An alternating iterative procedure for the Cauchy problem for the Helmholtz equation

Berntsson, Fredrik; Kozlov, Vladimir; Mpinganzima, Lydie; Turesson, Bengt-Ove

doi:10.1080/17415977.2013.827181

Cited by 38 publications

(39 citation statements)

References 25 publications

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“…This method has gained a lot of popularity since then. In this regard, we also mention the work of Avdonin, Kozlov and Maxwell [2] for a nonlinear elliptic equation and the work of Berntsson, Kozlov, Mpinganzima and Turesson [14] for the Helmholtz equation. Andrieux, Baranger and Ben Abda [1] have improved in some sense the algorithm of [58].…”

Section: Introductionmentioning

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: Introductionmentioning

confidence: 99%

Carleman estimates for the regularization of ill-posed Cauchy problems

Klibanov

2015

Applied Numerical Mathematics

127

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“…Let us finally prove that the operator A is non-expansive. If Ψ D ∈ Λ, the sequence (||u k || α,η ) k is decreasing; see [17,18]. We then obtain ||AΨ D || 0,ΓI = ||u 2 || α,η ≤ ||u 0 || α,η = ||Ψ D || 0,ΓI .…”

Section: The Optimized Dirichlet-order 2 Alternating Methodsmentioning

confidence: 99%

“…In fact, this procedure, which extends analogue results for elasticity problems (see [14,15]), has become very popular lately, especially in the context of obstacle detection [16], or coefficient reconstruction [3]. The alternating method has been applied successfully to the Cauchy Stokes problem and similar problems, see, for example [17,18,19]. A deep one tool relates the two approaches.…”

Section: Introductionmentioning

confidence: 99%

The sub-Cauchy–Stokes problem: Solvability issues and Lagrange multiplier methods with artificial boundary conditions

Ahmed

Abda

2018

Journal of Computational and Applied Mathematics

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In contrast to the conventional Cauchy Stokes problems, in which the velocity and the stress force data are given on the accessible boundary, in the present paper, we reduce the accessible boundary data information and we consider a problem which deals only with shear stress data. We refer to this problem as a sub-Cauchy Stokes problem. This problem is ill-posed because of severe instability and even uniqueness is unknown. We first address the uniqueness issues associated with this problem. Resorting to the domain decomposition techniques together with the duplication process of Vogelius [1], we propose new Lagrange multiplier methods to solve the sub-Cauchy Stokes problem. These methods consist in recasting the problem in terms of interfacial equations, by equalizing two solutions of the sub-Cauchy Stokes problem using matching conditions defined on the inaccessible boundary. The matching is based on second order conditions and depends on the equations used to match the values of the unknowns on the inaccessible boundary. The underlying interfacial problems are then solved by iterative procedures in which coefficients can be optimized to improve convergence rates. A complete analysis of the methods is presented, and various numerical results illustrate the effectiveness and the performance of the proposed approaches.

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“…Similarly, by definition, ( , ) = ( 0 , 2 ), and using the normal bending moment boundary condition for 2 on Γ 1 together with (13) and the symmetry of the bilinear form (⋅, ⋅),…”

Section: The Adjoint Of the Operatormentioning

confidence: 99%

An iterative regularizing method for an incomplete boundary data problem for the biharmonic equation

Chapko

Johansson

2018

Z Angew Math Mech

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An incomplete boundary data problem for the biharmonic equation is considered, where the displacement is known throughout the boundary of the solution domain whilst the normal derivative and bending moment are specified on only a portion of the boundary. For this inverse ill‐posed problem an iterative regularizing method is proposed for the stable data reconstruction on the underspecified boundary part. Convergence is proven by showing that the method can be written as a Landweber‐type procedure for an operator formulation of the incomplete data problem. This reformulation renders a stopping rule, the discrepancy principle, for terminating the iterations in the case of noisy data. Uniqueness of a solution to the considered problem is also shown.

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An alternating iterative procedure for the Cauchy problem for the Helmholtz equation

Cited by 38 publications

References 25 publications

Carleman estimates for the regularization of ill-posed Cauchy problems

Carleman estimates for the regularization of ill-posed Cauchy problems

The sub-Cauchy–Stokes problem: Solvability issues and Lagrange multiplier methods with artificial boundary conditions

An iterative regularizing method for an incomplete boundary data problem for the biharmonic equation

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