Detecting nonlinear corrosion by electrostatic measurements

Alessandrini, Giovanni; Sincich, Eva

doi:10.1080/00036810500277702

Cited by 22 publications

(42 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In view of the argument given in [2], the problem is conjectured to be exponentially ill-posed such that faster Hölder type rates, as obtained in [21,22] or [9,23,24], cannot be expected, see also [16]. Existence of a source function w in (1.5) is proven under mild assumptions on the regularity of f − f * and on the smoothness of the trace of u on Γ 1 .…”

Section: Introductionmentioning

confidence: 97%

“…We provide an identifiability result in case of exact data as well as logarithmic convergence rates, which were neither considered in [2,3] nor in [5], for the regularized solutions in dependence of the noise level of the data. Numerical examples demonstrate the ill-posedness of the problem and the practicability of our approach.…”

Section: Introductionmentioning

confidence: 98%

“…Hence, the problem at hand needs to be regularized in order to recover f in a stable manner. In [2,3] a logarithmic stability result under a priori assumptions on f as well as a reconstruction technique relying on the regularized inversion of compact operators were provided. In [5] a reconstruction algorithm based on a regularization by discretization has been described where the focus was on the choice of the discretization level in a data driven way.…”

Section: Introductionmentioning

confidence: 99%

“…for the underlying Cauchy problem from [2], where u f 1 and u f 2 are two voltage potentials corresponding to nonlinear profiles f 1 and f 2 , we provide the logarithmic convergence rate…”

Section: Introductionmentioning

confidence: 99%

“…In Section 3 we investigate the inverse problem. We recall from [2] an estimate in terms of the a priori data for the length of the unknown interval I † on which f can be uniquely recovered. Then, we formulate our reconstruction procedure based on the observation that the inverse problem can be considered as a nonlinear operator equation…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Logarithmic convergence rates for the identification of a nonlinear Robin coefficient

Kügler

Sincich

2009

Journal of Mathematical Analysis and Applications

Self Cite

View full text Add to dashboard Cite

We consider the identification of a nonlinear corrosion profile from single voltage boundary data and show injectivity of the parameter-to-output map. We demonstrate that Tikhonov regularization can be applied in order to solve the inverse problem in a stable manner despite the presence of noisy data. In combination with a logarithmic stability estimate for the underlying Cauchy problem, rates for the convergence of the regularized solutions are proven using a source condition that does not involve the Fréchet derivative of the parameter-to-output map. We present sufficient conditions for the existence of a source function and illustrate our approach by means of numerical examples.

show abstract

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

“…for the underlying Cauchy problem from [2], where u f 1 and u f 2 are two voltage potentials corresponding to nonlinear profiles f 1 and f 2 , we provide the logarithmic convergence rate…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Logarithmic convergence rates for the identification of a nonlinear Robin coefficient

Kügler

Sincich

2009

Journal of Mathematical Analysis and Applications

Self Cite

View full text Add to dashboard Cite

show abstract

Uniqueness and stable determination in the inverse Robin transmission problem with one electrostatic measurement

Belhachmi¹,

Meftahi

2013

Math Methods in App Sciences

View full text Add to dashboard Cite

In this paper, we consider the inverse Robin transmission problem with one electrostatic measurement. We prove a uniqueness result for the simultaneous determination of the Robin parameter p, the conductivity k, and the subdomain D, when D is a ball. When D and k are fixed, we prove a uniqueness result and a directional Lipschitz stability estimate for the Robin parameter p. When p and k are fixed, we give an upper bound to the subdomain D. For the reconstruction purposes of the Robin parameter p, we set the inverse problem under an optimization form for a Kohn-Vogelius cost functional. We prove the existence and the stability of the optimization problem. Finally, we show some numerical experiments that agree with the theoretical considerations.

show abstract

Discretized Tikhonov regularization for Robin boundaries localization

Cao

Pereverzev

Sincich

2014

Applied Mathematics and Computation

View full text Add to dashboard Cite

We deal with a boundary detection problem arising in nondestructive testing of materials. The problem consists in recovering an unknown portion of the boundary, where a Robin condition is satisfied, with the use of a Cauchy data pair collected on the accessible part of the boundary. We combine a linearization argument with a Tikhonov regularization approach for the local reconstruction of the unknown defect. Moreover, we discuss the regularization parameter choice by means of the so called balancing principle and we present some numerical tests that show the efficiency of our method.

show abstract

Detecting nonlinear corrosion by electrostatic measurements

Abstract: We deal with an inverse problem arising in corrosion detection. We prove a stability estimate for a nonlinear term on the inaccessible portion of the boundary by electrostatic boundary measurements on the accessible one.

Cited by 22 publications

References 18 publications

Logarithmic convergence rates for the identification of a nonlinear Robin coefficient

Logarithmic convergence rates for the identification of a nonlinear Robin coefficient

Uniqueness and stable determination in the inverse Robin transmission problem with one electrostatic measurement

Discretized Tikhonov regularization for Robin boundaries localization

Contact Info

Product

Resources

About