Lipschitz stability for the finite dimensional fractional Calderón problem with finite Cauchy data

Previous Article
Determining rough first order perturbations of the polyharmonic operator
IPI Home
This Issue
Next Article
Identifiability of diffusion coefficients for source terms of non-uniform sign

October 2019, 13(5): 1023-1044. doi: 10.3934/ipi.2019046

Lipschitz stability for the finite dimensional fractional Calderón problem with finite Cauchy data

Angkana Rüland ^1, and Eva Sincich ^2,,

1.	Max-Planck Institute for Mathematics in the Sciences, Inselstraße 22, 04103 Leipzig, Germany
2.	Dipartimento di Matematica e Geoscienze Università degli Studi di Trieste, via Valerio 12/1, 34127 Trieste, Italy

^* Corresponding author: Eva Sincich

Received November 2018 Revised April 2019 Published July 2019

In this note we discuss the conditional stability issue for the finite dimensional Calderón problem for the fractional Schrödinger equation with a finite number of measurements. More precisely, we assume that the unknown potential $ q \in L^{\infty}(\Omega) $ in the equation $ ((- \Delta)^s+ q)u = 0 \mbox{ in } \Omega\subset \mathbb{R}^n $ satisfies the a priori assumption that it is contained in a finite dimensional subspace of $ L^{\infty}(\Omega) $. Under this condition we prove Lipschitz stability estimates for the fractional Calderón problem by means of finitely many Cauchy data depending on $ q $. We allow for the possibility of zero being a Dirichlet eigenvalue of the associated fractional Schrödinger equation. Our result relies on the strong Runge approximation property of the fractional Schrödinger equation.

Keywords: Lipschitz stability, finite dimensional fractional Calderón problem, finite Cauchy data.

Mathematics Subject Classification: Primary: 35R30, 35A35, 35S15.

Citation: Angkana Rüland, Eva Sincich. Lipschitz stability for the finite dimensional fractional Calderón problem with finite Cauchy data. Inverse Problems and Imaging, 2019, 13 (5) : 1023-1044. doi: 10.3934/ipi.2019046

References:

