August  2019, 13(4): 863-878. doi: 10.3934/ipi.2019039

Reconstruction of rough potentials in the plane

Mathematics Department, Dublin City University, Dublin 9, Ireland

Received  December 2018 Revised  March 2019 Published  May 2019

Fund Project: 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

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.

Citation: Jorge Tejero. Reconstruction of rough potentials in the plane. Inverse Problems & Imaging, 2019, 13 (4) : 863-878. doi: 10.3934/ipi.2019039
References:
[1]

K. AstalaD. 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. BarcelóJ. BennettA. 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. BlåstenO. 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. LakshtanovR. 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. LakshtanovJ. 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.

show all references

References:
[1]

K. AstalaD. 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. BarcelóJ. BennettA. 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. BlåstenO. 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. LakshtanovR. 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. LakshtanovJ. 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.

Figure 5.  Shepp-Logan phantom $ \lambda = 100 $. Top: potential, standard main term. Bottom: mollifier, angular, combined
Figure 1.  Rectangles $ \lambda = 10 $. Top: potential, standard main term. Bottom: mollifier, angular, combined
Figure 2.  Ovals $ \lambda = 15 $. Top: potential, standard main term. Bottom: mollifier, angular, combined
Figure 3.  Circles spiral $ \lambda = 30 $. Top: potential, standard main term. Bottom: mollifier, angular, combined
Figure 4.  Geometric figures $ \lambda = 50 $. Top: potential, standard main term. Bottom: mollifier, angular, combined
Figure 6.  Error reduction in the $ L^1 $ norm
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