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doi: 10.3934/ipi.2022021
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Reconstruction of singular and degenerate inclusions in Calderón's problem

1. 

Department of Mathematics, Aarhus University, Ny Munkegade 118, 8000 Aarhus C, Denmark

2. 

Department of Mathematics and Systems Analysis, Aalto University, P.O. Box 11100, 00076 Helsinki, Finland

*Corresponding author

Received  November 2021 Revised  March 2022 Early access April 2022

We consider the reconstruction of the support of an unknown perturbation to a known conductivity coefficient in Calderón's problem. In a previous result by the authors on monotonicity-based reconstruction, the perturbed coefficient is allowed to simultaneously take the values $ 0 $ and $ \infty $ in some parts of the domain and values bounded away from $ 0 $ and $ \infty $ elsewhere. We generalise this result by allowing the unknown coefficient to be the restriction of an $ A_2 $-Muckenhoupt weight in parts of the domain, thereby including singular and degenerate behaviour in the governing equation. In particular, the coefficient may tend to $ 0 $ and $ \infty $ in a controlled manner, which goes beyond the standard setting of Calderón's problem. Our main result constructively characterises the outer shape of the support of such a general perturbation, based on a local Neumann-to-Dirichlet map defined on an open subset of the domain boundary.

Citation: Henrik Garde, Nuutti Hyvönen. Reconstruction of singular and degenerate inclusions in Calderón's problem. Inverse Problems and Imaging, doi: 10.3934/ipi.2022021
References:
[1]

K. AstalaM. Lassas and L. Päivärinta, The borderlines of invisibility and visibility in Calderón's inverse problem, Anal. PDE, 9 (2016), 43-98.  doi: 10.2140/apde.2016.9.43.

[2]

L. Borcea, Electrical impedance tomography, Inverse Problems, 18 (2002), 99-136.  doi: 10.1088/0266-5611/18/6/201.

[3]

L. Borcea, Addendum to "Electrical impedance tomography", Inverse Problems, 19 (2003), 997-998.  doi: 10.1088/0266-5611/19/4/501.

[4]

V. CandianiJ. DardéH. Garde and N. Hyvönen, Monotonicity-based reconstruction of extreme inclusions in electrical impedance tomography, SIAM J. Math. Anal., 52 (2020), 6234-6259.  doi: 10.1137/19M1299219.

[5]

C. Cârstea and J. N. Wang, Uniqueness for the two dimensional Calderón's problem with unbounded conductivities, Ann. Sc. Norm. Super. Pisa Cl. Sci., 18 (2018), 1459-1482. 

[6]

M. CheneyD. Isaacson and J. C. Newell, Electrical impedance tomography, SIAM Rev., 41 (1999), 85-101.  doi: 10.1137/S0036144598333613.

[7]

P. Ciarlet, The Finite Element Method for Elliptic Problems, Studies in Mathematics and its Applications, Vol. 4. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978.

[8]

I. Drelichman and R. G. Durán, Improved Poincaré inequalities with weights, J. Math. Anal. Appl., 347 (2008), 286-293.  doi: 10.1016/j.jmaa.2008.06.005.

[9]

R. G. Durán and F. L. García, Solutions of the divergence and analysis of the Stokes equations in planar Hölder-$\alpha$ domains, Math. Models Methods Appl. Sci., 20 (2010), 95-120.  doi: 10.1142/S0218202510004167.

[10]

A. C. Esposito, L. Faella, G. Piscitelli, R. Prakash and A. Tamburrino, Monotonicity Principle in tomography of nonlinear conducting materials, Inverse Problems, 37 (2021), Article ID 045012, 25 pp. doi: 10.1088/1361-6420/abd29a.

[11]

E. B. FabesC. E. Kenig and R. P. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. PDE, 7 (1982), 77-116.  doi: 10.1080/03605308208820218.

[12]

H. Garde, Reconstruction of piecewise constant layered conductivities in electrical impedance tomography, Comm. Partial Differential Equations, 45 (2020), 1118-1133.  doi: 10.1080/03605302.2020.1760884.

