January 2017, 11(1): 203-220. doi: 10.3934/ipi.2017010

Variational source conditions and stability estimates for inverse electromagnetic medium scattering problems

Institute for Numerical and Applied Mathematics, University of Göttingen, Lotzestr. 16-18, D-37083 Göttingen, Germany

* Corresponding author: Frederic Weidling

Received  December 2015 Revised  November 2016 Published  January 2017

This paper is concerned with the inverse problem to recover the scalar, complex-valued refractive index of a medium from measurements of scattered time-harmonic electromagnetic waves at a fixed frequency. The main results are two variational source conditions for near and far field data, which imply logarithmic rates of convergence of regularization methods, in particular Tikhonov regularization, as the noise level tends to 0. Moreover, these variational source conditions imply conditional stability estimates which improve and complement known stability estimates in the literature.

Citation: Frederic Weidling, Thorsten Hohage. Variational source conditions and stability estimates for inverse electromagnetic medium scattering problems. Inverse Problems & Imaging, 2017, 11 (1) : 203-220. doi: 10.3934/ipi.2017010
References:
[1]

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

[2]

S. W. Anzengruber, B. Hofmann and R. Ramlau, On the interplay of basis smoothness and specific range conditions occurring in sparsity regularization, Inverse Problems 29 (2013), 125002, 21pp. doi: 10.1088/0266-5611/29/12/125002.

[3]

M. Burger, J. Flemming and B. Hofmann, Convergence rates in $\ell^1$-regularization if the sparsity assumption fails, Inverse Problems 29 (2013), 025013, 16pp. doi: 10.1088/0266-5611/29/2/025013.

[4]

A. -P. Calderón, On an inverse boundary value problem, in Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980), Soc. Brasil. Mat. , Rio de Janeiro, 1980, 65-73.

[5]

P. Caro, Stable determination of the electromagnetic coefficients by boundary measurements, Inverse Problems 26 (2010), 105014, 25pp. doi: 10.1088/0266-5611/26/10/105014.

[6]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory vol. 93 of Applied Mathematical Sciences, 3rd edition, Springer, New York, 2013. doi: 10.1007/978-1-4614-4942-3.

[7]

D. Colton and L. Päivärinta, The uniqueness of a solution to an inverse scattering problem for electromagnetic waves, Arch. Rational Mech. Anal., 119 (1992), 59-70. doi: 10.1007/BF00376010.

[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. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems vol. 375 of Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht, 1996.

[10]

L. D. Faddeev, Increasing solutions of the Schrödinger equation, Sov. Phys. , Dokl. , 1033-1035, Translation from Dokl. Akad. Nauk SSSR, 165 (1965), 514-517.

[11]

J. Flemming, Generalized Tikhonov Regularization and Modern Convergence Rate Theory in Banach spaces Shaker, 2012.

[12]

J. Flemming and M. Hegland, Convergence rates in $\ell^1$-regularization when the basis is not smooth enough, Appl. Anal., 94 (2015), 464-476. doi: 10.1080/00036811.2014.886106.

[13]

M. Grasmair, Generalized Bregman distances and convergence rates for non-convex regularization methods, Inverse Problems 26 (2010), 115014, 16pp. doi: 10.1088/0266-5611/26/11/115014.

[14]

P. Hähner, A periodic Faddeev-type solution operator, J. Differential Equations, 128 (1996), 300-308. doi: 10.1006/jdeq.1996.0096.

[15]

P. Hähner, On Acoustic, Electromagnetic, and Elastic Scattering Problems in Inhomogenous Media, Habilitation thesis, Georg-August Universität, Göttingen, http://webdoc.sub.gwdg.de/ebook/e/2000/mathe-goe/haehner.pdf, 1998

[16]

P. Hähner, Stability of the inverse electromagnetic inhomogeneous medium problem, Inverse Problems, 16 (2000), 155-174. doi: 10.1088/0266-5611/16/1/313.

[17]

P. Hähner and T. Hohage, New stability estimates for the inverse acoustic inhomogeneous medium problem and applications, SIAM J. Math. Anal., 33 (2001), 670-685. doi: 10.1137/S0036141001383564.

[18]

B. HofmannB. KaltenbacherC. Pöschl and O. Scherzer, A convergence rates result for Tikhonov regularization in Banach spaces with non-smooth operators, Inverse Problems, 23 (2007), 987-1010. doi: 10.1088/0266-5611/23/3/009.

