Iterative methods for solving coefficient inverse problems of wave tomography in models with attenuation

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. 1 arXiv:1805.06025v1 [math.NA] 15 May 2018procedures, which would combine the currently used energy information with the estimates of dielectric constants. This combination, in turn might result in lower false alarm rates. Our targets are three dimensional ones of course. On the other hand, that radar can measure only one time dependent curve for each target. Thus, what can be done at most by any data inversion technique is to estimate a sort of an average of the dielectric constant for a target. We believe, however, that even these estimates might be useful for the goal of decreasing the false alarm rate. Thus, we model the wave propagation process by the 1-D Helmholtz equation.Convexification is the method, which constructs globally strictly convex weighted Tikhonov-like functionals either for Coefficient Inverse Problems (CIPs) or for ill-posed Cauchy problems for quasilinear PDEs. See our works [1,4,5,6,7,8,9] for CIPs and [10,11,12] for quasilinear PDEs. The key element of such a functional is the presence of the Carleman Weight Function (CWF). This is the weight function in the Carleman estimate for the principal part of the corresponding differential operator.Thus, convexification addresses the well known problem of multiple local minima and ravines of conventional Tikhonov-like functionals for CIPs, see, e.g. the paper [13] for a numerical example of multiple local minima. Convexification allows one to construct globally convergent numerical methods for CIPs. We call a numerical method for a CIP globally convergent if a theorem is proved, which claims that this method delivers at least one point in a sufficiently small neighborhood of the exact coefficient without any advanced knowledge of this neighborhood. Since conventional least squares Tikhonov functionals are non convex, then they usually have many local minima and ravines. This means that in order to obtain the correct solution, using such a functional, one needs to start the optimization process in a sufficiently small neighborhood of this solution, i.e. one should work with a locally convergent numerical method. However, such a small neighborhood is rarely known in applications.The convexification is a globally convergent numerical method, see Remark 4.1 in section 4. The idea of the convexification has roots in the method of Carleman estimates for CIPs. This method was originated in the work of Bukhgeim and Klibanov (1981) [14] as the tool of proofs of global uniqu...

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“…(m,n)=(0,0) . Then (27) implies that M N is an upper diagonal matrix with det (M N ) = k − k −N = 0. Thus, the inverse matrix M −1 N exists.…”

Section: Neither Classical Orthonormal Polynomials Nor the Basis Of Tmentioning

confidence: 99%

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

et al. 2018

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“…Note that the majority of known numerical methods of solutions of nonlinear illposed problems minimize conventional least squares cost functionals (see, e.g. [11,13,14]), which are usually non convex and have multiple local minima and ravines, see, e.g. [39] for a good numerical example of multiple local minima.…”

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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“…There exists a vast literature about numerical solutions of inverse scattering problems, which are close to the CIP we consider, see, e.g. [1,2,8,11,12,13,14,15,16,31,32,29,33] and references cited therein. They consider either problems of finding unknown coefficients or problems of finding shapes of obstacles.…”

Section: Introductionmentioning

confidence: 99%

Convexification for the Inversion of a Time Dependent Wave Front in a Heterogeneous Medium

Klibanov¹,

Li²,

Zhang³

2019

SIAM J. Appl. Math.

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An inverse scattering problem for the 3D acoustic equation in time domain is considered. The unknown spatially distributed speed of sound is the subject of the solution of this problem. A single location of the point source is used. Using a Carleman Weight Function, a globally strictly convex cost functional is constructed. A new Carleman estimate is proven. Global convergence of the gradient projection method is proven. Numerical experiments are conducted.

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Iterative methods for solving coefficient inverse problems of wave tomography in models with attenuation

Cited by 50 publications

References 43 publications

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

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

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

Convexification for the Inversion of a Time Dependent Wave Front in a Heterogeneous Medium

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