Showing papers in "Inverse Problems in 1997"
TL;DR: In this article, a simple algorithm for reconstructing the complex index of refraction of a bounded object immersed in a known background from a knowledge of how the object scatters known incident radiation is described.
Abstract: This paper describes a simple algorithm for reconstructing the complex index of refraction of a bounded object immersed in a known background from a knowledge of how the object scatters known incident radiation. The method described here is versatile accommodating both spatially and frequency varying incident fields and allowing a priori information about the scatterer to be introduced in a simple fashion. Numerical results show that this new algorithm outperforms the modified gradient approach which until now has been one of the most effective reconstruction algorithms available.
768 citations
TL;DR: In this paper, a Levenberg-Marquardt scheme for nonlinear inverse problems where the corresponding Lagrange (or regularization) parameter is chosen from an inexact Newton strategy is proposed.
Abstract: The first part of this paper studies a Levenberg - Marquardt scheme for nonlinear inverse problems where the corresponding Lagrange (or regularization) parameter is chosen from an inexact Newton strategy While the convergence analysis of standard implementations based on trust region strategies always requires the invertibility of the Frechet derivative of the nonlinear operator at the exact solution, the new Levenberg - Marquardt scheme is suitable for ill-posed problems as long as the Taylor remainder is of second order in the interpolating metric between the range and domain topologies Estimates of this type are established in the second part of the paper for ill-posed parameter identification problems arising in inverse groundwater hydrology Both transient and steady-state data are investigated Finally, the numerical performance of the new Levenberg - Marquardt scheme is studied and compared to a usual implementation on a realistic but synthetic two-dimensional model problem from the engineering literature
392 citations
TL;DR: In this article, a joint inversion of two different data sets with the assumption that the underlying models have a common structure is proposed. But the problem is nonlinear and is solved iteratively using Krylov space techniques.
Abstract: We develop a methodology to invert two different data sets with the assumption that the underlying models have a common structure. Structure is defined in terms of absolute value of curvature of the model and two models are said to have common structure if the changes occur at the same physical locations. The joint inversion is solved by defining an objective function which quantifies the difference in structure between two models, and then minimizing this objective function subject to satisfying the data constraints. The problem is nonlinear and is solved iteratively using Krylov space techniques. Testing the algorithm on synthetic data sets shows that the joint inversion is superior to individual inversions. In an application to field data we show that the data sets are consistent with models that are quite similar.
308 citations
TL;DR: In this paper, a simple inversion scheme was given for inverse scattering problems in the resonance region which is easy to implement and is relatively independent of the geometry and physical properties of the scatterer.
Abstract: This paper is a continuation of earlier research in which a simple inversion scheme was given for inverse scattering problems in the resonance region which is easy to implement and is relatively independent of the geometry and physical properties of the scatterer. The purpose of the paper is to give new and improved theorems establishing the mathematical basis of this method and to show how noisy data can be treated using Morozov's discrepancy principle where the regularization parameter is a function of an auxiliary parameter appearing in the inversion scheme.
301 citations
TL;DR: In this paper, the BC method is applied to the problem of recovering a density, including the case of inverse data given on part of a boundary, and the results of numerical testing are demonstrated.
Abstract: One of the approaches to inverse problems based upon their relations to boundary control theory (the so-called BC method) is presented. The method gives an efficient way to reconstruct a Riemannian manifold via its response operator (dynamical Dirichlet-to-Neumann map) or spectral data (a spectrum of the Beltrami - Laplace operator and traces of normal derivatives of the eigenfunctions). The approach is applied to the problem of recovering a density, including the case of inverse data given on part of a boundary. The results of the numerical testing are demonstrated.
223 citations
TL;DR: In this article, the convergence and logarithmic convergence rate of the Gauss-Newton method in a Hilbert space setting were proven provided a log-linear source condition is satisfied.
