Inverse problem on a tree-shaped network: unified approach for uniqueness

Baudouin, Lucie; Yamamoto, Masahiro

doi:10.1080/00036811.2014.985214

Cited by 7 publications

(4 citation statements)

References 34 publications

(113 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Since Bukhgeim and Klibanov [12], their methodology has been developed for various equations and we can refer to many papers on inverse problems of determining spatially varying coefficients and components of source terms. As a partial list of references on inverse problems for hyperbolic and parabolic equations by Carleman estimates, we refer to Baudouin and Yamamoto [3], Bellassoued [5], [6], Bellassoued and Yamamoto [8], Benabdallah, Cristofol, Gaitan and Yamamoto [11], Cristofol, Gaitan and Ramoul [16], Imanuvilov and Yamamoto [31] - [34], Klibanov [43], [44], Klibanov and Yamamoto [47], Yamamoto [57], Yuan and Yamamoto [60].…”

Section: Partial Observability Inequalitymentioning

confidence: 99%

Carleman estimate and application to an inverse source problem for a viscoelasticity model in anisotropic case

2017

Self Cite

View full text Add to dashboard Cite

We consider an anisotropic hyperbolic equation with memory term:for x ∈ Ω and t ∈ (0, T ) or ∈ (−T, T ), which is a model equation for viscoelasticity.First we establish a Carleman estimate for this equation with overdetermining boundary data on a suitable lateral subboundary Γ × (−T, T ). Second we apply the Carleman estimate to establish a both-sided estimate of u(·, 0) H 3 (Ω) by ∂ ν u| Γ×(0,T ) under the assumption that ∂ t u(·, 0) = 0 and T > 0 is sufficiently large, Γ ⊂ ∂Ω satisfies some geometric condition. Such an estimate is a kind of observability inequality and related to the exact controllability. Finally we apply the Carleman estimate to an inverse source problem of determining a spatial varying factor in F (x, t) and we establish a both-sided Lipschitz stability estimate.

show abstract

Section: Partial Observability Inequalitymentioning

confidence: 99%

Carleman estimate and application to an inverse source problem for a viscoelasticity model in anisotropic case

2017

Self Cite

View full text Add to dashboard Cite

show abstract

“…These results were improved more recently by Avdonin and Kurasov [10]. A unified approach that uses Carleman estimates and is applicable to various equations on tree networks was introduced by Baudouin and Yamamoto [11]. Also, [12] implies that it is possible to solve the problem in the presence of various different boundary conditions when doing the measurements.…”

Section: Introductionmentioning

confidence: 99%

Blockage detection in networks: The area reconstruction method†

Blåsten

Zouari

Louati

et al. 2019

Mathematics in Engineering

View full text Add to dashboard Cite

In this note we present a reconstructive algorithm for solving the cross-sectional pipe area from boundary measurements in a tree network with one inaccessbile end. This is equivalent to reconstructing the first order perturbation to a wave equation on a quantum graph from boundary measurements at all network ends except one. The method presented here is based on a time reversal boundary control method originally presented by Sondhi and Gopinath for one dimensional problems and later by Oksanen to higher dimensional manifolds. The algorithm is local, so is applicable to complicated networks if we are interested only in a part isomorphic to a tree. Moreover the numerical implementation requires only one matrix inversion or least squares minimization per discretization point in the physical network. We present a theoretical solution existence proof, a step-by-step algorithm, and a numerical implementation applied to two numerical experiments.

show abstract

“…Ignat et al in [31] worked on the inverse problem for the heat equation and the Schrödinger equation on a tree. Later on, Baudouin and Yamamoto [7] proposed a unified -and simpler -method to study the inverse problem of determining a coefficient. Results of stabilization and boundary controllability for KdV equation on star-shaped graphs was also proved by Ammari and Crépeau [3] and Cerpa et al [21,22].…”

Section: Introductionmentioning

confidence: 99%

Forcing operators on star graphs applied for the cubic fourth order Schrödinger equation

Capistrano–Filho

Cavalcante

Gallego

2022

DCDS-B

View full text Add to dashboard Cite

In a recent article [<xref ref-type="bibr" rid="b16">16</xref>], the authors gave a starting point of the study on a series of problems concerning the initial boundary value problem and control theory of Biharmonic NLS in some non-standard domains. In this direction, this article deals to present answers for some questions left in [<xref ref-type="bibr" rid="b16">16</xref>] concerning the study of the cubic fourth order Schrödinger equation in a star graph structure <inline-formula><tex-math id="M1">\begin{document}$ \mathcal{G} $\end{document}</tex-math></inline-formula>. Precisely, consider <inline-formula><tex-math id="M2">\begin{document}$ \mathcal{G} $\end{document}</tex-math></inline-formula> composed by <inline-formula><tex-math id="M3">\begin{document}$ N $\end{document}</tex-math></inline-formula> edges parameterized by half-lines <inline-formula><tex-math id="M4">\begin{document}$ (0,+\infty) $\end{document}</tex-math></inline-formula> attached with a common vertex <inline-formula><tex-math id="M5">\begin{document}$ \nu $\end{document}</tex-math></inline-formula>. With this structure the manuscript proposes to study the well-posedness of a dispersive model on star graphs with three appropriated vertex conditions by using the boundary forcing operator approach. More precisely, we give positive answer for the Cauchy problem in low regularity Sobolev spaces. We have noted that this approach seems very efficient, since this allows to use the tools of Harmonic Analysis, for instance, the Fourier restriction method, introduced by Bourgain, while for the other known standard methods to solve partial differential partial equations on star graphs are more complicated to capture the dispersive smoothing effect in low regularity. The arguments presented in this work have prospects to be applied for other nonlinear dispersive equations in the context of star graphs with unbounded edges.

show abstract

Inverse problem on a tree-shaped network: unified approach for uniqueness

Cited by 7 publications

References 34 publications

Carleman estimate and application to an inverse source problem for a viscoelasticity model in anisotropic case

Carleman estimate and application to an inverse source problem for a viscoelasticity model in anisotropic case

Blockage detection in networks: The area reconstruction method<sup>†</sup>

Forcing operators on star graphs applied for the cubic fourth order Schrödinger equation

Contact Info

Product

Resources

About