An iterative procedure for solving a Cauchy problem for second order elliptic equations

Johansson, Björn

doi:10.1002/mana.200310188

Cited by 26 publications

(27 citation statements)

References 4 publications

(8 reference statements)

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“…−β (Ω) satisfies (7), with N * v = r 2(β−3/2) ξ on Γ 0 . Indeed, assume first that η ∈ C ∞ 0 (Γ 1 ) and ξ ∈ C ∞ 0 (Γ 0 ).…”

Section: Proof Of Convergence Of the Iterative Proceduresmentioning

confidence: 99%

“…Another way is to use iterative methods which preserves this operator. A method of this kind for a Cauchy problem for second-order elliptic equations in bounded plain domains is given in [7].…”

Section: Introductionmentioning

confidence: 99%

“…The aim of this paper is to show that the method in [7] can be applied to higher dimensional domains. The regularizing character of the procedure is achieved by appropriate change of boundary conditions, see Section 2.3.…”

Section: Introductionmentioning

confidence: 99%

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An iterative method for reconstruction of temperature

Johansson¹

2006

j inv ill-posed problems

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“…−β (Ω) satisfies (7), with N * v = r 2(β−3/2) ξ on Γ 0 . Indeed, assume first that η ∈ C ∞ 0 (Γ 1 ) and ξ ∈ C ∞ 0 (Γ 0 ).…”

Section: Proof Of Convergence Of the Iterative Proceduresmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

An iterative method for reconstruction of temperature

Johansson¹

2006

j inv ill-posed problems

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“…In this paper, we propose a Landweber-Fridman procedure that solves mixed problems of the same type throughout the iterations and compare it with the results obtained in [4]. For bounded domains, Landweber-Fridman methods have been applied for Cauchy problems for Helmholtz, heat equation, Laplace equation, linear elasticity and the Stokes system, see, for example, [5], [6], [7], [8] and [9].…”

Section: Given the Temperature And Heat Flux On The Boundary Of A Plamentioning

confidence: 99%

“…We point out that different growth conditions at infinity can be introduced and these can be captured using appropriate weighted spaces, see [11] and [12]. Thus, it is possible to extend this work to such weighted spaces employing results from [7]. We then note that the procedure in Section 2 also works with inexact data since it is well-known that the Landweber-Fridman method with the so-called discrepancy principle is an order optimal regularization method, see for example Engl et al [10, p. 159].…”

Section: Convergence Of the Iterative Proceduresmentioning

confidence: 99%

An iterative method based on boundary integrals for elliptic Cauchy problems in semi-infinite domains

Chapko

Johansson²

2009

EJBE

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In this study, we investigate the problem of reconstruction of a stationary temperature field from given temperature and heat flux on a part of the boundary of a semi-infinite region containing an inclusion. This situation can be modelled as a Cauchy problem for the Laplace operator and it is an ill-posed problem in the sense of Hadamard. We propose and investigate a Landweber-Fridman type iterative method, which preserve the (stationary) heat operator, for the stable reconstruction of the temperature field on the boundary of the inclusion. In each iteration step, mixed boundary value problems for the Laplace operator are solved in the semi-infinite region. Well-posedness of these problems is investigated and convergence of the procedures is discussed. For the numerical implementation of these mixed problems an efficient boundary integral method is proposed which is based on the indirect variant of the boundary integral approach. Using this approach the mixed problems are reduced to integral equations over the (bounded) boundary of the inclusion. Numerical examples are included showing that stable and accurate reconstructions of the temperature field on the boundary of the inclusion can be obtained also in the case of noisy data. These results are compared with those obtained with the alternating iterative method.

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Determining the temperature from incomplete boundary data

Johansson

2007

Mathematische Nachrichten

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An iterative procedure for determining temperature fields from Cauchy data given on a part of the boundary is presented. At each iteration step, a series of mixed well-posed boundary value problems are solved for the heat operator and its adjoint. A convergence proof of this method in a weighted L 2 -space is included, as well as a stopping criteria for the case of noisy data. Moreover, a solvability result in a weighted Sobolev space for a parabolic initial boundary value problem of second order with mixed boundary conditions is presented. Regularity of the solution is proved.

show abstract

An iterative procedure for solving a Cauchy problem for second order elliptic equations

Cited by 26 publications

References 4 publications

An iterative method for reconstruction of temperature

An iterative method for reconstruction of temperature

An iterative method based on boundary integrals for elliptic Cauchy problems in semi-infinite domains

Determining the temperature from incomplete boundary data

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