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An iterative thresholding algorithm for linear inverse problems with multi-constraints and its applications. (English) Zbl 1238.65046

The minimization of functionals of the form \[ \Phi(f) = \| Kf-g \| ^2 + |\!|\!|f|\!|\!|_{W_1,p_1}^{p_1} + |\!|\!|f|\!|\!|_{W_2,p_2}^{p_2} + \cdots + |\!|\!|f|\!|\!|_{W_n,p_n}^{p_n}, \quad f \in \mathcal{H}, \] is considered, where \( K: \mathcal{H} \to \mathcal{H}^\prime \) is a bounded linear operator between Hilbert spaces \( \mathcal{H} \) and \( \mathcal{H}^\prime \), and \( g \in \mathcal{H}^\prime \). In addition we have \( 1 \leq p_i \leq 2 \) for \( i = 1,2,\dots, n \), and the notation \( |\!|\!|f|\!|\!|_{W_i,p_i} \allowbreak = (\sum_{\gamma\in \Gamma} w_{i,\gamma} | \langle f, \varphi_\gamma \rangle |^{p_i})^{1/p_i} \) for \( i = 1,2,\dots, n \) is used, where \( \{\varphi_\gamma\}_{\gamma \in \Gamma} \) denotes an orthonormal basis of the Hilbert space \( \mathcal{H} \), and \( W_i = \{w_{i,\gamma}\}_{\gamma \in \Gamma} \) denotes a system of nonnegative weights (\(i = 1,2,\dots, n\)). Under the assumption \( \| K \| < 1 \) and some other conditions, the author considers an iterative process of the form \( f^m = S(f^{m-1}+K^*(g-Kf^{m-1})), \;m = 1,2,\dots \), where \( f^0 \in \mathcal{H} \) is arbitrarily chosen, and \( K^* \) denotes the adjoint operator of \( K \), and \( S: \mathcal{H} \to \mathcal{H} \) is a nonexpansive operator specified in the paper. It is shown that this iterative process converges strongly to a minimizer of the considered functional \( \Phi \).
In addition, a multiparameter Tikhonov type regularization of the form \[ \Phi_{\alpha,g}(f) = \| Kf-g \| ^2 + \alpha_1 |\!|\!|f|\!|\!|_{W_1,p_1}^{p_1} + \alpha_2 |\!|\!|f|\!|\!|_{W_2,p_2}^{p_2} + \cdots + \alpha_n |\!|\!|f|\!|\!|_{W_n,p_n}^{p_n}, \quad f \in \mathcal{H}, \] is considered, where \( \alpha_1, \dots, \alpha_n \) are nonnegative regularization parameters, and the notation \( \alpha = (\alpha_1, \dots, \alpha_n) \) is used here. It is shown that under the conditions \( \lim_{\epsilon \to 0} \alpha_i(\epsilon) = 0, \;\lim_{\epsilon \to 0} \frac{\epsilon^2}{\alpha_i(\epsilon)} = 0 \) and \( \lim_{\epsilon \to 0} \frac{\alpha_i(\epsilon)}{\alpha_j(\epsilon)} = 1 \)
for each \( 1 \leq i, j \leq n \) and some other conditions specified in the paper, a regularizing scheme is obtained, i.e., for each \( f_0 \in \mathcal{H} \) one has \[ \lim_{\epsilon \to 0} \sup_{\| g - Kf_0\| < \epsilon} \| f_{\alpha(\epsilon),g}^*-f^\dagger \| = 0 . \] Here \( f^\dagger \) denotes the unique minimizer of the functional \(|\!|\!|\cdot|\!|\!|_{W_1,p_1}^{p_1} + \cdots + |\!|\!|\cdot|\!|\!|_{W_n,p_n}^{p_n} \) on \( N(K)+f_0 = \{f \in \mathcal{H} : Kf = Kf_0 \} \), and \( f_{\alpha,g}^* \) denotes the minimizer of the functional \( \Phi_{\alpha,g} \). Finally some applications are presented.

MSC:

65J10 Numerical solutions to equations with linear operators
65J22 Numerical solution to inverse problems in abstract spaces
42C15 General harmonic expansions, frames
49M30 Other numerical methods in calculus of variations (MSC2010)
65K10 Numerical optimization and variational techniques
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References:

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