Reconstruction of rough potentials in the plane

Jorge Tejero

doi:10.3934/ipi.2019039

Article Contents

2019, Volume 13, Issue 4: 863-878. Doi: 10.3934/ipi.2019039

This issue Previous Article Nash strategies for the inverse inclusion Cauchy-Stokes problem Next Article Two gesture-computing approaches by using electromagnetic waves

Reconstruction of rough potentials in the plane

Jorge Tejero

Mathematics Department, Dublin City University, Dublin 9, Ireland

Received: November 30, 2018

Revised: February 28, 2019

Published: May 2019

The work was partially supported by the ERC grant 834728, the MINECO grants MTM2014-57769-1-P, SEV-2015-0554 and MTM2017-85934-P (Spain) and the SFI grant 16/IA/4443.

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Abstract

We provide a reconstruction scheme for complex-valued potentials in $ H^s (\mathbb{R}^2) $ for $ s > 0 $. The procedure extends the method of Bukhgeim relying on quadratic exponential solutions.

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Mathematics Subject Classification: Primary: 31A25, 34A55, 34L25, 34M50.

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Figure 5. Shepp-Logan phantom $ \lambda = 100 $. Top: potential, standard main term. Bottom: mollifier, angular, combined

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Figure 1. Rectangles $ \lambda = 10 $. Top: potential, standard main term. Bottom: mollifier, angular, combined

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Figure 2. Ovals $ \lambda = 15 $. Top: potential, standard main term. Bottom: mollifier, angular, combined

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Figure 3. Circles spiral $ \lambda = 30 $. Top: potential, standard main term. Bottom: mollifier, angular, combined

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Figure 4. Geometric figures $ \lambda = 50 $. Top: potential, standard main term. Bottom: mollifier, angular, combined

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Figure 6. Error reduction in the $ L^1 $ norm

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References

[1]	K. Astala, D. Faraco and K. M. Rogers, Unbounded potential recovery in the plane, Annales Scientifiques de l'École Normale Supériure. Quatrième Série, 49 (2016), 1027-1051. doi: 10.24033/asens.2302.
[2]	K. Astala and L. Päivärinta, Calderón Inverse Conductivity problem in plane, Annals of Mathematics, 163 (2006), 265-299.
[3]	J. A. Barceló, J. Bennett, A. Carbery and K. M. Rogers, On the dimension of divergence sets of dispersive equations, Mathematische Annalen, 349 (2011), 599-622. doi: 10.1007/s00208-010-0529-z.
[4]	R. Beals and R. R. Coifman, The D-bar approach to inverse scattering and nonlinear evolutions, Physica D: Nonlinear Phenomena, 18 (1986), 242-249. doi: 10.1016/0167-2789(86)90184-3.
[5]	E. Blåsten, O. Yu. Imanuvilov and M. Yamamoto, Stability and uniqueness for a two-dimensional inverse boundary value problem for less regular potentials, Inverse Problems and Imaging, 9 (2015), 709-723. doi: 10.3934/ipi.2015.9.709.
[6]	R. M. Brown and G. A. Uhlmann, Uniqueness in the inverse conductivity problem for nonsmooth conductivities in two dimensions, Communications in Partial Differential Equations, 22 (1997), 1009-1027. doi: 10.1080/03605309708821292.
[7]	A. L. Bukhgeim, Recovering a potential from Cauchy data in the two-dimensional case, Journal of Inverse and Ill-Posed Problems, 16 (2008), 19-33. doi: 10.1515/jiip.2008.002.
[8]	A. P. Calderón, On an inverse boundary value problem, Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980), 65–73, Soc. Brasil. Mat., Rio de Janeiro, 1980.
[9]	P. Caro and K. M. Rogers, Global uniqueness for the Calderón problem with Lipschitz conductivities, Forum of Mathematics Pi, 4 (2016), e2, 28 pp. doi: 10.1017/fmp.2015.9.
[10]	I. Gel'fand, Some Aspects of Functional Analysis and Algebra, Proceedings of the International Congress of Mathematics, Amsterdam, 1954.
[11]	B. Haberman and D. Tataru, Uniqueness in Calderón's problem with Lipschitz conductivities, Duke Mathematical Journal, 162 (2013), 496-516. doi: 10.1215/00127094-2019591.
[12]	E. L. Lakshtanov, R. G. Novikov and B. R. Vainberg, A global Riemann-Hilbert problem for two-dimensional inverse scattering at fixed energy, Rendiconti dell'Istituto di Matematica dell'Università di Trieste, 38 (2016), 21-47.
[13]	E. L. Lakshtanov, J. Tejero and B. R. Vainberg, Uniqueness in the inverse conductivity problem for complex-valued Lipschitz conductivities in the plane, SIAM Journal on Mathematical Analysis, 49 (2017), 3766-3775. doi: 10.1137/17M1120981.
[14]	E. L. Lakshtanov and B. R. Vainberg, Recovery of $L^p$-potential in the plane, J. Inverse Ill-Posed Probl., 25 (2017), 633-651. doi: 10.1515/jiip-2016-0052.
[15]	A. I. Nachman, Reconstructions from boundary measurements, Annals of Mathematics, 128 (1988), 531-576. doi: 10.2307/1971435.
[16]	A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Annals of Mathematics, 143 (1996), 71-96. doi: 10.2307/2118653.
[17]	R. G. Novikov, Multidimensional inverse spectral problem for the equation $-\Delta \psi + (v(x) - Eu(x)) \psi = 0$, Functional Analysis and Its Applications, 22 (1988), 263-272. doi: 10.1007/BF01077418.
[18]	J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Annals of Mathematics, 125 (1987), 153-169. doi: 10.2307/1971291.
[19]	J. Tejero, Reconstruction and stability for piecewise smooth potentials in the plane, SIAM Journal on Mathematical Analysis, 49 (2017), 398-420. doi: 10.1137/16M1085048.