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doi: 10.3934/ipi.2022017
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Refined instability estimates for some inverse problems

1. 

Department of Mathematics and Statistics, P.O. Box 35 (MaD), FI-40014 University of Jyväskylä, Finland

2. 

Institute of Applied Mathematical Sciences, National Taiwan University, Taipei 106, Taiwan

*Corresponding author: Jenn-Nan Wang

Received  November 2021 Revised  March 2022 Early access April 2022

Many inverse problems are known to be ill-posed. The ill-posedness can be manifested by an instability estimate of exponential type, first derived by Mandache [29]. In this work, based on Mandache's idea, we refine the instability estimates for two inverse problems, including the inverse inclusion problem and the inverse scattering problem. Our aim is to derive explicitly the dependence of the instability estimates on key parameters.

The first result of this work is to show how the instability depends on the depth of the hidden inclusion and the conductivity of the background medium. This work can be regarded as a counterpart of the depth-dependent and conductivity-dependent stability estimate proved by Li, Wang, and Wang [28], or pure dependent stability estimate proved by Nagayasu, Uhlmann, and Wang [31]. We rigorously justify the intuition that the exponential instability becomes worse as the inclusion is hidden deeper inside a conductor or the conductivity is larger.

The second result is to justify the optimality of increasing stability in determining the near-field of a radiating solution of the Helmholtz equation from the far-field pattern. Isakov [16] showed that the stability of this inverse problem increases as the frequency increases in the sense that the stability estimate changes from a logarithmic type to a Hölder type. We prove in this work that the instability changes from an exponential type to a Hölder type as the frequency increases. This result is inspired by our recent work [25].

Citation: Pu-Zhao Kow, Jenn-Nan Wang. Refined instability estimates for some inverse problems. Inverse Problems and Imaging, doi: 10.3934/ipi.2022017
References:
[1]

G. Alessandrini, Stable determination of conductivity by boundary measurements, Appl. Anal., 27 (1988), 153-172.  doi: 10.1080/00036818808839730.

[2]

G. Alessandrini and M. Di Cristo, Stable determination of an inclusion by boundary measurements, SIAM J. Math. Anal., 37 (2005), 200-217.  doi: 10.1137/S003614100444191X.

[3]

G. Alessandrini and A. Scapin, Depth dependent resolution in electrical impedance tomography, J. Inverse Ill-Posed Probl., 25 (2017), 391-402.  doi: 10.1515/jiip-2017-0029.

[4]

J. H. Bramble and J. E. Pasciak, A note on the existence and uniqueness of solutions of frequency domain elastic wave problems: A priori estimates in $H^{1}$, J. Math. Anal. Appl., 345 (2008), 396-404.  doi: 10.1016/j.jmaa.2008.04.028.

[5]

A. P. Calderón, On an inverse boundary value problem, Comput. Appl. Math., 25 (2006), 133-138.  doi: 10.1590/S0101-82052006000200002.

[6]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Applied Mathematical Sciences, 93. Springer, Cham, 2019. doi: 10.1007/978-3-030-30351-8.

[7]

M. Di Cristo and L. Rondi, Examples of exponential instability for elliptic inverse problems, preprint, (2003), arXiv: math/0303126.

[8]

M. Di Cristo and L. Rondi, Examples of exponential instability for inverse inclusion and scattering problems, Inverse Problems, 19 (2003), 685-701.  doi: 10.1088/0266-5611/19/3/313.

[9]

H. Garde and N. Hyvönen, Optimal depth-dependent distinguishability bounds for electrical impedance tomography in arbitrary dimension, SIAM J. Appl. Math., 80 (2020), 20-43.  doi: 10.1137/19M1258761.

[10]

H. Garde and K. Knudsen, Distinguishability revisited: Depth dependent bounds on reconstruction quality in electrical impedance tomography, SIAM J. Appl. Math., 77 (2017), 697-720.  doi: 10.1137/16M1072991.