[1]	G. S. Alberti and M. Santacesaria, Calderón's Inverse Problem with a Finite Number of Measurements, arXiv: 1803.04224, 2018.
[2]	G. Alessandrini, M. V. de Hoop, R. Gaburro and E. Sincich, Lipschitz stability for the electrostatic inverse boundary value problem with piecewise linear conductivities, Journal de Mathématiques Pures et Appliquées, 107 (2016), 638-664. doi: 10.1016/j.matpur.2016.10.001.
[3]	G. Alessandrini, M. V. de Hoop and R. Gaburro, Romina and Eva Sincich, Lipschitz stability for a piecewise linear Schrödinger potential from local Cauchy data, Asymptotic Analysis, 108 (2018), 115-149. doi: 10.3233/ASY-171457.
[4]	G. Alessandrini and V. Isakov, Analiticity and uniqueness for the inverse conductivity problem, Rend. Ist. Mat. Univ. Trieste, 28 (1996), 351-369.
[5]	G. Alessandrini, L. Rondi, E. Rosset and S. Vessella, The stability for the Cauchy problem for elliptic equations, Inverse Problems, 25 (2009), 123004, 47pp. doi: 10.1088/0266-5611/25/12/123004.
[6]	G. Alessandrini and S. Vessella, Lipschitz stability for the inverse conductivity problem, Advances in Applied Mathematics, 35 (2005), 207-241. doi: 10.1016/j.aam.2004.12.002.
[7]	B. Barceló, E. Fabes and J. Keun Seo, The inverse conductivity problem with one measurement: Uniqueness for convex polyhedra, Proc. Amer. Math. Soc., 122 (1994), 183-189. doi: 10.1090/S0002-9939-1994-1195476-6.
[8]	E. Beretta, M. V. de Hoop and L. Qiu, Lipschitz stability for an inverse boundary value problem for a Schrödinger-type equation, SIAM Journal on Mathematical Analysis, 45 (2013), 679-699. doi: 10.1137/120869201.
[9]	E. Beretta and E. Francini, Lipschitz stability for the electrical impedance tomography problem: The complex case, Comm. Partial Differential Equations, 36 (2011), 1723-1749. doi: 10.1080/03605302.2011.552930.
[10]	E. Beretta, E. Francini and S. Vessella, Lipschitz stability estimates for polygonal conductivity inclusions from boundary measurements, arXiv: 1901.01152.
[11]	E. Blåsten and H. Liu, Recovering piecewise constant refractive indices by a single far-field pattern, arXiv: 1705.00815, 2017.
[12]	X. Cao and H. Liu, Determining a fractional Helmholtz system with unknown source and medium parameter, arXiv preprint, arXiv: 1803.09538, 2018.
[13]	X. Cao, Y.-H. Lin and H. Liu, Simultaneously recovering potentials and embedded obstacles for anisotropic fractional Schrödinger operators, AIMS, 13 (2019), 197–210, arXiv: 1712.00937. doi: 10.3934/ipi.2019011.
[14]	S. Dipierro, O. Savin and E. Valdinoci, All functions are locally s-harmonic up to a small error, J. Eur. Math. Soc. (JEMS), 19 (2017), 957-966. doi: 10.4171/JEMS/684.
[15]	E. Fabes, H. Kang and J. K. Seo, Inverse conductivity problem with one measurement: Error estimates and approximate identification for perturbed disks, SIAM J. Math. Anal., 30 (1999), 699-720. doi: 10.1137/S0036141097324958.
[16]	M. Moustapha Fall and V. Felli, Unique continuation property and local asymptotics of solutions to fractional elliptic equations, Communication in Partial Differential Equations, 39 (2014), 354-397. doi: 10.1080/03605302.2013.825918.
[17]	A. Friedman and V. Isakov, On the uniqueness in the inverse conductivity problem with one measurement, Indiana Univ. Math J., 38 (1989), 563-579. doi: 10.1512/iumj.1989.38.38027.
[18]	R. Gaburro and E. Sincich, Lipschitz stability for the inverse conductivity problem for a conformal class of anisotropic conductivities, Inverse Problems, 31 (2015), 015008, 26pp. doi: 10.1088/0266-5611/31/1/015008.
