Carleman estimates for the regularization of ill-posed Cauchy problems

Klibanov, Michael V.

doi:10.1016/j.apnum.2015.02.003

Cited by 86 publications

(133 citation statements)

References 79 publications

(249 reference statements)

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“…where ε ∈ (0, 1) is the regularization parameter. The quasi-reversibility method for problem (3.15)-(3.16) amounts to the following minimization problem: Conventionally, the convergence analysis of the quasi-reversibility method is performed on the basis of Carleman estimates [21]. However, since U (x) is a vector function rather than a 1D function and also since the matrix D (x) is likely not self adjoint, we cannot currently derive a proper Carleman estimate for the differential operator in the integrand of the right hand side of (4.1).…”

Section: The Quasi-reversibilitymentioning

confidence: 99%

See 1 more Smart Citation

PDE-based numerical method for a limited angle x-ray tomography

Klibanov¹,

Nguyen²

2019

Inverse Problems

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A new numerical method for X-ray tomography for a specific case of incomplete Radon data is proposed. Potential applications are in checking out bulky luggage in airports. This method is based on the analysis of the transport PDE governing the X-ray tomography rather than on the conventional integral formulation. The quasi-reversibility method is applied. Convergence analysis is performed using a new Carleman estimate. Numerical results are presented and compared with the inversion of the Radon transform using the well-known filtered back projection algorithm. In addition, it is shown how to use our method to study the inversion of the attenuated X-ray transform for the same case of incomplete data.

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Section: The Quasi-reversibilitymentioning

confidence: 99%

“…Next, the same Carleman estimate enables us to establish convergence rate of regularized solutions. We work with a semi discrete version of the quasi-reversibility method, which is more realistic than its conventional continuos version, see a survey in [21] for the continuos version.…”

Section: Reconstruction Errorsmentioning

confidence: 99%

PDE-based numerical method for a limited angle x-ray tomography

Klibanov¹,

Nguyen²

2019

Inverse Problems

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“…In section 2 of chapter 4 of [24] and later in [14] a Carleman estimate was used to obtain the Hölder stability estimate. However, that Hölder stability estimate is valid only on a sufficiently small time interval t ∈ (T − ε, T ) for a sufficiently small ε > 0.…”

Section: Introductionmentioning

confidence: 99%

“…A new Carleman estimate for a general parabolic operator of the second order with time reversed data is proven. This estimate works on an arbitrary time interval t ∈ (0, T ), unlike a sufficiently small interval of previous publications [7,14,24]. Results listed in items 2-4 below are based on this estimate.…”

Section: Introductionmentioning

confidence: 99%

Convergent numerical methods for parabolic equations with reversed time via a new Carleman estimate

Klibanov

Yagola

2019

Inverse Problems

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The key tool of this paper is a new Carleman estimate for an arbitrary parabolic operator of the second order for the case of reversed time data. This estimate works on an arbitrary time interval. On the other hand, the previously known Carleman estimate for the reversed time case works only on a sufficiently small time interval. First, a stability estimate is proven. Next, the quasi-reversibility numerical method is proposed for an arbitrary time interval for the linear case. This is unlike a sufficiently small time interval in the previous work. The convergence rate for the quasi-reversibility method is established. Finally, the quasilinear parabolic equation with reversed time is considered. A weighted globally strictly convex Tikhonovlike functional is constructed. The weight is the Carleman Weight Function which is involved in that Carleman estimate. The global convergence of the gradient projection method to the exact solution is proved for this functional.Key Words. linear and quasilinear parabolic equations, reversed time, Carleman estimate, stability estimate, convergent quasi-reversibility method for the linear case, globally convergent numerical method for the quasilinear case

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“…(1.1), see e.g. [3,4,5,6,7,9,10,13,14,16,17] to mention only a few. On the other hand, the Cauchy problem for nonlinear elliptic equations has been much less investigated, [11,21], and it is the purpose of this study to make advances into the semi-linear problem (1.1).…”

Section: Introductionmentioning

confidence: 99%

On the Cauchy problem for semilinear elliptic equations

Tuan

Binh

Viet

et al. 2015

Journal of Inverse and Ill-Posed Problems

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We study the Cauchy problem for non-linear (semilinear) elliptic partial differential equations in Hilbert spaces. The problem is severely ill-posed in the sense of Hadamard. Under a weak a priori assumption on the exact solution, we propose a new regularization method for stabilising the ill-posed problem. These new results extend some earlier works on Cauchy problems for nonlinear elliptic equations. Numerical results are presented and discussed.

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Carleman estimates for the regularization of ill-posed Cauchy problems

Cited by 86 publications

References 79 publications

PDE-based numerical method for a limited angle x-ray tomography

PDE-based numerical method for a limited angle x-ray tomography

Convergent numerical methods for parabolic equations with reversed time via a new Carleman estimate

On the Cauchy problem for semilinear elliptic equations

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