Divergence of finite element formulations for inverse problems treated as optimization problems

Rivas, Carlos E.; Barbone, Paul E.; Oberai, Assad A.

doi:10.1088/1742-6596/135/1/012088

Cited by 6 publications

(2 citation statements)

References 3 publications

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“…The first one, the most important, is the discretization of the unknown. In fact, discretizing the unknown σ 0 produces a discretization error that adds to the noise error on the measurements and it is very difficult to tackle, see for example [48]. The main issue is in finding a good balance between the desire of a good resolution (which requires a finer discretization) and that of stability (which requires a coarser discretization).…”

Section: Introductionmentioning

confidence: 99%

Full discretization and regularization for the Calderón problem

Felisi¹,

Rondi²

2021

Preprint

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We consider the inverse conductivity problem with discontinuous conductivities. We show in a rigorous way, by a convergence analysis, that one can construct a completely discrete minimization problem whose solution is a good approximation of a solution to the inverse problem. The minimization problem contains a regularization term which is given by a total variation penalization and is characterized by a regularization parameter. The discretization involves at the same time the boundary measurements, by the use of the complete electrode model, the unknown conductivity and the solution to the direct problem. The electrodes are characterized by a parameter related to their size, which in turn controls the number of electrodes to be used. The discretization of the unknown and of the solution to the direct problem is characterized by another parameter related to the size of the mesh involved. In our analysis we show how to precisely choose the regularization, electrodes size and mesh size parameters with respect to the noise level in such a way that the solution to the discrete regularized problem is meaningful. In particular we obtain that the electrodes and mesh size parameters should decay polynomially with respect to the noise level.

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Section: Introductionmentioning

confidence: 99%

Full discretization and regularization for the Calderón problem

Felisi¹,

Rondi²

2021

Preprint

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“…Again, the discrete regularised solution, that is, the solution to the regularised problem (1.3) with σ varying in such a discrete subset, should be a good approximation of the solution of the inverse problem. Actually, for inverse problems, this may not be necessarily so, as an example in [36] shows. Therefore, studying the effect of the discretisation when solving an inverse problem is not at all an easy task.…”

Section: Introductionmentioning

confidence: 99%

Discrete approximation and regularisation for the inverse conductivity problem

Rondi

2017

Preprint

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We study the inverse conductivity problem with discontinuous conductivities. We consider, simultaneously, a regularisation and a discretisation for a variational approach to solve the inverse problem. We show that, under suitable choices of the regularisation and discretisation parameters, the discrete regularised solutions converge, as the noise level on the measurements goes to zero, to the looked for solution of the inverse problem.

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A Review of the Mathematical and Computational Foundations of Biomechanical Imaging

Barbone

Oberai

2009

Computational Modeling in Biomechanics

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Divergence of finite element formulations for inverse problems treated as optimization problems

Cited by 6 publications

References 3 publications

Full discretization and regularization for the Calderón problem

Full discretization and regularization for the Calderón problem

Discrete approximation and regularisation for the inverse conductivity problem

A Review of the Mathematical and Computational Foundations of Biomechanical Imaging

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