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Michael V. Klibanov

Publications and source records attributed to Michael V. Klibanov.

At least 73 records · Page 4Linked to original sources
arXiv2018

A new version of the convexification method for a 1-D coefficient inverse problem with experimental data

A new version of the convexification method is developed analytically and tested numerically for a 1-D coefficient inverse problem in the frequency domain. Unlike the previous version, this one does not use the so-called "tail function", which is a complement of a certain truncated integral with respect to the wave number. Globally strictly convex cost functional is constructed with the Carleman Weight Function. Global convergence of the gradient projection method to the correct solution is proved. Numerical tests are conducted for both computationally simulated and experimental data.

math.NA↗
arXiv2018

Electrical Impedance Tomography with Restricted Dirichlet-to-Neumann Map Data

We propose a new numerical method to reconstruct the isotropic electrical conductivity from measured restricted Dirichlet-to-Neumann map data in electrical impedance tomography (EIT) model. "Restricted Dirichlet-to-Neumann (DtN) map data" means that the Dirichlet and Neumann boundary data for EIT are generated by a point source running either along an interval of a straight line or along a curve located outside of the domain of interest. We "convexify" the problem via constructing a globally strictly convex Tikhonov-like functional using a Carleman Weight Function. In particular, two new Carleman estimates are established. Global convergenceto the correct solution of the gradient projection method for this functional is proven. Numerical examples demonstrate a good performance of this numerical procedure.

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arXiv2018

A numerical method to solve a phaseless coefficient inverse problem from a single measurement of experimental data

We propose in this paper a globally numerical method to solve a phaseless coefficient inverse problem: how to reconstruct the spatially distributed refractive index of scatterers from the intensity (modulus square) of the full complex valued wave field at an array of light detectors located on a measurement board. The propagation of the wave field is governed by the 3D Helmholtz equation. Our method consists of two stages. On the first stage, we use asymptotic analysis to obtain an upper estimate for the modulus of the scattered wave field. This estimate allows us to approximately reconstruct the wave field at the measurement board using an inversion formula. This reduces the phaseless inverse scattering problem to the phased one. At the second stage, we apply a recently developed globally convergent numerical method to reconstruct the desired refractive index from the total wave obtained at the first stage. Unlike the optimization approach, the two-stage method described above is global in the sense that it does not require a good initial guess of the true solution. We test our numerical method on both computationally simulated and experimental data. Although experimental data are noisy, our method produces quite accurate numerical results.

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arXiv2018

Convexification of a 3-D coefficient inverse scattering problem

A version of the so-called "convexification" numerical method for a coefficient inverse scattering problem for the 3D Hemholtz equation is developed analytically and tested numerically. Backscattering data are used, which result from a single direction of the propagation of the incident plane wave on an interval of frequencies. The method converges globally. The idea is to construct a weighted Tikhonov-like functional. The key element of this functional is the presence of the so-called Carleman Weight Function (CWF). This is the function which is involved in the Carleman estimate for the Laplace operator. This functional is strictly convex on any appropriate ball in a Hilbert space for an appropriate choice of the parameters of the CWF. Thus, both the absence of local minima and convergence of minimizers to the exact solution are guaranteed. Numerical tests demonstrate a good performance of the resulting algorithm. Unlikeprevious the so-called tail functions globally convergent method, we neither do not impose the smallness assumption of the interval of wavenumbers, nor we do not iterate with respect to the so-called tail functions.

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arXiv2017

A coefficient inverse problem with a single measurement of phaseless scattering data

This paper is concerned with a numerical method for a 3D coefficient inverse problem with phaseless scattering data. These are multi-frequency data generated by a single direction of the incident plane wave. Our numerical procedure consists of two stages. The first stage aims to reconstruct the (approximate) scattered field at the plane of measurements from its intensity. We present an algorithm for the reconstruction process and prove a uniqueness result of this reconstruction. After obtaining the approximate scattered field, we exploit a newly developed globally convergent numerical method to solve the coefficient inverse problem with the phased scattering data. The latter is the second stage of our algorithm. Numerical examples are presented to demonstrate the performance of our method. Finally, we present a numerical study which aims to show that, under a certain assumption, the solution of the scattering problem for the 3D scalar Helmholtz equation can be used to approximate the component of the electric field which was originally incident upon the medium.

