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

Armin Lechleiter; Marcel Rennoch

doi:10.3934/ipi.2017008

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2017, Volume 11, Issue 1: 151-176. Doi: 10.3934/ipi.2017008

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

Armin Lechleiter and
Marcel Rennoch

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

Received: November 30, 2015

Revised: August 31, 2016

Published: March 2018

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

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Abstract

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 $q$.

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Mathematics Subject Classification: Primary: 47J06, 65J20, 65J22; Secondary: 47J25, 65F50.

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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 $q^{\mathrm{sc}\;(2)}$

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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.

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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.

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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.

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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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