Partial data inverse problems for Maxwell equations via Carleman estimates

Chung, Francis J.; Ola, Petri; Salo, Mikko; Tzou, Leo

doi:10.1016/j.anihpc.2017.06.005

Cited by 7 publications

(5 citation statements)

References 36 publications

(64 reference statements)

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“…These are elliptic systems that generalize the scalar Schrödinger equation (−△ + q)u = 0 and are very close to the time-harmonic Maxwell equations when n = 3. In fact, using the results of the present paper, we have finally been able to extend the partial data result of [KSU07] to the Maxwell system [COST15]. The main technical contribution of the present paper is a Carleman estimate for the Hodge Laplacian, with limiting Carleman weights, that has boundary terms involving the relative and absolute boundary values of graded forms.…”

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confidence: 64%

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Partial data inverse problems for the Hodge Laplacian

2017

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Abstract. We prove uniqueness results for a Calderón type inverse problem for the Hodge Laplacian acting on graded forms on certain manifolds in three dimensions. In particular, we show that partial measurements of the relative-to-absolute or absolute-to-relative boundary value maps uniquely determine a zeroth order potential. The method is based on Carleman estimates for the Hodge Laplacian with relative or absolute boundary conditions, and on the construction of complex geometrical optics solutions which reduce the Calderón type problem to a tomography problem for 2-tensors. The arguments in this paper allow to establish partial data results for elliptic systems that generalize the scalar results due to Kenig-Sjöstrand-Uhlmann.

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confidence: 64%

“…We expect that the methods developed in this paper open the way for obtaining partial data results via Carleman estimates for various elliptic systems. This has already been achieved for Maxwell equations [COST15].…”

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confidence: 76%

Partial data inverse problems for the Hodge Laplacian

2017

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“…In [43], an inverse boundary value problem for the time-harmonic Maxwell equations was considered. The authors showed that the electromagnetic material parameters were determined by boundary measurements where part of the boundary data is measured on a possibly very small set.…”

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confidence: 99%

Direct Method for Identification of Two Coefficients of Acoustic Equation

Novikov

Shishlenin

2023

Mathematics

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We consider the coefficient inverse problem for the 2D acoustic equation. The problem is recovering the speed of sound in the medium (which depends only on the depth) and the density (function of both variables). We describe the method, based on the Gelfand–Levitan–Krein approach, which allows us to obtain both functions by solving two sets of integral equations. The main advantage of the proposed approach is that the method does not use the multiple solution of direct problems, and thus has quite low CPU time requirements. We also consider the variation of the method for the 1D case, where the variation of the wave equation is considered. We illustrate the results with numerical experiments in the 1D and 2D case and study the efficiency and stability of the approach.

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“…Due to its versatility and robustness, this technique has since become the standard tool for solving partial data elliptic inverse problems. The review article [19] contains an excellent overview of recent work in partial data Calderón-type problems; examples for other elliptic inverse problems can be found in [31], [32], [22], [9], and [8].…”

Section: Introductionmentioning

confidence: 99%

Partial data inverse problem with 𝐿^{𝑛/2} potentials

Chung

Tzou

2020

Trans. Amer. Math. Soc. Ser. B

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We construct an explicit Green’s function for the conjugated Laplacian e − ω ⋅ x / h Δ e − ω ⋅ x / h e^{-\omega \cdot x/h}\Delta e^{-\omega \cdot x/h} , which lets us control our solutions on roughly half of the boundary. We apply the Green’s function to solve a partial data inverse problem for the Schrödinger equation with potential q ∈ L n / 2 q \in L^{n/2} . Separately, we also use this Green’s function to derive L p L^p Carleman estimates similar to the ones in Kenig-Ruiz-Sogge [Duke Math. J. 55 (1987), pp. 329–347], but for functions with support up to part of the boundary. Unlike many previous results, we did not obtain the partial data result from the boundary Carleman estimate—rather, both results stem from the same explicit construction of the Green’s function. This explicit Green’s function has potential future applications in obtaining direct numerical reconstruction algorithms for partial data Calderón problems which is presently only accessible with full data [Inverse Problems 27 (2011)].

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Partial data inverse problems for Maxwell equations via Carleman estimates

Cited by 7 publications

References 36 publications

Partial data inverse problems for the Hodge Laplacian

Partial data inverse problems for the Hodge Laplacian

Direct Method for Identification of Two Coefficients of Acoustic Equation

Partial data inverse problem with 𝐿^{𝑛/2} potentials

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