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January 2017, 11(1): 151-176. doi: 10.3934/ipi.2017008

Non-linear Tikhonov regularization in Banach spaces for inverse scattering from anisotropic penetrable media

Center for Industrial Mathematics, University of Bremen, 28359 Bremen, Germany

Received  December 2015 Revised  September 2016 Published  January 2017

Fund Project: The authors are supported by German Research Foundation (DFG) grant Le 2499/2-1.

We consider Tikhonov and sparsity-promoting regularization in Banach spaces for inverse scattering from penetrable anisotropic media. To this end, we equip an admissible set of material parameters with the $L^p$-topology and use Meyers' gradient estimate for solutions of elliptic equations to analyze the dependence of scattered fields and their Fréchet derivatives on the material parameter. This allows to show convergence of a non-linear Tikhonov regularization against a minimum-norm solution to the inverse problem, but also to set up sparsity-promoting versions of that regularization method. For both approaches, the discrepancy is defined via a $q$-Schatten norm or an $L^q$-norm with $1 < q < ∞$. Numerical reconstruction examples indicate the reconstruction quality of the method, as well as the qualitative dependence of the reconstructions on $q$.

Citation: Armin Lechleiter, Marcel Rennoch. Non-linear Tikhonov regularization in Banach spaces for inverse scattering from anisotropic penetrable media. Inverse Problems & Imaging, 2017, 11 (1) : 151-176. doi: 10.3934/ipi.2017008
References:
[1]

L. Armijo, Minimization of functions having Lipschitz continuous first partial derivatives, Pacific J. Math., 16 (1966), 1-3. doi: 10.2140/pjm.1966.16.1.

[2]

A. Chambolle and T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging, Journal of Mathematical Imaging and Vision, 40 (2011), 120-145. doi: 10.1007/s10851-010-0251-1.

[3] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 3rd edition, Springer, 2013. doi: 10.1007/978-1-4614-4942-3.
[4]

I. Daubechies, Orthonormal bases of compactly supported wavelets, Communications on Pure and Applied Mathematics, 41 (1988), 909-996. doi: 10.1002/cpa.3160410705.

[5] I. Daubechies, Ten Lectures on Wavelets, SIAM, 1992. doi: 10.1137/1.9781611970104.fm.
[6]

I. DaubechiesM. Defrise and C. De Mol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Comm. Pure Appl. Math., 57 (2004), 1413-1457. doi: 10.1002/cpa.20042.

[7]

B. Gramsch, Zum Einbettungssatz von Rellich bei Sobolevräumen, Math. Zeitschrift, 106 (1968), 81-87. doi: 10.1007/BF01110715.

[8]

A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires Mem. Amer. Math. Soc. 1955 (1955), 140pp. doi: 10.1090/memo/0016.

[9]

P. Hähner, On the uniqueness of the shape of a penetrable, anisotropic obstacle, Journal of Computational and Applied Mathematics, 116 (2000), 167-180. doi: 10.1016/S0377-0427(99)00323-4.

[10]

T. Hohage and C. Homann, A generalization of the Chambolle-Pock algorithm to Banach spaces with applications to inverse problems, preprint, arXiv: 1412.0126.

[11]

B. Jin and P. Maass, An analysis of electrical impedance tomography with applications to Tikhonov regularization, ESAIM: Control, Optimisation and Calculus of Variations, 18 (2012), 1027-1048. doi: 10.1051/cocv/2011193.

[12]

A. Kirsch, An integral equation for the scattering problem for an anisotropic medium and the factorization method, in Advanced Topics in Scattering and Biomedical Engineering, 2008, 57-70. doi: 10.1142/9789812814852_0007.

[13]

A. Lechleiter, K. S. Kazimierski and M. Karamehmedović, Tikhonov regularization in Lp applied to inverse medium scattering Inverse Problems 29 (2013), 075003, 19pp. doi: 10.1088/0266-5611/29/7/075003.