Abstract: Convergence and logarithmic convergence rates of the iteratively regularized Gauss - Newton method in a Hilbert space setting are proven provided a logarithmic source condition is satisfied. This method is applied to an inverse potential and an inverse scattering problem, and the source condition is interpreted as a smoothness condition in terms of Sobolev spaces for the case where the domain is a circle. Numerical experiments yield convergence and convergence rates of the form expected by our general convergence theorem.
198 citations
TL;DR: In this article, the problem of determining quantitative information about corrosion occurring on an inaccesible part of a specimen is considered and a regularized numerical method for obtaining approximate solutions to the problem is presented.
Abstract: We consider the problem of determining quantitative information about corrosion occurring on an inaccesible part of a specimen. The data for the problem consist of prescribed current flux and voltage measurements on an accessible part of the specimen boundary. The problem is modelled by Laplace's equation with an unknown term in the boundary conditions. Our goal is recovering from the data. We prove uniqueness under certain regularity assumptions and construct a regularized numerical method for obtaining approximate solutions to the problem. The numerical method, which is based on the assumption that the specimen is a thin plate, is tested in numerical experiments using synthetic data.
179 citations
TL;DR: In this article, a combination of Newton's method with linear Tikhonov regularization, linear Landweber iteration and truncated SVD is proposed for regularizing an abstract, nonlinear, ill-posed operator equation.
Abstract: In this paper we consider a combination of Newton's method with linear Tikhonov regularization, linear Landweber iteration and truncated SVD, for regularizing an abstract, nonlinear, ill-posed operator equation. We show that under certain smoothness conditions on the nonlinear operator, these methods converge locally. For perturbed data we propose an a priori stopping rule, that guarantees convergence of the iterates to a solution, as the noise level goes to zero. Under appropriate closeness and smoothness assumptions on the starting value and the solution, we obtain convergence rates.
169 citations
TL;DR: In this article, a continuous wavelet technique was used to locate and characterize homogeneous point sources from the field they generate measured in a distant hyperplane, and a class of wavelets was introduced on which the Poisson semi-group essentially acts as a dilation.
Abstract: It is shown how a continuous wavelet technique may be used to locate and characterize homogeneous point sources from the field they generate measured in a distant hyperplane. For this a class of wavelets is introduced on which the Poisson semi-group essentially acts as a dilation.
163 citations
TL;DR: This paper applies to this method the so-called preconditioning which is frequently used for increasing the efficiency of the conjugate gradient method and shows that it implies a modification of the original constrained least-squares problem.
Abstract: The projected Landweber method is an iterative method for solving constrained least-squares problems when the constraints are expressed in terms of a convex and closed set . The convergence properties of the method have been recently investigated. Moreover, it has important applications to many problems of signal processing and image restoration. The practical difficulty is that the convergence is too slow. In this paper we apply to this method the so-called preconditioning which is frequently used for increasing the efficiency of the conjugate gradient method. We discuss the significance of preconditioning in this case and we show that it implies a modification of the original constrained least-squares problem. However, when the original problem is ill-posed, the approximate solutions provided by the preconditioned method are similar to those provided by the standard method if the preconditioning is suitably chosen. Moreover, the number of iterations can be reduced by a factor of 10 and even more. A few applications to problems of image restoration are also discussed.
145 citations
TL;DR: In this article, the authors considered the impedance tomography problem of estimating the conductivity distribution within the body from static current/voltage measurements on the body's surface and presented a new method of implementing prior information of the conductivities in the optimization algorithm.
Abstract: In this paper, we consider the impedance tomography problem of estimating the conductivity distribution within the body from static current/voltage measurements on the body's surface. We present a new method of implementing prior information of the conductivities in the optimization algorithm. The method is based on the approximation of the prior covariance matrix by simulated samples of feasible conductivities. The reduction of the dimensionality of the optimization problem is performed by principal component analysis (PCA).
TL;DR: In this paper, the distribution of expected asset returns implied by current market prices of options with different strikes was investigated and it was shown that the S&P 500 Index is negatively skewed with a higher probability of the sudden decline of the US stock market.
