a novel structure expressed by a parametric representation in terms of the BKP τ-functions. A purely algebraic proof of the solution is given by establishing various identities among the τ-functions. The large time asymptotic of the solution recovers the formula for the phase shift which was derived in I by a different method. Finally, the structure of the N-soliton solution is discussed in comparison with that of the Camassa–Holm shallow water wave equation.

In the present paper, the experimental set-up of Institut Fresnel used to measure the scattered fields of different elongated objects is precisely described. Since the special issue on ‘Testing inversion algorithms against experimental data’, the modifications of this system, outlined here, have mostly been done to improve the synchronization of the apparatuses and the precision of our measurements. Due to a large number of requests from the inverse problem community, it has been decided to add new measurements to the Institut Fresnel's database. All the new targets presented here are two-dimensional inhomogeneous ones. They are made of different dielectrics or are mixing metal and dielectric parts. Both TE and TM polarizations are measured for each target, from 2 to 10 GHz and even 18 GHz for the most complex target. In the first part of this paper the set-up is described precisely. The second part is devoted to the presentation of the targets. Finally, some TE and TM comparisons of measurements and direct problem simulations are shown to accredit our experimental method and to give an idea of the accuracy of these measurements.

Local regularization methods allow for the application of sequential solution techniques for the solution of Volterra problems, retaining the causal structure of the original Volterra problem and leading to fast solution techniques. Stability and convergence of these methods was shown to hold on a large class of linear Volterra problems, i.e., the class of ν-smoothing problems for ν = 1, 2, ... in Lamm (2005 Inverse Problems 21 785–803). In this paper, we enlarge the family of convergent local regularization methods to include sequential versions of classical regularization methods such as sequential Tikhonov regularization. In fact, sequential Tikhonov regularization was considered earlier by Lamm and Elden (1997 SIAM J. Numer. Anal. 34 1432–50) but there the theory was limited to the class of discretized one-smoothing Volterra problems. An interesting feature of sequential classical regularization methods is that they involve two regularization parameters: the usual local regularization parameter r controls the size of the local problem while a second parameter α controls the amount of regularization to be applied in each subproblem. This approach suggests a wavelet type of regularization method with the parameter r controlling spatial resolution and α controlling frequency resolution. In this paper, we also show how the 'future polynomial regularization' method of Cinzori (2004 Inverse Problems 20 1791–806) can be viewed as a special case of the general framework of Lamm (2005) in the 1-smoothing case. In addition, we extend the results of Lamm (2005) to nonlinear Volterra problems of Hammerstein type and give numerical results to illustrate the effectiveness of the method in this case.

This paper extends the results of the previous paper designated I hereafter in which the one- and two-soiton solutions of the Degasperis-Procesi(DP) equation were obtained and their peakon limit was considered. Here, we present the general N-soliton solution of the DP equation and investigate its property. We show that it has a novel structure expressed by a parametric representation in terms of the BKP tau-functions. A purely algebraic proof of the solution is given by establishing various identities among the tau-functions. The large time asymptotic of the solution recovers the formula for the phase shift which was derived in I by a different method. Finally, the structure of the N-soliton solution is discussed in comparison with that of the Camassa-Holm shallow water wave equation.

