Landweber-Kaczmarz method in Banach spaces with inexact inner solvers (Q2832552)

The article deals with Landweber-Kaczmarz iterative method of approximate solving a nonlinear system NEWLINE\[NEWLINEF_i(x) = y_i, \qquad i = 0,1,\dots,N - 1,\tag{*}NEWLINE\]NEWLINE where, for each \(i = 0,1,\dots,N - 1\), \(F_i\) is a nonlinear operator between reflexive Banach spaces \({\mathcal X}\) and \({\mathcal Y}_i\) satisfying the condition NEWLINE\[NEWLINE\|F_i(\overline{x}) - F_(x) - L_i(x)(\overline{x} - x)\| \leq \gamma \|F_i(\overline{x}) - F_i(x)\|NEWLINE\]NEWLINE with \(L_i(x):\;{\mathcal X} \to {\mathcal Y}\) from \({\mathcal L}({\mathcal X},{\mathcal Y}_i)\) and \(0 \leq \gamma< 1\). It is assumed that the function \(\Theta:\;{\mathcal X} \to (-\infty,+\infty]\) is proper, lower semi-continuous and \(p\)-convex and such that, for any \(\varepsilon> 0\), and there is a procedure \(S_\varepsilon:\;{\mathcal X}^* \to {\mathcal X}\) for which NEWLINE\[NEWLINEx = \text{arg min}\, \{\Theta(z) - \langle \xi,z \rangle\}NEWLINE\]NEWLINE and NEWLINE\[NEWLINE\Theta(x) - \langle \xi,x \rangle \leq \min_{z \in {\mathcal X}}\;\{\Theta(z) - \langle \xi,z \rangle\} + \varepsilon.NEWLINE\]NEWLINE The author considers the following algorithm NEWLINE\[NEWLINE\xi_{n+1} = \xi_n - \mu_nL_{i_n}(x_n)^*J_s^{{\mathcal Y}_{i_n}}[F_{i_n}(x_n) - y_{i_n}], \quad x_{n+1} = S_{\varepsilon_{n+1}}(\xi_{n+1}), \quad n = 0,1,2,\ldotsNEWLINE\]NEWLINE (\(i_n = n(\text{mod}\, N)\), \(\mu_n\) are some exactly definite numbers) for solving (*) with exact data, the similar algorithm for solving (*) with noisy data, and some their modifications. And, further, he formulates conditions under which the following equations NEWLINE\[NEWLINE\lim_{n \to \infty}\;\|x_n - x_*\| = 0, \quad \lim_{n \to \infty}\;D_{\xi_n}^{\varepsilon_n}\Theta(x_*,x_n) = 0NEWLINE\]NEWLINE (\(x_8\) is an exact solution to (*)) hold. Some numerical simulations are presented in the end of the article.