A quasi-reversibility approach to solve the inverse obstacle problem

Bourgeois, Laurent; Dardé, Jérémi

doi:10.3934/ipi.2010.4.351

Cited by 45 publications

(105 citation statements)

References 22 publications

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“…This effort was initiated by Bourgeois in 2005 [15]. Papers [15,17,19,20,21,30] of these authors contain quite good results of numerical experiments. These results are obtained using the FEM.…”

Section: Published Resultsmentioning

confidence: 99%

Carleman estimates for the regularization of ill-posed Cauchy problems

Klibanov

2015

Applied Numerical Mathematics

127

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This is a survey, which is a continuation of the previous survey of the author about applications of Carleman estimates to Inverse Problems, J. Inverse and Ill-Posed Problems, 21, 477-560, 2013. It is shown here that Tikhonov functionals for some ill-posed Cauchy problems for linear PDEs can be generated by unbounded linear operators of those PDEs. These are those operators for which Carleman estimates are valid, e.g. elliptic, parabolic and hyperbolic operators of the second order. Convergence rates of minimizers are established using Carleman estimates. Generalizations to nonlinear inverse problems, such as problems of reconstructions of obstacles and coefficient inverse problems are discussed as well.

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Section: Published Resultsmentioning

confidence: 99%

Carleman estimates for the regularization of ill-posed Cauchy problems

Klibanov

2015

Applied Numerical Mathematics

127

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“…As some examples, we refer to, e.g. [1,4,5,7,8,9,10,11,16,17,18,20,23,26,30] and there are many more publications on this topic. However, all those works consider only linear PDEs.…”

Section: Introductionmentioning

confidence: 99%

Numerical solution of an ill-posed Cauchy problem for a quasilinear parabolic equation using a Carleman weight function

Klibanov

Koshev²,

et al. 2016

Journal of Inverse and Ill-Posed Problems

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This is the first publication in which an ill-posed Cauchy problem for a quasilinear PDE is solved numerically by a rigorous method. More precisely, we solve the side Cauchy problem for a 1-d quasilinear parabolc equation. The key idea is to minimize a strictly convex cost functional with the Carleman Weight Function in it. Previous publications about numerical solutions of ill-posed Cauchy problems were considering only linear equations.

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“…4 Numerical Analysis of the mixed formulations 4.1 Numerical approximation of the mixed formulation (6) We now proceed to the numerical analysis of the mixed formulation (6), assuming r > 0. We follow [13], to which we refer for the details.…”

Section: Remarkmentioning

confidence: 99%

Simultaneous reconstruction of the solution and the source of hyperbolic equations from boundary measurements: a robust numerical approach

Ĉındea

Münch

2016

Inverse Problems

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We introduce a direct method allowing to solve numerically inverse type problems for linear hyperbolic equations. We first consider the reconstruction of the full solution of the wave equation posed in Ω × (0, T ) -Ω a bounded subset of R N -from a partial boundary observation. We employ a least-squares technique and minimize the L 2 -norm of the distance from the observation to any solution. Taking the hyperbolic equation as the main constraint of the problem, the optimality conditions are reduced to a mixed formulation involving both the state to reconstruct and a Lagrange multiplier. Under usual geometric optic conditions, we show the well-posedness of this mixed formulation (in particular the inf-sup condition) and then introduce a numerical approximation based on space-time finite elements discretization. We prove the strong convergence of the approximation and then discuss several examples for N = 1 and N = 2. The problem of the reconstruction of both the state and the source term is also addressed.

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A quasi-reversibility approach to solve the inverse obstacle problem

Cited by 45 publications

References 22 publications

Carleman estimates for the regularization of ill-posed Cauchy problems

Carleman estimates for the regularization of ill-posed Cauchy problems

Numerical solution of an ill-posed Cauchy problem for a quasilinear parabolic equation using a Carleman weight function

Simultaneous reconstruction of the solution and the source of hyperbolic equations from boundary measurements: a robust numerical approach

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