[13]

H. Garde and S. Staboulis, Convergence and regularization for monotonicity-based shape reconstruction in electrical impedance tomography, Numer. Math., 135 (2017), 1221-1251.  doi: 10.1007/s00211-016-0830-1.

[14]

H. Garde and S. Staboulis, The regularized monotonicity method: Detecting irregular indefinite inclusions, Inverse Probl. Imag., 13 (2019), 93-116.  doi: 10.3934/ipi.2019006.

[15]

G. Grubb, Distributions and Operators, Springer, New York, 2009.

[16]

B. Harrach, Uniqueness and Lipschitz stability in electrical impedance tomography with finitely many electrodes, Inverse Problems, 35 (2019), Article ID 024005, 19 pp. doi: 10.1088/1361-6420/aaf6fc.

[17]

B. Harrach and J. K. Seo, Exact shape-reconstruction by one-step linearization in electrical impedance tomography, SIAM J. Math. Anal., 42 (2010), 1505-1518.  doi: 10.1137/090773970.

[18]

B. Harrach and M. Ullrich, Monotonicity-based shape reconstruction in electrical impedance tomography, SIAM J. Math. Anal., 45 (2013), 3382-3403.  doi: 10.1137/120886984.

[19]

B. Harrach and M. Ullrich, Resolution guarantees in electrical impedance tomography, IEEE T. Med. Imaging, 34 (2015), 1513-1521.  doi: 10.1109/TMI.2015.2404133.

[20]

J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Dover Publications, Inc., Mineola, NY, 2006.

[21]

M. Ikehata, Size estimation of inclusion, J. Inverse Ill-Posed Probl., 6 (1998), 127-140.  doi: 10.1515/jiip.1998.6.2.127.

[22]

H. KangJ. K. Seo and D. Sheen, The inverse conductivity problem with one measurement: Stability and estimation of size, SIAM J. Math. Anal., 28 (1997), 1389-1405.  doi: 10.1137/S0036141096299375.

[23]

A. NachmanI. Regev and D. Tataru, A nonlinear Plancherel theorem with applications to global well-posedness for the defocusing Davey-Stewartson equation and to the inverse boundary value problem of Calderón, Invent. Math., 220 (2020), 395-451.  doi: 10.1007/s00222-019-00930-0.

[24]

A. Tamburrino and G. Rubinacci, A new non-iterative inversion method for electrical resistance tomography, Inverse Problems, 18 (2002), 1809-1829.  doi: 10.1088/0266-5611/18/6/323.

[25]

G. Uhlmann, Electrical impedance tomography and Calderón's problem, Inverse Problems, 25 (2009), Article ID 123011, 39 pp. doi: 10.1088/0266-5611/25/12/123011.

show all references

References:
[1]

K. AstalaM. Lassas and L. Päivärinta, The borderlines of invisibility and visibility in Calderón's inverse problem, Anal. PDE, 9 (2016), 43-98.  doi: 10.2140/apde.2016.9.43.

[2]

L. Borcea, Electrical impedance tomography, Inverse Problems, 18 (2002), 99-136.  doi: 10.1088/0266-5611/18/6/201.

[3]

L. Borcea, Addendum to "Electrical impedance tomography", Inverse Problems, 19 (2003), 997-998.  doi: 10.1088/0266-5611/19/4/501.

[4]

V. CandianiJ. DardéH. Garde and N. Hyvönen, Monotonicity-based reconstruction of extreme inclusions in electrical impedance tomography, SIAM J. Math. Anal., 52 (2020), 6234-6259.  doi: 10.1137/19M1299219.

[5]

C. Cârstea and J. N. Wang, Uniqueness for the two dimensional Calderón's problem with unbounded conductivities, Ann. Sc. Norm. Super. Pisa Cl. Sci., 18 (2018), 1459-1482. 

[6]

M. CheneyD. Isaacson and J. C. Newell, Electrical impedance tomography, SIAM Rev., 41 (1999), 85-101.  doi: 10.1137/S0036144598333613.