The analysis of measured spectra, when it reduces to finding the positions and heights of spectral lines, becomes an extremely difficult problem when the lines are closely spaced and broadened to the point where overlapping occurs, and explicit models for the shape of the peaks are unknown. For the widely used nonlinear fitting methods, good starting values cannot be found by direct inspection of the spectra. Sometimes, it is even impossible to determine the number of overlapping spectral lines. Resolution enhancement procedures aim to undo the broadening in order to go back towards the original situation. Traditional resolution rules-of-thumb, such as the Rayleigh, Houston and Sparrow criteria, must be replaced by computational procedures which invoke appropriate assumptions about the broadening of the spectral lines. Measured spectra can be viewed as convolutions between known smooth kernels and discrete line spectra. In this way, the determination of positions and heights of the spectral lines can be recast as a deconvolution problem. Unfortunately, the theory which guarantees the effectiveness of deconvolution requires smoothness of the solution and, as a consequence, does not apply to discrete line spectra. However, discrete line spectra only need to be approximated by spectral peaks which satisfy the smoothness criteria required by the regularization theory in order to formulate a suitable deconvolution framework. In this paper, the corresponding deconvolution problem is solved effectively using a numerical differentiation procedure which plays a derivative spectroscopy enhancement role. When compared with similar deconvolution methods, its enhanced performance is confirmed by theoretical error analysis which identifies a sense in which this method has best possible convergence rates. In practice, the method provides good starting values for the positions of the lines for the nonlinear fitting methods. Sometimes, the 'starting values' are so good that a subsequent nonlinear fitting step is not necessary. By modelling the spectra as probability measures in an appropriate Hilbert space framework, estimates are derived that characterize the nature of the smoothing performed by the regularization recovery methodology proposed. The paper concludes with an examination, using the Hilbert space framework, of the trade-off between resolution and the various types of error that occur in a measured spectrum.

Local regularization methods for ill-posed linear Volterra equations have been shown to be efficient regularization procedures preserving the causal structure of the Volterra problem and allowing for sequential solution methods. However questions posed recently in Ring and Prix (2000 Inverse Problems 16 619–34) raise doubts as to whether such methods are convergent for problems which are more than just mildly ill-posed. In this paper we address these questions by reformulating the local regularization method via the use of signed Borel measures instead of the positive Borel measures used in earlier approaches. The result is a new theory for the local regularization of ν-smoothing Volterra problems for which stability and convergence is assured for all ν = 1, 2, .... In this paper we discuss this new local regularization theory for general finitely smoothing Volterra problems and demonstrate convergence and stability of the resulting method. In addition we indicate why using signed Borel measures instead of positive measures makes sense in the context of the Volterra problem and also has connections to the theory of mollification and approximate inverses (Louis A K 1996 Inverse Problems 12 175–90). Finally we include numerical examples which illustrate the improvement which comes from using signed measures instead of positive measures and which facilitates an examination of the role played by the regularization parameter.

This paper provides some new a priori choice strategy for regularization parameters in order to obtain convergence rates in Tikhonov regularization for solving ill-posed problems Af0 = g0, f0 X, g0 Y, with a linear operator A mapping in Hilbert spaces X and Y. Our choice requires only that the range of the adjoint operator A* includes a member of some variable Hilbert scale and is, in principle, applicable in the case of general f0 without source conditions imposed otherwise in the existing papers. For testing our strategies, we apply them to the determination of a wave source, to the Abel integral equation, to a backward heat equation and to the determination of initial temperature by boundary observation.

Designing an experiment for identifying a distributed parameter is a very challenging problem when the spatial variability of the parameter is unknown. First, we need to determine an appropriate structure complexity for the unknown parameter. A more complicated structure needs more data for identification. Second, we need to evaluate the sufficiency of an experiment design before it is actually conducted. In this paper, the complexity of parameter structure is determined by the accuracy requirement of model application, and the sufficiency of data is evaluated by solving a generalized inverse problem. The worst-case parameter (WCP) defined in this paper is the one that causes the maximum deviation in model application when its structure is simplified. The WCP of a structure can be found by solving a discrete optimization problem with the genetic algorithm. Quantitative relationships between structure complexity, parameter identifiability, data sufficiency and model reliability are derived. For example, it has been proven that if an experimental design is sufficient for identifying the WCP then it must be sufficient for identifying all other parameters with the same structure or simplified structures in the admissible region, and thus, it is a robust design. Based on the theoretical results and algorithms presented in this paper, we may develop a cost-effective methodology for constructing reliable distributed parameter models.

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.

