A parabolic inverse problem with mixed boundary data. Stability estimates for the unknown boundary and impedance

Bacchelli, Valeria; Cristo, Michele Di; Sincich, Eva; Vessella, Sergio

doi:10.1090/s0002-9947-2014-05807-8

Cited by 12 publications

(6 citation statements)

References 24 publications

(41 reference statements)

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“…Isakov proved that an additional measurement of the normal derivative on the known part of the boundary for large enough t f uniquely determines Γ and γ. Moreover, inverse problems involving a Robin condition in a parabolic equation were considered by Bacchelli, Di Cristo, Sinchich and Vessella [2]. They prove that two pairs of measurements guarantee uniqueness and stability of both Γ and γ.…”

Section: Introductionmentioning

confidence: 99%

Parameter identification for the linear wave equation with Robin boundary condition

Bacchelli

Pierotti

Micheletti

et al. 2018

Journal of Inverse and Ill-Posed Problems

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We consider an initial-boundary value problem for the classical linear wave equation, where mixed boundary conditions of Dirichlet and Neumann/Robin type are enforced at the endpoints of a bounded interval. First, by a careful application of the method of characteristics, we derive a closed-form representation of the solution for an impulsive Dirichlet data at the left endpoint, and valid for either a Neumann or a Robin data at the right endpoint. Then we devise a reconstruction procedure for identifying both the interval length and the Robin parameter. We provide a corresponding stability result and verify numerically its performance moving from a finite element discretization.

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Section: Introductionmentioning

confidence: 99%

Parameter identification for the linear wave equation with Robin boundary condition

Bacchelli

Pierotti

Micheletti

et al. 2018

Journal of Inverse and Ill-Posed Problems

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“…C([0,T];H) < +∞. • By C F ([0, T]; L 2 (Ω, H)) we denote the space of all H-valued F-adapted processes satisfying X(•) : [0, T] → L 2 FT (Ω; H) is continuous.…”

Section: Introductionmentioning

confidence: 99%

Stability estimate for an inverse stochastic parabolic problem of determining unknown time-varying boundary ^*

Liao,

Lü

2024

Inverse Problems

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Stochastic parabolic equations are widely used to model many random phenomena in natural sciences, such as the temperature distribution in a noisy medium, the dynamics of a chemical reaction in a noisy environment, or the evolution of the density of bacteria population. In many cases, the equation may involve an unknown moving boundary which could represent a change of phase, a reaction front, or an unknown population. In this paper, we focus on an inverse problem with the goal is to determine an unknown moving boundary based on data observed in a speciﬁc interior subdomain for the stochastic parabolic equation. The uniqueness of the solution of this problem is proved, and furthermore a stability estimate of log type is derived. This allows us, theoretically, to track and to monitor the behavior of the unknown boundary from observation in an arbitrary interior domain. The primary tool is a new Carleman estimate for stochastic parabolic equations. As a byproduct, we obtain a quantitative unique continuation property for stochastic parabolic equations.

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“…There are many theoretical results for the above inverse problem; see, for example, [3,[13][14][15]24], where some uniqueness results and stability estimates are established. It should be mentioned that in [14,15] the unknown inclusions can depend on time.…”

Section: Introductionmentioning

confidence: 99%

Numerical reconstruction of unknown Robin inclusions inside a heat conductor by a non-iterative method

Nakamura

Wang

2017

Inverse Problems

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Consider the problem of reconstructing unknown Robin inclusions inside a heat conductor from boundary measurements. This problem arises from active thermography and is formulated as an inverse boundary value problem for the heat equation. In our previous works, we proposed a sampling-type method for reconstructing the boundary of the Robin inclusion and gave its rigorous mathematical justification. This method is non-iterative and based on the characterization of the solution to the so-called Neumannto-Dirichlet map gap equation. In this paper, we give a further investigation of the reconstruction method from both the theoretical and numerical points of view. First, we clarify the solvability of the Neumann-to-Dirichlet map gap equation and establish a relation of its solution to the Green function associated with an initial-boundary value problem for the heat equation inside the Robin inclusion. This naturally provides a way of computing this Green function from the Neumann-to-Dirichlet map and explains what is the input for the linear sampling method. Assuming that the Neumann-to-Dirichlet map gap equation has a unique solution, we also show the convergence of our method for noisy measurements. Second, we give the numerical implementation of the reconstruction method for two-dimensional spatial domains. The measurements for our inverse problem are simulated by solving the forward problem via the boundary integral equation method. Numerical results are presented to illustrate the efficiency and stability of the proposed method. By using a finite sequence of transient input over a time interval, we propose a new sampling method over the time interval by single measurement which is most likely to be practical.

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A parabolic inverse problem with mixed boundary data. Stability estimates for the unknown boundary and impedance

Abstract: We consider the problem of determining an unaccessible part of the boundary of a conductor by means of thermal measurements. We study a problem of corrosion where a Robin type condition is prescribed on the damaged part and we prove logarithmic stability estimate.

Cited by 12 publications

References 24 publications

Parameter identification for the linear wave equation with Robin boundary condition

Parameter identification for the linear wave equation with Robin boundary condition

Stability estimate for an inverse stochastic parabolic problem of determining unknown time-varying boundary ^*

Numerical reconstruction of unknown Robin inclusions inside a heat conductor by a non-iterative method

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A parabolic inverse problem with mixed boundary data. Stability estimates for the unknown boundary and impedance

Abstract: We consider the problem of determining an unaccessible part of the boundary of a conductor by means of thermal measurements. We study a problem of corrosion where a Robin type condition is prescribed on the damaged part and we prove logarithmic stability estimate.

Cited by 12 publications

References 24 publications

Parameter identification for the linear wave equation with Robin boundary condition

Parameter identification for the linear wave equation with Robin boundary condition

Stability estimate for an inverse stochastic parabolic problem of determining unknown time-varying boundary *

Numerical reconstruction of unknown Robin inclusions inside a heat conductor by a non-iterative method

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Stability estimate for an inverse stochastic parabolic problem of determining unknown time-varying boundary ^*