[11]

T. Hrycak and V. Isakov, Increased stability in the continuation of solutions to the Helmholtz equation, Inverse Problems, 20 (2004), 697-712.  doi: 10.1088/0266-5611/20/3/004.

[12]

T. IdeH. IsozakiS. NakataS. Siltanen and G. Uhlmann, Probing for electrical inclusions with complex spherical waves, Comm. Pure Appl. Math., 60 (2007), 1415-1442.  doi: 10.1002/cpa.20194.

[13]

V. Isakov, On uniqueness of recovery of a discontinuous conductivity coefficient, Comm. Pure Appl. Math., 41 (1988), 865-877.  doi: 10.1002/cpa.3160410702.

[14]

V. Isakov, Increased stability in the continuation for the Helmholtz equation with variable coefficient, Control Methods in PDE-Dynamical Systems, Contemp. Math., Amer. Math. Soc., Providence, RI, 426 (2007), 255-267.  doi: 10.1090/conm/426/08192.

[15]

V. Isakov, Increasing stability for the Schrödinger potential from the Dirichlet-to-Neumann map, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 631-640.  doi: 10.3934/dcdss.2011.4.631.

[16]

V. Isakov, Increasing stability for near field from the scattering amplitude, Spectral Theory and Partial Differential Equations, Contemp. Math., Amer. Math. Soc., Providence, RI, 640 (2015), 59-70.  doi: 10.1090/conm/640/12839.

[17]

V. Isakov and S. Kindermann, Subspaces of stability in the Cauchy problem for the Helmholtz equation, Methods Appl. Anal., 18 (2011), 1-29.  doi: 10.4310/MAA.2011.v18.n1.a1.

[18]

V. IsakovS. Lu and B. Xu, Linearized inverse Schrödinger potential problem at a large wavenumber, SIAM J. Appl. Math., 80 (2020), 338-358.  doi: 10.1137/18M1226932.

[19]

V. IsakovS. NagayasuG. Uhlmann and J.-N. Wang, Increasing stability of the inverse boundary value problem for the Schrödinger equation, Inverse Problems and Applications, Contemp. Math., Amer. Math. Soc., Providence, RI, 615 (2014), 131-141.  doi: 10.1090/conm/615/12268.

[20]

E. Jahnke, F. Emde and F. Lösch, Tables of Higher Functions, 6th edition, McGraw-Hill Book Co., Inc., New York-Toronto-London; B. G. Teubner Verlagsgesellschaft, Stuttgart, 1960.

[21]

F. John, Continuous dependence on data for solutions of partial differential equations with a prescribed bound, Comm. Pure Appl. Math., 13 (1960), 551-585.  doi: 10.1002/cpa.3160130402.

[22] A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford Lecture Series in Mathematics and its Applications, 36. Oxford University Press, Oxford, 2008. 
[23]

H. Koch, A. Rüland and M. Salo, On instability mechanisms for inverse problems, Ars Inveniendi Analytica, Paper No. 7, 93 pp. doi: 10.15781/c93s-pk62.

[24]

A. N. Kolmogorov and V. M. Tikhomirov, $\varepsilon$-entropy and $\varepsilon$-capacity in functional spaces, Amer. Math. Soc. Transl., 17 (1961), 277-364.  doi: 10.1090/trans2/017/10.

[25]

P.-Z. KowG. Uhlmann and J.-N. Wang, Optimality of increasing stability for an inverse boundary value problem, SIAM J. Math. Anal., 53 (2021), 7062-7080.  doi: 10.1137/21M1402169.

[26]

K. Krupchyk and G. Uhlmann, Stability estimates for partial data inverse problems for Schrödinger operators in the high frequency limit, J. Math. Pures Appl., 126 (2019), 273-291.  doi: 10.1016/j.matpur.2019.02.017.

[27]

R.-Y. LaiQ. Li and G. Uhlmann, Inverse problems for the stationary transport equation in the diffusion scaling, SIAM J. Appl. Math., 79 (2019), 2340-2358.  doi: 10.1137/18M1207582.