Abstract: Valuation of options and other financial derivatives critically depends on the underlying stochastic process specified for a particular market. An inverse problem of option pricing is to determine the nature of this stochastic process, namely, the distribution of expected asset returns implied by current market prices of options with different strikes. We give a rigorous mathematical formulation of this inverse problem, establish uniqueness, and suggest an efficient numerical solution. We apply the method to the S&P 500 Index and conclude that the index is negatively skewed with a higher probability of the sudden decline of the US stock market.
TL;DR: In this paper, the anisotropic conductivity inverse boundary value problem is presented in a geometric formulation and a uniqueness result is proved, under two different hypotheses, for the case where the conductivity is known up to a multiplicative scalar field.
Abstract: The anisotropic conductivity inverse boundary value problem (or reconstruction problem for anisotropic electrical impedance tomography) is presented in a geometric formulation and a uniqueness result is proved, under two different hypotheses, for the case where the conductivity is known up to a multiplicative scalar field. The first of these results relies on the conductivity being determined by boundary measurements up to a diffeomorphism fixing points on the boundary, which has been shown for analytic conductivities in three and higher dimensions by Lee and Uhlmann and for conductivities close to constant by Sylvester. The apparatus of G-structures is then used to show that a conformal mapping of a Riemannian manifold which fixes all points on the boundary must be the identity. A second approach, which proves the result in the piecewise analytic category, is a straightforward extension of the work of Kohn and Vogelius on the isotropic problem.
TL;DR: In this article, the potential and boundary condition of the Sturm-Liouville problem were reconstructed from nodes of its eigenfunctions using the same method as McLaughlin.
Abstract: In this paper, we reconstruct the potential and the boundary condition of the Sturm - Liouville problem from nodes of its eigenfunctions. We also give the uniqueness for general boundary conditions using the same method as McLaughlin.
TL;DR: In this article, an iterative algorithm based on a conjugate gradient method was applied to the nonlinear problem of complex permittivity profile reconstruction of inhomogeneous objects from measured scattered field data.
Abstract: The reconstruction of the complex permittivity profile of inhomogeneous objects from measured scattered field data is a strongly nonlinear and ill-posed problem. Generally, the quality of the reconstruction from noisy data is enhanced by the introduction of a regularization scheme. Starting from an iterative algorithm based on a conjugate gradient method and applied to the nonlinear problem, this paper deals with a new regularization scheme, using edge-preserving (EP) potential functions. With this a priori information, the object to reconstruct is modelled with homogeneous areas separated by borderlike discontinuities. The enhancement is illustrated throughout some examples with noisy synthetic data.
TL;DR: In this article, the generalized Radon transform (GRT) inversion of seismic data is carried out in general anisotropic media and the inversion provides a reflectivity map and reflection/transmission coefficients as functions of scattering angles and azimuths.
Abstract: The resolution analysis of generalized Radon transform (GRT) inversion of seismic data is carried out in general anisotropic media. The GRT inversion formula is derived from the ray-Born approximation of the wave field for volume scatterers. However, by considering scattering surfaces in the resolution analysis, rather than parameter perturbations, we show that the inversion provides a reflectivity map and reflection/transmission coefficients as functions of scattering angles and azimuths. Those coefficients can be subjected to any type of amplitude versus angle (AVA) or amplitude versus offset (AVO) analysis. By applying the inversion to Kirchhoff approximate data rather than Born approximate data, we show that the output is actually linear in the reflection coefficients and, hence, a nonlinear function of the change in medium parameters across discontinuity surfaces - the reflectors of the medium.
TL;DR: In this article, a solution of the inverse problem consisting of reconstructing the reflector R from the following data: the position of the source O, its radiation intensity I, the object T, and prespecified energy pattern to be achieved on T was presented.
