Fast subspace optimization method for nonlinear inverse problems in Banach spaces with uniformly convex penalty terms

where the noise level δ > 0 is known. Due to the inherent ill-posedness of inverse problems, some regularization methods should be used to produce a stable approximate solution of (1). Landweber iteration is one of the most prominent regularization methods for solving nonlinear inverse problems due to its simplicity, see [1] and reference therein. In order to capture the special features of the sought solutions, such as sparsity and discontinuities, the penalty term is allowed to be non-smooth to include L and total variation (TV) like penalty functionals. Let θ : X → (−∞,∞] be a proper, lower semi-continuous, convex function, then the method in [1] has the form of

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