[28]

H. LiJ.-N. Wang and L. Wang, Refined stability estimates in electrical impedance tomography with multi-layer structure, Inverse Probl. Imaging, 16 (2022), 229-249.  doi: 10.3934/ipi.2021048.

[29]

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.

[30]

A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Ann. of Math., 143 (1996), 71-96.  doi: 10.2307/2118653.

[31]

S. Nagayasu, G. Uhlmann and J.-N. Wang, A depth-dependent stability estimate in electrical impedance tomography, Inverse Problems, 25 (2009), 075001, 14 pp. doi: 10.1088/0266-5611/25/7/075001.

[32]

S. Nagayasu, G. Uhlmann and J.-N. Wang, Increasing stability in an inverse problem for the acoustic equation, Inverse Problems, 29 (2013), 025012, 11 pp. doi: 10.1088/0266-5611/29/2/025012.

[33]

J.-C. Nédélec, Acoustic and Electromagnetic Equations. Integral Representations for Harmonic Problems, Applied Mathematical Sciences, 144. Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-4393-7.

[34]

L. Rondi and M. Sini, Stable determination of a scattered wave from its far-field pattern: The high frequency asymptotics, Arch. Ration. Mech. Anal., 218 (2015), 1-54.  doi: 10.1007/s00205-015-0855-0.

[35]

D. A. Subbarayappa and V. Isakov, On increased stability in the continuation of the Helmholtz equation, Inverse Problems, 23 (2007), 1689-1697.  doi: 10.1088/0266-5611/23/4/019.

[36]

D. A. Subbarayappa and V. Isakov, Increasing stability of the continuation for the Maxwell system, Inverse Problems, 26 (2010), 074005, 14 pp. doi: 10.1088/0266-5611/26/7/074005.

[37]

J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153-169.  doi: 10.2307/1971291.

[38]

M. E. Taylor, Partial Differential Equations. II. Qualitative Studies of Linear Equations, Applied Mathematical Sciences, 116. Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4757-4187-2.

[39]

G. Uhlmann and J.-N. Wang, Reconstructing discontinuities using complex geometrical optics solutions, SIAM J. Appl. Math., 68 (2008), 1026-1044.  doi: 10.1137/060676350.

[40]

G. UhlmannJ.-N. Wang and C.-T. Wu, Reconstruction of inclusions in an elastic body, J. Math. Pures Appl., 91 (2009), 569-582.  doi: 10.1016/j.matpur.2009.01.006.

[41] G. N. Watson, A Treatise on the Theory of Bessel Functions, 2 edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995. 
[42]

H. Zhao and Y. Zhong, Instability of an inverse problem for the stationary radiative transport near the diffusion limit, SIAM J. Math. Anal., 51 (2019), 3750-3768.  doi: 10.1137/18M1222582.

show all references

References:
[1]

G. Alessandrini, Stable determination of conductivity by boundary measurements, Appl. Anal., 27 (1988), 153-172.  doi: 10.1080/00036818808839730.

[2]

G. Alessandrini and M. Di Cristo, Stable determination of an inclusion by boundary measurements, SIAM J. Math. Anal., 37 (2005), 200-217.  doi: 10.1137/S003614100444191X.

[3]

G. Alessandrini and A. Scapin, Depth dependent resolution in electrical impedance tomography, J. Inverse Ill-Posed Probl., 25 (2017), 391-402.  doi: 10.1515/jiip-2017-0029.

[4]

J. H. Bramble and J. E. Pasciak, A note on the existence and uniqueness of solutions of frequency domain elastic wave problems: A priori estimates in $H^{1}$, J. Math. Anal. Appl., 345 (2008), 396-404.  doi: 10.1016/j.jmaa.2008.04.028.

[5]

A. P. Calderón, On an inverse boundary value problem, Comput. Appl. Math., 25 (2006), 133-138.  doi: 10.1590/S0101-82052006000200002.