Abstract: Consider a reflector system consisting of a reflecting surface R, a point light source O and some object T. Suppose that the source O, reflector R and object T are positioned so that the rays reflected off R are incident on T. Depending on the geometry of R, the energy, radiated by O and redirected by R, is distributed on T producing a certain illumination pattern. We present here a solution of the inverse problem consisting of reconstructing the reflector R from the following data: the position of the source O, its radiation intensity I, the object T, and prespecified energy pattern to be achieved on T. It is shown that under assumptions of geometric optics theory the problem admits a solution, provided the total input and output energies are equal, and some other geometric conditions are satisfied. In analytic formulation, the problem leads to an equation of Monge - Ampere type in a domain on a unit sphere. In this paper we formulate the problem in terms of certain associated measures and establish the existence of weak solutions.
TL;DR: In this paper, the inverse scattering problem by an open sound-hard arc is studied and a uniqueness theorem for this problem is proved, and regularized Newton-type methods for approximately solving the problem are described for a number of different ansatz functions and arcs.
Abstract: This paper studies the inverse scattering problem by an open sound-hard arc. We prove a uniqueness theorem for this problem. For the foundation of Newton methods we establish the Fr?chet differentiability of the far-field operator with respect to the open arc and show that the Fr?chet derivative can be obtained as the solution of a hypersingular integral equation. We describe regularized Newton-type methods for approximately solving the inverse scattering problem and present numerical results for a number of differently chosen ansatz functions and arcs.
TL;DR: In this paper, an inverse boundary-value problem arising in questions of nondestructive testing such as corrosion detection by electrostatic measurements was considered and it was shown that logarithmic stability is the best possible.
Abstract: We consider an inverse boundary-value problem arising in questions of nondestructive testing such as corrosion detection by electrostatic measurements. We construct examples which show that, for this problem, logarithmic stability is the best possible.
TL;DR: In this paper, the sideways heat equation is modeled as a Cauchy problem in the quarter plane, where data are given at x = 1 and a solution is sought in the interval 0 < x < 1.
Abstract: We consider a Cauchy problem for the heat equation in the quarter plane, where data are given at x = 1 and a solution is sought in the interval 0 < x < 1. This sideways heat equation is a model of a problem where one wants to determine the temperature on both sides of a thick wall, but where one side is inaccessible to measurements. The problem is ill-posed, in the sense that the solution (if it exists) does not depend continuously on the data. Meyer wavelets have the property that their Fourier transform has compact support. Therefore, by expanding the data and the solution in a basis of Meyer wavelets, high-frequency components can be filtered away. We show that using a wavelet - Galerkin approach, we restore continuous dependence on the data, and we give a recipe for choosing the coarse level resolution in the wavelet representation, depending on the noise level of the data. Furthermore, we solve the sideways problem numerically in the coarse level representation, as an ordinary differential equation in the space variable, where the time derivative is replaced by its wavelet representation. Numerical examples are given.
TL;DR: In this paper, a generalized diffraction tomographic (DT) algorithm is derived for subsurface imaging from multifrequency multi-monostatic ground penetrating radar (GPR) data.
Abstract: A generalized diffraction tomographic (DT) algorithm is derived for subsurface imaging from multifrequency multi-monostatic ground penetrating radar (GPR) data. The algorithm is based on the Born approximation for vector electromagnetic scattering and incorporates realistic nearfield models for the receiving and transmitting antennas. The forward scattering model is inverted analytically using the regularized pseudoinverse operator to yield an algorithm for imaging the underground region based on scattered field measurements at a set of receiving antennas. Whereas the usual inversion algorithms of DT require a lossless background medium and ideal point sources and receivers, the algorithm described here allows an attenuating background and arbitrary transmitting and receiving antennas. The algorithm places no restrictions on the radar frequency, and can thus include shallow imaging applications where the wavelengths are on the same order as the depth of buried objects of interest. Versions of the algorithm are given for both the three dimensional and the 2.5-dimensional cases. Results are given of computer simulations designed to test the algorithm.
TL;DR: In this article, the projected Landweber method is applied to the inversion of real data, yielding satisfactory results also in the case of quite complex events, such as very large earthquakes.
