Numerical Solution of a Linearized Travel Time Tomography Problem With Incomplete Data

Klibanov, Michael V.; Le, Thuy T.; Nguyen, Loc H.

doi:10.1137/19m1299487

Cited by 10 publications

(5 citation statements)

References 40 publications

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“…Remark 5.1. The basis {Ψ n } n≥1 was successfully used very often in our research group to solve a long list of inverse problems including the nonlinear coefficient inverse problems for elliptic equations [50] and parabolic equations [51,43,44,52], and ill-posed inverse source problems for elliptic equations [41] and parabolic equations [42], transport equations [45] and full transfer equations [53]. Another reason for us to employ this basis rather than the well-known basis of the Fourier series is that the first elements of this basis is a constant.…”

Section: Methodsmentioning

confidence: 99%

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The quasi-reversibility method to numerically solve an inverse source problem for hyperbolic equations

Le¹,

Nguyen²,

Nguyen³

et al. 2020

Preprint

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We propose a numerical method to solve an inverse source problem of computing the initial condition of hyperbolic equations from the measurements of Cauchy data. This problem arises in thermoand photo-acoustic tomography in a bounded cavity, in which the reflection of the wave makes the widely-used approaches, such as the time reversal method, not applicable. In order to solve this inverse source problem, we approximate the solution to the hyperbolic equation by its Fourier series with respect to a special orthonormal basis of L 2 . Then, we derive a coupled system of elliptic equations for the corresponding Fourier coefficients. We solve it by the quasi-reversibility method. The desired initial condition follows. We rigorously prove the convergence of the quasi-reversibility method as the noise level tends to 0. Some numerical examples are provided. In addition, we numerically prove that the use of the special basic above is significant.

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Section: Methodsmentioning

confidence: 99%

“…Our main contribution to this field is to relax a technical condition on the noise. In our previous works [41,42,43,44,45] and references therein, we assumed that the noise contained in the boundary data can be "smoothly extended" as a function defined on the domain Ω. This condition implies that the noise must be smooth.…”

Section: Introductionmentioning

confidence: 99%

The quasi-reversibility method to numerically solve an inverse source problem for hyperbolic equations

Le¹,

Nguyen²,

Nguyen³

et al. 2020

Preprint

Self Cite

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show abstract

“…Although the rigorous study of the asymptotic behavior of (2.6) as N large is missing, we do not experience any difficulty in our numerical study. We refer the readers to [15,30,32,23,34,39,35] for the successful use of similar approximations when the basis {Ψ n } is given in [19].…”

Section: An Approximate Cauchy Problem For Problem 11mentioning

confidence: 99%

The Carleman-based contraction principle to reconstruct the potential of nonlinear hyperbolic equations

Nguyen¹,

Nguyen²,

Truong³

2022

Preprint

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We develop an efficient and convergent numerical method for solving the inverse problem of determining the potential of nonlinear hyperbolic equations from lateral Cauchy data. In our numerical method we construct a sequence of linear Cauchy problems whose corresponding solutions converge to a function that can be used to efficiently compute an approximate solution to the inverse problem of interest. The convergence analysis is established by combining the contraction principle and Carleman estimates. We numerically solve the linear Cauchy problems using a quasi-reversibility method. Numerical examples are presented to illustrate the efficiency of the method.

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“…This basis was first introduced to solve the electrical impedance tomography problem with partial data in [9]. Afterward, it is widely used in our research group to solve a variety kinds of inverse problems, including ill-posed inverse source problems for elliptic equations [35], parabolic equations [25] [29] and hyperbolic equations [27], nonlinear coefficient inverse problems for elliptic equations [38], and parabolic equations [20,40,32,39,26], transport equations [14] and full transfer equations [37].…”

Section: A Special Orthonormal Basismentioning

confidence: 99%

Global reconstruction of initial conditions of nonlinear parabolic equations via the Carleman-contraction method

Le¹

2022

Preprint

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We propose a global convergent numerical method to reconstruct the initial condition of a nonlinear parabolic equation from the measurement of both Dirichlet and Neumann data on the boundary of a bounded domain. The first step in our method is to derive, from the nonlinear governing parabolic equation, a nonlinear systems of elliptic partial differential equations (PDEs) whose solution yields directly the solution of the inverse source problem. We then establish a contraction mapping-like iterative scheme to solve this system. The convergence of this iterative scheme is rigorously proved by employing a Carleman estimate and the argument in the proof of the traditional contraction mapping principle. This convergence is fast in both theoretical and numerical senses. Moreover, our method, unlike the methods based on optimization, does not require a good initial guess of the true solution. Numerical examples are presented to verify these results.

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Numerical Solution of a Linearized Travel Time Tomography Problem With Incomplete Data

Abstract: The first numerical solution of the 3-D travel time tomography problem is presented. The globally convergent convexification numerical method is applied.

Cited by 10 publications

References 40 publications

The quasi-reversibility method to numerically solve an inverse source problem for hyperbolic equations

The quasi-reversibility method to numerically solve an inverse source problem for hyperbolic equations

The Carleman-based contraction principle to reconstruct the potential of nonlinear hyperbolic equations

Global reconstruction of initial conditions of nonlinear parabolic equations via the Carleman-contraction method

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