[6]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Applied Mathematical Sciences, 93. Springer, Cham, 2019. doi: 10.1007/978-3-030-30351-8.

[7]

M. Di Cristo and L. Rondi, Examples of exponential instability for elliptic inverse problems, preprint, (2003), arXiv: math/0303126.

[8]

M. Di Cristo and L. Rondi, Examples of exponential instability for inverse inclusion and scattering problems, Inverse Problems, 19 (2003), 685-701.  doi: 10.1088/0266-5611/19/3/313.

[9]

H. Garde and N. Hyvönen, Optimal depth-dependent distinguishability bounds for electrical impedance tomography in arbitrary dimension, SIAM J. Appl. Math., 80 (2020), 20-43.  doi: 10.1137/19M1258761.

[10]

H. Garde and K. Knudsen, Distinguishability revisited: Depth dependent bounds on reconstruction quality in electrical impedance tomography, SIAM J. Appl. Math., 77 (2017), 697-720.  doi: 10.1137/16M1072991.

[11]

T. Hrycak and V. Isakov, Increased stability in the continuation of solutions to the Helmholtz equation, Inverse Problems, 20 (2004), 697-712.  doi: 10.1088/0266-5611/20/3/004.

[12]

T. IdeH. IsozakiS. NakataS. Siltanen and G. Uhlmann, Probing for electrical inclusions with complex spherical waves, Comm. Pure Appl. Math., 60 (2007), 1415-1442.  doi: 10.1002/cpa.20194.

[13]

V. Isakov, On uniqueness of recovery of a discontinuous conductivity coefficient, Comm. Pure Appl. Math., 41 (1988), 865-877.  doi: 10.1002/cpa.3160410702.

[14]

V. Isakov, Increased stability in the continuation for the Helmholtz equation with variable coefficient, Control Methods in PDE-Dynamical Systems, Contemp. Math., Amer. Math. Soc., Providence, RI, 426 (2007), 255-267.  doi: 10.1090/conm/426/08192.

[15]

V. Isakov, Increasing stability for the Schrödinger potential from the Dirichlet-to-Neumann map, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 631-640.  doi: 10.3934/dcdss.2011.4.631.

[16]

V. Isakov, Increasing stability for near field from the scattering amplitude, Spectral Theory and Partial Differential Equations, Contemp. Math., Amer. Math. Soc., Providence, RI, 640 (2015), 59-70.  doi: 10.1090/conm/640/12839.

[17]

V. Isakov and S. Kindermann, Subspaces of stability in the Cauchy problem for the Helmholtz equation, Methods Appl. Anal., 18 (2011), 1-29.  doi: 10.4310/MAA.2011.v18.n1.a1.

[18]

V. IsakovS. Lu and B. Xu, Linearized inverse Schrödinger potential problem at a large wavenumber, SIAM J. Appl. Math., 80 (2020), 338-358.  doi: 10.1137/18M1226932.

[19]

V. IsakovS. NagayasuG. Uhlmann and J.-N. Wang, Increasing stability of the inverse boundary value problem for the Schrödinger equation, Inverse Problems and Applications, Contemp. Math., Amer. Math. Soc., Providence, RI, 615 (2014), 131-141.  doi: 10.1090/conm/615/12268.

[20]

E. Jahnke, F. Emde and F. Lösch, Tables of Higher Functions, 6th edition, McGraw-Hill Book Co., Inc., New York-Toronto-London; B. G. Teubner Verlagsgesellschaft, Stuttgart, 1960.

[21]

F. John, Continuous dependence on data for solutions of partial differential equations with a prescribed bound, Comm. Pure Appl. Math., 13 (1960), 551-585.  doi: 10.1002/cpa.3160130402.

[22] A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford Lecture Series in Mathematics and its Applications, 36. Oxford University Press, Oxford, 2008. 
[23]

H. Koch, A. Rüland and M. Salo, On instability mechanisms for inverse problems, Ars Inveniendi Analytica, Paper No. 7, 93 pp. doi: 10.15781/c93s-pk62.