[19]	B. Gebauer, Localized potentials in electrical impedance tomography, AIMS, 2 (2018), 251-369. doi: 10.3934/ipi.2008.2.251.
[20]	T. Ghosh, Y.-H. Lin and J. Xiao, The Calderón problem for variable coefficients nonlocal elliptic operators, Communication in Partial Differential Equations, 42 (2017), 1923-1961. doi: 10.1080/03605302.2017.1390681.
[21]	T. Ghosh, A. Rüland, M. Salo and G. Uhlmann, Uniqueness and reconstruction for the fractional Calderón problem with a single measurement, arXiv: 1801.04449, 2018.
[22]	T. Ghosh, M. Salo and G. Uhlmann, The Calderón problem for the fractional Schrödinger equation, to appear in Analysis and PDE.
[23]	G. Grubb, Fractional Laplacians on domains, a development of Hörmander's theory of $\mu$-transmission pseudodifferential operators, Advances in Mathematics, 268 (2015), 478-528. doi: 10.1016/j.aim.2014.09.018.
[24]	B. Harrach and Y.-H. Lin, Monotonicity-based inversion of the fractional Schrödinger equation, arXiv: 1711.05641, 2017.
[25]	B. Harrach, V. Pojola and M. Salo, Monotonicity and local uniqueness for the Helmholtz equation, arXiv: 1709.08756, 2017.
[26]	M. Ikehata, On reconstruction in the inverse conductivity problem with one measurement, Inverse Problems, 16 (2000), 785-793. doi: 10.1088/0266-5611/16/3/314.
[27]	M. Ikehata, On reconstruction from a partial knowledge of the Neumann-to-Dirichlet operator, Inverse Problems, 17 (2001), 45-51. doi: 10.1088/0266-5611/17/1/304.
[28]	M. Ikehata, Extraction formulae for an inverse boundary value problem for the equation $\nabla\cdot(\sigma-i\omega\epsilon)\nabla u = 0$, Inverse Problems, 18 (2002), 1281-1290. doi: 10.1088/0266-5611/18/5/304.
[29]	V. Isakov and J. Powell, On the inverse conductivity problem with one measurement, Inverse Problems, 6 (1990), 311-318. doi: 10.1088/0266-5611/6/2/011.
[30]	H. Liu and C.-H. Tsou, Stable determination of polygonal inclusions in Calderón's problem by a single partial boundary measurement, arXiv: 1902.04462. 2019.
[31]	N. Mandache, Exponential instability in an inverse problem for the Schrödinger equation, Inverse Problems, 17 (2001), 1435-1444. doi: 10.1088/0266-5611/17/5/313.
[32]	W. McLean, Strongly Elliptic Systems and Boundary Integral Equations,, Cambridge University Press, Cambridge, 2000.
[33]	L. Rondi, A remark on a paper by Alessandrini and Vessella, Advances in Applied Mathematics, 36 (2006), 67-69. doi: 10.1016/j.aam.2004.12.003.
[34]	R.-Y. Lai and Y.-H. Lin, Global uniqueness for the semilinear fractional Schrödinger equation, Proceedings of the American Mathematical Society, 147 (2019), 1189-1199.
[35]	A. Rüland, Unique continuation for fractional Schrödinger equations with rough potentials, Communications In Partial Differential Equations, 40 (2015), 77-114. doi: 10.1080/03605302.2014.905594.
[36]	A. Rüland and M. Salo, The fractional Calderón problem: Low regularity and stability, to appear in Nonlinear Analysis.
[37]	A. Rüland and M. Salo, Exponential instability in the fractional Calderón problem, Inverse Problems, 34 (2018), 045003, 21pp. doi: 10.1088/1361-6420/aaac5a.
[38]	M. Salo, The fractional Calderón problem, Journés Équations aux Dérivées Partielles, 7 (2017), 8pp, arXiv: 1711.06103. doi: 10.5802/jedp.657.
[39]	J. K. Seo, On the uniqueness in the inverse conductivity problem, J. Fourier Anal. Appl., 2 (1996), 227-235. doi: 10.1007/s00041-001-4030-7.
[40]	H. Yu, Unique continuation for fractional orders of elliptic equations, Annals of PDE (2017) 3: 16. https://doi.org/10.1007/s40818-017-0033-9

