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A regularizing multilevel approach for nonlinear inverse problems. (English) Zbl 1403.65024
Summary: In this paper, we propose a multilevel method for solving nonlinear inverse problems $$F(x) = y$$ in Banach spaces. By minimizing the discretized version of the regularized functionals at different levels, we define a sequence of regularized approximations to the sought solution, which is shown to be stable and globally convergent. The penalty term $$\Theta$$ in regularized functionals is allowed to be non-smooth to include $$L^p - L^1$$ or $$L^p - \mathrm{TV}$$ (Total Variation) reconstructions, which are significant in reconstructing special features of solutions such as sparsity and discontinuities. Two parameter identification examples are presented to validate the theoretical analysis and to verify the effectiveness of the method.

##### MSC:
 65J15 Numerical solutions to equations with nonlinear operators (do not use 65Hxx) 65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization 47J06 Nonlinear ill-posed problems
TIGRA
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