Energy methods for Cauchy problems of evolutions equations

Andrieux, Stéphane; Baranger, Thouraya

doi:10.1088/1742-6596/135/1/012007

Cited by 13 publications

(17 citation statements)

References 6 publications

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“…As mentioned in [5] for elliptic PDEs and in [13,14] for parabolic PDEs, the data completion problem (3) is split into two well-posed ones. Thus, we introduce two distinct functions …”

Section: Energy Methodsmentioning

confidence: 99%

“…[1][2][3][4] We propose a specific regularization procedure for the Cauchy data and the identified data provided by the energy-like method. This latter was introduced in [5][6][7][8][9][10][11][12] for different applications of elliptic partial differential equations and in [13][14][15] for the parabolic heat equation and damped elastodynamic one. In this approach, two distinct fields were introduced, they are solutions of two wellposed problems, each of them meeting only one of the overspecified data and has an unknown boundary condition (Dirichlet or Neumann).…”

Section: Introductionmentioning

confidence: 99%

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Combined energy method and regularization to solve the Cauchy problem for the heat equation

Baranger

Andrieux²,

Rischette

2013

Inverse Problems in Science and Engineering

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International audienceThis paper deals with an energy method coupled with total variation regularization and an adequate stopping criterion in order to solve a Cauchy problem for the heat equation when using noisy data. First, the Cauchy problem is written as a data completion one, then it is split into two well-posed thermal problems. Therefore, a pseudo-energy functional measuring the gap between solutions of these two problems is introduced and minimized. The problem is then converted into one of constrained optimization; the computation of the gradient of this functional is given for the full time-space discrete problems by means of the adjoint method. In order to deal with noisy data, two regularization techniques were used, the first one which fits into the optimization procedure is an adequate stopping criterion depending on the noise level and it avoids numerical instability. The second one is a total variation regularization method which can be carried out a priori and/or a posteriori of the optimization procedure. Numerical experiments are performed on the noisy Cauchy data and/or the identified data. Numerical experiments highlight the efficiency and weakness of the coupled methods

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“…As mentioned in [5] for elliptic PDEs and in [13,14] for parabolic PDEs, the data completion problem (3) is split into two well-posed ones. Thus, we introduce two distinct functions …”

Section: Energy Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Combined energy method and regularization to solve the Cauchy problem for the heat equation

Baranger

Andrieux²,

Rischette

2013

Inverse Problems in Science and Engineering

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“…Each iteration involves, the solution of four linear systems (24), (25), (32) and (33). More details are given on this topic in Baranger et al [5,6], Andrieux et al [1][2][3][4] where this method is applied to 2D and 3D linear elasticity field to identify nonlinear boundary conditions and [12,13] where it is used in hydrogeology contex.…”

Section: The Energy Like Error Discrete Methodsmentioning

confidence: 99%

Misfit functional for recovering data in 2D ElectroCardioGraphy problems

Hariga-Tlatli¹,

Baranger²,

Erhel³

2010

Engineering Analysis with Boundary Elements

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“…Here, for each boundary inverse problem, we consider an approach based on an energy norm defined from two well-posed problems, which appears to be self-regularizing [2,3]. There are several ways to minimize an energy norm which use a least-squares formulation.…”

Section: Introductionmentioning

confidence: 99%

Crack identification for transient heat operator by using domain decomposition method

Khalfallah¹,

Hassin²

2017

ntmsci

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This work deals with cracks identification from over-determined boundary data. The consideration physical phenomena corresponds to the transient heat equation. we give a theoretical result of identifiability for the inverse problem under consideration. Then, we consider a recovering process based on coupling domain decomposition method and minimizing an energy-type error functional. The efficiency of the proposed approach is illustrated by several numerical results.

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Energy methods for Cauchy problems of evolutions equations

Cited by 13 publications

References 6 publications

Combined energy method and regularization to solve the Cauchy problem for the heat equation

Combined energy method and regularization to solve the Cauchy problem for the heat equation

Misfit functional for recovering data in 2D ElectroCardioGraphy problems

Crack identification for transient heat operator by using domain decomposition method

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