Linear inverse scattering method based on the Born approximation has found wide applications in the on-site detection of buried objects, such as in diffraction tomography. However, the Born approximation becomes invalid when the contrast of targets is high, in which the multiple-scattering effect within the target is more important. To avoid solving the nonlinear inverse scattering problem for large-contrast objects, quasi-linear and modified quasi-linear approximations are proposed based on the extended Born approximation. In this paper, new formulations are derived for the inversion of large-contrast dielectric objects buried in a lossy earth using the idea of backconditioner and lumped-mass approximation. We have shown that the zeroth-order form of such formulations is just the well-known Born approximation, and the first-order form is completely equivalent to the extended Born approximation-based quasi-linear approximation. Hence, high-order approximations are proposed in this paper, which are given in closed forms. Using such approximations, good resolution images can be obtained for large-contrast objects by only solving a linear inverse scattering integral equation. Reconstruction examples show the validity and efficiency of the proposed formulae.

The problem of imaging objects embedded in a transparent homogeneous medium is considered. It is assumed that the wavelength of the probing radiation is finite so that scattering effects need to be taken into consideration in the reconstruction process. This problem is commonly referred to as 'inverse scattering' in the literature. Many algorithms, including backpropagation-based algorithms, attempt to solve this imaging problem in either of two ways: (1) by assuming linearizing approximations such as the Born, Rytov or physical optics approximations which result in closed-form expressions for the inversion formula; or (2) by solving the nonlinear inverse scattering problem using an iterative algorithm, which is computationally more expensive. In this paper, a new method for inverse scattering is proposed. This method is based on the notion of the 'optimal topology' that solves the inverse scattering problem. To find this optimal topology, a function called the topological derivative is defined. This function quantifies the sensitivity of the scattered field to the introduction of a small scatterer at a point in the domain. Based on this definition, and the heuristic that the boundary of the objects can be considered as a group of point scatterers, we will identify high values of this function with the location of these boundaries. It is shown that the topological derivative can be calculated analytically so, as a result, the proposed reconstruction algorithm is not iterative. In addition, no approximations (such as the Born, Rytov or physical optics approximations) to the wavefield are made. The numerical examples shown in this paper demonstrate that this simple and efficient heuristic scheme can be used to accurately reconstruct the shape of scatterers.

The Generalized Minimal RESidual (GMRES) method and conjugate gradient method applied to the normal equations (CGNR) are popular iterative schemes for the solution of large linear systems of equations. GMRES requires the matrix of the linear system to be square and nonsingular, while CGNR also can be applied to overdetermined or underdetermined linear systems of equations. When equipped with a suitable stopping rule, both GMRES and CGNR are regularization methods for the solution of linear ill-posed problems. Many linear ill-posed problems that arise in the sciences and engineering have non-negative solutions. This paper describes iterative schemes, based on the GMRES or CGNR methods, for the computation of non-negative solutions of linear ill-posed problems. The computations with these schemes are terminated as soon as a non-negative approximate solution which satisfies the discrepancy principle has been found. Several computed examples illustrate that the schemes of this paper are able to compute non-negative approximate solutions of higher quality with less computational effort than several available numerical methods.

In this paper we describe a new approach to sharp boundary geophysical inversion. We demonstrate that regularized inversion with a minimum support stabilizer can be implemented by using a specially designed nonlinear parametrization of the model parameters. This parametrization plays the same role as transformation into the space of the weighted model parameters, introduced in the original papers on focusing inversion. It allows us to transform the nonquadratic minimum support stabilizer into the traditional quadratic minimum norm stabilizer, which simplifies the solution of the inverse problem. This transformation automatically ensures that the solution belongs to the class of models with a minimum support. The method is illustrated with synthetic examples of 3D magnetotelluric inversion for an earth conductivity structure. To simplify the calculations, in the initial stage of the iterative inversion we use the quasi-analytical approximation developed by Zhdanov and Hursan (2000 Inverse Problems 16 1297–322). However, to increase the accuracy of inversion, we apply rigorous forward modelling based on the integral equation method at the final stage of the inversion. To obtain a stable solution of a 3D inverse problem, we use the Tikhonov regularization method with a new nonlinear parametrization. This technique leads to the generation of a sharp image of anomalous conductivity distribution. The inversion is based on the regularized conjugate gradient method.