[7]

P. Ciarlet, The Finite Element Method for Elliptic Problems, Studies in Mathematics and its Applications, Vol. 4. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978.

[8]

I. Drelichman and R. G. Durán, Improved Poincaré inequalities with weights, J. Math. Anal. Appl., 347 (2008), 286-293.  doi: 10.1016/j.jmaa.2008.06.005.

[9]

R. G. Durán and F. L. García, Solutions of the divergence and analysis of the Stokes equations in planar Hölder-$\alpha$ domains, Math. Models Methods Appl. Sci., 20 (2010), 95-120.  doi: 10.1142/S0218202510004167.

[10]

A. C. Esposito, L. Faella, G. Piscitelli, R. Prakash and A. Tamburrino, Monotonicity Principle in tomography of nonlinear conducting materials, Inverse Problems, 37 (2021), Article ID 045012, 25 pp. doi: 10.1088/1361-6420/abd29a.

[11]

E. B. FabesC. E. Kenig and R. P. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. PDE, 7 (1982), 77-116.  doi: 10.1080/03605308208820218.

[12]

H. Garde, Reconstruction of piecewise constant layered conductivities in electrical impedance tomography, Comm. Partial Differential Equations, 45 (2020), 1118-1133.  doi: 10.1080/03605302.2020.1760884.

[13]

H. Garde and S. Staboulis, Convergence and regularization for monotonicity-based shape reconstruction in electrical impedance tomography, Numer. Math., 135 (2017), 1221-1251.  doi: 10.1007/s00211-016-0830-1.

[14]

H. Garde and S. Staboulis, The regularized monotonicity method: Detecting irregular indefinite inclusions, Inverse Probl. Imag., 13 (2019), 93-116.  doi: 10.3934/ipi.2019006.

[15]

G. Grubb, Distributions and Operators, Springer, New York, 2009.

[16]

B. Harrach, Uniqueness and Lipschitz stability in electrical impedance tomography with finitely many electrodes, Inverse Problems, 35 (2019), Article ID 024005, 19 pp. doi: 10.1088/1361-6420/aaf6fc.

[17]

B. Harrach and J. K. Seo, Exact shape-reconstruction by one-step linearization in electrical impedance tomography, SIAM J. Math. Anal., 42 (2010), 1505-1518.  doi: 10.1137/090773970.

[18]

B. Harrach and M. Ullrich, Monotonicity-based shape reconstruction in electrical impedance tomography, SIAM J. Math. Anal., 45 (2013), 3382-3403.  doi: 10.1137/120886984.

[19]

B. Harrach and M. Ullrich, Resolution guarantees in electrical impedance tomography, IEEE T. Med. Imaging, 34 (2015), 1513-1521.  doi: 10.1109/TMI.2015.2404133.

[20]

J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Dover Publications, Inc., Mineola, NY, 2006.

[21]

M. Ikehata, Size estimation of inclusion, J. Inverse Ill-Posed Probl., 6 (1998), 127-140.  doi: 10.1515/jiip.1998.6.2.127.

[22]

H. KangJ. K. Seo and D. Sheen, The inverse conductivity problem with one measurement: Stability and estimation of size, SIAM J. Math. Anal., 28 (1997), 1389-1405.  doi: 10.1137/S0036141096299375.

[23]

A. NachmanI. Regev and D. Tataru, A nonlinear Plancherel theorem with applications to global well-posedness for the defocusing Davey-Stewartson equation and to the inverse boundary value problem of Calderón, Invent. Math., 220 (2020), 395-451.  doi: 10.1007/s00222-019-00930-0.

[24]

A. Tamburrino and G. Rubinacci, A new non-iterative inversion method for electrical resistance tomography, Inverse Problems, 18 (2002), 1809-1829.  doi: 10.1088/0266-5611/18/6/323.

[25]

G. Uhlmann, Electrical impedance tomography and Calderón's problem, Inverse Problems, 25 (2009), Article ID 123011, 39 pp. doi: 10.1088/0266-5611/25/12/123011.

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