An inversion method for parabolic equations based on quasireversibility

Tadi, M.; Klibanov, Michael V.; Cai, Wei

doi:10.1016/s0898-1221(02)80003-7

Cited by 24 publications

(18 citation statements)

References 25 publications

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“…Moreover, an overdetermined inverse problem, as given by Eqs. 1, 3-6 and 8, has been investigated numerically in [16]. (ii) An inverse problem similar to IP2, as given by the Eqs.…”

Section: Remarkmentioning

confidence: 99%

Space-dependent perfusion coefficient identification in the transient bio-heat equation

2009

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The identification of the space-dependent perfusion coefficient in the one-dimensional transient bio-heat conduction equation is investigated. While Dirichlet boundary conditions are assumed, the additional measurement necessary to render a unique solution is either a heat-flux measurement or a time-average temperature measurement inside the space region of interest. A numerical approach based on the Crank-Nicolson finite-difference scheme combined with the first-order Tikhonov's regularization method is developed. Numerical results are presented and discussed.

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“…Moreover, an overdetermined inverse problem, as given by Eqs. 1, 3-6 and 8, has been investigated numerically in [16]. (ii) An inverse problem similar to IP2, as given by the Eqs.…”

Section: Remarkmentioning

confidence: 99%

Space-dependent perfusion coefficient identification in the transient bio-heat equation

2009

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“…The QRM is well suitable to handle overdetermined boundary conditions for PDEs [6,8,[12][13][14]. As to CIPs, in the past QRM was used for numerical solutions of 1-D CIPs for parabolic equations via locally convergent numerical methods [12,15,16]. However, it was not used for the global convergence.…”

Section: Introductionmentioning

confidence: 99%

Global convergence for a 1-D inverse problem with application to imaging of land mines

Kuzhuget

Klibanov

2010

Applicable Analysis

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A new globally convergent numerical method is developed for a 1-D coefficient inverse problem for a hyperbolic partial differential equation (PDE). The back reflected data are used. A version of the quasi-reversibility method is proposed. A global convergence theorem is proven via a Carleman estimate. The results of numerical experiments are presented. IntroductionIn this article, a new globally convergent numerical method for a one-dimensional (1-D) coefficient inverse problem (CIP) for a hyperbolic partial differential equation (PDE) is presented. We modify here the idea of [1] for our case (also, see follow-up publications [2][3][4]). This CIP has applications in acoustics and electromagnetics. More specifically, we consider here an application to the problem of imaging of antipersonnel plastic land mines. It is well known that plastic land mines are hard to detect by ground penetrating radars, because they do not have a significant metal component in them. So, our idea is to image the spatial distribution of the relative dielectric constant in them. A similar idea was carried out in [5] by the globally convergent numerical method of the first generation, the so-called convexification algorithm [6]. However, [1] represents the second generation of such methods.We point out that the 1-D problem is considered here only as a preliminary step before applying similar ideas to 2-D and 3-D cases. In other words, the goal of this article is to develop a methodology for our future studies of 2-D and 3-D problems.

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“…has been investigated by several authors, e.g., in [2,3,8,10,12]. In [3], the Hölder space method is applied to determine the unknown coefficient q(x).…”

Section: Introductionmentioning

confidence: 99%

“…Existence and uniqueness for the determination of q(x) are derived in [8] by using the contraction mapping principle. In [10], the method of quasireversibility is applied to obtain the numerical solution of q(x). In [2,12], motivated by heuristic arguments, the optimization method is applied to stabilize the inverse problem.…”

Section: Introductionmentioning

confidence: 99%

Identifying the radiative coefficient of heat conduction equations from discrete measurement data

Deng

Yang

Jiang

2009

Applied Mathematics Letters

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An inversion method for parabolic equations based on quasireversibility

Cited by 24 publications

References 25 publications

Space-dependent perfusion coefficient identification in the transient bio-heat equation

Space-dependent perfusion coefficient identification in the transient bio-heat equation

Global convergence for a 1-D inverse problem with application to imaging of land mines

Identifying the radiative coefficient of heat conduction equations from discrete measurement data

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