Abstract: The empirical Green function (EGF) model, which is used in this paper for the analysis of the waveforms of low-energy earthquakes, consists in assuming that the propagating medium and the recording instrument can be treated as a linear system and that the impulse response function of the system can be approximated by the waveform of a very small earthquake. The deconvolution of the Green function event from the waveform of a larger one, located at approximately the same position, provides information about the source time function (STF) of the latter. Linear inversion methods do not yield satisfactory estimations of the STF which must be positive and causal. Moreover, an estimate of the duration (support) of the STF should be desirable. In this paper we apply to this problem the so-called projected Landweber method, which is an iterative nonlinear method allowing for the introduction of constraints on the solution. The implementation of the method is easy and efficient. We first validate the method by means of synthetic data, generated by the use of waveforms of a seismic swarm that occurred in the Ligurian Alps (north-western Italy) during July 1993. Then, taking into account the indications provided by the simulations, the method has been applied to the inversion of real data, yielding satisfactory results also in the case of quite complex events.
TL;DR: In this article, the authors considered a non-isospectral scattering problem having as its spatial part an energy-dependent Schrodinger operator and gave a completely integrable multicomponent system of equations in (2 + 1) dimensions.
Abstract: We consider a non-isospectral scattering problem having as its spatial part an energy-dependent Schrodinger operator. This gives rise to new completely integrable multicomponent systems of equations in (2 + 1) dimensions. Their reductions to systems in (1 + 1) dimensions have isospectral scattering problems and include multicomponent extensions of the AKNS equation and also a generalization of the Dym equation. An extension of the Fuchssteiner - Fokas - Camassa - Holm equation to (2 + 1) dimensions is also presented.
TL;DR: In this article, the Earth's subsurface elastic properties from reflection seismic data are estimated using a posteriori probabilistic information about models, which may very well include features in the null space of the forward problem.
Abstract: In Bayesian inference, probabilistic information about models is posited a priori. This information, which may very well include features in the null space of the forward problem, affects both the computed models and the resulting resolution estimates. In Occam's inversion, on the other hand, the goal is to construct the smoothest model consistent with the data. This is not to say that one believes a priori that models are really smooth, but rather that a more conservative interpretation of the data ought to be made by eliminating features of the model that are not required to fit the data. The length scale associated with the smoothing is an indirect measure of resolution. In some cases the mathematical machinery of Bayesian inference resembles that of Occam's inversion, but the goals and interpretations of the two methods are rather different. To understand better the similarities and differences of these two approaches, we show an application of both methods to the problem of inferring the Earth's subsurface elastic properties from reflection seismic data. On the one hand, by deriving a priori information about the Earth's layering from fine-scale borehole measurements, coupled with information about the noise in the data and the elastic forward modelling operator, we are able to compute the Bayesian a posteriori probability distribution on the space of models. Models pseudo-randomly simulated from this a posteriori probability will exhibit features that are implied by the a priori information as well as the data, even if the former are not well resolved by the data. Then we solve the Occam's inversion problem by determining the maximum model smoothness that allows for the data to be fit, without incorporating a priori information about the models. In this case we estimate the resolution in terms of the degree of model smoothness implied by the data. The main conclusions for the numerical experiments considered in this work are that the subsurface models derived from both techniques are quite similar but error estimates associated with such models are rather different, reflecting the role of the a priori information in the inverse calculation.
TL;DR: In this article, the singular manifold method was generalized in such a way to a pair of equations in 2 + 1 dimensions and applied to the problem of finding the Darboux transformations for the equations.
Abstract: The interesting result obtained in this paper involves using the generalized singular manifold method to determine the Darboux transformations for the equations. It allows us to establish an iterative procedure to obtain multisolitonic solutions. This procedure is closely related to the Hirota -function method. In this paper, we report how to improve the singular manifold method when the equation has more than one Painlev? branch. The singular manifold method generalized in such a way is applied to a pair of equations in 2 + 1 dimensions
TL;DR: In this paper, the inverse scattering problem was studied to recover a periodic structure by scattered waves measured above the structure. And it was shown that a finite number of incident plane waves is sufficient to identify the structure by a monotonicity principle for the eigenvalues of the Laplacian.
