Determining a fractional Helmholtz equation with unknown source and scattering potential

Cao, Xinlin; Liu, Hongyu

doi:10.4310/cms.2019.v17.n7.a5

Cited by 44 publications

(29 citation statements)

References 14 publications

(22 reference statements)

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“…the discussion below). Moreover, these techniques have been extended to other nonlocal problems [RS17b,CL18] in a slightly different context. Also, for positive potentials, monotonicity inversion formulas have been successfully discovered in [HL17].…”

Section: Introductionmentioning

confidence: 99%

The Calderón problem for the fractional Schrödinger equation with drift

2020

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We investigate the Calderón problem for the fractional Schrödinger equation with drift, proving that the unknown drift and potential in a bounded domain can be determined simultaneously and uniquely by an infinite number of exterior measurements. In particular, in contrast to its local analogue, this nonlocal problem does not enjoy a gauge invariance. The uniqueness result is complemented by an associated logarithmic stability estimate under suitable apriori assumptions. Also uniqueness under finitely many generic measurements is discussed. Here the genericity is obtained through singularity theory which might also be interesting in the context of hybrid inverse problems. Combined with the results from [GRSU18], this yields a finite measurements constructive reconstruction algorithm for the fractional Calderón problem with drift. The inverse problem is formulated as a partial data type nonlocal problem and it is considered in any dimension n ≥ 1.

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Section: Introductionmentioning

confidence: 99%

The Calderón problem for the fractional Schrödinger equation with drift

2020

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“…Recently, [62] investigated the Calderón problem for a space-time fractional parabolic equation. We also refer readers to [16,17] for further studies on the simultaneous determination of parameters in fractional inverse problems.…”

Section: Introductionmentioning

confidence: 99%

Monotonicity-Based Inversion of the Fractional Schrödinger Equation II. General Potentials and Stability

Harrach

Lin²

2020

SIAM J. Math. Anal.

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In this work, we use monotonicity-based methods for the fractional Schrödinger equation with general potentials q ∈ L ∞ (Ω) in a Lipschitz bounded open set Ω ⊂ R n in any dimension n ∈ N. We demonstrate that if-and-only-if monotonicity relations between potentials and the Dirichlet-to-Neumann map hold up to a finite dimensional subspace. Based on these if-and-only-if monotonicity relations, we derive a constructive global uniqueness result for the fractional Calderón problem and its linearized version. We also derive a reconstruction method for unknown obstacles in a given domain that only requires the background solution of the fractional Schrödinger equation, and we prove uniqueness and Lipschitz stability from finitely many measurements for potentials lying in an a-priori known bounded set in a finite dimensional subset of L ∞ (Ω).

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“…By virtue of the constructions in [25], this dependence is optimal. We refer to [8,13,17,19,24,26] and the references therein for further developments on the fractional Calderón and related problems.…”

Section: Introductionmentioning

confidence: 99%

Lipschitz stability for the finite dimensional fractional Calderón problem with finite Cauchy data

Rüland

Sincich²

2019

Inverse Problems &Amp; Imaging

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In this note we discuss the conditional stability issue for the finite dimensional Calderón problem for the fractional Schrödinger equation with a finite number of measurements. More precisely, we assume that the unknown potential q ∈ L ∞ (Ω) in the equation ((−∆) s + q)u = 0 in Ω ⊂ R n satisfies the a priori assumption that it is contained in a finite dimensional subspace of L ∞ (Ω). Under this condition we prove Lipschitz stability estimates for the fractional Calderón problem by means of finitely many Cauchy data depending on q. We allow for the possibility of zero being a Dirichlet eigenvalue of the associated fractional Schrödinger equation. Our result relies on the strong Runge approximation property of the fractional Schrödinger equation.

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Determining a fractional Helmholtz equation with unknown source and scattering potential

Cited by 44 publications

References 14 publications

The Calderón problem for the fractional Schrödinger equation with drift

The Calderón problem for the fractional Schrödinger equation with drift

Monotonicity-Based Inversion of the Fractional Schrödinger Equation II. General Potentials and Stability

Lipschitz stability for the finite dimensional fractional Calderón problem with finite Cauchy data

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