show all references

References:

[1]	G. S. Alberti and M. Santacesaria, Calderón's Inverse Problem with a Finite Number of Measurements, arXiv: 1803.04224, 2018.
[2]	G. Alessandrini, M. V. de Hoop, R. Gaburro and E. Sincich, Lipschitz stability for the electrostatic inverse boundary value problem with piecewise linear conductivities, Journal de Mathématiques Pures et Appliquées, 107 (2016), 638-664. doi: 10.1016/j.matpur.2016.10.001.
[3]	G. Alessandrini, M. V. de Hoop and R. Gaburro, Romina and Eva Sincich, Lipschitz stability for a piecewise linear Schrödinger potential from local Cauchy data, Asymptotic Analysis, 108 (2018), 115-149. doi: 10.3233/ASY-171457.
[4]	G. Alessandrini and V. Isakov, Analiticity and uniqueness for the inverse conductivity problem, Rend. Ist. Mat. Univ. Trieste, 28 (1996), 351-369.
[5]	G. Alessandrini, L. Rondi, E. Rosset and S. Vessella, The stability for the Cauchy problem for elliptic equations, Inverse Problems, 25 (2009), 123004, 47pp. doi: 10.1088/0266-5611/25/12/123004.
[6]	G. Alessandrini and S. Vessella, Lipschitz stability for the inverse conductivity problem, Advances in Applied Mathematics, 35 (2005), 207-241. doi: 10.1016/j.aam.2004.12.002.
[7]	B. Barceló, E. Fabes and J. Keun Seo, The inverse conductivity problem with one measurement: Uniqueness for convex polyhedra, Proc. Amer. Math. Soc., 122 (1994), 183-189. doi: 10.1090/S0002-9939-1994-1195476-6.
[8]	E. Beretta, M. V. de Hoop and L. Qiu, Lipschitz stability for an inverse boundary value problem for a Schrödinger-type equation, SIAM Journal on Mathematical Analysis, 45 (2013), 679-699. doi: 10.1137/120869201.
[9]	E. Beretta and E. Francini, Lipschitz stability for the electrical impedance tomography problem: The complex case, Comm. Partial Differential Equations, 36 (2011), 1723-1749. doi: 10.1080/03605302.2011.552930.
[10]	E. Beretta, E. Francini and S. Vessella, Lipschitz stability estimates for polygonal conductivity inclusions from boundary measurements, arXiv: 1901.01152.
[11]	E. Blåsten and H. Liu, Recovering piecewise constant refractive indices by a single far-field pattern, arXiv: 1705.00815, 2017.
[12]	X. Cao and H. Liu, Determining a fractional Helmholtz system with unknown source and medium parameter, arXiv preprint, arXiv: 1803.09538, 2018.
[13]	X. Cao, Y.-H. Lin and H. Liu, Simultaneously recovering potentials and embedded obstacles for anisotropic fractional Schrödinger operators, AIMS, 13 (2019), 197–210, arXiv: 1712.00937. doi: 10.3934/ipi.2019011.
[14]	S. Dipierro, O. Savin and E. Valdinoci, All functions are locally s-harmonic up to a small error, J. Eur. Math. Soc. (JEMS), 19 (2017), 957-966. doi: 10.4171/JEMS/684.
[15]	E. Fabes, H. Kang and J. K. Seo, Inverse conductivity problem with one measurement: Error estimates and approximate identification for perturbed disks, SIAM J. Math. Anal., 30 (1999), 699-720. doi: 10.1137/S0036141097324958.
[16]	M. Moustapha Fall and V. Felli, Unique continuation property and local asymptotics of solutions to fractional elliptic equations, Communication in Partial Differential Equations, 39 (2014), 354-397. doi: 10.1080/03605302.2013.825918.
[17]	A. Friedman and V. Isakov, On the uniqueness in the inverse conductivity problem with one measurement, Indiana Univ. Math J., 38 (1989), 563-579. doi: 10.1512/iumj.1989.38.38027.
[18]	R. Gaburro and E. Sincich, Lipschitz stability for the inverse conductivity problem for a conformal class of anisotropic conductivities, Inverse Problems, 31 (2015), 015008, 26pp. doi: 10.1088/0266-5611/31/1/015008.