Abstract: This paper is devoted to the inverse scattering problem to recover a periodic structure by scattered waves measured above the structure. It is shown that a finite number of incident plane waves is sufficient to identify the structure. Additionally by a monotonicity principle for the eigenvalues of the Laplacian some upper bounds of the required number of wavenumbers are presented if a priori information on the height of the structure is available.
TL;DR: In this paper, an n-dimensional (n = 2,3) inverse problem for the parabolic/diffusion equation,,, is considered, where the problem consists of determining the function a(x) inside of a bounded domain given the values of the solution u(x,t) for a single source location on a set of detectors where is the boundary of.
Abstract: An n-dimensional (n = 2,3) inverse problem for the parabolic/diffusion equation , , , is considered. The problem consists of determining the function a(x) inside of a bounded domain given the values of the solution u(x,t) for a single source location on a set of detectors , where is the boundary of . A novel numerical method is derived and tested. Numerical tests are conducted for n = 2 and for ranges of parameters which are realistic for applications to early breast cancer diagnosis and the search for mines in murky shallow water using ultrafast laser pulses. The main innovation of this method lies in a new approach for a novel linearized problem (LP). Such a LP is derived and reduced to a well-posed boundary-value problem for a coupled system of elliptic partial differential equations. A principal advantage of this technique is in its speed and accuracy, since it leads to the factorization of well conditioned, sparse matrices with non-zero entries clustered in a narrow band near the diagonal. The authors call this approach the elliptic systems method (ESM). The ESM can be extended to other imaging modalities.
TL;DR: Using the matrix Riemann - Hilbert factorization approach for nonlinear evolution equations (NLEEs) integrable in the sense of the inverse scattering method, this paper obtained, in the solitonless sector, the leading-order asymptotics as of the solution to the Cauchy initial-value problem for the modified nonlinear Schrodinger equation.
Abstract: Using the matrix Riemann - Hilbert factorization approach for nonlinear evolution equations (NLEEs) integrable in the sense of the inverse scattering method, we obtain, in the solitonless sector, the leading-order asymptotics as of the solution to the Cauchy initial-value problem for the modified nonlinear Schrodinger equation, , : also obtained are analogous results for two gauge-equivalent NLEEs; in particular, the derivative nonlinear Schrodinger equation, .
TL;DR: In this paper, a semi-explicit algorithm is proposed to reconstruct two-dimensional (2D) segment cracks, or three-dimensional planar cracks, in the framework of overspecified boundary data.
Abstract: This paper deals with a semi-explicit algorithm to reconstruct two-dimensional (2D) segment cracks, or three-dimensional (3D) planar cracks, in the framework of overspecified boundary data. The algorithm is based on the reciprocity gap concept, introduced by Andrieux and Ben Abda, which provides explicitly the line (or the plane) support of the cracks. A numerical reconstruction of the cracks, which are actually the support of the solution jump across this plane, is then performed by computing the Fourier expansion of the solution jump itself. After the numerical analysis of the method, some numerical results are presented and commented on.
TL;DR: In this paper, the authors revisited integrable discretizations for the nonlinear Schrodinger equation due to Ablowitz and Ladik and demonstrated how their main drawback, the non-locality, can be overcome.
Abstract: We revisit integrable discretizations for the nonlinear Schrodinger equation due to Ablowitz and Ladik. We demonstrate how their main drawback, the non-locality, can be overcome. Namely, we factorize the non-local difference scheme into the product of local ones. This must improve the performance of the scheme in the numerical computations dramatically. Using the equivalence of the Ablowitz - Ladik and the relativistic Toda hierarchies, we find the interpolating Hamiltonians for the local schemes and show how to solve them in terms of matrix factorizations.