# American Institute of Mathematical Sciences

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. Google Scholar [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. Google Scholar [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. Google Scholar [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.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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.Google Scholar [11] J. Flemming, Generalized Tikhonov Regularization and Modern Convergence Rate Theory in Banach spaces Shaker, 2012.Google Scholar [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. Google Scholar [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. Google Scholar [14] P. Hähner, A periodic Faddeev-type solution operator, J. Differential Equations, 128 (1996), 300-308. doi: 10.1006/jdeq.1996.0096. Google Scholar [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, 1998Google Scholar [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. Google Scholar [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. Google Scholar [18] B. Hofmann, B. Kaltenbacher, C. 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [22] V. Isakov, R.-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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [29] G. Uhlmann, Electrical impedance tomography and CalderÃ³n's problem, Inverse Problems 25 (2009), 123011, 39pp. doi: 10.1088/0266-5611/25/12/123011. Google Scholar [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. Google Scholar

show all references

##### References:
 [1] G. Alessandrini, Stable determination of conductivity by boundary measurements, Appl. Anal., 27 (1988), 153-172. doi: 10.1080/00036818808839730. Google Scholar [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. Google Scholar [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. Google Scholar [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.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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.Google Scholar [11] J. Flemming, Generalized Tikhonov Regularization and Modern Convergence Rate Theory in Banach spaces Shaker, 2012.Google Scholar [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. Google Scholar [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. Google Scholar [14] P. Hähner, A periodic Faddeev-type solution operator, J. Differential Equations, 128 (1996), 300-308. doi: 10.1006/jdeq.1996.0096. Google Scholar [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, 1998Google Scholar [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. Google Scholar [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. Google Scholar [18] B. Hofmann, B. Kaltenbacher, C. 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [22] V. Isakov, R.-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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [29] G. Uhlmann, Electrical impedance tomography and CalderÃ³n's problem, Inverse Problems 25 (2009), 123011, 39pp. doi: 10.1088/0266-5611/25/12/123011. Google Scholar [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. Google Scholar
Comparison between different stability estimates
 new Hähner [16] Caro [5] Lai et al [22] data near/far field far field Cauchy Cauchy validity global local anywhere global local 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, $<\frac{1}{3}$ $\leq1$ special strong norm in image space Hölder-logarithmic
 new Hähner [16] Caro [5] Lai et al [22] data near/far field far field Cauchy Cauchy validity global local anywhere global local 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, $<\frac{1}{3}$ $\leq1$ special strong norm in image space Hölder-logarithmic
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