Monotonicity-based Inversion of the Fractional Schrödinger Equation I. Positive Potentials

We consider the inverse problems of for the fractional Schr\"odinger equation by using monotonicity formulas. We provide if-and-only-if monotonicity relations between positive bounded potentials and their associated nonlocal Dirichlet-to-Neumann maps. Based on the monotonicity relation, we can prove uniqueness for the nonlocal Calder\'on problem in a constructive manner. Secondly, we offer a reconstruction method for an unknown obstacles in a given domain. Our method is independent of the dimension $n\geq 2$ and only requires the background solution of the fractional Schr\"odinger equation.

[1]  Michael V. Klibanov,et al.  Stability estimates for ill-posed Cauchy problems involving hyperbolic equations and inequalities , 1993 .

[2]  Antonello Tamburrino,et al.  A new non-iterative inversion method for electrical resistance tomography , 2002 .

[3]  Elena Beretta,et al.  Lipschitz Stability for the Electrical Impedance Tomography Problem: The Complex Case , 2010, 1008.4046.

[4]  Yi-Hsuan Lin,et al.  Simultaneously recovering potentials and embedded obstacles for anisotropic fractional Schrödinger operators , 2017, Inverse Problems & Imaging.

[5]  Y. Y. Belov,et al.  Inverse Problems for Partial Differential Equations , 2002 .

[6]  Sergio Vessella,et al.  Lipschitz stability for the inverse conductivity problem , 2005, Adv. Appl. Math..

[7]  Gunther Uhlmann,et al.  The Calderón problem for the fractional Schrödinger equation , 2016, 1609.09248.

[8]  Mourad Sini,et al.  On the reconstruction of interfaces using complex geometrical optics solutions for the acoustic case , 2012 .

[9]  Maarten V. de Hoop,et al.  Lipschitz stability for a piecewise linear Schrödinger potential from local Cauchy data , 2017, Asymptot. Anal..

[10]  Lalita Udpa,et al.  Monotonicity Based Imaging Method in Time Domain Eddy Current Testing , 2016 .

[11]  Gen Nakamura,et al.  Inverse Modeling An introduction to the theory and methods of inverse problems and data assimilation , 2015 .

[12]  Masahiro Yamamoto,et al.  Lipschitz stability in inverse problems for a Kirchhoff plate equation , 2007, Asymptot. Anal..

[13]  Bastian von Harrach,et al.  Monotonicity and Enclosure Methods for the p-Laplace Equation , 2017, SIAM J. Appl. Math..

[14]  Gunther Uhlmann,et al.  Inverse problems for the anisotropic Maxwell equations , 2009, 0905.3275.

[15]  Gen Nakamura,et al.  Oscillating-Decaying Solutions for Elliptic Systems , 2006 .

[16]  Jin Keun Seo,et al.  The inverse conductivity problem with one measurement: stability and estimation of size , 1997 .

[17]  Sergio Vessella,et al.  Lipschitz stability for a stationary 2D inverse problem with unknown polygonal boundary , 2006 .

[18]  B. Harrach,et al.  Combining frequency-difference and ultrasound modulated electrical impedance tomography , 2015, 1810.04392.

[19]  L. Beilina,et al.  Lipschitz stability for an inverse hyperbolic problem of determining two coefficients by a finite number of observations , 2017, 1707.03785.

[20]  Antonello Tamburrino,et al.  A Novel Technique for Evaluating the Effective Permittivity of Inhomogeneous Interconnects Based on the Monotonicity Property , 2016, IEEE Transactions on Components, Packaging and Manufacturing Technology.

[21]  Angkana Ruland,et al.  Lipschitz stability for the finite dimensional fractional Calderón problem with finite Cauchy data , 2018, Inverse Problems & Imaging.

[22]  B. Harrach.  On uniqueness in diffuse optical tomography , 2009 .

[23]  E. Valdinoci,et al.  Hitchhiker's guide to the fractional Sobolev spaces , 2011, 1104.4345.

[24]  Victor Isakov,et al.  On uniqueness of recovery of a discontinuous conductivity coefficient , 1988 .

[25]  Bastian Harrach,et al.  Monotonicity and local uniqueness for the Helmholtz equation , 2017 .

[26]  Bastian Harrach,et al.  Enhancing residual-based techniques with shape reconstruction features in electrical impedance tomography , 2015, 1511.07079.

[27]  Bastian Harrach,et al.  Local uniqueness for an inverse boundary value problem with partial data , 2016, 1810.05834.

[28]  Tommi Brander,et al.  Enclosure method for the p-Laplace equation , 2014, Inverse Problems.

[29]  Gunther Uhlmann,et al.  Uniqueness and reconstruction for the fractional Calderón problem with a single measurement , 2018, Journal of Functional Analysis.

