| 1. |
- Choi, Doosung, et al.
(författare)
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Inverse Problem for a Planar Conductivity Inclusion*
- 2023
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Ingår i: SIAM Journal on Imaging Sciences. - 1936-4954. ; 16:2, s. 969-995
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Tidskriftsartikel (refereegranskat)abstract
- This paper concerns the inverse problem of determining a planar conductivity inclusion. Our aim is to analytically recover from the generalized polarization tensors (GPTs), which can be obtained from exterior measurements, a homogeneous inclusion with arbitrary constant conductivity. The primary outcome of recovering a homogeneous inclusion is an inversion formula in terms of the GPTs for conformal mapping coefficients associated with the inclusion. To prove the formula, we establish matrix factorizations for the GPTs.
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| 2. |
- Choi, Doo Sung, et al.
(författare)
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Corner effects on the perturbation of an electric potential
- 2018
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Ingår i: SIAM Journal on Applied Mathematics. - 0036-1399. ; 78:3, s. 1577-1601
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Tidskriftsartikel (refereegranskat)abstract
- We consider the perturbation of an electric potential due to an insulating inclusion with corners. This perturbation is known to admit a multipole expansion whose coeffcients are linear combinations of generalized polarization tensors. We define new geometric factors of a simple planar domain in terms of a conformal mapping associated with the domain. The geometric factors share properties of the generalized polarization tensors and are the Fourier series coeffcients of a generalized external angle of the inclusion boundary. Since the generalized external angle contains the Dirac delta singularity at corner points, we can determine a criteria for the existence of corner points on the inclusion boundary in terms of the geometric factors. We illustrate and validate our results with numerical examples computed to a high degree of precision using integral equation techniques, the Nystrom discretization, and recursively compressed inverse preconditioning.
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| 3. |
- Helsing, Johan, et al.
(författare)
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Classification of spectra of the Neumann–Poincaré operator on planar domains with corners by resonance
- 2017
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Ingår i: Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis. - : European Mathematical Society - EMS - Publishing House GmbH. - 0294-1449 .- 1873-1430. ; 34:4, s. 991-1011
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Tidskriftsartikel (refereegranskat)abstract
- We study spectral properties of the Neumann–Poincaré operator on planar domains with corners with particular emphasis on existence of continuous spectrum and pure point spectrum. We show that the rate of resonance at continuous spectrum is different from that at eigenvalues, and then derive a method to distinguish continuous spectrum from eigenvalues. We perform computational experiments using the method to see whether continuous spectrum and pure point spectrum appear on domains with corners. For the computations we use a modification of the Nyström method which makes it possible to construct high-order convergent discretizations of the Neumann–Poincaré operator on domains with corners. The results of experiments show that all three possible spectra, absolutely continuous spectrum, singularly continuous spectrum, and pure point spectrum, may appear depending on domains. We also prove rigorously two properties of spectrum which are suggested by numerical experiments: symmetry of spectrum (including continuous spectrum), and existence of eigenvalues on rectangles of high aspect ratio.
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