[14]

A. Lechleiter and D.-L. Nguyen, A trigonometric galerkin method for volume integral equations arising in TM grating scattering, Adv. Compt. Math., 40 (2014), 1-25. doi: 10.1007/s10444-013-9295-2.

[15] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, 2000. doi: 10.1017/S0013091501244435.
[16]

N. G. Meyers, An Lp-estimate for the gradient of solutions of second order elliptic divergence equations, Annali della Scuola Norm. Sup. Pisa, 17 (1963), 189-206.

[17] J. -C. Nédélec, Acoustic and Electromagnetic Equations, Springer, New York etc, 2001. doi: 10.1007/978-1-4757-4393-7.
[18] R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NY, 1997.
[19]

O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier and F. Lenzen, Variational Methods in Imaging vol. 167 of Applied Mathematical Sciences, Springer, 2009. doi: 10.1007/978-0-387-69277-7.

[20]

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, De Gruyter, 2012.

[21]

H. Triebel, Theory of Function Spaces Ⅲ Monographs in mathematics, Birkhäuser Verlag, Basel, Boston, Berlin, 2006. doi: 10.1007/3-7643-7582-5.

[22]

G. Vainikko, Fast solvers of the Lippmann-Schwinger equation, in Direct and Inverse Problems of Mathematical Physics (eds. R. P. Gilbert, J. Kajiwara and Y. S. Xu), vol. 5 of International Society for Analysis, Applications and Computation, Springer US, 2000,423-440. doi: 10.1002/cpa.3160410705.

[23] E. Zeidler, Nonlinear Functional Analysis and its Applications. Ⅰ Fixed-Point Theorems, Springer, 1986.

show all references

References:
[1]

L. Armijo, Minimization of functions having Lipschitz continuous first partial derivatives, Pacific J. Math., 16 (1966), 1-3. doi: 10.2140/pjm.1966.16.1.

[2]

A. Chambolle and T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging, Journal of Mathematical Imaging and Vision, 40 (2011), 120-145. doi: 10.1007/s10851-010-0251-1.

[3] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 3rd edition, Springer, 2013. doi: 10.1007/978-1-4614-4942-3.
[4]

I. Daubechies, Orthonormal bases of compactly supported wavelets, Communications on Pure and Applied Mathematics, 41 (1988), 909-996. doi: 10.1002/cpa.3160410705.

[5] I. Daubechies, Ten Lectures on Wavelets, SIAM, 1992. doi: 10.1137/1.9781611970104.fm.
[6]

I. DaubechiesM. Defrise and C. De Mol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Comm. Pure Appl. Math., 57 (2004), 1413-1457. doi: 10.1002/cpa.20042.

[7]

B. Gramsch, Zum Einbettungssatz von Rellich bei Sobolevräumen, Math. Zeitschrift, 106 (1968), 81-87. doi: 10.1007/BF01110715.

[8]

A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires Mem. Amer. Math. Soc. 1955 (1955), 140pp. doi: 10.1090/memo/0016.

[9]

P. Hähner, On the uniqueness of the shape of a penetrable, anisotropic obstacle, Journal of Computational and Applied Mathematics, 116 (2000), 167-180. doi: 10.1016/S0377-0427(99)00323-4.

[10]

T. Hohage and C. Homann, A generalization of the Chambolle-Pock algorithm to Banach spaces with applications to inverse problems, preprint, arXiv: 1412.0126.

[11]

B. Jin and P. Maass, An analysis of electrical impedance tomography with applications to Tikhonov regularization, ESAIM: Control, Optimisation and Calculus of Variations, 18 (2012), 1027-1048. doi: 10.1051/cocv/2011193.

[12]

A. Kirsch, An integral equation for the scattering problem for an anisotropic medium and the factorization method, in Advanced Topics in Scattering and Biomedical Engineering, 2008, 57-70. doi: 10.1142/9789812814852_0007.

[13]

A. Lechleiter, K. S. Kazimierski and M. Karamehmedović, Tikhonov regularization in Lp applied to inverse medium scattering Inverse Problems 29 (2013), 075003, 19pp. doi: 10.1088/0266-5611/29/7/075003.