[19]

T. Hohage and F. Weidling, Verification of a variational source condition for acoustic inverse medium scattering problems, Inverse Problems 31 (2015), 075006, 14pp. doi: 10.1088/0266-5611/31/7/075006.

[20]

M. I. Isaev and R. G. Novikov, New global stability estimates for monochromatic inverse acoustic scattering, SIAM J. Math. Anal., 45 (2013), 1495-1504. doi: 10.1137/120897833.

[21]

M. I. Isaev and R. G. Novikov, Effectivized Hölder-logarithmic stability estimates for the Gel'fand inverse problem, Inverse Problems 30 (2014), 095006, 18pp. doi: 10.1088/0266-5611/30/9/095006.

[22]

V. IsakovR.-Y. Lai and J.-N. Wang, Increasing stability for the conductivity and attenuation coefficients, SIAM J. Math. Anal., 48 (2016), 569-594. doi: 10.1137/15M1019052.

[23]

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.

[24]

R. G. Novikov, A multidimensional inverse spectral problem for the equation −∆ψ + (v(x) − Eu(x))ψ = 0, Funktsional. Anal. i Prilozhen., 22 (1988), 11–22, 96, URLhttp://dx.doi.org/10.1007/BF01077418. doi: 10.1007/BF0107741.

[25]

P. Ola and E. Somersalo, Electromagnetic inverse problems and generalized Sommerfeld potentials, SIAM J. Appl. Math., 56 (1996), 1129-1145. doi: 10.1137/S0036139995283948.

[26]

T. Schuster, B. Kaltenbacher, B. Hofmann and K. S. Kazimierski, Regularization Methods in Banach Spaces vol. 10 of Radon Series on Computational and Applied Mathematics, Walter de Gruyter GmbH & Co. KG, Berlin, 2012. doi: 10.1515/9783110255720.

[27]

P. Stefanov, Stability of the inverse problem in potential scattering at fixed energy, Ann. Inst. Fourier (Grenoble), 40 (1990), 867-884 (1991). doi: 10.5802/aif.1239.

[28]

J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math. (2), 125 (1987), 153-169. doi: 10.2307/1971291.

[29]

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

[30]

F. Werner and T. Hohage, Convergence rates in expectation for Tikhonov-type regularization of inverse problems with Poisson data, Inverse Problems 28 (2012), 104004, 15pp. doi: 10.1088/0266-5611/28/10/104004.

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]

S. W. Anzengruber, B. Hofmann and R. Ramlau, On the interplay of basis smoothness and specific range conditions occurring in sparsity regularization, Inverse Problems 29 (2013), 125002, 21pp. doi: 10.1088/0266-5611/29/12/125002.

[3]

M. Burger, J. Flemming and B. Hofmann, Convergence rates in $\ell^1$-regularization if the sparsity assumption fails, Inverse Problems 29 (2013), 025013, 16pp. doi: 10.1088/0266-5611/29/2/025013.

[4]

A. -P. Calderón, On an inverse boundary value problem, in Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980), Soc. Brasil. Mat. , Rio de Janeiro, 1980, 65-73.

[5]

P. Caro, Stable determination of the electromagnetic coefficients by boundary measurements, Inverse Problems 26 (2010), 105014, 25pp. doi: 10.1088/0266-5611/26/10/105014.

[6]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory vol. 93 of Applied Mathematical Sciences, 3rd edition, Springer, New York, 2013. doi: 10.1007/978-1-4614-4942-3.

[7]

D. Colton and L. Päivärinta, The uniqueness of a solution to an inverse scattering problem for electromagnetic waves, Arch. Rational Mech. Anal., 119 (1992), 59-70. doi: 10.1007/BF00376010.

[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. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems vol. 375 of Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht, 1996.

[10]

L. D. Faddeev, Increasing solutions of the Schrödinger equation, Sov. Phys. , Dokl. , 1033-1035, Translation from Dokl. Akad. Nauk SSSR, 165 (1965), 514-517.

[11]

J. Flemming, Generalized Tikhonov Regularization and Modern Convergence Rate Theory in Banach spaces Shaker, 2012.

[12]

J. Flemming and M. Hegland, Convergence rates in $\ell^1$-regularization when the basis is not smooth enough, Appl. Anal., 94 (2015), 464-476. doi: 10.1080/00036811.2014.886106.

[13]

M. Grasmair, Generalized Bregman distances and convergence rates for non-convex regularization methods, Inverse Problems 26 (2010), 115014, 16pp. doi: 10.1088/0266-5611/26/11/115014.

[14]

P. Hähner, A periodic Faddeev-type solution operator, J. Differential Equations, 128 (1996), 300-308. doi: 10.1006/jdeq.1996.0096.