[19]	B. Gebauer, Localized potentials in electrical impedance tomography, AIMS, 2 (2018), 251-369. doi: 10.3934/ipi.2008.2.251.
[20]	T. Ghosh, Y.-H. Lin and J. Xiao, The Calderón problem for variable coefficients nonlocal elliptic operators, Communication in Partial Differential Equations, 42 (2017), 1923-1961. doi: 10.1080/03605302.2017.1390681.
[21]	T. Ghosh, A. Rüland, M. Salo and G. Uhlmann, Uniqueness and reconstruction for the fractional Calderón problem with a single measurement, arXiv: 1801.04449, 2018.
[22]	T. Ghosh, M. Salo and G. Uhlmann, The Calderón problem for the fractional Schrödinger equation, to appear in Analysis and PDE.
[23]	G. Grubb, Fractional Laplacians on domains, a development of Hörmander's theory of $\mu$-transmission pseudodifferential operators, Advances in Mathematics, 268 (2015), 478-528. doi: 10.1016/j.aim.2014.09.018.
[24]	B. Harrach and Y.-H. Lin, Monotonicity-based inversion of the fractional Schrödinger equation, arXiv: 1711.05641, 2017.
[25]	B. Harrach, V. Pojola and M. Salo, Monotonicity and local uniqueness for the Helmholtz equation, arXiv: 1709.08756, 2017.
[26]	M. Ikehata, On reconstruction in the inverse conductivity problem with one measurement, Inverse Problems, 16 (2000), 785-793. doi: 10.1088/0266-5611/16/3/314.
[27]	M. Ikehata, On reconstruction from a partial knowledge of the Neumann-to-Dirichlet operator, Inverse Problems, 17 (2001), 45-51. doi: 10.1088/0266-5611/17/1/304.
[28]	M. Ikehata, Extraction formulae for an inverse boundary value problem for the equation $\nabla\cdot(\sigma-i\omega\epsilon)\nabla u = 0$, Inverse Problems, 18 (2002), 1281-1290. doi: 10.1088/0266-5611/18/5/304.
[29]	V. Isakov and J. Powell, On the inverse conductivity problem with one measurement, Inverse Problems, 6 (1990), 311-318. doi: 10.1088/0266-5611/6/2/011.
[30]	H. Liu and C.-H. Tsou, Stable determination of polygonal inclusions in Calderón's problem by a single partial boundary measurement, arXiv: 1902.04462. 2019.
[31]	N. Mandache, Exponential instability in an inverse problem for the Schrödinger equation, Inverse Problems, 17 (2001), 1435-1444. doi: 10.1088/0266-5611/17/5/313.
[32]	W. McLean, Strongly Elliptic Systems and Boundary Integral Equations,, Cambridge University Press, Cambridge, 2000.
[33]	L. Rondi, A remark on a paper by Alessandrini and Vessella, Advances in Applied Mathematics, 36 (2006), 67-69. doi: 10.1016/j.aam.2004.12.003.
[34]	R.-Y. Lai and Y.-H. Lin, Global uniqueness for the semilinear fractional Schrödinger equation, Proceedings of the American Mathematical Society, 147 (2019), 1189-1199.
[35]	A. Rüland, Unique continuation for fractional Schrödinger equations with rough potentials, Communications In Partial Differential Equations, 40 (2015), 77-114. doi: 10.1080/03605302.2014.905594.
[36]	A. Rüland and M. Salo, The fractional Calderón problem: Low regularity and stability, to appear in Nonlinear Analysis.
[37]	A. Rüland and M. Salo, Exponential instability in the fractional Calderón problem, Inverse Problems, 34 (2018), 045003, 21pp. doi: 10.1088/1361-6420/aaac5a.
[38]	M. Salo, The fractional Calderón problem, Journés Équations aux Dérivées Partielles, 7 (2017), 8pp, arXiv: 1711.06103. doi: 10.5802/jedp.657.
[39]	J. K. Seo, On the uniqueness in the inverse conductivity problem, J. Fourier Anal. Appl., 2 (1996), 227-235. doi: 10.1007/s00041-001-4030-7.
[40]	H. Yu, Unique continuation for fractional orders of elliptic equations, Annals of PDE (2017) 3: 16. https://doi.org/10.1007/s40818-017-0033-9

