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

Brander, Tommi; Harrach, Bastian; Kar, Manas; Salo, Mikko

doi:10.1137/17m1128599

Cited by 50 publications

(48 citation statements)

References 36 publications

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“…An application of the monotonicity method to an inverse crack detection problem for the Helmholtz equation has recently been considered in [10]. For further recent contributions on monotonicity based reconstruction methods for various inverse problems for partial differential equations we refer to [1,2,20,21,32,36,38].…”

Section: Introductionmentioning

confidence: 99%

Monotonicity in inverse obstacle scattering on unbounded domains

Albicker

Griesmaier

2020

Inverse Problems

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We consider an inverse obstacle scattering problem for the Helmholtz equation with obstacles that carry mixed Dirichlet and Neumann boundary conditions. We discuss far field operators that map superpositions of plane wave incident fields to far field patterns of scattered waves, and we derive monotonicity relations for the eigenvalues of suitable modifications of these operators. These monotonicity relations are then used to establish a novel characterization of the support of mixed obstacles in terms of the corresponding far field operators. We apply this characterization in reconstruction schemes for shape detection and object classification, and we present numerical results to illustrate our theoretical findings.

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Section: Introductionmentioning

confidence: 99%

Monotonicity in inverse obstacle scattering on unbounded domains

Albicker

Griesmaier

2020

Inverse Problems

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“…. , u K ∈ H 1 (Ω) solve (22)- (25) with Neumann boundary data g and Robin transmission coefficients γ (1) , . .…”

Section: Lemma 10mentioning

confidence: 99%

“…On the methodological side, this work builds upon [48,53] and stems from the theory of combining monotonicity estimates with localized potentials, cf. [9,13,25,35,43,45,46,[50][51][52][56][57][58][59]61,94] for related works, and [29,[37][38][39][40]49,54,55,60,85,97,99,100,102,106] for practical monotonicity-based reconstruction methods. In this work, the monotonicity and convexity of the forward function is based on the so-called monotonicity relation which goes back to Ikehata, Kang, Seo, and Sheen [65,70].…”

Section: Introductionmentioning

confidence: 99%

Uniqueness, stability and global convergence for a discrete inverse elliptic Robin transmission problem

Harrach

2020

Numer. Math.

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We derive a simple criterion that ensures uniqueness, Lipschitz stability and global convergence of Newton’s method for the finite dimensional zero-finding problem of a continuously differentiable, pointwise convex and monotonic function. Our criterion merely requires to evaluate the directional derivative of the forward function at finitely many evaluation points and for finitely many directions. We then demonstrate that this result can be used to prove uniqueness, stability and global convergence for an inverse coefficient problem with finitely many measurements. We consider the problem of determining an unknown inverse Robin transmission coefficient in an elliptic PDE. Using a relation to monotonicity and localized potentials techniques, we show that a piecewise-constant coefficient on an a-priori known partition with a-priori known bounds is uniquely determined by finitely many boundary measurements and that it can be uniquely and stably reconstructed by a globally convergent Newton iteration. We derive a constructive method to identify these boundary measurements, calculate the stability constant and give a numerical example.

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“…A recent result by Brander, Kar and Salo [10] shows that one can detect the convex hull of an inclusion with conductivity bounded away from zero and infinity. Further results related to the enclosure method and the monotonicity method for the p-Laplace equation can be found in [11]. Very recently Guo, Kar and Salo [28] proved that under a monotonicity assumption the DN map is injective for Lipschitz conductivites when n = 2 for general p, and when n ≥ 3 when one of the conductivities is almost constant.…”

Section: Introductionmentioning

confidence: 99%

Superconductive and insulating inclusions for linear and non-linear conductivity equations

Brander

Ilmavirta

Kar

2018

Inverse Problems &Amp; Imaging

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We detect an inclusion with infinite conductivity from boundary measurements represented by the Dirichlet-to-Neumann map for the conductivity equation. We use both the enclosure method and the probe method. We use the enclosure method to prove partial results when the underlying equation is the quasilinear p-Laplace equation. Further, we rigorously treat the forward problem for the partial differential equation div(σ|∇u| p−2 ∇u) = 0 where the measurable conductivity σ : Ω → [0, ∞] is zero or infinity in large sets and 1 < p < ∞.

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

Cited by 50 publications

References 36 publications

Monotonicity in inverse obstacle scattering on unbounded domains

Monotonicity in inverse obstacle scattering on unbounded domains

Uniqueness, stability and global convergence for a discrete inverse elliptic Robin transmission problem

Superconductive and insulating inclusions for linear and non-linear conductivity equations

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