A duality-based method of quasi-reversibility to solve the Cauchy problem in the presence of noisy data

Bourgeois, Laurent; Dardé, Jérémi

doi:10.1088/0266-5611/26/9/095016

Cited by 70 publications

(90 citation statements)

References 19 publications

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“…This method was first introduced by Lattès and Lions [28] for numerical solutions of ill-posed problems for PDEs. It has been studied intensively since then, see e.g., [2,8,9,11,12,18,20,32]. A recent survey on this method can be found in [21].…”

Section: Introductionmentioning

confidence: 99%

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: Introductionmentioning

confidence: 99%

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

Klibanov¹,

Nguyen²

2019

Inverse Problems

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“…We also refer to, e.g. [7,8,12,23] for this method. The second author has shown in the survey paper [19] that as long as a proper Carleman estimate for an ill-posed problem for a linear PDE is available, the convergent QRM can be constructed for this problem.…”

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confidence: 99%

On an Inverse Source Problem for the Full Radiative Transfer Equation with Incomplete Data

Smirnov¹,

Klibanov²,

Nguyen³

2019

SIAM J. Sci. Comput.

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A new numerical method to solve an inverse source problem for the radiative transfer equation involving the absorption and scattering terms, with incomplete data, is proposed. No restrictive assumption on those absorption and scattering coefficients is imposed. The original inverse source problem is reduced to boundary value problem for a system of coupled partial differential equations of the first order. The unknown source function is not a part of this system. Next, we write this system in the fully discrete form of finite differences. That discrete problem is solved via the quasi-reversibility method. We prove the existence and uniqueness of the regularized solution. Especially, we prove the convergence of regularized solutions to the exact one as the noise level in the data tends to zero via a new discrete Carleman estimate. Numerical simulations demonstrate good performance of this method even when the data is highly noisy.

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“…Herein we will advocate a different approach based on discretization of the ill-posed physical model in an optimization framework, followed by regularization of the discrete problem. This primal-dual approach was first introduced by Burman in the papers [11,13,12,14], drawing on previous work by Bourgeois and Dardé on quasi reversibility methods [4,5,7,8] and further developed for elliptic data assimilation problems [17], for parabolic data reconstruction problems in [20,18] and finally for unique continuation for Helmholtz equation [19]. For a related method using finite element spaces with C 1 -regularity see [22] and for methods designed for well-posed, but indefinite problems, we refer to [9] and for second order elliptic problems on non-divergence form see [38] and [39].…”

Section: Introductionmentioning

confidence: 99%

Primal-Dual Mixed Finite Element Methods for the Elliptic Cauchy Problem

Burman¹,

Larson²,

Oksanen³

2018

SIAM J. Numer. Anal.

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We consider primal-dual mixed finite element methods for the solution of the elliptic Cauchy problem, or other related data assimilation problems. The method has a local conservation property. We derive a priori error estimates using known conditional stability estimates and determine the minimal amount of weakly consistent stabilization and Tikhonov regularization that yields optimal convergence for smooth exact solutions. The effect of perturbations in data is also accounted for. A reduced version of the method, obtained by choosing a special stabilization of the dual variable, can be viewed as a variant of the least squares mixed finite element method introduced by Dardé, Hannukainen and Hyvönen in An H div -based mixed quasi-reversibility method for solving elliptic Cauchy problems, SIAM J. Numer. Anal., 51(4) 2013. The main difference is that our choice of regularization does not depend on auxiliary parameters, the mesh size being the only asymptotic parameter. Finally, we show that the reduced method can be used for defect correction iteration to determine the solution of the full method. The theory is illustrated by some numerical examples.

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A duality-based method of quasi-reversibility to solve the Cauchy problem in the presence of noisy data

Cited by 70 publications

References 19 publications

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

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

On an Inverse Source Problem for the Full Radiative Transfer Equation with Incomplete Data

Primal-Dual Mixed Finite Element Methods for the Elliptic Cauchy Problem

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