[1]	María Ángeles García-Ferrero, Angkana Rüland, Wiktoria Zatoń. Runge approximation and stability improvement for a partial data Calderón problem for the acoustic Helmholtz equation. Inverse Problems and Imaging, 2022, 16 (1) : 251-281. doi: 10.3934/ipi.2021049
[2]	Pedro Caro, Mikko Salo. Stability of the Calderón problem in admissible geometries. Inverse Problems and Imaging, 2014, 8 (4) : 939-957. doi: 10.3934/ipi.2014.8.939
[3]	Yernat M. Assylbekov. Reconstruction in the partial data Calderón problem on admissible manifolds. Inverse Problems and Imaging, 2017, 11 (3) : 455-476. doi: 10.3934/ipi.2017021
[4]	Kim Knudsen, Aksel Kaastrup Rasmussen. Direct regularized reconstruction for the three-dimensional Calderón problem. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022002
[5]	Albert Clop, Daniel Faraco, Alberto Ruiz. Stability of Calderón's inverse conductivity problem in the plane for discontinuous conductivities. Inverse Problems and Imaging, 2010, 4 (1) : 49-91. doi: 10.3934/ipi.2010.4.49
[6]	Petteri Piiroinen, Martin Simon. Probabilistic interpretation of the Calderón problem. Inverse Problems and Imaging, 2017, 11 (3) : 553-575. doi: 10.3934/ipi.2017026
[7]	Javier A. Almonacid, Gabriel N. Gatica, Ricardo Oyarzúa, Ricardo Ruiz-Baier. A new mixed finite element method for the n-dimensional Boussinesq problem with temperature-dependent viscosity. Networks and Heterogeneous Media, 2020, 15 (2) : 215-245. doi: 10.3934/nhm.2020010
[8]	Li Liang. Increasing stability for the inverse problem of the Schrödinger equation with the partial Cauchy data. Inverse Problems and Imaging, 2015, 9 (2) : 469-478. doi: 10.3934/ipi.2015.9.469
[9]	Jinye Shen, Xian-Ming Gu. Two finite difference methods based on an H2N2 interpolation for two-dimensional time fractional mixed diffusion and diffusion-wave equations. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 1179-1207. doi: 10.3934/dcdsb.2021086
[10]	Michael L. Frankel, Victor Roytburd. A Finite-dimensional attractor for a nonequilibrium Stefan problem with heat losses. Discrete and Continuous Dynamical Systems, 2005, 13 (1) : 35-62. doi: 10.3934/dcds.2005.13.35
[11]	Matteo Santacesaria. Note on Calderón's inverse problem for measurable conductivities. Inverse Problems and Imaging, 2019, 13 (1) : 149-157. doi: 10.3934/ipi.2019008
[12]	Oleg Yu. Imanuvilov, Masahiro Yamamoto. Calderón problem for Maxwell's equations in cylindrical domain. Inverse Problems and Imaging, 2014, 8 (4) : 1117-1137. doi: 10.3934/ipi.2014.8.1117
[13]	Henrik Garde, Nuutti Hyvönen. Reconstruction of singular and degenerate inclusions in Calderón's problem. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022021
[14]	Messoud Efendiev, Alain Miranville. Finite dimensional attractors for reaction-diffusion equations in $R^n$ with a strong nonlinearity. Discrete and Continuous Dynamical Systems, 1999, 5 (2) : 399-424. doi: 10.3934/dcds.1999.5.399
[15]	Shuyang Dai, Fengru Wang, Jerry Zhijian Yang, Cheng Yuan. On the Cauchy-Born approximation at finite temperature for alloys. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3131-3153. doi: 10.3934/dcdsb.2021176
[16]	Zhan-Dong Mei, Jigen Peng, Yang Zhang. On general fractional abstract Cauchy problem. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2753-2772. doi: 10.3934/cpaa.2013.12.2753
[17]	Eddye Bustamante, José Jiménez Urrea, Jorge Mejía. The Cauchy problem for a family of two-dimensional fractional Benjamin-Ono equations. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1177-1203. doi: 10.3934/cpaa.2019057
[18]	Brahim Alouini. Finite dimensional global attractor for a damped fractional anisotropic Schrödinger type equation with harmonic potential. Communications on Pure and Applied Analysis, 2020, 19 (9) : 4545-4573. doi: 10.3934/cpaa.2020206
[19]	Brahim Alouini. Finite dimensional global attractor for a class of two-coupled nonlinear fractional Schrödinger equations. Evolution Equations and Control Theory, 2022, 11 (2) : 559-581. doi: 10.3934/eect.2021013
[20]	Xin Zhong. Global well-posedness to the cauchy problem of two-dimensional density-dependent boussinesq equations with large initial data and vacuum. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6713-6745. doi: 10.3934/dcds.2019292

2020 Impact Factor: 1.639

Tools

Metrics

Other articles
by authors

on AIMS
Angkana Rüland Eva Sincich
on Google Scholar
Angkana Rüland Eva Sincich

2020 Impact Factor: 1.639

Tools

Article outline

2020 Impact Factor: 1.639

Tools

Metrics

Other articles
by authors

on AIMS
Angkana Rüland Eva Sincich
on Google Scholar
Angkana Rüland Eva Sincich

[Back to Top]