In this paper, we address the problem of the reconstruction of the shape of a 3D perfectly conducting object. This object is assumed to be immersed in a homogeneous lossy embedding medium in a cross borehole configuration. The method is based on an optimization process performed with the help of a conjugate gradient method. A boundary integral formalism and an adjoint state method are used in order to compute the field scattered by an estimated object and its derivatives with respect to small deformations of the shape. In this paper we also extend to the 3D case a regularization process in which we optimize a growing number of Fourier coefficients describing the shape of the estimation. The method is tested against the reconstruction of an ellipsoid, and the influence of noise is also discussed.

Inverse problems in option pricing are frequently regarded as simple and resolved if a formula of Black–Scholes type defines the forward operator. However, precisely because the structure of such problems is straightforward, they may serve as benchmark problems for studying the nature of ill-posedness and the impact of data smoothness and no arbitrage on solution properties. In this paper, we analyse the inverse problem (IP) of calibrating a purely time-dependent volatility function from a term-structure of option prices by solving an ill-posed nonlinear operator equation in spaces of continuous and power-integrable functions over a finite interval. The forward operator of the IP under consideration is decomposed into an inner linear convolution operator and an outer nonlinear Nemytskii operator given by a Black–Scholes function. The inversion of the outer operator leads to an ill-posedness effect localized at small times, whereas the inner differentiation problem is ill posed in a global manner. Several aspects of regularization and their properties are discussed. In particular, a detailed analysis of local ill-posedness and Tikhonov regularization of the complete IP including convergence rates is given in a Hilbert space setting. A brief numerical case study on synthetic data illustrates and completes the paper.

The inverse scattering problem for objects characterized by compactly supported scattering potentials embedded in a known inhomogeneous background medium given single-frequency, multistatic data is addressed within the distorted wave Born approximation (DWBA). The DWBA formulation of the multistatic data matrix is found to be in the form of a linear mapping from the Hilbert space of compactly supported scattering potentials to the finite-dimensional space CN of complex N-tuples where N = Nβ × Nα is the product of the number of receiver elements Nβ with the number of transmitter elements Nα. A pseudo-inverse of this mapping is shown to be representable as a series of products of complex conjugate background Green functions whose expansion coefficients are readily found by inverting a set of N simultaneous linear algebraic equations in N unknowns. An expression for the point spread function of the inversion algorithm is computed that is object independent and only a function of the background Green function and the measurement geometry, and thus allows the user to access 'image quality' of the algorithm as a function of these quantities. The paper includes a number of computer simulations illustrating the reconstruction algorithm developed in the paper.

In this paper, the inverse problem of resistivity retrieval is addressed in the frame of electrical resistance tomography (ERT). The ERT data is a set of measurements of the dc resistances between pairs of electrodes in contact with the conductor under investigation. This paper is focused on a non-iterative inversion method based on the monotonicity, of the resistance matrix (and of its numerical approximations). The main features of the proposed inversion method are its low computational cost requiring the solution of O(n) direct problems, where n is the number of parameters used to represent the unknown resistivity, and its very simple numerical implementation.

This paper presents a rigorous mathematical justification of the Helmholtz equation least squares (HELS) method for reconstructing acoustic radiation from an arbitrary source. It is shown that the acoustic pressure radiated from a non-spherical structure can be approximated using the spherical wavefunctions and spherical harmonics, and errors involved in this approximation are bounded. This explains why previously the HELS method could be used to produce satisfactory reconstruction of acoustic radiation from various types of structures. In the present paper, these analytical studies are further supported by experimental validation of the reconstruction of acoustic radiation from a typical filing cabinet in both exterior and interior regions.

In this paper dynamic inverse problems are studied, where the investigated object is allowed to change during the measurement procedure. In order to achieve reasonable results, temporal a priori information will be considered. Here, 'temporal smoothness' is used as a quite general, but for many applications sufficient, a priori information. This is justified in the case of slight movements during an x-ray scan in computerized tomography, or in the field of current density reconstruction, where one wants to conclude from electrical measurements on the surface of the head, the locations of brain activity. First, the notion of a dynamic inverse problem is introduced, then we describe how temporal smoothness can be incorporated in the regularization of the problem, and finally an efficient solver and some regularization properties of this solver are presented. This theory will be exploited in three practically relevant applications in a following paper.