math.NA↗
arXiv2017

Convexification of Restricted Dirichlet-to-Neumann Map

By our definition, "restricted Dirichlet-to-Neumann map" (DN) means that the Dirichlet and Neumann boundary data for a Coefficient Inverse Problem (CIP) are generated by a point source running along an interval of a straight line. On the other hand, the conventional DN data can be generated, at least sometimes, by a point source running along a hypersurface. CIPs with the restricted DN data are non-overdetermined in the $n-$D case with $n \geq 2$. We develop, in a unified way, a general and a radically new numerical concept for CIPs with restricted DN data for a broad class of PDEs of the second order, such as, e.g. elliptic, parabolic and hyperbolic ones. Namely, using Carleman Weight Functions, we construct globally convergent numerical methods. Hölder stability and uniqueness are also proved. The price we pay for these features is a well acceptable one in the Numerical Analysis: we truncate a certain Fourier-like series with respect to some functions depending only on the position of that point source. At least three applications are: imaging of land mines, crosswell imaging and electrical impedance tomography.

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arXiv2017

Imaging of buried objects from multi-frequency experimental data using a globally convergent inversion method

This paper is concerned with the numerical solution to a 3D coefficient inverse problem for buried objects with multi-frequency experimental data. The measured data, which are associated with a single direction of an incident plane wave, are backscatter data for targets buried in a sandbox. These raw scattering data were collected using a microwave scattering facility at the University of North Carolina at Charlotte. We develop a data preprocessing procedure and exploit a newly developed globally convergent inversion method for solving the inverse problem with these preprocessed data. It is shown that dielectric constants of the buried targets as well as their locations are reconstructed with a very good accuracy. We also prove a new analytical result which rigorously justifies an important step of the so-called "data propagation" procedure.

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arXiv2017

Globally strictly convex cost functional for a 1-D inverse medium scattering problem with experimental data

A new numerical method is proposed for a 1-D inverse medium scattering problem with multi-frequency data. This method is based on the construction of a weighted cost functional. The weight is a Carleman Weight Function (CWF). In other words, this is the function, which is present in the Carleman estimate for the undelying differential operator. The presence of the CWF makes this functional strictly convex on any a priori chosen ball with the center at $\left\{ 0\right\} $ in an appropriate Hilbert space. Convergence of the gradient minimization method to the exact solution starting from any point of that ball is proven. Computational results for both computationally simulated and experimental data show a good accuracy of this method.

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arXiv2017

Uniqueness of a 3-D coefficient inverse scattering problem without the phase information

We use a new method to prove uniqueness theorem for a coefficient inverse scattering problem without the phase information for the 3-D Helmholtz equation. We consider the case when only the modulus of the scattered wave field is measured and the phase is not measured. The spatially distributed refractive index is the subject of the interest in this problem. Applications of this problem are in imaging of nanostructures and biological cells.

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arXiv2017

An inverse problem without the phase information

We prove a new uniqueness theorem for an inverse scattering problem without the phase information for the 3-D Helmholtz equation. The spatially distributed dielectric constant is the subject of the interest in this problem. We consider the case when the modulus of the scattered wave field |u_{sc}| is measured. The phase is not measured.

math-ph↗
arXiv2016

A globally convergent numerical method for a 3D coefficient inverse problem with a single measurement of multi-frequency data

The goal of this paper is to reconstruct spatially distributed dielectric constants from complex-valued scattered wave field by solving a 3D coefficient inverse problem for the Helmholtz equation at multi-frequencies. The data are generated by only a single direction of the incident plane wave. To solve this inverse problem, a globally convergent algorithm is analytically developed. We prove that this algorithm provides a good approximation for the exact coefficient without any \textit{a priori} knowledge of any point in a small neighborhood of that coefficient. This is the main advantage of our method, compared with classical approaches using optimization schemes. Numerical results are presented for both computationally simulated data and experimental data. Potential applications of this problem are in detection and identification of explosive-like targets.

math.NA↗
arXiv2016

Numerical Solution of a Coefficient Inverse Problem with Multi-Frequency Experimental Raw Data by a Globally Convergent Algorithm

