A convergent data completion algorithm using surface integral equations

Boukari, Yosra; Haddar, Houssem

doi:10.1088/0266-5611/31/3/035011

Cited by 18 publications

(15 citation statements)

References 26 publications

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“…Since Γ 0 is assumed to be known, we can use a data completion algorithm to recover u ( f ) and ∂ ν u ( f ) on the inner boundary. Recently in previous studies, data completion algorithms are derived using boundary integral equations. This implies that the mapping

(f, Λ f) |_{{normalΓ}_{1}} \mapsto (u (f), \partial_{ν} u (f)) |_{{normalΓ}_{0}}

is known.…”

Section: Uniqueness Of the Inverse Impedance Problemmentioning

confidence: 99%

Detecting inclusions with a generalized impedance condition from electrostatic data via sampling

Harris

2019

Math Methods in App Sciences

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In this paper, we derive a sampling method to solve the inverse shape problem of recovering an inclusion with a generalized impedance condition from electrostatic Cauchy data. The generalized impedance condition is a second order differential operator applied to the boundary of the inclusion. We assume that the Dirichlet-to-Neumann mapping is given from measuring the current on the outer boundary from an imposed voltage. A simple numerical example is given to show the effectiveness of the proposed inversion method for recovering the inclusion. We also consider the inverse impedance problem of determining the impedance parameters for a known material from the Dirichlet-to-Neumann mapping assuming the inclusion has been reconstructed where uniqueness for the reconstruction of the coefficients is proven. KEYWORDS inverse boundary value problems, sampling methods, second order boundary condition, shape reconstruction MSC CLASSIFICATION 35J05; 31A25; 78A30

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(f, Λ f) |_{{normalΓ}_{1}} \mapsto (u (f), \partial_{ν} u (f)) |_{{normalΓ}_{0}}

is known.…”

Section: Uniqueness Of the Inverse Impedance Problemmentioning

confidence: 99%

Detecting inclusions with a generalized impedance condition from electrostatic data via sampling

Harris

2019

Math Methods in App Sciences

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“…Several regularization techniques has been proposed to tackle Problem (1.1). Without being exhaustive, we may mention methods based on surface integral equations [12,23], Lavrentiev regularization [10,11], stabilized finite elements methods [15][16][17], quasi-reversibility method [13,18,26,35,36], fading regularization method [24,27], etc.…”

Section: Introductionmentioning

confidence: 99%

A dual approach to Kohn–Vogelius regularization applied to data completion problem

Caubet¹,

Dardé²

2020

Inverse Problems

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This paper focuses on a dual approach in order to study the data completion problem. A classical method to solve this problem is to minimize the so-called regularized Kohn-Vogelius functional. However this method needs to choose an appropriate parameter of regularization to ensure its efficiency in the numerical reconstruction. To avoid this difficulty, we propose to study the inverse problem through a dual problem. Using some well-chosen functional spaces and establishing theoretical results in a abstract setting, we prove the well-posedness of the dual minimization problem and the convergence of our regularized solution to the exact solution when the amount of noise on the data goes to 0. Moreover we prove that the regularized solution satisfies the well-known Morozov discrepancy principle. Then we establish that the minimization of the dual functional permits not only to stably obtain a good reconstruction of the missing data of the Cauchy problem but also to determine the value of a suitable parameter of regularization in the Kohn-Vogelius strategy. We finally present numerical results, in two and three dimensions, to underline the efficiency of the proposed method.

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“…Boundary integrals of the kind considered in the present work can be used to solve direct mixed problems and such problems can be used iteratively to obtain a solution to the Cauchy problem, see [8] and [22], and references therein. An analysis of a layer approach for the Cauchy problem for the Helmholtz equation was recently presented in [5]. The reader can get a glimpse of other works and results by consulting, for example, [23,24,7,4,9,13].…”

Section: Introductionmentioning

confidence: 99%

A boundary integral equation method for numerical solution of parabolic and hyperbolic Cauchy problems

Chapko

Johansson

2018

Applied Numerical Mathematics

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We present a unified boundary integral approach for the stable numerical solution of the ill-posed Cauchy problem for the heat and wave equation. The method is based on a transformation in time (semi-discretisation) using either the method of Rothe or the Laguerre transform, to generate a Cauchy problem for a sequence of inhomogenous elliptic equations; the total entity of sequences is termed an elliptic system. For this stationary system, following a recent integral approach for the Cauchy problem for the Laplace equation, the solution is represented as a sequence of single-layer potentials invoking what is known as a fundamental sequence of the elliptic system thereby avoiding the use of volume potentials and domain discretisation. Matching the given data, a system of boundary integral equations is obtained for finding a sequence of layer densities. Full discretisation is obtained via a Nyström method together with the use of Tikhonov regularization for the obtained linear systems. Numerical results are included both for the heat and wave equation confirming the practical usefulness, in terms of accuracy and resourceful use of computational effort, of the proposed approach.

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A convergent data completion algorithm using surface integral equations

Cited by 18 publications

References 26 publications

Detecting inclusions with a generalized impedance condition from electrostatic data via sampling

Detecting inclusions with a generalized impedance condition from electrostatic data via sampling

A dual approach to Kohn–Vogelius regularization applied to data completion problem

A boundary integral equation method for numerical solution of parabolic and hyperbolic Cauchy problems

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