An Explicit Update Scheme for Inverse Parameter and Interface Estimation of Piecewise Constant Coefficients in Linear Elliptic PDEs

Hegemann, Jan; Cantarero, Alejandro; Richardson, Casey L.; Teran, Joseph

doi:10.1137/110834500

Cited by 6 publications

(4 citation statements)

References 66 publications

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“…We refer to [16,24,32,36] for numerical implementations. The use of level set descriptions of the interfaces in the context of perimeter regularizations is described in [3,26,27]. Related to this is the use of total variation of a regularized Heaviside function with argument being a level set function, [22,40].…”

Section: Other Approachesmentioning

confidence: 99%

Double obstacle phase field approach to an inverse problem for a discontinuous diffusion coefficient

2016

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We propose a double obstacle phase field approach to the recovery of piece-wise constant diffusion coefficients for elliptic partial differential equations. The approach to this inverse problem is that of optimal control in which we have a quadratic fidelity term to which we add a perimeter regularisation weighted by a parameter σ. This yields a functional which is optimised over a set of diffusion coefficients subject to a state equation which is the underlying elliptic PDE. In order to derive a problem which is amenable to computation the perimeter functional is relaxed using a gradient energy functional together with an obstacle potential in which there is an interface parameter . This phase field approach is justified by proving Γ−convergence to the functional with perimeter regularisation as → 0. The computational approach is based on a finite element approximation. This discretisation is shown to converge in an appropriate way to the solution of the phase field problem. We derive an iterative method which is shown to yield an energy decreasing sequence converging to a discrete critical point. The efficacy of the approach is illustrated with numerical experiments.

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Section: Other Approachesmentioning

confidence: 99%

Double obstacle phase field approach to an inverse problem for a discontinuous diffusion coefficient

2016

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“…The inverse problem of reconstructing piecewise constants Lamé parameters λ, µ and their jump sets Γ simultaneously has been considered in [12] and the question of stability is not so wellstudied.…”

Section: Introductionmentioning

confidence: 99%

Quantitative stability estimate for the inverse coefficients problem in linear elasticity

Meftahi,

Rezgui

2024

Revue Africaine De Recherche en Informatique Et Mathématiques Appliquées

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In this article we consider the inverse problem of reconstructing piece-wise Lamé coefficients from boundary measurements. We reformulate the inverse problem into a minimization one using a Kohn-Vogelius type functional. We study the stability of the parameters when the jump of the discontinuity is perturbed. Using tools of shape calculus, we give a quantitative stability result for local optimal solution. Dans cet article, nous considérons le problème inverse de reconstruction des coefficients de Lamé constants par morceaux à partir de mesures au bord. Nous reformulons le problème inverse en un problème de minimisation utilisant une fonctionnelle de type Kohn-Vogelius. Nous étudions la stabilité des paramètres lorsque le saut de la discontinuité est perturbé. En utilisant les outils du calcul de forme, nous donnons un résultat de stabilité quantitative pour une la solution optimale locale.

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“…[u k ]| Γ := u k,+ − u k,− = 0, [β∇u k · n]| Γ := β + ∇u k,+ − β − ∇u k,− · n = 0, n is the normal of Γ, 1 ≤ k ≤ K, (1.2) in which u k,s = u k | Ω s , β(X) = β s for X ∈ Ω s , s = −, +. An important inverse problem related to the typical second order elliptic equation is to identify the coefficient β where one needs to either identify the physical properties of materials, i.e., the values (the parameter estimation problem) and/or detect the location and shape of inclusions/interfaces (the inverse geometric problem) using the data measured for u k , 1 ≤ k ≤ K on a subset of the domain or on a subset of the boundary ∂Ω [20,35,40]. This type of inverse problems arise from many applications in engineering and sciences, such as the electrical impedance tomography (EIT) [12,37] and groundwater or oil reservoir simulation [23,73].…”

Section: Introductionmentioning

confidence: 99%

“…It is well known that these inverse problems are usually ill-posed especially when the available data is rather limited. Numerical methods based on the output-least-squares formulation are commonly used to handle these types of inverse problems, see [15,18,20,35,39] and references therein.…”

Section: Introductionmentioning

confidence: 99%

A Fixed Mesh Method with Immersed Finite Elements for Solving Interface Inverse Problems

Guo

Lin

2018

J Sci Comput

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We present a new fixed mesh algorithm for solving a class of interface inverse problems for the typical elliptic interface problems. These interface inverse problems are formulated as shape optimization problems whose objective functionals depend on the shape of the interface. Regardless of the location of the interface, both the governing partial differential equations and the objective functional are discretized optimally, with respect to the involved polynomial space, by an immersed finite element (IFE) method on a fixed mesh. Furthermore, the formula for the gradient of the descritized objective function is derived within the IFE framework that can be computed accurately and efficiently through the discretized adjoint procedure. Features of this proposed IFE method based on a fixed mesh are demonstrated by its applications to three representative interface inverse problems: the interface inverse problem with an internal measurement on a sub-domain, a Dirichlet-Neumann type inverse problem whose data is given on the boundary, and a heat dissipation design problem.

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An Explicit Update Scheme for Inverse Parameter and Interface Estimation of Piecewise Constant Coefficients in Linear Elliptic PDEs

Cited by 6 publications

References 66 publications

Double obstacle phase field approach to an inverse problem for a discontinuous diffusion coefficient

Double obstacle phase field approach to an inverse problem for a discontinuous diffusion coefficient

Quantitative stability estimate for the inverse coefficients problem in linear elasticity

A Fixed Mesh Method with Immersed Finite Elements for Solving Interface Inverse Problems

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