[15]

P. Hähner, On Acoustic, Electromagnetic, and Elastic Scattering Problems in Inhomogenous Media, Habilitation thesis, Georg-August Universität, Göttingen, http://webdoc.sub.gwdg.de/ebook/e/2000/mathe-goe/haehner.pdf, 1998

[16]

P. Hähner, Stability of the inverse electromagnetic inhomogeneous medium problem, Inverse Problems, 16 (2000), 155-174. doi: 10.1088/0266-5611/16/1/313.

[17]

P. Hähner and T. Hohage, New stability estimates for the inverse acoustic inhomogeneous medium problem and applications, SIAM J. Math. Anal., 33 (2001), 670-685. doi: 10.1137/S0036141001383564.

[18]

B. HofmannB. KaltenbacherC. Pöschl and O. Scherzer, A convergence rates result for Tikhonov regularization in Banach spaces with non-smooth operators, Inverse Problems, 23 (2007), 987-1010. doi: 10.1088/0266-5611/23/3/009.

[19]

T. Hohage and F. Weidling, Verification of a variational source condition for acoustic inverse medium scattering problems, Inverse Problems 31 (2015), 075006, 14pp. doi: 10.1088/0266-5611/31/7/075006.

[20]

M. I. Isaev and R. G. Novikov, New global stability estimates for monochromatic inverse acoustic scattering, SIAM J. Math. Anal., 45 (2013), 1495-1504. doi: 10.1137/120897833.

[21]

M. I. Isaev and R. G. Novikov, Effectivized Hölder-logarithmic stability estimates for the Gel'fand inverse problem, Inverse Problems 30 (2014), 095006, 18pp. doi: 10.1088/0266-5611/30/9/095006.

[22]

V. IsakovR.-Y. Lai and J.-N. Wang, Increasing stability for the conductivity and attenuation coefficients, SIAM J. Math. Anal., 48 (2016), 569-594. doi: 10.1137/15M1019052.

[23]

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.

[24]

R. G. Novikov, A multidimensional inverse spectral problem for the equation −∆ψ + (v(x) − Eu(x))ψ = 0, Funktsional. Anal. i Prilozhen., 22 (1988), 11–22, 96, URLhttp://dx.doi.org/10.1007/BF01077418. doi: 10.1007/BF0107741.

[25]

P. Ola and E. Somersalo, Electromagnetic inverse problems and generalized Sommerfeld potentials, SIAM J. Appl. Math., 56 (1996), 1129-1145. doi: 10.1137/S0036139995283948.

[26]

T. Schuster, B. Kaltenbacher, B. Hofmann and K. S. Kazimierski, Regularization Methods in Banach Spaces vol. 10 of Radon Series on Computational and Applied Mathematics, Walter de Gruyter GmbH & Co. KG, Berlin, 2012. doi: 10.1515/9783110255720.

[27]

P. Stefanov, Stability of the inverse problem in potential scattering at fixed energy, Ann. Inst. Fourier (Grenoble), 40 (1990), 867-884 (1991). doi: 10.5802/aif.1239.

[28]

J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math. (2), 125 (1987), 153-169. doi: 10.2307/1971291.

[29]

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

[30]

F. Werner and T. Hohage, Convergence rates in expectation for Tikhonov-type regularization of inverse problems with Poisson data, Inverse Problems 28 (2012), 104004, 15pp. doi: 10.1088/0266-5611/28/10/104004.

Table 1.  Comparison between different stability estimates
new Hähner [16]Caro [5] Lai et al [22]
datanear/far fieldfar fieldCauchyCauchy
validitygloballocal anywheregloballocal around $0$
stability of$\sigma, \epsilon$$\sigma, \epsilon$$\sigma, \epsilon, \mu$$\sigma$
norm$H^m$$L^\infty$$H^1$$H^{-s}$
exponent$<1$$1/15$unknown, $<\frac{1}{3}$ $\leq1$
specialstrong norm in image spaceHölder-logarithmic
new Hähner [16]Caro [5] Lai et al [22]
datanear/far fieldfar fieldCauchyCauchy
validitygloballocal anywheregloballocal around $0$
stability of$\sigma, \epsilon$$\sigma, \epsilon$$\sigma, \epsilon, \mu$$\sigma$
norm$H^m$$L^\infty$$H^1$$H^{-s}$
exponent$<1$$1/15$unknown, $<\frac{1}{3}$ $\leq1$
specialstrong norm in image spaceHölder-logarithmic
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