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Michael V. Klibanov

Publications and source records attributed to Michael V. Klibanov.

99 records · Page 6Linked to original sources
arXiv2013

Reconstruction of the refractive index from experimental backscattering data using a globally convergent inverse method

The problem to be studied in this work is within the context of coefficient identification problems for the wave equation. More precisely, we consider the problem of reconstruction of the refractive index (or equivalently, the dielectric constant) of an inhomogeneous medium using one backscattering boundary measurement. The goal of this paper is to analyze the performance of a globally convergent algorithm of Beilina and Klibanov on experimental data acquired in the Microwave Laboratory at University of North Carolina at Charlotte. The main challenge working with experimental data is the the huge misfit between these data and computationally simulated data. We present data pre-processing steps to make the former somehow look similar to the latter. Results of both non-blind and blind targets are shown indicating good reconstructions even for high contrasts between the targets and the background medium.

math-ph↗
arXiv2013

Uniqueness of two phaseless inverse acoustics problems in 3-d

Uniqueness is proven for two 3-d inverse problems of the determination of the spatially distributed sound speed in the frequency dependent acoustic PDE. The main new point is the assumption that only the modulus of the scattered complex valued wave field is measured on a certain set.

math-ph↗
arXiv2013

Phaseless inverse scattering problems in 3-d

Uniqueness theorems are proved for 3-d inverse scattering problems in the frequency domain under the assumption that only the modulus of the complex valued wave field is measured, while the phase is unknown.

math-ph↗
arXiv2012

The adaptivity refines approximate solutions of ill-posed problems due to the relaxation property

Adaptive Finite Element Method (adaptivity) is known to be an effective numerical tool for some ill-posed problems. The key advantage of the adaptivity is the image improvement with local mesh refinements. A rigorous proof of this property is the central part of this paper. In terms of Coefficient Inverse Problems with single measurement data, the authors consider the adaptivity as the second stage of a two-stage numerical procedure. The first stage delivers a good approximation of the exact coefficient without an advanced knowledge of a small neighborhood of that coefficient. This is a necessary element for the adaptivity to start iterations from. Numerical results for the two-stage procedure are presented for both computationally simulated and experimental data.

math-ph↗
arXiv2012

A new approximate mathematical model for global convergence for a coefficient inverse problem with backscattering data

An approximately globally convergent numerical method for a 3d Coefficient Inverse Problem for a hyperbolic equation with backscattering data is presented. A new approximate mathematical model is presented. An approximation is used only on the first iteration and amounts to the truncation of a certain asymptotic series. A significantly new element of the convergence analysis is that the so-called "tail functions" are estimated. Numerical results in 2d and 3d cases are presented, including the one for a quite heterogeneous medium.

math-ph↗
arXiv2012

Thermoacoustic tomography with an arbitrary elliptic operator

Thermoacoustic tomography is a term for the inverse problem of determining of one of initial conditions of a hyperbolic equation from boundary measurements. In the past publications both stability estimates and convergent numerical methods for this problem were obtained only under some restrictive conditions imposed on the principal part of the elliptic operator. In this paper logarithmic stability estimates are obatined for an arbitrary variable principal part of that operator. Convergence of the Quasi-Reversibility Method to the exact solution is also established for this case. Both complete and incomplete data collection cases are considered.

math-ph↗
arXiv2012

Optical imaging of phantoms from real data by an approximately globally convergent inverse algorithm

A numerical method for an inverse problem for an elliptic equation with the running source at multiple positions is presented. This algorithm does not rely on a good first guess for the solution. The so-called "approximate global convergence" property of this method is shown here. The performance of the algorithm is verified on real data for Diffusion Optical Tomography. Direct applications are in near-infrared laser imaging technology for stroke detection in brains of small animals.

math-ph↗