We analyze in this paper the performance of a newly developed globally convergent numerical method for a coefficient inverse problem for the case of multi-frequency experimental backscatter data associated to a single incident wave. These data were collected using a microwave scattering facility at the University of North Carolina at Charlotte. The challenges for the inverse problem under the consideration are not only from its high nonlinearity and severe ill-posedness but also from the facts that the amount of the measured data is minimal and that these raw data are contaminated by a significant amount of noise, due to a non-ideal experimental setup. This setup is motivated by our target application in detecting and identifying explosives. We show in this paper how the raw data can be preprocessed and successfully inverted using our inversion method. More precisely, we are able to reconstruct the dielectric constants and the locations of the scattering objects with a good accuracy, without using any advanced \emph{a priori} knowledge of their physical and geometrical properties.

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arXiv2016

Single measurement experimental data for an inverse medium problem inverted by a multi-frequency globally convergent numerical method

The recently developed globally convergent numerical method for an inverse medium problem for the Helmholtz equation is tested on experimental data. The data were originally collected in the time domain, whereas the method works in the frequency domain with the multi-frequency data. Due to a huge discrepancy between the collected and computationally simulated data, the straightforward Fourier transform of the experimental data does not work. Hence, it is necessary to develop a heuristic data preprocessing procedure. This procedure is described. The preprocessed data are used as the input for the inversion algorithm. Numerical results demonstrate good accuracy in the reconstruction of both refracive indices and locations of targets. Furthermore, the reconstruction errors for refractive indices of dielectric targets are significantly less than errors of a posteriori direct measurements.

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arXiv2016

Carleman weight functions for a globally convergent numerical method for ill-posed Cauchy problems for some quasilinear PDEs

In a series of publications of the second author, including some with coauthors, globally strictly convex Tikhonov-like functionals were constructed for some nonlinear ill-posed problems. The main element of such a functional is the presence of the Carleman Weight Function. Compared with previous publications, the main novelty of this paper is that the existence of the regularized solution (i.e. the minimizer) is proved rather than assumed. The method works for both ill-posed Cauchy problems for some quasilinear PDEs of the second order and for some Coefficient Inverse Problems. However, to simplify the presentation, we focus here only on ill-posed Cauchy problems. Along with the theory, numerical results are presented for the case of a 1-D quasilinear parabolic PDE with the lateral Cauchy data given on one edge of the interval (0,1).

math.AP↗
arXiv2016

A globally convergent method for a 3-D inverse medium problem for the generalized Helmholtz equation

A 3-D inverse medium problem in the frequency domain is considered. Another name for this problem is Coefficient Inverse Problem. The goal is to reconstruct spatially distributed dielectric constants from scattering data. Potential applications are in detection and identification of explosive-like targets. A single incident plane wave and multiple frequencies are used. A new numerical method is proposed. A theorem is proved, which claims that a small neigborhood of the exact solution of that problem is reached by this method without any advanced knowledge of that neighborhood. We call this property of that numerical method "global convergence". Results of numerical experiments for the case of the backscattering data are presented.

math.NA↗
arXiv2016

Numerical solution of an ill-posed Cauchy problem for a quasilinear parabolic equation using a Carleman weight function

This is the first publication in which an ill-posed Cauchy problem for a quasi- linear PDE is solved numerically by a rigorous method. More precisely, we solve the side Cauchy problem for a 1-d quasilinear parabolc equation. The key idea is to minimize a strictly convex cost functional with the Carleman Weight Function in it. Previous publications about numerical solutions of ill-posed Cauchy problems were considering only linear equations.

math-ph↗
arXiv2016

A globally convergent numerical method for a 1-d inverse medium problem with experimental data

In this paper, a reconstruction method for the spatially distributed dielectric constant of a medium from the back scattering wave field in the frequency domain is considered. Our approach is to propose a globally convergent algorithm, which does not require any knowledge of a small neighborhood of the solution of the inverse problem in advance. The Quasi-Reversibility Method (QRM) is used in the algorithm. The convergence of the QRM is proved via a Carleman estimate. The method is tested on both computationally simulated and experimental data.

math.AP↗