[14]

A. Lechleiter and D.-L. Nguyen, A trigonometric galerkin method for volume integral equations arising in TM grating scattering, Adv. Compt. Math., 40 (2014), 1-25. doi: 10.1007/s10444-013-9295-2.

[15] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, 2000. doi: 10.1017/S0013091501244435.
[16]

N. G. Meyers, An Lp-estimate for the gradient of solutions of second order elliptic divergence equations, Annali della Scuola Norm. Sup. Pisa, 17 (1963), 189-206.

[17] J. -C. Nédélec, Acoustic and Electromagnetic Equations, Springer, New York etc, 2001. doi: 10.1007/978-1-4757-4393-7.
[18] R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NY, 1997.
[19]

O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier and F. Lenzen, Variational Methods in Imaging vol. 167 of Applied Mathematical Sciences, Springer, 2009. doi: 10.1007/978-0-387-69277-7.

[20]

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, De Gruyter, 2012.

[21]

H. Triebel, Theory of Function Spaces Ⅲ Monographs in mathematics, Birkhäuser Verlag, Basel, Boston, Berlin, 2006. doi: 10.1007/3-7643-7582-5.

[22]

G. Vainikko, Fast solvers of the Lippmann-Schwinger equation, in Direct and Inverse Problems of Mathematical Physics (eds. R. P. Gilbert, J. Kajiwara and Y. S. Xu), vol. 5 of International Society for Analysis, Applications and Computation, Springer US, 2000,423-440. doi: 10.1002/cpa.3160410705.

[23] E. Zeidler, Nonlinear Functional Analysis and its Applications. Ⅰ Fixed-Point Theorems, Springer, 1986.
Figure 1.  Contrasts plotted in $[-0.4,0.4)^2$. (a) Real part of $q^{\mathrm{sc}\;(1)}$ (b) Imaginary part of $q^{\mathrm{sc}\;(1)}$ (c) Real-valued contrast $q^{\mathrm{sc}\;(2)}$
Figure 2.  Reconstructions of $q^{\mathrm{sc}\;(1)}$ by shrinked Landweber method, plotted in $[-0.4,0.4)^2$ (real parts in top row, imaginary parts in bottom row). (a/d) $\varepsilon=0.01$, 500 iter., 2145 min., rel. error=0.533 (b/e) $\varepsilon=0.05$, 300 iter., 748 min., rel. error=0.565 (c/f) $\varepsilon=0.1$, 57 iter., 126 min., rel. error=0.677.
Figure 3.  Reconstructions of $q^{\mathrm{sc}\;(2)}$ by shrinked Landweber method, plotted in $[-0.4,0.4)^2$ (real parts in top row, imaginary parts in bottom row). (a/d) $\varepsilon=0.01$, 200 iter., 390 min., rel. error=0.653 (b/e) $\varepsilon=0.05$, 48 iter., 87 min., rel. error=0.665 (c/f) $\varepsilon=0.1$, 20 iter., 38 min., rel. error=0.703.
Figure 4.  Reconstructions of $q^{\mathrm{sc}\;(2)}$ rotated by $25^\circ$ by shrinked Landweber method, plotted in $[-0.4,0.4)^2$ (real parts in top row, imaginary parts in bottom row). (a/d) $\varepsilon=0.01$, 300 iter., rel. error=0.668 (b/e) $\varepsilon=0.05$, 445 iter., rel. error=0.669 (c/f) $\varepsilon=0.1$, 99 iter., rel. error=0.734.
Figure 5.  Real part of reconstructions of $q^{\mathrm{sc}\;(2)}$ by primal-dual algorithm for different discrepancy norms $\| \cdot \|_q^q/q$ (see Remark 7) and fixed artificial noise level $\varepsilon=0.01$, plotted on $[-0.4,0.4)^2$. (a) $q=2$, 5 iter., 12 min., rel. error=0.658 (b) $q=3$, 2 iter., 4 min., rel. error=0.738 (c) $q=1.6$, 41 iter., 82 min., rel. error=0.763.
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