[24]

A. N. Kolmogorov and V. M. Tikhomirov, $\varepsilon$-entropy and $\varepsilon$-capacity in functional spaces, Amer. Math. Soc. Transl., 17 (1961), 277-364.  doi: 10.1090/trans2/017/10.

[25]

P.-Z. KowG. Uhlmann and J.-N. Wang, Optimality of increasing stability for an inverse boundary value problem, SIAM J. Math. Anal., 53 (2021), 7062-7080.  doi: 10.1137/21M1402169.

[26]

K. Krupchyk and G. Uhlmann, Stability estimates for partial data inverse problems for Schrödinger operators in the high frequency limit, J. Math. Pures Appl., 126 (2019), 273-291.  doi: 10.1016/j.matpur.2019.02.017.

[27]

R.-Y. LaiQ. Li and G. Uhlmann, Inverse problems for the stationary transport equation in the diffusion scaling, SIAM J. Appl. Math., 79 (2019), 2340-2358.  doi: 10.1137/18M1207582.

[28]

H. LiJ.-N. Wang and L. Wang, Refined stability estimates in electrical impedance tomography with multi-layer structure, Inverse Probl. Imaging, 16 (2022), 229-249.  doi: 10.3934/ipi.2021048.

[29]

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.

[30]

A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Ann. of Math., 143 (1996), 71-96.  doi: 10.2307/2118653.

[31]

S. Nagayasu, G. Uhlmann and J.-N. Wang, A depth-dependent stability estimate in electrical impedance tomography, Inverse Problems, 25 (2009), 075001, 14 pp. doi: 10.1088/0266-5611/25/7/075001.

[32]

S. Nagayasu, G. Uhlmann and J.-N. Wang, Increasing stability in an inverse problem for the acoustic equation, Inverse Problems, 29 (2013), 025012, 11 pp. doi: 10.1088/0266-5611/29/2/025012.

[33]

J.-C. Nédélec, Acoustic and Electromagnetic Equations. Integral Representations for Harmonic Problems, Applied Mathematical Sciences, 144. Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-4393-7.

[34]

L. Rondi and M. Sini, Stable determination of a scattered wave from its far-field pattern: The high frequency asymptotics, Arch. Ration. Mech. Anal., 218 (2015), 1-54.  doi: 10.1007/s00205-015-0855-0.

[35]

D. A. Subbarayappa and V. Isakov, On increased stability in the continuation of the Helmholtz equation, Inverse Problems, 23 (2007), 1689-1697.  doi: 10.1088/0266-5611/23/4/019.

[36]

D. A. Subbarayappa and V. Isakov, Increasing stability of the continuation for the Maxwell system, Inverse Problems, 26 (2010), 074005, 14 pp. doi: 10.1088/0266-5611/26/7/074005.

[37]

J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153-169.  doi: 10.2307/1971291.

[38]

M. E. Taylor, Partial Differential Equations. II. Qualitative Studies of Linear Equations, Applied Mathematical Sciences, 116. Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4757-4187-2.

[39]

G. Uhlmann and J.-N. Wang, Reconstructing discontinuities using complex geometrical optics solutions, SIAM J. Appl. Math., 68 (2008), 1026-1044.  doi: 10.1137/060676350.

[40]

G. UhlmannJ.-N. Wang and C.-T. Wu, Reconstruction of inclusions in an elastic body, J. Math. Pures Appl., 91 (2009), 569-582.  doi: 10.1016/j.matpur.2009.01.006.

[41] G. N. Watson, A Treatise on the Theory of Bessel Functions, 2 edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995. 
[42]

H. Zhao and Y. Zhong, Instability of an inverse problem for the stationary radiative transport near the diffusion limit, SIAM J. Math. Anal., 51 (2019), 3750-3768.  doi: 10.1137/18M1222582.

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