[30]  Bastian von Harrach,et al.  Resolution Guarantees in Electrical Impedance Tomography , 2015, IEEE Transactions on Medical Imaging.

[31]  Masahiro Yamamoto,et al.  Global Lipschitz stability in an inverse hyperbolic problem by interior observations , 2001 .

[32]  Henrik Garde,et al.  The regularized monotonicity method: Detecting irregular indefinite inclusions , 2017, Inverse Problems & Imaging.

[33]  Mikko Salo,et al.  Complex spherical waves and inverse problems in unbounded domains , 2006 .

[34]  Michael V. Klibanov,et al.  Lipschitz stability of a non-standard problem for the non-stationary transport equation via a Carleman estimate , 2006 .

[35]  Xinlin Cao,et al.  Determining a fractional Helmholtz system with unknown source and medium parameter , 2018 .

[36]  Tsuyoshi Murata,et al.  {m , 1934, ACML.

[37]  Bastian Harrach,et al.  Monotonicity in Inverse Medium Scattering on Unbounded Domains , 2018, SIAM J. Appl. Math..

[38]  Enrico Valdinoci,et al.  Local and global minimizers for a variational energy involving a fractional norm , 2011, 1104.1725.

[39]  Antonello Tamburrino,et al.  Design of a Real-Time Eddy Current Tomography System , 2017, IEEE Transactions on Magnetics.

[40]  P. Cochat,et al.  Et al , 2008, Archives de pediatrie : organe officiel de la Societe francaise de pediatrie.

[41]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[42]  Yi-Hsuan Lin.  Reconstruction of penetrable obstacles in the anisotropic acoustic scattering , 2014, 1402.3574.

[43]  Victor Isakov,et al.  Lipschitz stability in the lateral Cauchy problem for elasticity system , 2003 .

[44]  Xavier Ros-Oton,et al.  Nonlocal elliptic equations in bounded domains: a survey , 2015, 1504.04099.

[45]  Gen Nakamura,et al.  Oscillating-decaying solutions, Runge approximation property for the anisotropic elasticity system and their applications to inverse problems , 2005 .

[46]  L. Evans.  Measure theory and fine properties of functions , 1992 .

[47]  Bastian Harrach,et al.  Dimension Bounds in Monotonicity Methods for the Helmholtz Equation , 2019, SIAM J. Math. Anal..

[48]  Hyunjoong Kim,et al.  Functional Analysis I , 2017 .

[49]  Bastian Harrach,et al.  Uniqueness and Lipschitz stability in electrical impedance tomography with finitely many electrodes , 2018, Inverse Problems.

[50]  Giovanni Covi.  Inverse problems for a fractional conductivity equation , 2018, Nonlinear Analysis.

[51]  Yi-Hsuan Lin,et al.  The Calderón problem for the fractional Schrödinger equation with drift , 2020, Calculus of Variations and Partial Differential Equations.

[52]  Bastian Harrach,et al.  Monotonicity-based regularization for phantom experiment data in Electrical Impedance Tomography , 2016, 1610.05718.

[53]  Otmar Scherzer,et al.  Detecting Interfaces in a Parabolic-Elliptic Problem from Surface Measurements , 2007, SIAM J. Numer. Anal..

[54]  Andrea Barth,et al.  Detecting stochastic inclusions in electrical impedance tomography , 2017, 1706.03962.

[55]  Robert V. Kohn,et al.  Determining conductivity by boundary measurements , 1984 .

[56]  Maarten V. de Hoop,et al.  Uniqueness and Lipschitz stability of an inverse boundary value problem for time-harmonic elastic waves , 2014, 1412.3465.

[57]  Bastian von Harrach,et al.  Monotonicity-Based Shape Reconstruction in Electrical Impedance Tomography , 2013, SIAM J. Math. Anal..

[58]  Masahiro Yamamoto,et al.  Lipschitz stability in determining density and two Lame coefficients , 2007 .

[59]  Eva Sincich,et al.  Lipschitz stability for the inverse Robin problem , 2007 .

[60]  Mourad Sini,et al.  Reconstruction of interfaces using CGO solutions for the Maxwell equations , 2013 .

[61]  Bastian Harrach,et al.  Simultaneous determination of the diffusion and absorption coefficient from boundary data , 2012 .

[62]  Henrik Garde,et al.  Convergence and regularization for monotonicity-based shape reconstruction in electrical impedance tomography , 2015, Numerische Mathematik.

[63]  Bastian Gebauer,et al.  Localized potentials in electrical impedance tomography , 2008 .

[64]  Michael V. Klibanov,et al.  Lipschitz stability of an inverse problem for an acoustic equation , 2006 .

[65]  Masaru Ikehata,et al.  Size estimation of inclusion , 1998 .

