Numerical solution of a coefficient inverse problem with multi-frequency experimental raw data by a globally convergent algorithm

Nguyen, Dinh-Liem; Klibanov, Michael V.; Nguyen, Loc H.; Kolesov, Aleksandr E.; Fiddy, Michael A.; Liu, Hui

doi:10.1016/j.jcp.2017.05.015

Cited by 21 publications

(70 citation statements)

References 34 publications

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“…On the other hand, the globally convergent numerical inversion method of [32] provides very accurate locations and contrasts of abnormalities for a single measurement case. The latter was consistently demonstrated on both computationally simulated [32] and experimental data [33,41] including the case when unknown targets were buried in a sandbox [40]. The arguments in this paragraph are the reasons of our choice of the two-stage procedure.…”

Section: Introductionmentioning

confidence: 63%

A Coefficient Inverse Problem with a Single Measurement of Phaseless Scattering Data

Klibanov¹,

Nguyen²,

Nguyen³

2019

SIAM J. Appl. Math.

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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.

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Section: Introductionmentioning

confidence: 63%

A Coefficient Inverse Problem with a Single Measurement of Phaseless Scattering Data

Klibanov¹,

Nguyen²,

Nguyen³

2019

SIAM J. Appl. Math.

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“…The numerical method we develop in this paper can be considered as the second type of GCMs, which has certain advantages compared with the first type of GCMs in [6,35,36]. More precisely, we do not impose in the convergence analysis here the assumption on a small interval of wavenumbers.…”

Section: Introductionmentioning

confidence: 99%

Convexification Method for an Inverse Scattering Problem and Its Performance for Experimental Backscatter Data for Buried Targets

Klibanov¹,

Kolesov²,

Nguyen³

2019

SIAM J. Imaging Sci.

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We present in this paper a novel numerical reconstruction method for solving a 3D coefficient inverse problem with scattering data generated by a single direction of the incident plane wave. This inverse problem is well-known to be a highly nonlinear and ill-posed problem. Therefore, optimization-based reconstruction methods for solving this problem would typically suffer from the local-minima trapping and require strong a priori information of the solution. To avoid these problems, in our numerical method, we aim to construct a cost functional with a globally strictly convex property, whose minimizer can provide a good approximation for the exact solution of the inverse problem. The key ingredients for the construction of such functional are an integro-differential formulation of the inverse problem and a Carleman weight function. Under a (partial) finite difference approximation, the global strict convexity is proven using the tool of Carleman estimates. The global convergence of the gradient projection method to the exact solution is proven as well. We demonstrate the efficiency of our reconstruction method via a numerical study of experimental backscatter data for buried objects.

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“…This problem is solved below by the globally convergent numerical method, which was developed in [38]. As it was mentioned in Section 1, this method was successfully tested on microwave experimental data in [46,54,53].…”

Section: The Phased Coefficient Inverse Scattering Problemmentioning

confidence: 99%

“…It is on the second stage when we reconstruct locations and refractive indices of those microspheres. The numerical solution of the phased CIP is found using the globally convergent numerical method, which was recently developed in [38], also, see [46,54,53] for the performance of this method on microwave experimental backscattering data Our interest in phaseless CIPs is motivated by applications to optical imaging of such small objects as, e.g. biological cells and microspheres.…”

Section: Introductionmentioning

confidence: 99%

A Numerical Method to Solve a Phaseless Coefficient Inverse Problem from a Single Measurement of Experimental Data

Klibanov¹,

Koshev²,

Nguyen³

et al. 2018

SIAM J. Imaging Sci.

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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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Numerical solution of a coefficient inverse problem with multi-frequency experimental raw data by a globally convergent algorithm

Cited by 21 publications

References 34 publications

A Coefficient Inverse Problem with a Single Measurement of Phaseless Scattering Data

A Coefficient Inverse Problem with a Single Measurement of Phaseless Scattering Data

Convexification Method for an Inverse Scattering Problem and Its Performance for Experimental Backscatter Data for Buried Targets

A Numerical Method to Solve a Phaseless Coefficient Inverse Problem from a Single Measurement of Experimental Data

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