Local uniqueness for an inverse boundary value problem with partial data

Harrach, Bastian; Ullrich, Marcel

doi:10.1090/proc/12991

Cited by 27 publications

(20 citation statements)

References 26 publications

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“…Local uniqueness for the Helmholtz equation. We are now able to prove the first main result in this work, announced as theorem 1.1 in the introduction, and extend the local uniqueness result in [HU17] to the case of negative potentials, and n ≥ 2. with q = q 1 , resp., q = q 2 , and let k > 0 be not a resonance, neither for q 1 nor q 2 .…”

Section: Localized Potentials and Runge Approximationmentioning

confidence: 59%

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Monotonicity and local uniqueness for the Helmholtz equation

2019

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This work extends monotonicity-based methods in inverse problems to the case of the Helmholtz (or stationary Schrödinger) equation (∆ + k 2 q)u = 0 in a bounded domain for fixed non-resonance frequency k > 0 and real-valued scattering coefficient function q. We show a monotonicity relation between the scattering coefficient q and the local Neumann-Dirichlet operator that holds up to finitely many eigenvalues. Combining this with the method of localized potentials, or Runge approximation, adapted to the case where finitely many constraints are present, we derive a constructive monotonicity-based characterization of scatterers from partial boundary data. We also obtain the local uniqueness result that two coefficient functions q 1 and q 2 can be distinguished by partial boundary data if there is a neighborhood of the boundary part where q 1 ≥ q 2 and q 1 ≡ q 2 . and the local (or partial) Neumann-to-Dirichlet (NtD) operatorwhere u ∈ H 1 (Ω) solves (1.1) with Neumann data ∂ ν u| ∂Ω = g on Σ, 0 else.Here Σ ⊆ ∂Ω is assumed to be an arbitrary non-empty relatively open subset of ∂Ω.Since k is a non-resonance frequency, Λ(q) is well defined and is easily shown to be a self-adjoint compact operator.We will show that2. The Helmholtz equation in a bounded domain. We start by summarizing some properties of the Neumann-to-Dirichlet-operators, discuss well-posedness and the role of resonance frequencies, and state a unique continuation result for the Helmholtz equation in a bounded domain.

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Section: Localized Potentials and Runge Approximationmentioning

confidence: 59%

“…Theorem 1.1 will be proven in section 5. Note that this result removes the assumption q 1 , q 2 ∈ L ∞ + (Ω) from the local uniqueness result in [HU17], and that it implies global uniqueness if q 1 −q 2 is piecewise-analytic, cf. corollary 5.2.…”

Section: Introductionmentioning

confidence: 83%

Monotonicity and local uniqueness for the Helmholtz equation

2019

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“…To show that test sets B outside the true inclusion will not give false positive results in the monotonicity method, one requires a non-trivial converse of the implication (1.2) which has been shown by Harrach and Ullrich [17] using the concept of localized potentials [10]. Monotonicitybased arguments have been used to prove theoretical uniqueness results in [12,16,13,1,18,19], and several recent works study monotonicity-based reconstruction methods, cf. e.g., [14,29,32,9,15,23,33].…”

Section: Theorem 12mentioning

confidence: 99%

Monotonicity and Enclosure Methods for the $p$-Laplace Equation

Brander¹,

Harrach

Kar

et al. 2018

SIAM J. Appl. Math.

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We show that the convex hull of a monotone perturbation of a homogeneous background conductivity in the p-conductivity equation is determined by knowledge of the nonlinear Dirichlet-Neumann operator. We give two independent proofs, one of which is based on the monotonicity method and the other on the enclosure method. Our results are constructive and require no jump or smoothness properties on the conductivity perturbation or its support.

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“…The feature of this method is to understand the inclusion relation of an unknown onject and artificial one by comparing the data operator with some operator corresponding to an artificial object. For recent works of the monotonicity method, we refer to [2,3,4,5,6,10].…”

Section: Introductionmentioning

confidence: 99%

The monotonicity method for the inverse crack scattering problem

Daimon

Furuya

Saiin

2020

Inverse Problems in Science and Engineering

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The monotonicity method for the inverse acoustic scattering problem is to understand the inclusion relation between an unknown object and artificial one by comparing the far field operator with artificial operator. This paper introduces the development of this method to the inverse crack scattering problem. Our aim is to give the following two indicators: One (Theorem 1.1) is to determine whether an artificial small arc is contained in the unknown arc. The other one (Theorem 1.2) is whether an artificial large domain contain the unknown one. Finally, numerical examples based on Theorem 1.1 are given.

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Local uniqueness for an inverse boundary value problem with partial data

Cited by 27 publications

References 26 publications

Monotonicity and local uniqueness for the Helmholtz equation

Monotonicity and local uniqueness for the Helmholtz equation

Monotonicity and Enclosure Methods for the $p$-Laplace Equation

The monotonicity method for the inverse crack scattering problem

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