[66]  Lalita Udpa,et al.  Monotonicity principle in pulsed eddy current testing and its application to defect sizing , 2017, 2017 International Applied Computational Electromagnetics Society Symposium - Italy (ACES).

[67]  M. Kwasnicki,et al.  Ten equivalent definitions of the fractional laplace operator , 2015, 1507.07356.

[68]  Maarten V. de Hoop,et al.  Lipschitz stability for the electrostatic inverse boundary value problem with piecewise linear conductivities , 2015, 1509.06277.

[69]  Patricio Guzmán Meléndez.  Lipschitz stability in an inverse problem for the main coefficient of a Kuramoto–Sivashinsky type equation , 2013 .

[70]  Henrik Garde,et al.  Comparison of linear and non-linear monotonicity-based shape reconstruction using exact matrix characterizations , 2016, 1602.04053.

[71]  Giovanni Alessandrini,et al.  Singular solutions of elliptic equations and the determination of conductivity by boundary measurements , 1990 .

[72]  Bastian Harrach,et al.  Global Uniqueness and Lipschitz-Stability for the Inverse Robin Transmission Problem , 2018, SIAM J. Appl. Math..

[73]  Masahiro Yamamoto,et al.  Lipschitz stability in inverse parabolic problems by the Carleman estimate , 1998 .

[74]  Jin Keun Seo,et al.  Monotonicity-based electrical impedance tomography for lung imaging , 2017, 1702.02563.

[75]  Elena Beretta,et al.  Determining linear cracks by boundary measurements: Lipschitz stability , 1996 .

[76]  Mikko Salo,et al.  The fractional Calderón problem: Low regularity and stability , 2017, Nonlinear Analysis.

[77]  Yi-Hsuan Lin,et al.  The Enclosure Method for the Anisotropic Maxwell System , 2015, SIAM J. Math. Anal..

[78]  Maarten V. de Hoop,et al.  Lipschitz Stability of an Inverse Boundary Value Problem for a Schrödinger-Type Equation , 2012, SIAM J. Math. Anal..

[79]  Hongyu Liu,et al.  On Localizing and Concentrating Electromagnetic Fields , 2018, SIAM J. Appl. Math..

[80]  Gen Nakamura,et al.  Identification of a non-convex obstacle for acoustical scattering , 2007 .

[81]  Bastian von Harrach,et al.  Recent Progress on the Factorization Method for Electrical Impedance Tomography , 2013, Comput. Math. Methods Medicine.

[82]  V. Isakov.  Appendix -- Function Spaces , 2017 .

[83]  J. Cooper.  SINGULAR INTEGRALS AND DIFFERENTIABILITY PROPERTIES OF FUNCTIONS , 1973 .

[84]  Identification of the shape of the inclusion having essentially bounded conductivity , 1999 .

[85]  E. Valdinoci,et al.  Nonlocal Diffusion and Applications , 2015, 1504.08292.

[86]  Yi-Hsuan Lin,et al.  The Calderón problem for variable coefficients nonlocal elliptic operators , 2017, 1708.00654.

[87]  Mikko Salo,et al.  Exponential instability in the fractional Calderón problem , 2017, 1711.04799.

[88]  R. Kohn,et al.  Determining conductivity by boundary measurements II. Interior results , 1985 .

[89]  Jin Keun Seo,et al.  A Learning-Based Method for Solving Ill-Posed Nonlinear Inverse Problems: A Simulation Study of Lung EIT , 2018, SIAM J. Imaging Sci..

[90]  Kellen Petersen August.  Real Analysis , 2009 .

[91]  Bastian Harrach,et al.  Unique shape detection in transient eddy current problems , 2013 .

[92]  Jenn-Nan Wang,et al.  Reconstructing Discontinuities Using Complex Geometrical Optics Solutions , 2008, SIAM J. Appl. Math..

[93]  Yi-Hsuan Lin,et al.  Monotonicity-based inversion of the fractional Schr\"odinger equation , 2017, 1903.08771.

[94]  Masahiro Yamamoto,et al.  Lipschitz stability for a hyperbolic inverse problem by finite local boundary data , 2006 .

[95]  Yi-Hsuan Lin,et al.  Global uniqueness for the fractional semilinear Schrödinger equation , 2017, Proceedings of the American Mathematical Society.

[96]  Masahiro Yamamoto,et al.  Lipschitz stability in the determination of the principal part of a parabolic equation , 2009 .

[97]  Bastian Harrach,et al.  Lipschitz stability estimate and reconstruction of Lamé parameters in linear elasticity , 2019, Inverse Problems in Science and Engineering.

[98]  Jin Keun Seo,et al.  Exact Shape-Reconstruction by One-Step Linearization in Electrical Impedance Tomography , 2010, SIAM J. Math. Anal..

[99]  Mikko Salo,et al.  The fractional Calderón problem , 2017 .